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Model Uncertainty and Robust Duality in Finance
Bernt ØksendalCMA, University of Oslo, Norway
andNorwegian School of Economics (NHH),Bergen, Norway
Joint work with Agnès Sulem (INRIA Paris-Rocquencourt)
Conference on Stochastic Analysis and Applications,Hammamet,
Tunisia, October 2013
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Introduction
In the latest years, partly due to the financial crisis, there
has beenan increased focus on model uncertainty in mathematical
finance.Here model uncertainty is understood in the sense of
uncertaintywith respect to the choice of the underlying probability
measure.This is sometimes called Knightian uncertainty, after the
Universityof Chicago economist Frank Knight (1885-1972).
Thecorresponding model uncertainty stochastic control problem
isoften called robust control.
Optimizing under model uncertainty is also of interest because
it isrelated to risk minimization problems, where risk is
interpreted inthe setting of a convex risk measure (Föllmer &
Schied (2002),Frittelli & Rosazza-Gianin (2002)).
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Outline
I Duality method in portfolio optimization; use of
dualityrelationships in the space of convex functions
andsemimartingales.
I Itô - Lévy market case
I Maximum principles for stochastic control
I Optimal portfolio and optimal scenario
I Model uncertainty and robust control
I Extension to robust duality
I Illustrating examples
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Financial marketLet S(t) ; 0 ≤ t ≤ T represent discounted unit
price of a riskyasset at time t. We assume that S(t) is a
semimartingale on afiltered probability space (Ω,F ,
{Ft}t≥0,P).
T is a finite time horizon.
Let ϕ(t) be an Ft-predictable S-integrable portfolio process,
givingthe number of units held of the risky asset at time t.If ϕ(t)
is self-financing, the corresponding wealth processX (t) = X xϕ(t)
is given by
X (t) = x +
∫ t0ϕ(s)dS(s) ; 0 ≤ t ≤ T , (0.1)
where x ≥ 0 is the initial value of the wealth.We say that ϕ is
admissible and write ϕ ∈ A if Xϕ(t) ≥ 0 for allt ∈ [0,T ] a.s.
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Let M be the set of probability measures Q which are
equivalentlocal martingale measures (ELMM), in the sense that Q ∼ P
andS(t) is a local martingale under Q.We assume that
M 6= ∅
which is related to absence of arbitrage opportunities on
thesecurity market.
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Let U : [0,∞]→ R be a utility function, strictly increasing,
strictlyconcave, C1 and satisfying the Inada conditions:
U ′(0) = limx→0+
U ′(x) =∞ U ′(∞) = limx→∞
U ′(x) = 0.
Let V be the conjugate function of U:
V (y) := supx>0{U(x)− xy} ; y > 0
V is strictly convex, decreasing, C1 and satisfies
V ′(0) = −∞, V ′(∞) = 0,V (0) = U(∞), and V (∞) = U(0).
Moreover,
U(x) = infy>0{V (y) + xy} ; x > 0, and
U ′(x) = y ⇔ x = −V ′(y).
i.e. U ′ is the inverse function of −V ′.
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Duality approach for optimal portfolio problemThe primal problem
is to find ϕ∗ ∈ A such that
u(x) := supϕ∈A
E [U(X xϕ(T ))] = E [U(Xxϕ∗(T ))]. (0.2)
The dual problem to (0.2) is for given y > 0 to find Q∗ ∈M
s.t.
v(y) = infQ∈M
E
[V
(ydQ
dP
)]= E
[V
(ydQ∗
dP
)]. (0.3)
Kramkov and Schachermayer (2003) prove that, under
someconditions, ϕ∗ and Q∗ exist and are related by
U ′(X xϕ∗(T )) = ydQ∗
dPwith y = u′(x) (0.4)
i.e.
X xϕ∗(T ) = −V ′(ydQ∗
dP
)with x = −v ′(y). (0.5)
We extend this result by using stochastic control theory when
therisky asset price S(t) is an Itô-Lévy process.
