Page 1
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
On the global Lipschitz assumption in Computational
Stochastics
Arnulf Jentzen
Joint work with Martin Hutzenthaler
Faculty of Mathematics
Bielefeld University
16th August 2010
Arnulf Jentzen Global Lipschitz assumption
Page 2
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Overview
1 Stochastic differential equations (SDEs)
2 Computational problem and the Monte Carlo Euler method
3 Convergence for SDEs with globally Lipschitz continuous coefficients
4 Convergence for SDEs with superlinearly growing coefficients
Arnulf Jentzen Global Lipschitz assumption
Page 3
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Overview
1 Stochastic differential equations (SDEs)
2 Computational problem and the Monte Carlo Euler method
3 Convergence for SDEs with globally Lipschitz continuous coefficients
4 Convergence for SDEs with superlinearly growing coefficients
Arnulf Jentzen Global Lipschitz assumption
Page 4
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 5
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 6
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 7
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 8
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 9
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 10
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 11
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 12
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Consider • a probability space (Ω,F , P) with a normal filtration (Ft)t∈[0,T ] and T > 0
• a standard (Ft)t∈[0,T ]-Brownian motion W : [0, T ] × Ω → R
• continuous functions µ, σ : R → R and
• a F0/B(R)-measurable mapping ξ : Ω → R with E|ξ|p <∞∀ p ∈ [1,∞).
Then let X : [0, T ] × Ω → R be an adapted stochastic process with
continuous sample paths which fulfills
Xt = ξ +
∫ t
0
µ(Xs) ds +
∫ t
o
σ(Xs) dWs P-a.s.
for all t ∈ [0, T ]. Short form:
dXt = µ(Xt) dt + σ(Xt) dWt , X0 = ξ, t ∈ [0, T ].
Arnulf Jentzen Global Lipschitz assumption
Page 13
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
Arnulf Jentzen Global Lipschitz assumption
Page 14
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
Arnulf Jentzen Global Lipschitz assumption
Page 15
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
Arnulf Jentzen Global Lipschitz assumption
Page 16
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
Arnulf Jentzen Global Lipschitz assumption
Page 17
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs II
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
A Feller diffusion with logistic growth with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt +√
Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 18
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs II
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
A Feller diffusion with logistic growth with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt +√
Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 19
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Overview
1 Stochastic differential equations (SDEs)
2 Computational problem and the Monte Carlo Euler method
3 Convergence for SDEs with globally Lipschitz continuous coefficients
4 Convergence for SDEs with superlinearly growing coefficients
Arnulf Jentzen Global Lipschitz assumption
Page 20
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Weak approximation problem of the SDE (see, e.g., Kloeden & Platen (1992))
Suppose we want to compute
E
[
f(XT )]
for a given smooth function f : R → R whose derivatives grow at most
polynomially.
For instance, f(x) = x2 for all x ∈ R and we want to compute
E
[
(XT )2]
the second moment of the SDE.
Arnulf Jentzen Global Lipschitz assumption
Page 21
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Weak approximation problem of the SDE (see, e.g., Kloeden & Platen (1992))
Suppose we want to compute
E
[
f(XT )]
for a given smooth function f : R → R whose derivatives grow at most
polynomially.
For instance, f(x) = x2 for all x ∈ R and we want to compute
E
[
(XT )2]
the second moment of the SDE.
Arnulf Jentzen Global Lipschitz assumption
Page 22
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Weak approximation problem of the SDE (see, e.g., Kloeden & Platen (1992))
Suppose we want to compute
E
[
f(XT )]
for a given smooth function f : R → R whose derivatives grow at most
polynomially.
For instance, f(x) = x2 for all x ∈ R and we want to compute
E
[
(XT )2]
the second moment of the SDE.
Arnulf Jentzen Global Lipschitz assumption
Page 23
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Weak approximation problem of the SDE (see, e.g., Kloeden & Platen (1992))
Suppose we want to compute
E
[
f(XT )]
for a given smooth function f : R → R whose derivatives grow at most
polynomially.
For instance, f(x) = x2 for all x ∈ R and we want to compute
E
[
(XT )2]
the second moment of the SDE.
Arnulf Jentzen Global Lipschitz assumption
Page 24
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Weak approximation problem of the SDE (see, e.g., Kloeden & Platen (1992))
Suppose we want to compute
E
[
f(XT )]
for a given smooth function f : R → R whose derivatives grow at most
polynomially.
For instance, f(x) = x2 for all x ∈ R and we want to compute
E
[
(XT )2]
the second moment of the SDE.
Arnulf Jentzen Global Lipschitz assumption
Page 25
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 26
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 27
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 28
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 29
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 30
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 31
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 32
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 33
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)TN
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and M ∈ N Monte Carlo runs is then given by
1
M
(M∑
m=1
f(YN,mN )
)
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 34
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)T
N
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and N2 ∈ N Monte Carlo runs is then given by
1
N2
N2∑
m=1
f(YN,mN )
≈ E
[
f(Y NN )]
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 35
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Approximation of E[f (XT)
]
The stochastic Euler scheme Y Nk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, is
given by Y N0 = ξ and
Y Nk+1 = Y N
k +T
N· µ(Y N
k
)+ σ
(Y N
k
)·(
W (k+1)T
N
− W kTN
)
for all k ∈ 0, 1, . . . , N − 1 and all N ∈ N.
Let YN,mk : Ω → R, k ∈ 0, 1, . . . , N, N ∈ N, for m ∈ N be independent
copies of the Euler approximations. The Monte Carlo Euler approximation
with N ∈ N time steps and N2 ∈ N Monte Carlo runs is then given by
1
N2
N2∑
m=1
f(YN,mN )
≈ E
[
f(XT )]
.
