Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion Wiener-Hopf factorization and distribution of extrema for a family of L´ evy processes Alexey Kuznetsov Department of Mathematics and Statistics York University June 20, 2009 Research supported by the Natural Sciences and Engineering Research Council of Canada W-H factorization and distribution of extrema Alexey Kuznetsov 0/29
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Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Wiener-Hopf factorization and distribution ofextrema for a family of Levy processes
Alexey Kuznetsov
Department of Mathematics and StatisticsYork University
June 20, 2009
Research supported by the Natural Sciences and Engineering Research Council of Canada
W-H factorization and distribution of extrema Alexey Kuznetsov 0/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
2 Processes with meromorphic characteristic exponentCompound Poisson processProcess with jumps of infinite variation (analogue of NIGprocess)A β-family of Levy processes
3 Numerical results
4 Conclusion and future work
W-H factorization and distribution of extrema Alexey Kuznetsov 0/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Wiener-Hopf factorization
Review of Wiener-Hopf factorization
The characteristic exponent Ψ(z) is
E[eizXt
]= exp(−tΨ(z)),
and by the Levy-Khintchine representation
Ψ(z) = −iµz +σ2z2
2−∫R
(eizx − 1− izx
)ν(x)dx
We define extrema processes Mt = sup{Xs : s ≤ t} andNt = inf{Xs : s ≤ t}, introduce an exponential random variableτ = τ(q) with parameter q > 0, which is independent of the processXt, and use the following notation for characteristic functions of Mτ ,Nτ :
φ+q (z) = E
[eizMτ
], φ−q (z) = E
[eizNτ
]W-H factorization and distribution of extrema Alexey Kuznetsov 1/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Wiener-Hopf factorization
Review of Wiener-Hopf factorization
TheoremRandom variables Mτ and Xτ −Mτ are independent, randomvariables Nτ and Xτ −Mτ have the same distribution, and for z ∈ Rwe have
q
Ψ(z) + q= E
[eizXτ
]= E
[eizMτ
]E[eiz(Xτ−Mτ )
]= φ+
q (z)φ−q (z)
Moreover, random variable Mτ [Nτ ] is infinitely divisible, positive[negative] and has zero drift.
W-H factorization and distribution of extrema Alexey Kuznetsov 2/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
2 Processes with meromorphic characteristic exponentCompound Poisson processProcess with jumps of infinite variation (analogue of NIGprocess)A β-family of Levy processes
3 Numerical results
4 Conclusion and future work
W-H factorization and distribution of extrema Alexey Kuznetsov 2/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Well-known examples
WH for Brownian motion with drift
Let Xt = Wt + µt. Then Ψ(z) = z2
2 − iµz and Ψ(z) + q = 0 has twosolutions
z1,2 = i(µ±√µ2 + 2q)
Thus function q(Ψ(z) + q)−1 can be factorized as
q
Ψ(z) + q=
qz2
2 − iµz + q
=µ+
õ2 + 2q
iz + µ+√µ2 + 2q
× µ−√µ2 + 2q
iz + µ−√µ2 + 2q
W-H factorization and distribution of extrema Alexey Kuznetsov 3/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Well-known examples
WH for Brownian motion with drift
The main idea: since the random variable Mτ [Nτ ] is positive[negative], its characteristic function must be analytic and have nozeros in C+ [C−], where
C+ = {z ∈ C : Im(z) > 0}, C− = {z ∈ C : Im(z) < 0}, C± = C± ∪ R.
Thus
φ+q (z) =
µ−√µ2 + 2q
iz + µ−√µ2 + 2q
and Mτ is an exponential random variable with parameter√µ2 + 2q − µ.
W-H factorization and distribution of extrema Alexey Kuznetsov 4/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Well-known examples
Computing density of Mt
Let pt(x) be the density function of Mt and pM (q, x) the densityfunction of Mτ(q). Then
pM (q, x) = E[pτ(q)(x)] = q
∞∫0
e−qtpt(x)dt
and therefore
pt(x) =1
2πi
∫q0+iR
pM (q, x)eqtdq
q
In our example we have
pt(x) =1
2πi
∫q0+iR
(√µ2 + 2q − µ
)e−(√µ2+2q−µ)x+qt dq
q
W-H factorization and distribution of extrema Alexey Kuznetsov 5/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Well-known examples
Kou model: double exponential jump diffusion model
Xt is a Levy process with jumps defined by
ν(x) = a1e−b1xI{x>0} + a2e
b2xI{x<0}
Then the characteristic exponent is
Ψ(z) =σ2z2
2− iµz − a1
b1 − iz− a2
b2 + iz+a1
b1+a2
b2
Thus equation Ψ(z) + q = 0 is a fourth degree polynomial equation,and we have explicit solutions and exact WH factorization.
