SPDES driven by Poisson Random Measures and their numerical Approximation

Transcription

1 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

2 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

3 A typical Example Let O be a bounded domain in R d with smooth boundary. The Equation: ( ) du(t,ξ) dt = 2 ξ 2 u(t,ξ) + α u(t,ξ) + g(u(t,ξ)) L(t) + f(u(t,ξ)), ξ O, t > 0; u(0,ξ) = u 0 (ξ) ξ O; u(t,ξ) = u(t,ξ) = 0, t 0, ξ O; where u 0 L p (O), p 1, g a certain mapping and L(t) is a space time Lévy noise specified later. Problem: To find a function u : [0, ) O R solving Equation ( ). SPDES driven by Poisson Random Measures and their numerical Approximation p.3

4 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; F; P) if the 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 + ; L has a.s. càdlàg paths; SPDES driven by Poisson Random Measures and their numerical Approximation p.4

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 e i L(1),a = exp { i y,a λ + E (e iλ y,a 1 iλy1 { y 1} ) where a E, y E and ν : B(E) R + is a Lévy measure. } ν(dy), SPDES driven by Poisson Random Measures and their numerical Approximation p.5

6 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 e i L(1),a = exp { i y,a λ + E (e iλ y,a 1 iλy1 { y 1} ) where a E, y E and ν : B(E) R + is a Lévy measure. } ν(dy), If E = R, then the set of Lévy measures L(R) is given by L(R) = { ν : B(R) [0, );ν({0}) = 0; R } z 2 ν(dz) <. SPDES driven by Poisson Random Measures and their numerical Approximation p.5

7 Poisson Random Measure Remark 1 Let L be a real valued Lévy process and let A B(R). 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; F; P); N(t,A) Poisson(tν(A)) and N(t, ) = 0; For any disjoint sets A 1,...,A n, the random variables N(t,A 1 ),...,N(t,A n ) are pairwise independent; SPDES driven by Poisson Random Measures and their numerical Approximation p.6

8 Poisson Random Measure Definition 2 Let (S, S) be a measurable space and (Ω, A, P) a probability space. A random measure on (S, S) is a family η = {η(ω, ),ω Ω} of non-negative measures η(ω, ) : S R +, such that the mapping is measurable. η : (Ω, A) (M + (S, S), M + (S, S)) a 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 i B : M(S, S) ν ν(b) [0, ), B S. SPDES driven by Poisson Random Measures and their numerical Approximation p.7

9 Poisson Random Measure Definition 3 Let (S, S) be a measurable space and (Ω, A, P) a probability space. An integer valued Poisson random measure on (S, S) is a family η = {η(ω, ),ω Ω} of non-negative measures η(ω, ) : S IN, such that η(, ) = 0 a.s. η is a.s. σ additive. η is a.s. independently scattered. for each A S such that Eη(,A) is finite, η(,a) is a Poisson random variable with parameter E η(, A). SPDES driven by Poisson Random Measures and their numerical Approximation p.8

10 Poisson Random Measure Remark 2 The mapping S A ν(a) := Eη(,A) R is a measure on (S, S). Assume S is a separable Banach space. Then the integral is X := S s η(ds) 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

11 Poisson Random Measure Definition 4 A law ρ on S is infinite divisible, iff for any n IN there exist a law ρ n on S and a constant b n S such that ρ = ρ (n) n δ bn. Definition 5 A random variable X in S has infinite divisible law, iff for any n IN there exist a family {X n i : i = 1,...,n} of independent and identical distributed random vectors and a constant b n S such that L(X) = L(X n 1 + X n X n n + b n ). SPDES driven by Poisson Random Measures and their numerical Approximation p.10