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Itô-Lévy marketWe consider a financial market model, where the
discounted unitprice S(t) of the risky asset is given by a jump
diffusion of the formdS(t) = S(t−)
(btdt + σtdBt +
∫Rγ(t, ζ)Ñ(dt, dζ)
); 0 ≤ t ≤ T
S(0) > 0
(0.6)where bt , σt and γ(t, ζ) are predictable processes
satisfyingγ(t, ζ) > −1 and
E
[∫ T0
{|bt |+ σ2t +
∫Rγ2(t, ζ)ν(dζ)
}dt
]
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Primal ProblemLet ϕ(t) be a self-financing portfolio, giving the
number of unitsheld of the risky asset at time t and let X (t) = X
xϕ(t) be thecorresponding wealth process given by{dX (t) =
ϕ(t)S(t−)
[btdt + σtdBt +
∫R γ(t, ζ)Ñ(dt, dζ)
]; 0 ≤ t ≤ T
X (0) = x > 0.
We say that ϕ is admissible and write ϕ ∈ A ifI ϕ(t) is an
Ft-predictable S-integrable processI X (t) > 0 for all t ∈ [0,T
] a.s.I E
[∫ T0 ϕ(t)
2S(t)2{b2t + σ
2t +
∫R γ
2(t, ζ)ν(dζ)}dt] 0 s.t. E [∫ T
0 |X (t)|2+�dt]
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Primal problem : Find ϕ∗ ∈ A such that
u(x) := supϕ∈A
E [U(X xϕ(T ))] = E [U(Xxϕ∗(T ))]. (0.8)
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Dual ProblemWe represent the set M of ELMM by the family of
positivemeasures Qθ of the form
dQθ(ω) = Gθ(T )dP(ω) on FT , where{dGθ(t) = Gθ(t
−)[θ0(t)dBt +
∫R θ1(t, ζ)Ñ(dt, dζ)
]; 0 ≤ t ≤ T
Gθ(0) = y > 0.
and θ = (θ0, θ1) is a predictable process satisfying the
conditions
E
[∫ T0
{θ20(t) +
∫Rθ21(t, ζ)ν(dζ)
}dt
]
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Let Θ denote the set of processes θ satisfying the above
conditions.
Dual problem : For given y > 0 , find θ∗y ∈ Θ and v(y) s.
t.
− v(y) := supθ∈Θ
E [−V (G yθ (T ))] = E [−V (Gθ∗y (T ))]. (0.10)
We will use the maximum principle for stochastic control to
studythe primal problem and relate it to the dual problem.
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Stochastic Control of Jump DiffusionsThere are two main methods
for solving stochastic controlproblems:
(i) Dynamic programming and the HJB equation(R. Bellman ...)
This method is very efficient when it works, but itassumes that the
system is Markovian.
(ii) The maximum principle with the associated BSDE(Pontryagin,
Bismut, Bensoussan, Pardoux-Peng, Framstad-Ø.-Sulem,...) This
method is more robust; it does not assume thatthe system is
Markovian and it applies even to partial information,to systems
with delay, to FBSDEs and to SPDEs. The drawback isthat it involves
complicated BSDEs, ABSDEs, BSPDEs, ...
Thus, in our general non-Markovian setting it is natural to use
themaximum principle, and we will now use it to study the primal
anddual problem above.
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Maximum Principles for Stochastic Control
Consider the following general stochastic control problem:
supu∈A
E
[∫ T0
f (t,X (t), u(t), ω)dt + φ(X (T ), ω)
](0.11)
where A is a family of admissible F-predictable controls and
dX (t) = b(t,X (t), u(t), ω)dt + σ(t,X (t), u(t), ω)dBt
(0.12)
+
∫Rγ(t,X (t), u(t), ω, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T ; X (0) = x ∈
R
Assume b, σ, γ, f , φ ∈ C 1 and
E [
∫ T0
(|∇b|2 + |∇σ|2 + ‖∇γ‖2 + |∇f |2)(t)dt]
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For u ∈ A we assume the solution X u(t) of (0.12) exists, is
uniqueand satisfies, for some � > 0, E [
∫ T0 |X
u(t)|2+�dt]
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Theorem (Sufficient Maximum Principle, Ø. & Sulem,
JOTA2012)
Let û ∈ A with corresponding solutions X̂ , p̂, q̂, r̂ be such
that
supv∈U
H(t, X̂ (t), v , p̂(t), q̂(t), r̂(t, ·)) = H(t, X̂ (t), û(t),
p̂(t), q̂(t), r̂(t, ·)).