Arnulf Jentzen Global Lipschitz assumption
Page 36
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Overview
1 Stochastic differential equations (SDEs)
2 Computational problem and the Monte Carlo Euler method
3 Convergence for SDEs with globally Lipschitz continuous coefficients
4 Convergence for SDEs with superlinearly growing coefficients
Arnulf Jentzen Global Lipschitz assumption
Page 37
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The triangle inequality shows
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method
≤∣∣∣E
[
f(
XT
)]
− E
[
f(Y N
N
)]∣∣∣
︸ ︷︷ ︸
time discretization error
+∣∣∣E
[
f(
Y NN
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
statistical error
for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if
limN→∞
∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣ = 0
holds for every smooth function f : R → R whose derivatives have at most
polynomial growth (see, e.g., Kloeden & Platen (1992), Milstein (1995),
Talay (1996), Higham (2001), Rössler (2003)).Arnulf Jentzen Global Lipschitz assumption
Page 38
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The triangle inequality shows
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method
≤∣∣∣E
[
f(
XT
)]
− E
[
f(Y N
N
)]∣∣∣
︸ ︷︷ ︸
time discretization error
+∣∣∣E
[
f(
Y NN
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
statistical error
for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if
limN→∞
∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣ = 0
holds for every smooth function f : R → R whose derivatives have at most
polynomial growth (see, e.g., Kloeden & Platen (1992), Milstein (1995),
Talay (1996), Higham (2001), Rössler (2003)).Arnulf Jentzen Global Lipschitz assumption
Page 39
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The triangle inequality shows
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method
≤∣∣∣E
[
f(
XT
)]
− E
[
f(Y N
N
)]∣∣∣
︸ ︷︷ ︸
time discretization error
+∣∣∣E
[
f(
Y NN
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
statistical error
for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if
limN→∞
∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣ = 0
holds for every smooth function f : R → R whose derivatives have at most
polynomial growth (see, e.g., Kloeden & Platen (1992), Milstein (1995),
Talay (1996), Higham (2001), Rössler (2003)).Arnulf Jentzen Global Lipschitz assumption
Page 40
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The triangle inequality shows
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method
≤∣∣∣E
[
f(
XT
)]
− E
[
f(Y N
N
)]∣∣∣
︸ ︷︷ ︸
time discretization error
+∣∣∣E
[
f(
Y NN
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
statistical error
for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if
limN→∞
∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣ = 0
holds for every smooth function f : R → R whose derivatives have at most
polynomial growth (see, e.g., Kloeden & Platen (1992), Milstein (1995),
Talay (1996), Higham (2001), Rössler (2003)).Arnulf Jentzen Global Lipschitz assumption
Page 41
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The triangle inequality shows
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method
≤∣∣∣E
[
f(
XT
)]
− E
[
f(Y N
N
)]∣∣∣
︸ ︷︷ ︸
time discretization error
+∣∣∣E
[
f(
Y NN
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
statistical error
for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if
limN→∞
∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣ = 0
holds for every smooth function f : R → R whose derivatives have at most
polynomial growth (see, e.g., Kloeden & Platen (1992), Milstein (1995),
Talay (1996), Higham (2001), Rössler (2003)).Arnulf Jentzen Global Lipschitz assumption
Page 42
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The triangle inequality shows
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method
≤∣∣∣E
[
f(
XT
)]
− E
[
f(Y N
N
)]∣∣∣
︸ ︷︷ ︸
time discretization error
+∣∣∣E
[
f(
Y NN
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
statistical error
for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if
limN→∞
∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣ = 0
holds for every smooth function f : R → R whose derivatives have at most
polynomial growth (see, e.g., Kloeden & Platen (1992), Milstein (1995),
Talay (1996), Higham (2001), Rössler (2003)).Arnulf Jentzen Global Lipschitz assumption
Page 43
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence
Theorem (see, e.g., Kloeden & Platen (1992))
Let µ, σ, f : R → R be four times continuously differentiable with at most
polynomially growing derivatives. Moreover, let µ, σ : R → R be globally
Lipschitz continuous. Then there is a real number C > 0 such that
∣∣∣∣∣E
[
f(XT )]
−E
[
f(Y NN )]∣∣∣∣∣≤ C · 1
N
holds for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if the
coefficients of the SDE are smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 44
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence
Theorem (see, e.g., Kloeden & Platen (1992))
Let µ, σ, f : R → R be four times continuously differentiable with at most
polynomially growing derivatives. Moreover, let µ, σ : R → R be globally
Lipschitz continuous. Then there is a real number C > 0 such that
∣∣∣∣∣E
[
f(XT )]
−E
[
f(Y NN )]∣∣∣∣∣≤ C · 1
N
holds for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if the
coefficients of the SDE are smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 45
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence
Theorem (see, e.g., Kloeden & Platen (1992))
Let µ, σ, f : R → R be four times continuously differentiable with at most
polynomially growing derivatives. Moreover, let µ, σ : R → R be globally
Lipschitz continuous. Then there is a real number C > 0 such that
∣∣∣∣∣E
[
f(XT )]
−E
[
f(Y NN )]∣∣∣∣∣≤ C · 1
N
holds for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if the
coefficients of the SDE are smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 46
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence
Theorem (see, e.g., Kloeden & Platen (1992))
Let µ, σ, f : R → R be four times continuously differentiable with at most
polynomially growing derivatives. Moreover, let µ, σ : R → R be globally
Lipschitz continuous. Then there is a real number C > 0 such that
∣∣∣∣∣E
[
f(XT )]
−E
[
f(Y NN )]∣∣∣∣∣≤ C · 1
N
holds for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if the
coefficients of the SDE are smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 47
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence
Theorem (see, e.g., Kloeden & Platen (1992))
Let µ, σ, f : R → R be four times continuously differentiable with at most
polynomially growing derivatives. Moreover, let µ, σ : R → R be globally
Lipschitz continuous. Then there is a real number C > 0 such that
∣∣∣∣∣E
[
f(XT )]
−E
[
f(Y NN )]∣∣∣∣∣≤ C · 1
N
holds for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if the
coefficients of the SDE are smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 48
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence
Theorem (see, e.g., Kloeden & Platen (1992))
Let µ, σ, f : R → R be four times continuously differentiable with at most
polynomially growing derivatives. Moreover, let µ, σ : R → R be globally
Lipschitz continuous. Then there is a real number C > 0 such that
∣∣∣∣∣E
[
f(XT )]
−E
[
f(Y NN )]∣∣∣∣∣≤ C · 1
N
holds for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if the
coefficients of the SDE are smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 49
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence
Theorem (see, e.g., Kloeden & Platen (1992))
Let µ, σ, f : R → R be four times continuously differentiable with at most
polynomially growing derivatives. Moreover, let µ, σ : R → R be globally
Lipschitz continuous. Then there is a real number C > 0 such that
∣∣∣∣∣E
[
f(XT )]
−E
[
f(Y NN )]∣∣∣∣∣≤ C · 1
N
holds for all N ∈ N.