12 Poisson Random Measure Assume S = R and let X be given by X := s η(ds) Then the law of X is infinitely divisible and given by S lim sup ǫ 0 e ν([ǫ, )) k=0 ν([ǫ, )) k ν (k) ǫ k! where for ǫ > 0, the probability measure ν ǫ is defined by ν ǫ : B(R) A ν([ǫ, ) A) ν([ǫ, )) [0,1]. SPDES driven by Poisson Random Measures and their numerical Approximation p.11

13 Poisson Random Measure Let (E, E) be a measurable space. If S = E R +, S = E ˆ B(R + ), then a Poisson random measure on (S, S) is called Poisson point process. Remark 3 Let ν be a Lévy measure on a Banach space E and S = E R + S = B(E)ˆ B(R + ) ν = ν λ (λ is the Lebesgue measure). Then there exists a time homogeneous Poisson random measure η : Ω B(E) B(R + ) IN such that E η( A I) = ν(a)λ(i), A B(E),I B(R + ), ν is called the intensity of η. SPDES driven by Poisson Random Measures and their numerical Approximation p.12

14 Poisson Random Measure Definition 6 Let η : Ω B(E) B(R + ) R + be a Poisson random measure over (Ω; F; P) and {F t,0 t < } the filtration induced by η. Then the predictable measure γ : Ω B(E) B(R + ) R + is called compensator of η, if for any A B(E) the process η(a (0,t]) := η(a (0,t]) γ(a [0,t]) is a local martingale over (Ω; F; P). 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 B(E). SPDES driven by Poisson Random Measures and their numerical Approximation p.13

15 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 0 E z η(dz,ds) is a Lévy process with Lévy measure ν. SPDES driven by Poisson Random Measures and their numerical Approximation p.14

16 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 β = inf { α 0 : lim x 0 ν((x, ))x α < { and β 0 : lim ν((, x))x α < x 0 { x >1} x βν(dx) < }. } SPDES driven by Poisson Random Measures and their numerical Approximation p.15

17 The Numerical Approximation of a Lévy walk SPDES driven by Poisson Random Measures and their numerical Approximation p.16

18 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 [0, ) t L(t) := t 0 R z η(dz, ds) R. SPDES driven by Poisson Random Measures and their numerical Approximation p.17

19 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 Question: [0, ) t L(t) := t 0 R z η(dz, ds) R. SPDES driven by Poisson Random Measures and their numerical Approximation p.17

20 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 [0, ) t L(t) := t 0 R z η(dz, ds) R. Question: Given the characteristic measure of a Lévy process, how to simulate the Lévy walk? That means, how to simulate ( 0 τ L, 1 τl, 2 τl,..., k τl,... ), where k τl := L((k + 1)τ) L(kτ), k IN. SPDES driven by Poisson Random Measures and their numerical Approximation p.17

21 The Numerical Approximation of a Lévy walk Different approaches: SPDES driven by Poisson Random Measures and their numerical Approximation p.18

22 The Numerical Approximation of a Lévy walk Different approaches: Direct generation: in case of stable, exponential, gamma, geometric and negative binomial distribution. Generation from compound Poisson Process (Rubenthaler (2003), Amussen and Rósinski, Fornier (2010)) generation with shot noise representation (Bondesson (1980), Rosinski (2001), Kawai and Imai (2007)) Approximation of the density function by series (Eberlein,...) SPDES driven by Poisson Random Measures and their numerical Approximation p.18

23 The Numerical Approximation of a Lévy walk Generation from compound Poisson Process cutting off the small jumps at ǫ > 0, simulating a Poisson distributed Random variable T with parameter ν([ǫ, )), simulating Z = L( T) L( T ), i.e. simulating a random variable Z with distribution ν ǫ defined by B(R) A ν ǫ (A) := ν(a (, ǫ] [ǫ, )) ; ν((, ǫ] [ǫ, )) optional: simulating the small jumps by a Wiener process. SPDES driven by Poisson Random Measures and their numerical Approximation p.19