Assume
I The function x 7→ φ(x) is concaveI (The Arrow condition) The
function
H(x) := supv∈U
H(t, x , v , p̂(t), q̂(t), r̂(t, ·))
is concave for all t ∈ [0,T ].Then û is an optimal control for
the problem (0.11).
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Theorem (Necessary Maximum Principle, Ø. & Sulem,
JOTA2012)
Assume
I For all t0 ∈ [0,T ] and all bounded Ft0-measurable
randomvariables α(ω), the control β := 1[t0,T ]α belongs to A.
I For all u, β ∈ A with β bounded, there exists δ > 0 s.t.ũ
:= u + aβ belongs to A for all a ∈ (−δ, δ).
I The derivative process xt :=ddaX
u+aβ(t) |a=0, exists andbelongs to L2(dm × dP).
Then, if u∗ ∈ A is optimal,
∂H
∂u(t,X ∗(t), u∗, p∗(t), q∗(t), r∗(t, ·)) = 0 for all t ∈ [0,T
].
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To illustrate the method, we first prove two useful auxiliary
results:
LemmaPrimal problem.Let ϕ̂(t) ∈ A. Then ϕ̂(t) is optimal for the
primal problem (0.8) ifand only if the (unique) solution (p̂1, q̂1,
r̂1) of the BSDEdp̂1(t) = q̂1(t)dB(t) +
∫Rr̂1(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T
p̂1(T ) = U′(X xϕ̂(T )).
(0.14)satisfies the equation
b(t)p̂1(t) + σ(t)q̂1(t) +
∫Rγ(t, ζ)r̂1(t, ζ)ν(dζ) = 0 ; t ∈ [0,T ].
(0.15)
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Proof. (Sketch)(i) First assume that ϕ̂ ∈ A is optimal for the
primal problem(0.8). Then by the necessary maximum principle,
thecorresponding Hamiltonian, given by
H1(t, x , ϕ, p, q, r) = ϕS(t−)(b(t)p+σq+
∫Rγ(t, ζ)r(ζ)Ñ(dt, dζ))
(0.16)satisfies
∂H1∂ϕ
(t, x , ϕ, p̂1(t), q̂1(t), r̂1(t, ·)) |ϕ=ϕ̂(t)= 0,
where (p̂1, q̂1, r̂1) satisfies (0.14) since∂H1∂x = 0. This
implies
(0.15).
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(ii) Conversely, suppose the solution (p̂1, q̂1, r̂1) of the
BSDE(0.14) satisfies (0.15). Then ϕ̂, with the associated (p̂1,
q̂1, r̂1)satisfies the conditions for the sufficient maximum
principle, andhence ϕ̂ is optimal. �
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We now turn to the dual problem (0.10). In the following
weassume that
σ(t) 6= 0 for all t ∈ [0,T ]. (0.17)This is more because of
convenience and notational simplicity thanof necessity.
LemmaDual problem.Let θ̂ ∈ Θ. Then θ̂ is an optimal scenario for
the dual problem(0.10) if and only if the solution (p̂2, q̂2, r̂2)
of the BSDEdp̂2(t) =
q̂2(t)
σ(t)b(t)dt + q̂2(t)dB(t) +
∫Rr̂2(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T
p̂2(T ) = −V ′(G yθ̂ (T )).(0.18)
also satisfies
− q̂2(t)σ(t)
γ(t, ζ) + r̂2(t, ζ) = 0 ; 0 ≤ t ≤ T . (0.19)
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Proof.The Hamiltonian H2 associated to (0.10) is
H2(t, g , θ0, θ1, p, q, r) = gθ0q + g
∫Rθ1(ζ)r(ζ)ν(dζ). (0.20)
By (0.17), the constraint (0.9) can be written
θ0(t) = θ̃0(t) = −1
σ(t)
{b(t) +
∫Rγ(t, ζ)θ1(t, ζ)ν(dζ)
}; t ∈ [0,T ].