W-H factorization and distribution of extrema Alexey Kuznetsov 6/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Well-known examples
Phase-type distributed jumps
DefinitionThe distribution of the first passage time of the finite state continuoustime Markov chain is called phase-type distribution.
q(x) = p0exLe1 =
N∑k=1
aiebix
where bi are eigenvalues of the Markov generator L. Thus if Xt hasphase-type jumps, its characteristic exponent Ψ(z) is a rationalfunction, and Ψ(z) + q = 0 is reduced to a polynomial equation, andthe Wiener-Hopf factors are given in closed form (in terms of roots ofthis polynomial equation).
W-H factorization and distribution of extrema Alexey Kuznetsov 7/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
2 Processes with meromorphic characteristic exponentCompound Poisson processProcess with jumps of infinite variation (analogue of NIGprocess)A β-family of Levy processes
3 Numerical results
4 Conclusion and future work
W-H factorization and distribution of extrema Alexey Kuznetsov 7/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Compound Poisson process
Definition
Let Xt be a compound Poisson process defined by
ν(x) =eαx
cosh(x)
The characteristic exponent of Xt is given by
Ψ(z) = −∫R
(eixz − 1
)ν(x)dx =
π
cos(π2α) − π
cosh(π2 (z − iα)
)
W-H factorization and distribution of extrema Alexey Kuznetsov 8/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Compound Poisson process
WH factorization
Theorem
Assume that q > 0. Define
η =2π
arccos
(π
q + π sec(π2α))
p0 =Γ(
14 (1− α)
)Γ(
14 (3− α)
)Γ(
14 (η − α)
)Γ(
14 (4− η − α)
)Then for Im(z) > i(α− η) we have
φ+q (z) = p0
Γ(
14 (η − α− iz)
)Γ(
14 (4− η − α− iz)
)Γ(
14 (1− α− iz)
)Γ(
14 (3− α− iz)
)We have P(Mτ = 0) = p0 and the density of Mτ is given by explicitformula in terms of 2F1(a, b; c; z) – the Gauss hypergeometricfunction.
W-H factorization and distribution of extrema Alexey Kuznetsov 9/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Compound Poisson process
Uniquenes of Wiener-Hopf factorization
Lemma
Assume we have two functions f+(z) and f−(z), such that f±(0) = 1,f±(z) is analytic in C±, continuous and has no roots in C± andz−1 ln(f±(z))→ 0 as z →∞, z ∈ C±. If
q
Ψ(z) + q= f+(z)f−(z), z ∈ R
then f±(z) = φ±q (z).
W-H factorization and distribution of extrema Alexey Kuznetsov 10/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
2 Processes with meromorphic characteristic exponentCompound Poisson processProcess with jumps of infinite variation (analogue of NIGprocess)A β-family of Levy processes
3 Numerical results
4 Conclusion and future work
W-H factorization and distribution of extrema Alexey Kuznetsov 10/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Process with jumps of infinite variation
Definition
Let Xt be defined by a triple (µ, σ, ν), where the jump measure ν(x)is given by
ν(x) =eαx[
sinh(x2 )]2
with | α |< 1.
Proposition
The characteristic exponent of Xt is given by
Ψ(z) =σ2z2
2+ iρz + 4π(z − iα) coth (π(z − iα))− 4γ,
where
γ = πα cot (πα) , ρ = 4π2α+4γ(γ − 1)
α− µ
W-H factorization and distribution of extrema Alexey Kuznetsov 11/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Process with jumps of infinite variation
Theorem on zeros of Ψ(z) + q
TheoremAssume that q > 0.(i) Equation Ψ(iζ) + q = 0 has infinitely many solutions, all of which
2 Processes with meromorphic characteristic exponentCompound Poisson processProcess with jumps of infinite variation (analogue of NIGprocess)A β-family of Levy processes
3 Numerical results
4 Conclusion and future work
W-H factorization and distribution of extrema Alexey Kuznetsov 16/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
A β-family of Levy processes
Definition of the β-family
DefinitionWe define a β-family of Levy processes by the generating triple(µ, σ, ν), where the jump density is
ν(x) = c1e−α1β1x
(1− e−β1x)λ1I{x>0} + c2
eα2β2x
(1− eβ2x)λ2I{x<0}
and parameters satisfy αi > 0, βi > 0, ci ≥ 0 and λi ∈ (0, 3).