The stochastic Euler scheme converges in the numerically weak sense if the
coefficients of the SDE are smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 50
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence yields
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
≤ C · 1
N+ Cε ·
1
N(1−ε)≤ (C + Cε) ·
1
N(1−ε)P-a.s.
for all N ∈ N and all ε ∈ (0, 1) with an appropriate constant C ∈ (0,∞)and appropriate random variables Cε : Ω → [0,∞), ε ∈ (0, 1).
The Monte Carlo Euler method converges if the coefficients of the SDE are
smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 51
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence yields
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
≤ C · 1
N+ Cε ·
1
N(1−ε)≤ (C + Cε) ·
1
N(1−ε)P-a.s.
for all N ∈ N and all ε ∈ (0, 1) with an appropriate constant C ∈ (0,∞)and appropriate random variables Cε : Ω → [0,∞), ε ∈ (0, 1).
The Monte Carlo Euler method converges if the coefficients of the SDE are
smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 52
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence yields
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
≤ C · 1
N+ Cε ·
1
N(1−ε)≤ (C + Cε) ·
1
N(1−ε)P-a.s.
for all N ∈ N and all ε ∈ (0, 1) with an appropriate constant C ∈ (0,∞)and appropriate random variables Cε : Ω → [0,∞), ε ∈ (0, 1).
The Monte Carlo Euler method converges if the coefficients of the SDE are
smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 53
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence yields
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
≤ C · 1
N+ Cε ·
1
N(1−ε)≤ (C + Cε) ·
1
N(1−ε)P-a.s.
for all N ∈ N and all ε ∈ (0, 1) with an appropriate constant C ∈ (0,∞)and appropriate random variables Cε : Ω → [0,∞), ε ∈ (0, 1).
The Monte Carlo Euler method converges if the coefficients of the SDE are
smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 54
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Numerically weak convergence yields
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
≤ C · 1
N+ Cε ·
1
N(1−ε)≤ (C + Cε) ·
1
N(1−ε)P-a.s.
for all N ∈ N and all ε ∈ (0, 1) with an appropriate constant C ∈ (0,∞)and appropriate random variables Cε : Ω → [0,∞), ε ∈ (0, 1).
The Monte Carlo Euler method converges if the coefficients of the SDE are
smooth and globally Lipschitz continuous.
Arnulf Jentzen Global Lipschitz assumption
Page 55
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
The global Lipschitz assumption on the coefficients of the SDE is a serious
shortcoming:
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 56
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
The global Lipschitz assumption on the coefficients of the SDE is a serious
shortcoming:
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 57
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Overview
1 Stochastic differential equations (SDEs)
2 Computational problem and the Monte Carlo Euler method
3 Convergence for SDEs with globally Lipschitz continuous coefficients
4 Convergence for SDEs with superlinearly growing coefficients
Arnulf Jentzen Global Lipschitz assumption
Page 58
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Open problem
Convergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0
for SDEs with superlinearly growing coefficients such as
a SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
remained an open problem.
Arnulf Jentzen Global Lipschitz assumption
Page 59
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Open problem
Convergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0
for SDEs with superlinearly growing coefficients such as
a SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
remained an open problem.
Arnulf Jentzen Global Lipschitz assumption
Page 60
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Open problem
Convergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0
for SDEs with superlinearly growing coefficients such as
a SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
remained an open problem.
Arnulf Jentzen Global Lipschitz assumption
Page 61
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Open problem
Convergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0
for SDEs with superlinearly growing coefficients such as
a SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
remained an open problem.
Arnulf Jentzen Global Lipschitz assumption
Page 62
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Open problem
Convergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0
for SDEs with superlinearly growing coefficients such as
a SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
remained an open problem.
Arnulf Jentzen Global Lipschitz assumption
Page 63
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Open problem
Convergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0
for SDEs with superlinearly growing coefficients such as
a SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
remained an open problem.
Arnulf Jentzen Global Lipschitz assumption
Page 64
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Open problem
Convergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0
for SDEs with superlinearly growing coefficients such as
a SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
remained an open problem.
Arnulf Jentzen Global Lipschitz assumption
Page 65
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 66
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 67
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 68
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 69
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 70
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 71
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 72
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 73
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 74
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 75
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Gyöngy (1998) established pathwise convergence, i.e.
limN→∞
∣∣XT − Y N
N
∣∣ = 0 P-a.s..
Higham, Mao and Stuart (2002) showed a conditional result: If Euler’s
method has bounded moments
supN∈N
E
[
sup0≤n≤N
∣∣Y N
n
∣∣(2+ε)
]
< ∞
for some ε > 0, then Euler’s method converges in the sense
limN→∞
E∣∣XT − Y N
N
∣∣ = 0, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = 0.
