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Poisson Random Measures and Their Numerical Approximation

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Poisson Random Measures and Their Numerical Approximation SPDES driven by Poisson Random Measures and their numerical Approximation Erika Hausenblas joint work with Thomas Dunst and Andreas Prohl University of Salzburg, Austria SPDES driven by Poisson Random Measures and their numerical Approximation – p.1 Outline Lévy processes - Poisson Random Measure Stochastic Integration in Banach spaces SPDEs driven by Poisson Random Measure Their Numerical Approximation SPDES driven by Poisson Random Measures and their numerical Approximation – p.2 A typical Example Let be a bounded domain in Rd with smooth boundary. O The Equation: du(t,ξ) ∂2 = 2 u(t, ξ)+ α u(t, ξ)+ g(u(t, ξ))L˙ (t) dt ∂ξ ∇ + f(u(t, ξ)), ξ , t> 0; ∈ O (⋆) u(0, ξ) = u (ξ) ξ ; 0 ∈ O u(t, ξ) = u(t, ξ) = 0, t 0, ξ ∂ ; ≥ ∈ O where u Lp( ), p 1, g a certain mapping and L(t) is a space time 0 ∈ O ≥ Lévy noise specified later. Problem: To find a function u : [0, ) R ∞ ×O −→ solving Equation (⋆). SPDES driven by Poisson Random Measures and their numerical Approximation – p.3 The Lévy Process L Definition 1 Let E be a Banach space. A stochastic process L = L(t), 0 t< is an E–valued Lévy process over (Ω; ; P) if the { ≤ ∞} F following conditions are satisfied: L(0) = 0; L has independent and stationary increments; L is stochastically continuous, i.e. for φ bounded and measurable, the function t Eφ(L(t)) is continuous on R+; 7→ L has a.s. càdlàg paths; SPDES driven by Poisson Random Measures and their numerical Approximation – p.4 The Lévy Process L E denotes a separable Banach space and E′ the dual on E. The Fourier Transform of L is given by the Lévy - Hinchin - Formula (see Linde (1986)): E ihL(1),ai iλhy,ai e = exp i y,a λ + e 1 iλy1{|y|≤1} ν(dy) , h i E − − Z where a E′, y E and ν : (E) R+ is a Lévy measure. ∈ ∈ B → SPDES driven by Poisson Random Measures and their numerical Approximation – p.5 The Lévy Process L E denotes a separable Banach space and E′ the dual on E. The Fourier Transform of L is given by the Lévy - Hinchin - Formula (see Linde (1986)): E ihL(1),ai iλhy,ai e = exp i y,a λ + e 1 iλy1{|y|≤1} ν(dy) , h i E − − Z where a E′, y E and ν : (E) R+ is a Lévy measure. ∈ ∈ B → If E = R, then the set of Lévy measures (R) is given by L (R)= ν : (R) [0, ); ν( 0 ) = 0; z 2ν(dz) < . L B → ∞ { } R | | ∞ Z SPDES driven by Poisson Random Measures and their numerical Approximation – p.5 Poisson Random Measure Remark 1 Let L be a real valued Lévy process and let A (R). ∈B Defining the counting measure N(t, A)= ♯ s (0, t] : ∆L(s)= L(s) L(s ) A IN { ∈ − − ∈ }∈ ∪ {∞} one can show, that N(t, A) is a random variable over (Ω; ; P); F N(t, A) Poisson (tν(A)) and N(t, ) = 0; ∼ ∅ For any disjoint sets A1,...,An, the random variables N(t, A1),...,N(t, An) are pairwise independent; SPDES driven by Poisson Random Measures and their numerical Approximation – p.6 Poisson Random Measure Definition 2 Let (S, ) be a measurable space and (Ω, , P) a probability S A space. A random measure on (S, ) is a family S η = η(ω, ), ω Ω { ∈ } of non-negative measures η(ω, ) : R+, such that the mapping S → η : (Ω, ) (M (S, ), (S, ))a A → + S M+ S is measurable. a M+(S, S) denotes the set of all non negative measures on (S, S) equipped with the weak topology and M+(S, S) is the σ field on M(S, S) generated by all functions iB : M(S, S) ∋ ν 7→ ν(B) ∈ [0, ∞) , B ∈ S. SPDES driven by Poisson Random Measures and their numerical Approximation – p.7 Poisson Random Measure Definition 3 Let (S, ) be a measurable space and (Ω, , P) a probability S A space. An integer valued Poisson random measure on (S, ) is a family S η = η(ω, ), ω Ω { ∈ } of non-negative measures η(ω, ) : IN, such that S → η(, ) = 0 a.s. ∅ η is a.s. σ–additive. η is a.s. independently scattered. for each A such that E η( , A) is finite, η( , A) is a Poisson random ∈ S · · variable with parameter E η( , A). · SPDES driven by Poisson Random Measures and their numerical Approximation – p.8 Poisson Random Measure Remark 2 The mapping A ν(A) := E η( , A) R S ∋ 7→ · ∈ is a measure on (S, ). S Assume S is a separable