24 The Numerical Approximation of a Lévy walk Generation with shot noise representation Assume U is a random variable on [0, T] the Lévy measure can be written as ν(b) = 0 P (H(r, U) B) dr, B B(R). {E k : k IN} sequence of iid exponential random variables with unit mean; {Γ k : k IN} sequence of standard Poisson arrivials time, in particular {Γ k Γ k 1 : k IN} are exponential distributed with certain parameter; {U k : k IN} sequence of iid uniform distributed random variables on [0, 1] SPDES driven by Poisson Random Measures and their numerical Approximation p.20

25 The Numerical Approximation Our suggestions: (Hausenblas and Marchis (2007), Hausenblas Prohl and Dunst (2010)) Cutting off the small jumps; Approximating the characteristic measure by a discrete measure; Generating a random variable ˆ k τl ǫ which approximates k τl ǫ ; Approximating the small jumps by a Wiener process (optional); SPDES driven by Poisson Random Measures and their numerical Approximation p.21

26 The Numerical Approximation of a Lévy walk Cutting off the small jumps; Approximating the characteristic measure by a discrete measure; Generating a random variable ˆ k τl ǫ which approximates k τl ǫ ; Approximating the small jumps by a Wiener process (optional); SPDES driven by Poisson Random Measures and their numerical Approximation p.22

27 The Numerical Approximation of a Lévy walk fix a cut off parameter ǫ > 0; define the new characteristic measure ν cǫ : B(R) B ν (B ((, ǫ] [ǫ, ))) ; Let η cǫ be a Poisson random measure corresponding where the supports does not include [ ǫ, ǫ], i.e. η cǫ : B(R) B(R + ) B I 1 B (z) η(dz, dt) IN { }; (, ǫ] [ǫ, ) I Let L ǫ = {L(t) : 0 t < } the Lévy process which arises by cutting off the small jumps, i.e. L ǫ (t) := t 0 R ζ η cǫ (dζ, ds), t 0. SPDES driven by Poisson Random Measures and their numerical Approximation p.23

28 The Numerical Approximation of a Lévy walk Figure 1: Cutting of the small jumps SPDES driven by Poisson Random Measures and their numerical Approximation p.24

29 The Numerical Approximation Cutting off the small jumps; Approximating the characteristic measure by a discrete measure; Generating a random variable ˆ k τl ǫ which approximates k τl ǫ ; Approximating the small jumps by a Wiener process (optional); SPDES driven by Poisson Random Measures and their numerical Approximation p.25

30 Approximating the characteristic measure by a discrete measure Approximate the characteristic measure ν cǫ by a characteristic measure ˆν cǫ having the following form: Let D 1,...,D J be disjoint sets with J j=1 D j (, ǫ] [ǫ, ); let c 1,,c J be points belonging to R such that c j D j, j = 1...J; ν cǫ is approximated as ˆν cǫ := J j=1 ν(d j)δ cj. Example 2 Put D j = [x j,x j+1 ), j = 1,...,J/2 1, D J/2 = [x J/2, ), D J/2+j = [ x j, x j+1 ), j = 1,...,J/2 1, such that ν(b j ) < γ; Put D J = [ x J 1, ), where x 0 = ǫ < x 1 < < x J/2 < ; Put c j := x j. SPDES driven by Poisson Random Measures and their numerical Approximation p.26

31 The Numerical Approximation of a Lévy walk ˆν cǫ := J j=1 ν(d j)δ cj Figure 2: Discretization of the Lévy measure SPDES driven by Poisson Random Measures and their numerical Approximation p.27

32 The Numerical Approximation Different Discretization: equally weighted mesh: Let D 1 = [ǫ, x 2 ), D J = [x J, ), and D i = [x i, x i+1 ) such that ν(d i ) = ν(d j ) for 1 i, j J O(ǫ 1 ). equally spaced mesh: Let D 1 = [ǫ, x 2 ), D J = [x J, ), and D i = [x i, x i+1 ) such that λ(d i ) = λ(d j ) for 1 i, j J O(ǫ 1 ). strechted mesh: Let D 1 = [ǫ 2, x 2 ), D J = [x J, ), and choose ǫ j := λ(d j ) according to jǫ 2 if x j < 1, ǫ j := j γ ǫ if x j 1, for some 0 < γ < 1. SPDES driven by Poisson Random Measures and their numerical Approximation p.28