“In general, it is not clear when such moment bounds can be expected
to hold for explicit methods with f , g ∈ C1.“
Arnulf Jentzen Global Lipschitz assumption
Page 76
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 77
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 78
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 79
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 80
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 81
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 82
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 83
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 84
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose P[σ(ξ) 6= 0
]> 0 and let α, C > 1 be such that
|µ(x)| ≥ |x|αC
and |σ(x)| ≤ C|x|
holds for all |x| ≥ C. If the exact solution of the SDE satisfies
E
[
|XT |p]
< ∞ for one p ∈ [1,∞), then
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
holds.
Strong and numerically weak convergence fails to hold if the diffusion
coefficient grows at most linearly and the drift coefficient grows superlinearly.
Arnulf Jentzen Global Lipschitz assumption
Page 85
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
Divergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = ∞, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = ∞
holds for:
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
Arnulf Jentzen Global Lipschitz assumption
Page 86
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
Divergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = ∞, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = ∞
holds for:
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
Arnulf Jentzen Global Lipschitz assumption
Page 87
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs I
Divergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = ∞, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = ∞
holds for:
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
Arnulf Jentzen Global Lipschitz assumption
Page 88
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs II
Divergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = ∞, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = ∞
holds for:
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
A Feller diffusion with logistic growth with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt +√
Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 89
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs II
Divergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = ∞, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = ∞
holds for:
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
A Feller diffusion with logistic growth with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt +√
Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 90
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Examples of SDEs II
Divergence of Euler’s method
limN→∞
E∣∣XT − Y N
N
∣∣ = ∞, lim
N→∞
∣∣∣E
[
(XT )2]
− E
[(Y N
N
)2]∣∣∣ = ∞
holds for:
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
A Feller diffusion with logistic growth with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt +√
Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 91
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof of divergence of Euler’s method in the numerically weak
sense
For simplicity we restrict our attention to the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
and show
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
for every p ∈ [1,∞). Simple observation: It is sufficient to show
limN→∞
E∣∣Y N
N
∣∣ = ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 92
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof of divergence of Euler’s method in the numerically weak
sense
For simplicity we restrict our attention to the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
and show
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
for every p ∈ [1,∞). Simple observation: It is sufficient to show
limN→∞
E∣∣Y N
N
∣∣ = ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 93
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof of divergence of Euler’s method in the numerically weak
sense
For simplicity we restrict our attention to the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
and show
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
for every p ∈ [1,∞). Simple observation: It is sufficient to show
limN→∞
E∣∣Y N
N
∣∣ = ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 94
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof of divergence of Euler’s method in the numerically weak
sense
For simplicity we restrict our attention to the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
and show
limN→∞
E
[∣∣XT − Y N
N
∣∣p]
= ∞, limN→∞
∣∣∣E
[
|XT |p]
− E
[∣∣Y N
N
∣∣p]∣∣∣ = ∞
for every p ∈ [1,∞). Simple observation: It is sufficient to show
limN→∞
E∣∣Y N
N
∣∣ = ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 95
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 96
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 97
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 98
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 99
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 100
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 101
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 102
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 103
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 104
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 105
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Proof: Define “event of instability”
ΩN :=
ω ∈ Ω
∣∣∣∣
supk∈1,2,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
for every N ∈ N. Claim:∣∣YN
k (ω)∣∣ ≥ (3N)(2(k−1)) ∀ k ∈ 1, 2, . . . , N (1)
for every ω ∈ ΩN and every N ∈ N.
We fix N ∈ N, ω ∈ ΩN and show (1) by induction on k ∈ 1, 2, . . . , N.
∣∣Y N
1 (ω)∣∣ =
∣∣∣∣Y N
0 (ω) − 1
N
(Y N
0 (ω))3
+(
W 1N(ω) − W0(ω)
)∣∣∣∣
=∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
Arnulf Jentzen Global Lipschitz assumption
Page 106
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 107
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 108
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 109
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 110
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 111
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 112
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 113
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 114
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 115