(0.21)
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Substituting this into (0.20) we get
H̃2(t, g , θ1, p2, q2, r2) := H2(t, g , θ̃0, θ1, p2, q2, r2)
= g
(− q2σ(t)
{b(t) +
∫Rγ(t, ζ)θ1(ζ)ν(dζ)
}+
∫Rθ1(ζ)r2(ζ)ν(dζ)
).
(0.22)
The equation for the adjoint processes (p2, q2, r2) is thus
thefollowing BSDE:
dp2(t) =
[q2(t)
σ(t)b(t) +
∫Rθ1(t, ζ)
(q2(t)
σ(t)γ(t, ζ)− r2(t, ζ)
)ν(dζ)
]dt
+q2(t)dB(t) +
∫Rr2(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T
p2(T ) = −V ′(Gθ(T )).(0.23)
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If there exists a maximiser θ̂1 for H̃2 then
(∇θ1H̃2)θ1=θ̂1 = 0, (0.24)
i.e.
− q̂2(t)σ(t)
γ(t, ζ) + r̂2(t, ζ) = 0 ; 0 ≤ t ≤ T , (0.25)
where (p̂2, q̂2, r̂2) is the solution of (0.23) corresponding to
θ = θ̂.We thus get (0.18) and this ends the necessary part.
The sufficient part follows from the fact that the functionsg →
−V (g) and
g → supθ1
H̃2(t, g , θ1, p̂2(t), q̂2(t), r̂2(t, ·)) = −gq̂2(t)
σ(t)b(t)
are concave. �
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Optimal Scenario and Optimal Portfolio
We now use the above general machinery of stochastic control
toobtain an explicit connection between an optimal portfolio ϕ̂ ∈
Afor the primal problem and an optimal θ̂ ∈ Θ for the dual
problem:
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Theorem 1 a) Suppose ϕ̂ ∈ A is optimal for the primal
problem
supϕ∈A
E [U(X xϕ(T ))].
Let (p1, q1, r1) be the associated adjoint processes, solution
of theBSDEdp1(t) = q1(t)dB(t) +
∫Rr1(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T
p1(T ) = U′(X xϕ̂(T )).
Define
θ̂0(t) :=q1(t)
p1(t−); θ̂1(t, ζ) :=
r1(t, ζ)
p1(t−).
Suppose E [∫ T
0 {θ̂20(t) +
∫R θ̂
21(t, ζ)ν(dζ)}dt] −1.
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Then θ̂ = (θ̂0, θ̂1) ∈ Θ is optimal for the dual problem
supθ∈Θ
E [−V (G yθ (T ))]
with initial value y = p1(0) .
Moreover, the optimal density process G yθ̂
starting at y = p1(0)coincides with the optimal adjoint process
p1 for the primalproblem, i.e.
G yθ̂
(t) = p1(t); 0 ≤ t ≤ T . (0.26)
In particular,G yθ̂
(T ) = U ′(X xϕ̂(T )). (0.27)
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b) Conversely, suppose θ̂ = (θ̂0, θ̂1) ∈ Θ is optimal for the
dualproblem
supθ∈Θ
E [−V (G yθ (T ))]
Let (p2, q2, r2) be the associated adjoint processes, solution
of theBSDEdp2(t) =
q2(t)
σ(t)b(t)dt + q2(t)dBt +
∫Rr2(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T
p2(T ) = −V ′(G yθ̂ (T )).
Then
ϕ̂(t) :=q2(t)
σtS(t−)1σt 6=0 +
r2(t, ζ)
γ(t, ζ)S(t−)1σt=0,γ(t,ζ)6=0
is an optimal portfolio (if it is admissible) for the primal
problem
supϕ∈A
E [U(X xϕ(T ))]
with initial value x = p2(0).
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Moreover, the optimal wealth process X xϕ̂ starting at x =
p2(0)coincides with the optimal adjoint process p2 for the dual
problem,i.e.
X xϕ̂(t) = p2(t); 0 ≤ t ≤ T . (0.28)
In particularX xϕ̂(T ) = −V ′(G
y
θ̂(T )). (0.29)
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ExampleNo jumps (N = 0). Then Θ has just one element θ̂ given
by
θ̂(t) = −btσt.
and
G yθ̂
(T ) = y exp(−∫ T
0
bsσs
dBs −1
2
∫ T0
b2sσ2s
ds).