33 The Numerical Approximation There Errors: In general, the error in L p -mean is given by Err ǫ := ξ p (ν ˆν ǫ )(dξ). In particular, we have for the R\( ǫ,ǫ) equally weighted mesh: Err ǫ C ǫ 1 α β (β p) for 0 α < 1, equally spaced mesh: Err ǫ C max(ǫ p α, ǫ) strechted mesh: Err ǫ C max(ǫ 2(p α), ǫ 1 2γ, ǫ γ(β p) ) SPDES driven by Poisson Random Measures and their numerical Approximation p.29

34 The Numerical Approximation Cutting off the small jumps; Approximating the characteristic measure by a discrete measure; Generating a random variable ˆ k τl ǫ which approximates k τl ǫ ; Approximating the small jumps by a Wiener process (optional); SPDES driven by Poisson Random Measures and their numerical Approximation p.30

35 Generating a random variable ˆ k τl ǫ Remember (Independently scattered property) The family {η (D j [t k 1,t k )) : j = 1,...,n,k = 1,...,K} of random variables over (Ω, F, P) is mutually independent; for any D I {D j [t k 1,t k ) : j = 1,...,n,k = 1,...,K} the random variable η(d I) is Poisson distributed with Parameter ν(d) λ(i). SPDES driven by Poisson Random Measures and their numerical Approximation p.31

36 Generating a random variable ˆ k τl ǫ At each time step k generate a family {q 1 k,...,qj k } of independent random variables, where we have for each j = 1,...,J ) P (q j k = l = exp( τν(d j )) τl ν(d j ) l, l 0. l! The approximation ˆ k τl will be given as follows. ˆ k τl ǫ := J j=1 ( ) q j k τν(d j) x j, k 0. The Lévy random walk is approximated by ( ˆ 0 τ L ǫ, ˆ 1 τl ǫ, ˆ 2 τl ǫ,..., ˆ k τl ǫ,...), SPDES driven by Poisson Random Measures and their numerical Approximation p.32

37 Simulation des Lévy walks SPDES driven by Poisson Random Measures and their numerical Approximation p.33

38 Question At each time step k 1 is it sufficient to simulate a family {q 1 k,..., qj k } of independent random variables, where ) P (q j k = 0 = exp( τν(d j )) j = 1,...,k and ) P (q j k = 1 = 1 exp( τν(d j )) j = 1,...,k. The approximation is now given as follows. ˆ k τl ǫ := J j=1 ( ) q j k τν(d j) c j, k 0. The Lévy random walk is approximated by (ˆ 0 τ L ǫ, ˆ 1 τl ǫ, ˆ 2 τl ǫ,..., ˆ k τl ǫ,...), SPDES driven by Poisson Random Measures and their numerical Approximation p.34

39 Approximating the small jumps (optional); At each time step k generate a Gaussian random variables k τw, where L( k τw ǫ ) = σ(ǫ) N (0,τ) and ǫ σ(ǫ) := ǫ ζ 2 ν(dζ). The Lévy random walk is approximated by ( ˆ 0 τl ǫ + 0 τw ǫ, ˆ 1 τl ǫ + 1 τw ǫ, ˆ 2 τl ǫ + 2 τw ǫ,..., ˆ k τl ǫ + k τw ǫ,...). SPDES driven by Poisson Random Measures and their numerical Approximation p.35