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 116
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Induction hypothesis |YNk (ω)| ≥ (3N)(2(k−1)) for one k ∈ 1, 2, . . . , N:
∣∣Y N
k+1(ω)∣∣ =
∣∣∣∣Y N
k (ω) − 1
N
(Y N
k (ω))3
+(
W k+1N
(ω) − W kN(ω))∣∣∣∣
≥∣∣∣∣
1
N
(Y N
k (ω))3
∣∣∣∣−∣∣Y N
k (ω)∣∣−∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣
≥ 1
N
∣∣Y N
k (ω)∣∣3 −
∣∣Y N
k (ω)∣∣− 1
≥ 1
N
∣∣Y N
k (ω)∣∣3 − 2
∣∣Y N
k (ω)∣∣2
=∣∣Y N
k (ω)∣∣2(
1
N
∣∣Y N
k (ω)∣∣− 2
)
≥∣∣Y N
k (ω)∣∣2(
1
N3N − 2
)
=∣∣Y N
k (ω)∣∣2
≥(
(3N)(2k−1))2
= (3N)(2k )
Arnulf Jentzen Global Lipschitz assumption
Page 117
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
In particular, we obtain
∣∣Y N
N (ω)∣∣ ≥ (3N)(2(N−1)) (2)
for all ω ∈ ΩN and all N ∈ N. Recall that
ΩN =
ω ∈ Ω
∣∣∣∣
supk∈1,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
holds and therefore
P[ΩN
]≥ e−cN2
(3)
for all N ∈ N with c ∈ (0,∞) appropriate. Combining (2) and (3) shows
E∣∣Y N
N
∣∣ ≥ P
[ΩN
]· (3N)(2(N−1)) ≥ e−cN2 · (3N)(2(N−1)) N→∞−−−→ ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 118
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
In particular, we obtain
∣∣Y N
N (ω)∣∣ ≥ (3N)(2(N−1)) (2)
for all ω ∈ ΩN and all N ∈ N. Recall that
ΩN =
ω ∈ Ω
∣∣∣∣
supk∈1,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
holds and therefore
P[ΩN
]≥ e−cN2
(3)
for all N ∈ N with c ∈ (0,∞) appropriate. Combining (2) and (3) shows
E∣∣Y N
N
∣∣ ≥ P
[ΩN
]· (3N)(2(N−1)) ≥ e−cN2 · (3N)(2(N−1)) N→∞−−−→ ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 119
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
In particular, we obtain
∣∣Y N
N (ω)∣∣ ≥ (3N)(2(N−1)) (2)
for all ω ∈ ΩN and all N ∈ N. Recall that
ΩN =
ω ∈ Ω
∣∣∣∣
supk∈1,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
holds and therefore
P[ΩN
]≥ e−cN2
(3)
for all N ∈ N with c ∈ (0,∞) appropriate. Combining (2) and (3) shows
E∣∣Y N
N
∣∣ ≥ P
[ΩN
]· (3N)(2(N−1)) ≥ e−cN2 · (3N)(2(N−1)) N→∞−−−→ ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 120
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
In particular, we obtain
∣∣Y N
N (ω)∣∣ ≥ (3N)(2(N−1)) (2)
for all ω ∈ ΩN and all N ∈ N. Recall that
ΩN =
ω ∈ Ω
∣∣∣∣
supk∈1,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
holds and therefore
P[ΩN
]≥ e−cN2
(3)
for all N ∈ N with c ∈ (0,∞) appropriate. Combining (2) and (3) shows
E∣∣Y N
N
∣∣ ≥ P
[ΩN
]· (3N)(2(N−1)) ≥ e−cN2 · (3N)(2(N−1)) N→∞−−−→ ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 121
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
In particular, we obtain
∣∣Y N
N (ω)∣∣ ≥ (3N)(2(N−1)) (2)
for all ω ∈ ΩN and all N ∈ N. Recall that
ΩN =
ω ∈ Ω
∣∣∣∣
supk∈1,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
holds and therefore
P[ΩN
]≥ e−cN2
(3)
for all N ∈ N with c ∈ (0,∞) appropriate. Combining (2) and (3) shows
E∣∣Y N
N
∣∣ ≥ P
[ΩN
]· (3N)(2(N−1)) ≥ e−cN2 · (3N)(2(N−1)) N→∞−−−→ ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 122
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
In particular, we obtain
∣∣Y N
N (ω)∣∣ ≥ (3N)(2(N−1)) (2)
for all ω ∈ ΩN and all N ∈ N. Recall that
ΩN =
ω ∈ Ω
∣∣∣∣
supk∈1,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
holds and therefore
P[ΩN
]≥ e−cN2
(3)
for all N ∈ N with c ∈ (0,∞) appropriate. Combining (2) and (3) shows
E∣∣Y N
N
∣∣ ≥ P
[ΩN
]· (3N)(2(N−1)) ≥ e−cN2 · (3N)(2(N−1)) N→∞−−−→ ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 123
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
In particular, we obtain
∣∣Y N
N (ω)∣∣ ≥ (3N)(2(N−1)) (2)
for all ω ∈ ΩN and all N ∈ N. Recall that
ΩN =
ω ∈ Ω
∣∣∣∣
supk∈1,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
holds and therefore
P[ΩN
]≥ e−cN2
(3)
for all N ∈ N with c ∈ (0,∞) appropriate. Combining (2) and (3) shows
E∣∣Y N
N
∣∣ ≥ P
[ΩN
]· (3N)(2(N−1)) ≥ e−cN2 · (3N)(2(N−1)) N→∞−−−→ ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 124
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
In particular, we obtain
∣∣Y N
N (ω)∣∣ ≥ (3N)(2(N−1)) (2)
for all ω ∈ ΩN and all N ∈ N. Recall that
ΩN =
ω ∈ Ω
∣∣∣∣
supk∈1,...,N−1
∣∣∣W k+1
N(ω) − W k
N(ω)∣∣∣ ≤ 1,
∣∣∣W 1
N(ω) − W0(ω)
∣∣∣ ≥ 3N
holds and therefore
P[ΩN
]≥ e−cN2
(3)
for all N ∈ N with c ∈ (0,∞) appropriate. Combining (2) and (3) shows
E∣∣Y N
N
∣∣ ≥ P
[ΩN
]· (3N)(2(N−1)) ≥ e−cN2 · (3N)(2(N−1)) N→∞−−−→ ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 125
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations of the first absolute moment of the solution of a SDE
Consider the SDE
dXt = −10 sgn(Xt) |Xt |1.1 dt + 4 dWt , X0 = 0, t ∈ [0, 10].
The first absolute moment of XT with T = 10 satisfies
E
[
|X10|]
≈ 0.7141 .
Arnulf Jentzen Global Lipschitz assumption
Page 126
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations of the first absolute moment of the solution of a SDE
Consider the SDE
dXt = −10 sgn(Xt) |Xt |1.1 dt + 4 dWt , X0 = 0, t ∈ [0, 10].
The first absolute moment of XT with T = 10 satisfies
E
[
|X10|]
≈ 0.7141 .
Arnulf Jentzen Global Lipschitz assumption
Page 127
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations of the first absolute moment of the solution of a SDE
Consider the SDE
dXt = −10 sgn(Xt) |Xt |1.1 dt + 4 dWt , X0 = 0, t ∈ [0, 10].
The first absolute moment of XT with T = 10 satisfies
E
[
|X10|]
≈ 0.7141 .
Arnulf Jentzen Global Lipschitz assumption
Page 128
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
0 5 10 15 20 25 30 35 40 45 5010
0
1050
10100
10150
10200
10250
10300
Number of time steps N
Firs
t abs
olut
e m
omen
t of t
he E
uler
sch
eme
Absolute moment E|YNN
| of the Euler scheme versus N=1,2,3,...,50
Arnulf Jentzen Global Lipschitz assumption
Page 129
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 130
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 131
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 132
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 133
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 134
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 135
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 136
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 137
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 138
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 139
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 140
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 141
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 142
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and multiplicative noise
Consider the SDE
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3].