Therefore, by Theorem 2 b), if (p2, q2) is the solution of the
BSDEdp2(t) =q2(t)
σtbtdt + q2(t)dBt ; 0 ≤ t ≤ T
p2(T ) = −V ′(Gθ̂(T )),(0.30)
then ϕ̂(t) :=q2(t)
σ(t)S(t−)is an optimal portfolio for the problem
supϕ∈A
E [U(X xϕ(T ))]
with initial value x = p2(0).
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In particular, if U(x) = ln x , then V (y) = − ln y − 1 andV
′(y) = −1
y. So the BSDE (0.30) becomes
dp2(t) =q2(t)
σtbtdt + q2(t)dBt ; 0 ≤ t ≤ T
p2(T ) =1
yexp
(∫ T0
bsσs
dBs +1
2
∫ T0
b2sσ2s
ds
).
(0.31)
To solve this equation we try q2(t) = p2(t)btσt. Then
dp2(t) = p2(t)
[b2tσ2t
dt +btσt
dBt
], (0.32)
which has the solution
p2(t) =1
yexp
(∫ t0
bsσs
dBs +1
2
∫ t0
b2sσ2s
ds
); 0 ≤ t ≤ T . (0.33)
Hence (0.31) holds.
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We conclude that the optimal portfolio for primal problem
withinitial value x = 1y is
ϕ̂(t) = p2(t)bt
σ2t S(t−). (0.34)
Note that with this portfolio we get
dXϕ̂(t) = p2(t)btσ2t
[btdt + σtdBt ]
= p2(t)
[b2tσ2t
dt +btσt
dBt
]= dp2(t). (0.35)
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Therefore
ϕ̂(t) = Xϕ̂(t)bt
σ2t S(t−)
(0.36)
which means that the optimal fraction of wealth to be placed
inthe risky asset is
π̂(t) =ϕ̂(t)S(t−)
Xϕ̂(t)=
btσ2t, (0.37)
which agrees with the classical result of Merton.
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Model uncertainty setupFor a different approach to robust
duality see Gushkin (2011)[2].See also the survey in Föllmer et al
(2009) [1].
To get a representation of model uncertainty, we consider a
familyof probability measures R = Rκ ∼ P, with
Radon-Nikodymderivative on Ft given by
d(Rκ | Ft)d(P | Ft)
= Zκt
where, for 0 ≤ t ≤ T , Zκt is a martingale of the form
dZκt = Zκt− [κ0(t)dBt +
∫Rκ1(t, ζ)Ñ(dt, dζ)] ; Z
κ0 = 1.
Let K denote a given set of admissible scenario controlsκ = (κ0,
κ1), Ft-predictable, s.t. κ1(t, z) ≥ −1 + �, andE [∫ T
0 {|κ20(t)|+
∫R κ
21(t, z)ν(dz)}dt]
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By the Girsanov theorem, using the measure Rκ in stead of
theoriginal measure P in the computations involving the price
processS(t), is equivalent to using the original measure P in
thecomputations involving the perturbed price process Sµ(t) in
steadof P(t), where Sµ(t) is given by
dSµ(t) = Sµ(t−)[(bt + µtσt)dt + σtdBt +∫Rγ(t, ζ)Ñ(dt, dζ)]
Sµ(0) > 0,
(0.38)with
µtσt = −σtκ0(t)−∫Rγ(t, ζ)κ1(t, ζ)ν(dζ)dt (0.39)
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Recall that a measure Rκ is an Equivalent Local
MartingaleMeasure (ELMM) iff κ = (κ0, κ1) are such that
bt + σtκ0(t) +
∫Rγt(ζ)κ1(t, ζ)ν(dζ) = 0.
Note that we do not assume a priori that the measures Rκ for κ
∈K are ELMMs.