40 Simulation des Lévy walks Assume, that the intensity is of type α, α (0, 2], the time-step size τ > 0 and that the truncation parameter ǫ > 0 satisfies ǫ Cτ 1/α. Then for all p > 0 where k τl c ǫ := k τl k τl ǫ. E k τl c ǫ k τw ǫ p Cτ p 3 ǫ p p 3 α, k N SPDES driven by Poisson Random Measures and their numerical Approximation p.36

41 Simulation des Lévy walks Idea of the proof: Let X be a random variable and ˆX its Fourier transform. Then P ( X x) x 2 2 x ( ( )) 1 Re ˆX(λ) dλ. 2 x Straightforward calulations and the Taylor expansion. SPDES driven by Poisson Random Measures and their numerical Approximation p.37

42 Space - Time - White - Noise Let us recall the Definition of a Gaussian white noise (Dalang 2003): Definition 8 Let (Ω, F, P) be a complete probability space and (S, S,σ) a measure space. Then a Gaussian white noise on S based on σ is a measurable mapping W : (Ω, F) (M(E), M(E)) a For A S, W(A) is a real valued Gaussian random variable with mean 0 and variance σ(a), provided σ(a) < ; if A and B S are disjoint, then the random variables W(A) and W(B) are independent and W(A B) = W(A) + W(B). a M(S) denotes the set of all measures from S into R, i.e. M(S) := {µ : S R} and M(S) is the σ-field on M(S) generated by functions i B : M(S) µ µ(b) R, B S. SPDES driven by Poisson Random Measures and their numerical Approximation p.38

43 Space - Time - White - Noise Put O R d be a bounded domain with smooth boundary. E = O [0, ), E = B(O) B([0, )) σ = λ d+1a. The measure valued process t W( [0,t)) defines a space time Gaussian white noise. a λ d+1 denotes the Lebesgue measure in R d. SPDES driven by Poisson Random Measures and their numerical Approximation p.39

44 Space - Time - White - Noise SPDES driven by Poisson Random Measures and their numerical Approximation p.40

45 Space - Time - White - Noise Definition 9 Let (Ω, F, P) be a complete probability space and let (S, S, σ) be a measurable space. Then the Lévy white noise on S based on σ with characteristic jump size measure ν L(R) is a measurable mapping such that L : (Ω, F) (M(S), M(S)) a For A S, L(A) is a real valued infinite divisible random variables with characteristic exponent provided σ(a) <. e iθl(a) = exp ( σ(a) R ( 1 e iθx isin(θx) ) ν(dx) ), if A and B S are disjoint, then the random variables L(A) and L(B) are independent and L(A B) = L(A) + L(B). a M(E) denotes the set of all measures from E into R, i.e. M(S) := {µ : S R} and M(S) is the σ-field on M(S) generated by functions i B : M(S) µ µ(b) R, B S. SPDES driven by Poisson Random Measures and their numerical Approximation p.41

46 Space - Time - White - Noise Again put O R d be a bounded domain with smooth boundary. E = O [0, ), E = B(O) B([0, )) σ = λ d+1. The measure valued process t L( [0,t)) defines a space time Lévy white noise, for more details we refer to Peszat and Zabczyk (2007). SPDES driven by Poisson Random Measures and their numerical Approximation p.42

47 Space - Time - White - Noise SPDES driven by Poisson Random Measures and their numerical Approximation p.43

48 Space - Time - White - Noise Definition 10 Let (Ω, F, P) be a complete probability space and let (S, S, σ) be a measurable space. Then the Poisson white noise on S based on σ with characteristic jump size measure ν L(R) is a measurable mapping such that η : (Ω, F) (M (M I (S R)), M (M I (S R))) for A B S B(R), η(a B) is a Poisson random variable with parameter σ(a) ν(b), provided σ(a)ν(b) < ; if the sets A 1 B 1 S B(R) and A 2 B 2 S B(R) are disjoint, then the random variables η(a 1 B 1 ) and η(a 2 B 2 ) are independent and η ((A 1 B 1 ) (A 2 B 2 ) ) = η(a 1 B 1 ) + η(a 2 B 2 ). SPDES driven by Poisson Random Measures and their numerical Approximation p.44