The second moment of XT with T = 3 satisfies
E[(X3)
2]≈ 1.5423 .
Different simulation values of the Monte Carlo Euler method with 300 time
steps and 10 000 Monte Carlo runs:
NaN 0.5097 NaN 0.5378 0.5197
0.5243 NaN NaN 0.5475 NaN
Arnulf Jentzen Global Lipschitz assumption
Page 143
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 144
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 145
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 146
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 147
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 148
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 149
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 150
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 151
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 152
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 153
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 154
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Do we need new numerical methods which converge in the numerically
weak sense?
Central observation: Numerically weak convergence fails to hold, i.e.
limN→∞
E
[
(Y NN )2]
= ∞
but the Monte Carlo Euler method
limN→∞
1
N2
N2∑
m=1
(YN,mN )2
= E
[
(XT )2]
P-a.s.
nevertheless converges for a large class of SDEs.
Arnulf Jentzen Global Lipschitz assumption
Page 155
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose that µ, σ, f : R → R are four times continuously differentiable
functions with at most polynomially growing derivatives. Moreover, let σ be
globally Lipschitz continuous and let µ be globally one-sided Lipschitz
continuous, i.e.
(x − y) · (µ(x) − µ(y)) ≤ L (x − y)2
holds for all x, y ∈ R where L ∈ (0,∞) is a fixed constant. Then there are
F/B([0,∞))-measurable mappings Cε : Ω → [0,∞), ε ∈ (0, 1), and a
set Ω ∈ F with P[Ω] = 1 such that
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(YN,mN (ω))
∣∣∣∣∣∣
≤ Cε(ω) · 1
N(1−ε)
holds for every ω ∈ Ω, N ∈ N and every ε ∈ (0, 1).
Arnulf Jentzen Global Lipschitz assumption
Page 156
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose that µ, σ, f : R → R are four times continuously differentiable
functions with at most polynomially growing derivatives. Moreover, let σ be
globally Lipschitz continuous and let µ be globally one-sided Lipschitz
continuous, i.e.
(x − y) · (µ(x) − µ(y)) ≤ L (x − y)2
holds for all x, y ∈ R where L ∈ (0,∞) is a fixed constant. Then there are
F/B([0,∞))-measurable mappings Cε : Ω → [0,∞), ε ∈ (0, 1), and a
set Ω ∈ F with P[Ω] = 1 such that
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(YN,mN (ω))
∣∣∣∣∣∣
≤ Cε(ω) · 1
N(1−ε)
holds for every ω ∈ Ω, N ∈ N and every ε ∈ (0, 1).
Arnulf Jentzen Global Lipschitz assumption
Page 157
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose that µ, σ, f : R → R are four times continuously differentiable
functions with at most polynomially growing derivatives. Moreover, let σ be
globally Lipschitz continuous and let µ be globally one-sided Lipschitz
continuous, i.e.
(x − y) · (µ(x) − µ(y)) ≤ L (x − y)2
holds for all x, y ∈ R where L ∈ (0,∞) is a fixed constant. Then there are
F/B([0,∞))-measurable mappings Cε : Ω → [0,∞), ε ∈ (0, 1), and a
set Ω ∈ F with P[Ω] = 1 such that
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(YN,mN (ω))
∣∣∣∣∣∣
≤ Cε(ω) · 1
N(1−ε)
holds for every ω ∈ Ω, N ∈ N and every ε ∈ (0, 1).
Arnulf Jentzen Global Lipschitz assumption
Page 158
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose that µ, σ, f : R → R are four times continuously differentiable
functions with at most polynomially growing derivatives. Moreover, let σ be
globally Lipschitz continuous and let µ be globally one-sided Lipschitz
continuous, i.e.
(x − y) · (µ(x) − µ(y)) ≤ L (x − y)2
holds for all x, y ∈ R where L ∈ (0,∞) is a fixed constant. Then there are
F/B([0,∞))-measurable mappings Cε : Ω → [0,∞), ε ∈ (0, 1), and a
set Ω ∈ F with P[Ω] = 1 such that
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(YN,mN (ω))
∣∣∣∣∣∣
≤ Cε(ω) · 1
N(1−ε)
holds for every ω ∈ Ω, N ∈ N and every ε ∈ (0, 1).
Arnulf Jentzen Global Lipschitz assumption
Page 159
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose that µ, σ, f : R → R are four times continuously differentiable
functions with at most polynomially growing derivatives. Moreover, let σ be
globally Lipschitz continuous and let µ be globally one-sided Lipschitz
continuous, i.e.
(x − y) · (µ(x) − µ(y)) ≤ L (x − y)2
holds for all x, y ∈ R where L ∈ (0,∞) is a fixed constant. Then there are
F/B([0,∞))-measurable mappings Cε : Ω → [0,∞), ε ∈ (0, 1), and a
set Ω ∈ F with P[Ω] = 1 such that
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(YN,mN (ω))
∣∣∣∣∣∣
≤ Cε(ω) · 1
N(1−ε)
holds for every ω ∈ Ω, N ∈ N and every ε ∈ (0, 1).
Arnulf Jentzen Global Lipschitz assumption
Page 160
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose that µ, σ, f : R → R are four times continuously differentiable
functions with at most polynomially growing derivatives. Moreover, let σ be
globally Lipschitz continuous and let µ be globally one-sided Lipschitz
continuous, i.e.
(x − y) · (µ(x) − µ(y)) ≤ L (x − y)2
holds for all x, y ∈ R where L ∈ (0,∞) is a fixed constant. Then there are
F/B([0,∞))-measurable mappings Cε : Ω → [0,∞), ε ∈ (0, 1), and a
set Ω ∈ F with P[Ω] = 1 such that
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(YN,mN (ω))
∣∣∣∣∣∣
≤ Cε(ω) · 1
N(1−ε)
holds for every ω ∈ Ω, N ∈ N and every ε ∈ (0, 1).