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Robust Stochastic Control
Accordingly, we now replace the price process S(t) by
theperturbed processdSµ(t) = Sµ(t−)[(bt + µtσt)dt + σtdBt +
∫Rγ(t, ζ)Ñ(dt, dζ)]
Sµ(0) > 0,
(0.40)for some predictable perturbation process µt , assumed to
satisfy
E[∫ T
0 |µtσt |dt] 0.
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Let A denote the set of portfolios ϕt such that
E
[∫ T0ϕ2tSµ(t)
2
{(bt + µtσt)
2 + σ2t +
∫Rγ2(t, ζ)ν(dζ)
}dt
] 0 for all t ∈ [0,T ], a.s.
and ∃� > 0 s.t.
E [
∫ T0|X (t)|2+�dt]
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Let ρ : R→ R be a convex penalty function, assumed to be C1.The
robust primal problem is to find (ϕ̂, µ̂) ∈ A×M such that
infµ∈M
supϕ∈A
I (ϕ, µ) = I (ϕ̂, µ̂) = supϕ∈A
infµ∈M
I (ϕ, µ), (0.41)
where
I (ϕ, µ) = E
[U(Xϕ,µ(T )) +
∫ T0ρ(µt)dt
]. (0.42)
This a stochastic differential game that we handle by
usingmaximum principle. Define the Hamiltonian by
H1(t, x , ϕ, µ, p, q, r) = ρ(µ)
+ ϕSµ(t−)
[(bt + µσt)p + σtq +
∫Rγ(t, ζ)r(ζ)ν(dζ)
].
Since ∂H1∂x = 0, the BSDE for the adjoint processes (p1, q1, r1)
is
dp1(t) = q1(t)dBt +
∫Rr1(t, ζ)Ñ(dt, dζ); p1(T ) = U
′(Xϕ,µ(T )).
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First order conditions for a maximum ϕ̂ and a minimum µ̂:
(bt + µ̂tσt)p1(t) + σtq1(t) +
∫Rγ(t, ζ)r1(t, ζ)ν(dζ) = 0 (0.43)
ρ′(µ̂t) + ϕ̂tσtp1(t)Sµ(t−) = 0 (0.44)
Since H1 is concave with respect to ϕ and convex with respect
toµ, these first order conditions are also sufficient.Therefore we
obtain the following characterization of a saddlepoint of
(0.41):
Theorem 2: A pair (ϕ̂, µ̂) ∈ A×M is a solution of the
robustprimal game problem (0.41) iff the solution (p1, q1, r1) of
the BSDEdp1(t) = q1(t)dBt +
∫Rr1(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T
p1(T ) = U′(Xϕ̂,µ̂(T )).
(0.45)satisfies (0.43), (0.44).
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The robust dual problem is to find θ̂ ∈ Θ, µ̂ ∈M such that
supµ∈M
supθ∈Θ
J(θ, µ) = J(θ̂, µ̂) = supθ∈Θ
supµ∈M
J(θ, µ) (0.46)
where
J(θ, µ) = E
[−V (Gθ(T ))−
∫ T0ρ(µ(t))dt
], (0.47)
V is the conjugate function of U and G (t) = Gθ,µ(t) is given
bydG (t) = G (t−)[θ0(t)dBt +
∫Rθ1(t, ζ)Ñ(dt, dζ)
]; 0 ≤ t ≤ T
G (0) = y > 0,
(0.48)with the constraint that if y = 1, then the measure Qθ,µ
defined by
dQθ,µ = G (T )dP on FT
is an ELMM for the perturbed price process Sµ(t).
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By the Girsanov theorem, this is equivalent to requiring
that(θ0, θ1) satisfies
bt + µtσt + σtθ0(t) +
∫Rγ(t, ζ)θ1(t, ζ)ν(dζ) = 0 (0.49)
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Substituting
θ0(t) = −1
σt
[bt + µtσt +
∫Rγ(t, ζ)θ1(t, ζ)ν(d , ζ)
](0.50)
into (0.48) we getdG (t) = G (t−)
(− 1σt
[bt + µtσt +
∫Rγ(t, ζ)θ1(t, ζ)ν(dζ)
]dBt
+
∫Rθ1(t, ζ)Ñ(dt, dζ)
); 0 ≤ t ≤ T
G (0) = y > 0.