49 Space - Time - White - Noise Again put O R d be a bounded domain with smooth boundary. S = O [0, ), S = B(O) B([0, )) σ = λ d+1. Then, by definition, the space time Poisson white noise is the measure valued process process t η( [0,t)); (for more details we refer to Breźniak and Hausenblas (2009) or Peszat and Zabczyk (2007)). SPDES driven by Poisson Random Measures and their numerical Approximation p.45

50 The Approximation of a space time Lévy noise Let O be a bounded domain and let T be a subdivision with a set of element domains K = {K i : i = 1...,l}, a set of nodals variable N = {N i, i = 1...,l} and a set of space functions P = {φ i : i = 1,...,l} a Voronoi decomposition J = {J i ; i = 1,...,l} induced by N Let L = {L i : i = 1,...l} be a family of independent Lévy processes where L i has characteristic ρ i ν, ρ i = λ(j i ), i = 1,...,l. The measure valued process L = { L(t) : t 0} given by L(t) : B(O) A N c i i=1 A φ i (x) dxl i (t), t 0, where c i = ( φ i 1 L 1 ), is an approximation in space of the space time Lévy noise. SPDES driven by Poisson Random Measures and their numerical Approximation p.46

51 Space - Time - White - Noise /home/erica/desktop/bilder-vortrga/levysheet.eps SPDES driven by Poisson Random Measures and their numerical Approximation p.47

52 Space - Time - White - Noise SPDES driven by Poisson Random Measures and their numerical Approximation p.48

53 Our Equation The Equation: ( ) du(t,ξ) dt = 2 ξ 2 u(t,ξ) + α u(t,ξ) + g(u(t,ξ)) L(t) +f(u(t,ξ)), ξ O, t > 0; u(0,ξ) = u 0 (ξ) ξ O; u(t,ξ) = u(t,ξ) = 0, t 0, ξ O; where u 0 L p (0,1), p 1, g a certain mapping and L(t) is a Lévy process taking values in a certain Banach space Z. A mild solution of Equation ( ) is an adapted Banach-valued càdlàg process u = {u(t) : t [0,T]} such that for t 0 we have P-a.s. u(t) = S(t)u 0 + t 0 S(t s) f(u(s)) dt + t 0 Z S(t s)g(u(s);z) η(ds,dz) SPDES driven by Poisson Random Measures and their numerical Approximation p.49

54 Banach spaces of M type p Definition 11 a Let 1 p <. A Banach space E is of M type p, iff there exists a constant C = C(E; p), such that for each discrete E-valued martingale M = (M 1, M 2,...) one has sup n 1 E M n p E C n 1 E M n M n 1 p E. a see Pisier (1986), Maurey, Schwartz. SPDES driven by Poisson Random Measures and their numerical Approximation p.50

55 Banach spaces of M type p Definition 11 Let 1 p <. A Banach space E is of M type p, iff there exists a constant C = C(E; p), such that for each discrete E-valued martingale M = (M 1, M 2,...) one has sup n 1 E M n p E C n 1 E M n M n 1 p E. see Pisier (1986), Maurey, Schwartz. SPDES driven by Poisson Random Measures and their numerical Approximation p.50

56 The Itô Stochastic Integral Let h be a progressively measurable step function with representation h(t) = n H i 1 (ti,t i+1 ](t), t R +, i=1 where 0 = t 0 t n = T and H i : Ω L p (Z,ν;E) is F ti measurable for i = 1,...,n. Definition 12 The stochastic integral of h with respect to η is defined by n I(h) := H i (s) η(ds;(t i,t i+1 ]). ( ) i=1 S SPDES driven by Poisson Random Measures and their numerical Approximation p.51