Arnulf Jentzen Global Lipschitz assumption
Page 161
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose that µ, σ, f : R → R are four times continuously differentiable
functions with at most polynomially growing derivatives. Moreover, let σ be
globally Lipschitz continuous and let µ be globally one-sided Lipschitz
continuous, i.e.
(x − y) · (µ(x) − µ(y)) ≤ L (x − y)2
holds for all x, y ∈ R where L ∈ (0,∞) is a fixed constant. Then there are
F/B([0,∞))-measurable mappings Cε : Ω → [0,∞), ε ∈ (0, 1), and a
set Ω ∈ F with P[Ω] = 1 such that
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(YN,mN (ω))
∣∣∣∣∣∣
≤ Cε(ω) · 1
N(1−ε)
holds for every ω ∈ Ω, N ∈ N and every ε ∈ (0, 1).
Arnulf Jentzen Global Lipschitz assumption
Page 162
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Theorem (Hutzenthaler & J (2009))
Suppose that µ, σ, f : R → R are four times continuously differentiable
functions with at most polynomially growing derivatives. Moreover, let σ be
globally Lipschitz continuous and let µ be globally one-sided Lipschitz
continuous, i.e.
(x − y) · (µ(x) − µ(y)) ≤ L (x − y)2
holds for all x, y ∈ R where L ∈ (0,∞) is a fixed constant. Then there are
F/B([0,∞))-measurable mappings Cε : Ω → [0,∞), ε ∈ (0, 1), and a
set Ω ∈ F with P[Ω] = 1 such that
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(YN,mN (ω))
∣∣∣∣∣∣
≤ Cε(ω) · 1
N(1−ε)
holds for every ω ∈ Ω, N ∈ N and every ε ∈ (0, 1).
Arnulf Jentzen Global Lipschitz assumption
Page 163
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The theorem applies to ...
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 164
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The theorem applies to ...
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 165
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The theorem applies to ...
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 166
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The theorem applies to ...
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 167
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
The theorem applies to ...
Black-Scholes model with µ, σ, x0 ∈ (0,∞):
dXt = µ Xt dt + σ Xt dWt , X0 = x0, t ∈ [0, T ]
A SDE with a cubic drift and additive noise:
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1]
A SDE with a cubic drift and multiplicative noise:
dXt = −X 3t dt + 6 Xt dWt , X0 = 1, t ∈ [0, 3]
A stochastic Verhulst equation with η, x0 ∈ (0,∞):
dXt = Xt (η − Xt) dt + Xt dWt , X0 = x0, t ∈ [0, T ]
Arnulf Jentzen Global Lipschitz assumption
Page 168
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 169
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 170
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 171
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 172
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 173
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 174
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 175
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 176
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 177
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 178
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 179
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 180
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 181
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 182
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 183
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Simulations for a SDE with a cubic drift and additive noise
Consider the SDE
dXt = −X 3t dt + dWt , X0 = 0, t ∈ [0, 1].
The second moment of XT with T = 1 satisfies
E[(X3)
2]≈ 0.4529 .
Different simulation values of the Monte Carlo Euler method:
N = 20 N = 21 N = 22 N = 23 N = 24
1.4516 0.5166 0.4329 0.5308 0.4285
N = 25 N = 26 N = 27 N = 28 N = 29
0.4452 0.4602 0.4517 0.4548 0.4537
Arnulf Jentzen Global Lipschitz assumption
Page 184
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Summary
Counterexamples of numerically weak convergence of the
stochastic Euler scheme if the coefficients of the SDE grow
superlinearly.
The Monte Carlo Euler method nevertheless converges if the drift
function is globally one-sided Lipschitz continuous, the diffusion function
is globally Lipschitz continuous and both the drift and diffusion function
are smooth with at most polynomially growing derivatives.
Arnulf Jentzen Global Lipschitz assumption
Page 185
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Summary
Counterexamples of numerically weak convergence of the
stochastic Euler scheme if the coefficients of the SDE grow
superlinearly.
The Monte Carlo Euler method nevertheless converges if the drift
function is globally one-sided Lipschitz continuous, the diffusion function
is globally Lipschitz continuous and both the drift and diffusion function
are smooth with at most polynomially growing derivatives.
Arnulf Jentzen Global Lipschitz assumption
Page 186
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Summary
Counterexamples of numerically weak convergence of the
stochastic Euler scheme if the coefficients of the SDE grow
superlinearly.
The Monte Carlo Euler method nevertheless converges if the drift
function is globally one-sided Lipschitz continuous, the diffusion function
is globally Lipschitz continuous and both the drift and diffusion function
are smooth with at most polynomially growing derivatives.
Arnulf Jentzen Global Lipschitz assumption
Page 187
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Summary
Counterexamples of numerically weak convergence of the
stochastic Euler scheme if the coefficients of the SDE grow
superlinearly.
The Monte Carlo Euler method nevertheless converges if the drift
function is globally one-sided Lipschitz continuous, the diffusion function
is globally Lipschitz continuous and both the drift and diffusion function
are smooth with at most polynomially growing derivatives.
Arnulf Jentzen Global Lipschitz assumption
Page 188
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Summary
Counterexamples of numerically weak convergence of the
stochastic Euler scheme if the coefficients of the SDE grow
superlinearly.
The Monte Carlo Euler method nevertheless converges if the drift
function is globally one-sided Lipschitz continuous, the diffusion function
is globally Lipschitz continuous and both the drift and diffusion function
are smooth with at most polynomially growing derivatives.