(0.51)The Hamiltonian becomes
H2(t, g , θ1, µ, p, q, r) = −ρ(µ)−gq
σt
[bt + µσt +
∫Rγ(t, ζ)θ1(ζ)ν(dζ)
]+ g
∫Rθ1(ζ)r(ζ)ν(dζ). (0.52)
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The BSDE for the adjoint processes (p2, q2, r2) is
dp2(t) = (q2(t)
σt[bt + µtσt +
∫Rγt(ζ)θ1(t, ζ)ν(dζ)]−
∫Rθ1(t, ζ)r2(t, ζ)ν(dζ))dt
+ q2(t)dBt +
∫Rr2(t, ζ)Ñ(dt, dζ) ; p2(T ) = −V ′(G (T )).
(0.53)
The first order conditions for a maximum point (θ̃, µ̃) for H2
are
(∇θ1H2 =)−q2(t)
σtγt(ζ) + r2(t, ζ) = 0 (0.54)(
∂H2∂µ
=
)ρ′(µ̃t) + Gθ̃,µ̃(t)q2(t) = 0. (0.55)
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Substituting (0.54) into (0.53) we getdp2(t) =q2(t)
σt[b2(t) + µ̃tσt ]dt + q2(t)dBt +
∫Rr2(t, ζ)Ñ(dt, dζ) ; t ∈ [0,T ]
p2(T ) = −V ′(Gθ̃,µ̃(T )).(0.56)
Theorem 4: A pair (θ̃, µ̃) ∈ Θ×M is a solution of the robust
dualproblem (0.46)-(0.47) if and only the solution (p2, q2, r2) of
theBSDE (0.56) also satisfies (0.54)-(0.55), i.e.
−q2(t)σt
γt(ζ) + r2(t, ζ) = 0
ρ′(µ̃t) + Gθ̃,µ̃(t)q2(t) = 0
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From robust primal to robust dual
We now use the characterizations above of the solutions(ϕ̂, µ̂)
∈ A×M and (θ̃, µ̃) ∈ Θ×M of the robust primal and dualproblems, to
find the relations between them.
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Theorem : Assume (ϕ̂, µ̂) ∈ A×M is a solution of the
robustprimal problem and let (p1, q1, r1) be the associated
adjointprocesses. Then,
µ̃ = µ̂ (0.57)
θ̃0(t) =q1(t)
p1(t−); θ̃1(t, ζ) =
r1(t, ζ)
p1(t−). (0.58)
are optimal for dual problem with initial value y =
p1(0).Moreover the optimal adjoint process p1 for the robust
primalproblem coincides with the optimal density process Gθ̃,µ̃ for
therobust dual problem, i.e.
p1(t) = Gθ̃,µ̃(t); 0 ≤ t ≤ T . (0.59)
In particular,U ′(Xϕ̂,µ̂(T )) = Gθ̃,µ̃(T ). (0.60)
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From robust dual to robust primalTheorem :Let (θ̃, µ̃) ∈ Θ×M be
optimal for the robust dual problem and let(p2, q2, r2) be the
associated adjoint processes. Then
µ̂ := µ̃
ϕ̂t :=q2(t)
σtS(t−)1σt 6=0 +
r2(t, ζ)
γ(t, ζ)S(t−)1σt=0,γ(t,ζ) 6=0 ; t ∈ [0,T ].
are optimal for primal problem with initial value x =
p2(0).Moreover, the optimal adjoint process p2 for the robust
dualproblem coincides with the optimal state process Xϕ̂,µ̂ for
therobust primal problem, i.e.