57 Definition of the Integral Let E be a Banach space { of martingale type p and M p (ν) ([0, ); E) := h : Ω [0, ) L p (Z, ν; E), } h is progressively measurable and E h(s, z) p ν(dz) ds a < Theorem 1 b There exists a linear bounded operator R + I : M p ([0, T]; E) L p (Ω, F, P; E), which is a unique bounded extension of the operator defined in ( ). If h M p ([0, T]; E) and t > 0 then we put t 0 Z h(s)η(ds, dz) := I(1 (0,t]h) and we call the LHS the Itô integral of the process h up to time t. a ν is the intensity of η b p = 1, 2 B. Rüdiger (2005), p (1,2], EH (2005), EH and Brzeźniak (2008), Filipovic and Tappe (2008) for Burkholder Davies Gundy type inequalities see Preprint E. Hausenblas (2009) and Röckner, Marinelli (2009). SPDES driven by Poisson Random Measures and their numerical Approximation p.52 Z

58 SPDEs - Existence and Uniqueness Theorem 2 (EH, 2005 EJP) Assume that there exist some δ g < 1 p and δ f,δ I < 1 such that u 0 satisfies E ( A) δ Iu 0 p < ; ( A) δ ff : E E is Lipschitz continuous; ( A) δ g g : E L(Z,E) satisfies Z ( A) δ g [g(x,z) g(y,z)] p E ν(dz) C x y p E, x,y E. Then, there exists a unique mild solution to Problem (1), such that for any T > 0 T 0 E u(s) p E ds <, and ( A) γ u L 0 (Ω;ID([0,T];E)), where γ > 1 p. SPDES driven by Poisson Random Measures and their numerical Approximation p.53

59 SPDEs - Existence and Uniqueness Theorem 3 (EH, 2005 EJP) Assume that there exist some δ g < 1 p and δ f,δ I < 1 such that u 0 satisfies E ( A) δ Iu 0 p < ; ( A) δ ff : E E is Lipschitz continuous; ( A) δ g g : E L(Z,E) satisfies Z ( A) δ g [g(x,z) g(y,z)] p E ν(dz) C x y p E, x,y E. Then, there exists a unique mild solution to Problem (1), such that for any T > 0 T 0 E u(s) p E ds <, and ( A) γ u L 0 (Ω;ID([0,T];E)), where γ > 1 p. SPDES driven by Poisson Random Measures and their numerical Approximation p.54

60 The Besov space B α p, (O) for α = d p d Let ψ(x) [0, 1], if x (1, 3 ) and ψ(x) = 2 1, if x 1 0, if x 3 2. Put ϕ 0 (x) = φ(x) ϕ 1 (x) = ψ( x 2 ) ψ(x) ϕ j (x) = ψ(2 j+1 x), j = 2, 3,.... Let s R and 0 < q. If 0 < p, then Bp,q s := { f S : f Bp,q s = 2 sj F 1 [ϕ j Ff]( ) l q (L p ) < }. Therefore, if q =, then { Bp, s := f S : f Bp,q s = sup j>0 } 2 sj F 1 [ϕ j Ff] ( ) L p <. SPDES driven by Poisson Random Measures and their numerical Approximation p.55

61 Space Time Poissonian Noise Corresponding Banach-valued random measure: d p d Z = Bp, (O) (Besov space), and S = Z R + ; O ξ f(ξ) := δ ξ Z ; ˆη : B(Z R + ) IN { } be a Poisson random measure with intensity ˆν defined by B(Z) B ˆν(B) := χ B (rx)σ(dr)λ f (dx), R\{0} {z Z, z =1} where λ f (B) := λ(f 1 (B)), B B({z Z, z = 1}). SPDES driven by Poisson Random Measures and their numerical Approximation p.56