Arnulf Jentzen Global Lipschitz assumption
Page 189
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
We should not split the error of the Monte Carlo Euler method
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method →0
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣
︸ ︷︷ ︸
time discretization error →∞
+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
︸ ︷︷ ︸
statistical error →∞
P-a.s. as N → ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 190
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
We should not split the error of the Monte Carlo Euler method
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method →0
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣
︸ ︷︷ ︸
time discretization error →∞
+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
︸ ︷︷ ︸
statistical error →∞
P-a.s. as N → ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 191
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
We should not split the error of the Monte Carlo Euler method
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method →0
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣
︸ ︷︷ ︸
time discretization error →∞
+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
︸ ︷︷ ︸
statistical error →∞
P-a.s. as N → ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 192
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
We should not split the error of the Monte Carlo Euler method
∣∣∣E
[
f(
XT
)]
− 1
N2
N2∑
m=1
f(
YN,mN
)∣∣∣
︸ ︷︷ ︸
error of the Monte Carlo Euler method →0
≤∣∣∣E
[
f(XT
)]
− E
[
f(
Y NN
)]∣∣∣
︸ ︷︷ ︸
time discretization error →∞
+∣∣∣E
[
f(Y N
N
)]
− 1
N2
N2∑
m=1
f(Y
N,mN
)∣∣∣
︸ ︷︷ ︸
statistical error →∞
P-a.s. as N → ∞.
Arnulf Jentzen Global Lipschitz assumption
Page 193
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
Strong and numerically weak error estimates are convenient since stochastic
calculus is an L2-calculus (Itô isometry, etc.).
But, if Euler’s method is used to solve one of the nonlinear problems above,
then one needs different concepts such as
∣∣XT − Y N
N
∣∣ N→∞−−−→ 0 P-a.s.
for the strong approximation problem (Gyöngy (1998)) and
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(Y NN )
∣∣∣∣∣∣
N→∞−−−→ 0 P-a.s.
for the weak approximation problem (Hutzenthaler & J (2009)).
Arnulf Jentzen Global Lipschitz assumption
Page 194
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
Strong and numerically weak error estimates are convenient since stochastic
calculus is an L2-calculus (Itô isometry, etc.).
But, if Euler’s method is used to solve one of the nonlinear problems above,
then one needs different concepts such as
∣∣XT − Y N
N
∣∣ N→∞−−−→ 0 P-a.s.
for the strong approximation problem (Gyöngy (1998)) and
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(Y NN )
∣∣∣∣∣∣
N→∞−−−→ 0 P-a.s.
for the weak approximation problem (Hutzenthaler & J (2009)).
Arnulf Jentzen Global Lipschitz assumption
Page 195
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
Strong and numerically weak error estimates are convenient since stochastic
calculus is an L2-calculus (Itô isometry, etc.).
But, if Euler’s method is used to solve one of the nonlinear problems above,
then one needs different concepts such as
∣∣XT − Y N
N
∣∣ N→∞−−−→ 0 P-a.s.
for the strong approximation problem (Gyöngy (1998)) and
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(Y NN )
∣∣∣∣∣∣
N→∞−−−→ 0 P-a.s.
for the weak approximation problem (Hutzenthaler & J (2009)).
Arnulf Jentzen Global Lipschitz assumption
Page 196
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
Strong and numerically weak error estimates are convenient since stochastic
calculus is an L2-calculus (Itô isometry, etc.).
But, if Euler’s method is used to solve one of the nonlinear problems above,
then one needs different concepts such as
∣∣XT − Y N
N
∣∣ N→∞−−−→ 0 P-a.s.
for the strong approximation problem (Gyöngy (1998)) and
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(Y NN )
∣∣∣∣∣∣
N→∞−−−→ 0 P-a.s.
for the weak approximation problem (Hutzenthaler & J (2009)).
Arnulf Jentzen Global Lipschitz assumption
Page 197
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
Strong and numerically weak error estimates are convenient since stochastic
calculus is an L2-calculus (Itô isometry, etc.).
But, if Euler’s method is used to solve one of the nonlinear problems above,
then one needs different concepts such as
∣∣XT − Y N
N
∣∣ N→∞−−−→ 0 P-a.s.
for the strong approximation problem (Gyöngy (1998)) and
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(Y NN )
∣∣∣∣∣∣
N→∞−−−→ 0 P-a.s.
for the weak approximation problem (Hutzenthaler & J (2009)).
Arnulf Jentzen Global Lipschitz assumption
Page 198
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
Conclusion
Strong and numerically weak error estimates are convenient since stochastic
calculus is an L2-calculus (Itô isometry, etc.).
But, if Euler’s method is used to solve one of the nonlinear problems above,
then one needs different concepts such as
∣∣XT − Y N
N
∣∣ N→∞−−−→ 0 P-a.s.
for the strong approximation problem (Gyöngy (1998)) and
∣∣∣∣∣∣
E
[
f(XT )
]
− 1
N2
N2∑
m=1
f(Y NN )
∣∣∣∣∣∣
N→∞−−−→ 0 P-a.s.
for the weak approximation problem (Hutzenthaler & J (2009)).
Arnulf Jentzen Global Lipschitz assumption
Page 199
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
References
Hutzenthaler and J (2009), Non-globally Lipschitz Counterexamples for
the stochastic Euler scheme.
Hutzenthaler and J (2009), Convergence of the stochastic Euler
scheme for locally Lipschitz coefficients.
Arnulf Jentzen Global Lipschitz assumption
Page 200
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
References
Hutzenthaler and J (2009), Non-globally Lipschitz Counterexamples for
the stochastic Euler scheme.
Hutzenthaler and J (2009), Convergence of the stochastic Euler
scheme for locally Lipschitz coefficients.
Arnulf Jentzen Global Lipschitz assumption
Page 201
Stochastic differential equations (SDEs)Computational problem and the Monte Carlo Euler method
Convergence for SDEs with globally Lipschitz continuous coefficientsConvergence for SDEs with superlinearly growing coefficients
References
Hutzenthaler and J (2009), Non-globally Lipschitz Counterexamples for
the stochastic Euler scheme.
Hutzenthaler and J (2009), Convergence of the stochastic Euler
scheme for locally Lipschitz coefficients.
Arnulf Jentzen Global Lipschitz assumption