p2(t) = Xϕ̂,µ̂, 0 ≤ t ≤ T (0.61)
In particular− V ′(Gθ̃,µ̃(T )) = Xϕ̂,µ̂(T ). (0.62)
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Example
We study the robust primal problem
infµ∈M
supϕ∈A
E
[U(Xϕ,µ(T )) +
∫ T0ρ(µt)dt
]. (0.63)
with no jumps.Then there is only one ELMM for the price
processSµ, and the corresponding robust dual problem simplifies
to
supµ∈M
E
[−V (Gµ(T ))−
∫ T0ρ(µt)dt
], where (0.64)
dGµ(t) = −Gµ(t−)[btσt
+ µt ]dBt ; Gµ(0) = y > 0 (0.65)
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The first order conditions for the Hamiltonian reduce to:
µ̃t = ρ′−1(−Gµ̃(t)q2(t)) (0.66)
which substituted into the adjoint BSDE gives:dp2(t) =
q2(t)[btσt
+ ρ′−1
(−Gµ̃(t)q2(t))]dt + q2(t)dBt ; t ∈ [0,T ]
p2(T ) = −V ′(Gµ̃(T )).(0.67)
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Now assume that
U(x) = ln x and ρ(µ) =1
2µ2. (0.68)
Then V (y) = − ln y − 1.If bt and σt are deterministic, we can
use dynamic programmingand this leads to the solution
µ̃t = −bt
2σt; t ∈ [0,T ] (0.69)
We now check that this also holds if bt and σt are
Ft-adaptedprocesses, by verifying that the system (0.65)-(0.67) is
consistent:
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This system is now
Gµ̃(t) = y exp(−∫ t
0
bs2σs
dBs −1
2(bs
2σs)2ds) (0.70)
q2(t) =1
Gµ̃(t).bt
2σt(0.71)
dp2(t) =1
Gµ̃(t)
[bt
2σtdBt + (
bt2σt
)2dt
]; p2(T ) =
1
Gµ̃(T )(0.72)
which gives
d(1
Gµ̃(t)) =
1
Gµ̃(t)
[bt
2σtdBt + (
bt2σt
)2dt
]. (0.73)
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We see that (0.72) is in agreement with (0.73), so
µ̃t = −bt
2σt
is indeed optimal.The corresponding optimal portfolio for
therobust primal is
ϕ̂t =bt
Gµ̃(t)2σtSµ̃(t−); t ∈ [0,T ]. (0.74)
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We have proved:
TheoremSuppose (0.68) hold. Then the optimal scenario µ̂ = µ̃
andoptimal ϕ̂ for the robust primal problem (0.63) are given by
µ̃t = −bt
2σt; t ∈ [0,T ] (0.75)
and
ϕ̂t =bt
Gµ̃(t)2σtSµ̃(t−); t ∈ [0,T ]. (0.76)
respectively, with Gµ̃(t) as in (0.70).
It is interesting to compare this result with the solution we
found inthe corresponding previous example in the non-robust case
(ρ = 0):
ϕ̂(t) = Xϕ̂(t)bt
σ2t S(t−)
(0.77)
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Summary
I The purpose of this presentation has been to use
stochasticcontrol theory to obtain new results and new proofs of
resultsin portfolio optimization both with and without
modeluncertainty. In the case with no model uncertainty, then
someof the results have been proved earlier by using convex
dualitytheory.
I The advantage of this approach is that it gives an
explicitrelation between the optimal measure in the dual
formulationand the optimal portfolio in the primal formulation,
both inthe cases with and without model uncertainty. BSDEs play
acrucial role in this connection.
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Föllmer, H., Schied, A., Weber, S.: Robust preferences
androbust portfolio choice, In: Mathematical Modelling andNumerical
Methods in Finance. In: Ciarlet, P., Bensoussan, A.,Zhang, Q.
(eds): Handbook of Numerical Analysis 15, pp.29-88 (2009)
Gushkin, A. A.: Dual characterization of the value function
inthe robust utility maximization problem. Theory Probab. Appl.55
(2011), 611-630.
Hu, Y. and Peng, S.: Solution of forward-backward
stochasticdifferential equations, Probab.Theory Rel. Fields
(103),273-283 (1995).
Kramkov, D. and Schachermayer, W.: Necessary and
sufficientconditions in the problem of optimal investment in
incompletemarkets. Ann. Appl. Probab. 13 (2003), 1504-1516.
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Øksendal, B., Sulem, A.: Forward-backward stochasticdifferential
games and stochastic control under modeluncertainty. J. Optim.
Theory Appl.(2012).
Quenez, M.C., Sulem, A.: BSDEs with jumps, optimizationand
applications to dynamic risk measures. Stoch. Proc. andtheir Appl.
(2013)
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