W-H factorization and distribution of extrema Alexey Kuznetsov 17/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
2 Processes with meromorphic characteristic exponentCompound Poisson processProcess with jumps of infinite variation (analogue of NIGprocess)A β-family of Levy processes
3 Numerical results
4 Conclusion and future work
W-H factorization and distribution of extrema Alexey Kuznetsov 23/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Computing density of Mτ
(i) compute ζ±0 , ζn - Newton’s method for large n. For small n –localization, bisection and then Newton’s method.
(ii) compute coefficients c−k – use asymptotic expansion of ζn toaccelerate convergence of products.
(iii) series for density of Mτ is exponentially converging
W-H factorization and distribution of extrema Alexey Kuznetsov 24/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Computing density of Mt
As before, pt(x) can be recovered as the inverse Laplace transform ofpM (q, x)/q. Thus
pt(x) =1
2πi
∫q0+iR
eqtpM (q, x)dq
q=eq0t
π
∞∫0
Re[pM (q0 + iu, x)
q0 + iu
]cos(tu)du
Question: How do we solve Ψ(z) + q = 0 for q ∈ C ?
W-H factorization and distribution of extrema Alexey Kuznetsov 25/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Solving Ψ(z) + q = 0 for q ∈ C: ODE method
We need to compute the solutions of equation Ψ(iζ) + q = 0 for allq ∈ [q0, q0 + iu0]. First we compute the initial values: the roots ζ±0 , ζnfor real value of q = q0 using the method discussed above. Next weconsider each root as an implicit function of u: ζn(u) is defined as
Ψ(iζn(u)) + (q0 + iu) = 0, ζn(0) = ζn
Using implicit differentiation we obtain a first order differentialequation
dζn(u)du
= − 1Ψ′(iζn(u))
W-H factorization and distribution of extrema Alexey Kuznetsov 26/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Solving ODE: Runge-Kutta & Newton’s method
W-H factorization and distribution of extrema Alexey Kuznetsov 27/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Results: surface plot of pM(q, x) and pt(x)
W-H factorization and distribution of extrema Alexey Kuznetsov 28/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
2 Processes with meromorphic characteristic exponentCompound Poisson processProcess with jumps of infinite variation (analogue of NIGprocess)A β-family of Levy processes
3 Numerical results
4 Conclusion and future work
W-H factorization and distribution of extrema Alexey Kuznetsov 28/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Conclusion and future work
Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞
We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance
W-H factorization and distribution of extrema Alexey Kuznetsov 29/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Conclusion and future work
Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)
Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance
W-H factorization and distribution of extrema Alexey Kuznetsov 29/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Conclusion and future work
Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniques
To find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance
W-H factorization and distribution of extrema Alexey Kuznetsov 29/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Conclusion and future work
Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methods
β-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance
W-H factorization and distribution of extrema Alexey Kuznetsov 29/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Conclusion and future work
Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limit
Future work: applying these processes to price variousderivatives in Mathematical Finance
W-H factorization and distribution of extrema Alexey Kuznetsov 29/29
Introduction Processes with meromorphic Ψ(z) Numerical results Conclusion
Conclusion and future work
Wiener-Hopf factors can be identified explicitly for certain Levyprocesses, if Ψ(z) is meromorphic and if we know its asymptoticsas z →∞We have a lot of information about solutions of Ψ(z) + q = 0(localization, asymptotics)Infinite products can be computed efficiently using convergenceacceleration techniquesTo find solutions of Ψ(z) + q = 0 for complex values of q we canuse ODE/Newton’s methodsβ-family of Levy processes is very general, many existing modelscan be obtained as a limitFuture work: applying these processes to price variousderivatives in Mathematical Finance
W-H factorization and distribution of extrema Alexey Kuznetsov 29/29