Banach space. Then the integral is X := s η˜(ds) ZS is well defined, iff ν is a Lévy measure. Moreover, the law of X is infinite divisible. SPDES driven by Poisson Random Measures and their numerical Approximation – p.9 Poisson Random Measure Definition 4 A law ρ on S is infinite divisible, iff for any n IN there ∈ exist a law ρ on S and a constant b S such that n n ∈ ρ = ρ∗(n) δ . n ∗ bn Definition 5 A random variable X in S has infinite divisible law, iff for any n IN there exist a family Xn : i = 1,...,n of independent and ∈ { i } identical distributed random vectors and a constant b S such that n ∈ (X)= (Xn + Xn + + Xn + b ). L L 1 2 · · · n n SPDES driven by Poisson Random Measures and their numerical Approximation – p.10 Poisson Random Measure Assume S = R and let X be given by X := s η˜(ds) ZS Then the law of X is infinitely divisible and given by ∞ k ∗(k) ν([ǫ, )) νǫ lim sup e−ν([ǫ,∞)) ∞ ǫ→0 k! Xk=0 where for ǫ> 0, the probability measure νǫ is defined by ν([ǫ, ) A) ν : (R) A ∞ ∩ [0, 1]. ǫ B ∋ 7→ ν([ǫ, )) ∈ ∞ SPDES driven by Poisson Random Measures and their numerical Approximation – p.11 Poisson Random Measure Let (E, ) be a measurable space. If S = E R+, = ˆ (R+), then a E × S E×B Poisson random measure on (S, ) is called Poisson point process. S Remark 3 Let ν be a Lévy measure on a Banach space E and S = E R+ • × = (E) ˆ (R+) • S B ×B ν′ = ν λ (λ is the Lebesgue measure). • × Then there exists a time homogeneous Poisson random measure η : Ω (E) (R+) IN ×B ×B → such that E η( A I)= ν(A)λ(I), A (E),I (R+), × × ∈B ∈B ν is called the intensity of η. SPDES driven by Poisson Random Measures and their numerical Approximation – p.12 Poisson Random Measure Definition 6 Let η : Ω (E) (R+) R+ ×B ×B → be a Poisson random measure over (Ω; ; P) and , 0 t< the F {Ft ≤ ∞} filtration induced by η. Then the predictable measure γ : Ω (E) (R+) R+ ×B ×B → is called compensator of η, if for any A (E) the process ∈B η˜(A (0, t]) := η(A (0, t]) γ(A [0, t]) × × − × is a local martingale over (Ω; ; P). F Remark 4 The compensator is unique up to a P-zero set and in case of a time homogeneous Poisson random measure given by γ(A [0, t]) = t ν(A), A (E). × ∈B SPDES driven by Poisson Random Measures and their numerical Approximation – p.13 Poisson Random Measure Example 1 Let η be a time homogeneous Poisson random measure over a Banach space E with intensity ν, where ν is a Lévy measure. Then, the stochastic process t t z η˜(dz, ds) 7→ Z0 ZE is a Lévy process with Lévy measure ν. SPDES driven by Poisson Random Measures and their numerical Approximation – p.14 Poisson Random Measure Definition 7 We call a R–valued Lévy process of type (α,β) for α [0, 2] and β 0, iff ∈ ≥ α = inf α˜ 0 : lim ν((x, )) xα˜ < ≥ x→0 ∞ ∞ n and lim ν(( , x)) xα˜ < x→0 −∞ − ∞ o ˜ β = inf β˜ 0 : x βν(dx) < . ≥ | | ∞ ( Z{|x|>1} ) SPDES driven by Poisson Random Measures and their numerical Approximation – p.15 The Numerical Approximation of a Lévy walk SPDES driven by Poisson Random Measures and their numerical Approximation – p.16 The Numerical Approximation of a Lévy walk Let η be a Poisson random measure on R with intensity ν, i.e. let L = L(t) : 0 t< be the corresponding Lévy process with characteristic { ≤ ∞} measure ν defined by t [0, ) t L(t) := z η˜(dz,ds) R. ∞ ∋ 7→ R ∈ Z0 Z SPDES driven by Poisson Random Measures and their numerical Approximation – p.17 The Numerical Approximation of a Lévy walk Let η be a Poisson random measure on R with intensity ν, i.e. let L = L(t) : 0 t< be the corresponding Lévy process with characteristic { ≤ ∞} measure ν defined by t [0, ) t L(t) := z η˜(dz,ds) R. ∞ ∋ 7→ R ∈ Z0 Z Question: SPDES driven by Poisson Random Measures and their numerical Approximation – p.17 The Numerical Approximation of a Lévy walk Let η be a Poisson random measure on R with intensity ν, i.e. let L = L(t) : 0 t< be the corresponding Lévy process with characteristic { ≤ ∞} measure ν defined by t [0, ) t L(t) := z η˜(dz,ds) R. ∞ ∋ 7→ R ∈ Z0 Z Question: Given the characteristic measure of a Lévy process, how to simulate the Lévy walk ? That means, how to simulate 0 1 2 k ∆τ L, ∆τ L, ∆τ L, ..., ∆τ L, ... , where ∆kL := L((k + 1)τ) L(kτ), k IN.
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