62 Space Time Poissonian Noise The following inequality holds for all u,v L p (O) Thus, if Z (u v) z p B d p d p, d d p < 2 1 p ˆν(dz) C u v p L p. p < 2 + d d, then Theorem (2) gives Existence and Uniqueness of solution to Equation (1). Corollary 1 Let A be the Laplace operator. If there exists a p [1,2] with 1 < p < 2 d + 1, then there exists a solution to the SPDE above with space time Lévy noise. SPDES driven by Poisson Random Measures and their numerical Approximation p.57

63 The Numerical Approximation SPDES driven by Poisson Random Measures and their numerical Approximation p.58

64 The Approximation of a space time Lévy noise Let O be a bounded domain and let T be a subdivision with a set of element domains K = {K i : i = 1...,l}, a set of nodals variable N = {N i, i = 1...,l} and a set of space functions P = {φ i : i = 1,...,l} a Voronoi decomposition J = {J i ; i = 1,...,l} induced by N Let L = {L i : i = 1,...l} be a family of independent Lévy processes where L i has characteristic ρ i ν, ρ i = λ(j i ), i = 1,...,l. The measure valued process L = { L(t) : t 0} given by L(t) : B(O) A N c i i=1 A φ i (x) dxl i (t), t 0, where c i = ( φ i 1 L 1 ), is an approximation in space of the space time Lévy noise. SPDES driven by Poisson Random Measures and their numerical Approximation p.59

65 Space - Time - Poissonian Noise SPDES driven by Poisson Random Measures and their numerical Approximation p.60

66 Numeric of SPDEs The implicit Euler scheme. Here u n (t + τ n ) u n (t) τ n Au n (t + τ n ) f(u n (t)) + g(u n (t)) n L(t), Again, if v k n denotes the approximation of u n (kτ n ), then v k+1 n = (1 τ n A n ) 1 v k n + τ n f(v k n) + τ n g(u n (t)) ξ n k, v 0 n = x. Between the points kτ n and (k + 1)τ n the solution is linear interpolated. SPDES driven by Poisson Random Measures and their numerical Approximation p.61

67 Numerical Analysis Theorem 4 (E.H. 2008) Let u = {u(t); 0 t < } be the mild solution of an SPDE wit space time Lévy noise Let û = {û k, k IN} be the approximation of u Discretization in space: affine finite Elements with length h; Discretization in time: implicit Euler scheme with time step τ h. The, the error is give by [ E u(kτ h ) û k h p L p (O) where α < 2 p d + d p and δ < (2 + d 2 d p ] 1 p )( C 0 τ α h + C 1 h δ. ) 2 p d + d p. SPDES driven by Poisson Random Measures and their numerical Approximation p.62

68 The Numerical Approximation (Dunst, Prohl, E.H. (2010)) Assume in addition that the space time Lévy white noise is approximated as described before. Then The following holds. For any M IN there exist constants C 1 and C 2 such that sup 0 m M E v h m ˆv ǫ m p C h L p (O) 1ǫ p α h + C 2 ζ R\( ǫ h,ǫ h ) ζ p d(ˆν ǫ h ν ǫ h )(dζ). If the stability condition ( ) is satisfied, then there exist constants C 1 and C 2 such that for any θ > 0, and ǫ h with ǫ h τ θ h, sup 0 m M E vh m ˆv w,m p C 1 τ p 3 L p (O) h,ǫ + C 2 ζ R\( ǫ h,ǫ h ) h ǫ p(3 α) 3 h ζ p d(ˆν c,ǫ h ν ǫ h )(dζ). SPDES driven by Poisson Random Measures and their numerical Approximation p.63

69 Numerical Experiments SPDES driven by Poisson Random Measures and their numerical Approximation p.64

70 Numerical Experiments SPDES driven by Poisson Random Measures and their numerical Approximation p.65

71 Numerical Experiments SPDES driven by Poisson Random Measures and their numerical Approximation p.66

72 Nearly the End real numerical simulations SPDES driven by Poisson Random Measures and their numerical Approximation p.67