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 A. Prohl and T. Dunst from Tübingen, D Montana University of Leoben, Austria SPDES driven by Poisson Random Measures and their numerical Approximation p.1

2 Outline SPDEs - a short Example Simulating a Lévy walk Simulating an space time Lévy noise; Simulating an SPDEs driven by a Poisson Random Measure 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 = u(t,ξ)+α u(t,ξ)+g(u(t,ξ)) L(t,ξ) u(0,ξ) = u 0 (ξ) ξ O; +f(u(t,ξ)), ξ O, t > 0; 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 defined over a probability space (Ω,F,(F) t 0,P) specified later. Problem: To find an approximation of the stochastic process solving Equation ( ). u : Ω [0, ) O R a (see e.g. St. Lupert Bié, Albeverio, Wu and Zhang, Applebaum, Brzezniak and Hausenblas,...) a. SPDES driven by Poisson Random Measures and their numerical Approximation p.3

4 Motivations SPDEs versus PDEs Change of the dynamical behaviour of the system, e.g. White noise eliminates instability [Crauel (2000)] Stochastic Resonance [Schimansky-Geier,...] Uniqueness of the solution (Lecture notes of St. Flour by Flandoli) Uniqueness of the Invariant Measures, etc. [Flandoli, Mattingly, Rozovskii, Shirykan and Kuksin,...] Better description of the real world system E.g. in thin-film models, SPDEs leads to a better description of data s gained by experiments [Grüne, Mecke, Rauscher (2006)] E.g. Falkovich, Kolokolov, Lebedev, Mezentsev, and Turitsyn (2004) uses stochastic nonlinear Schrödinger equation to describe certain parameters in optical soliton transmission. SPDES driven by Poisson Random Measures and their numerical Approximation p.4

5 Motivations Lévy processes versus Wiener processes Systems with jumps have another dynamical behaviour [see Imkeller and Pavlyukevich (2006)] In Climatology, e.g. the time in which the temperature is changing is short compared to the time scale [see Imkeller and Pavlyukevich (2006)] Finance Mathematics (Eberlein and his coworkers) In case of semilinear SPDEs driven by a Lévy process the conditions of the diffusion term g can be weakened. SPDES driven by Poisson Random Measures and their numerical Approximation p.5

6 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 continuous and bounded, 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.6

7 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 iλ y,a 1 iλy1 { y 1} ν(dy), where a E, y E and ν : B(E) R + is a Lévy measure. E SPDES driven by Poisson Random Measures and their numerical Approximation p.7

8 The Lévy Process L In what follows E denotes a separable Banach space, B(E) denotes the Borel-σ algebra on E and E the dual on E. Definition 2 (see Linde (1986), Section 5.4) A σ finite symmetric a Borel-measure ν : B(E) R + is called a Lévy measure if ν({0}) = 0 and the function ( ) E a exp (cos( x,a ) 1) ν(dx) E C is a characteristic function of a certain Radon measure on E. An arbitrary σ-finite Borel measure ν is a Lèvy measure if its symmetrization ν +ν is a symmetric Lévy measure. a ν(a) = ν( A) for alla B(E) SPDES driven by Poisson Random Measures and their numerical Approximation p.8

9 Poisson Random Measure Remark 1 Let L be a real valued Lévy process over a filtered probability space (Ω,F,{F t } t 0,P) 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 for any A B(R) N(t,A) is a random variable over (Ω;F;P); N(t,A) Poisson(tν(A)) and N(t, ) = 0; (independently scattered) For any disjoint sets A 1,...,A n, the random variables N(t,A 1 ),...,N(t,A n ) are mutually independent; SPDES driven by Poisson Random Measures and their numerical Approximation p.9

10 Poisson Random Measure Definition 3 Let (Z, Z) be a measurable space. A Poisson random measure η on (Z,Z) over (Ω,F,F,P) is a measurable function η : (Ω,F) (M I (Z R + ),M I (Z R + )), such that (i) for each B Z B(R + ), η(b) := i B η : Ω N is a Poisson random variable with parameter a Eη(B); (ii) η is independently scattered, i.e. if the sets B j Z B(R + ), j = 1,,n, are disjoint, then the random variables η(b j ), j = 1,,n, are independent; (iii) for each U Z, the N-valued process (N(t,U)) t 0 defined by N(t,U) := η(u (0,t]), t 0 is {F t } t 0 -adapted and its increments are independent of the past, i.e. if t > s 0, then N(t,U) N(s,U) = η(u (s,t]) is independent of F s = {collection of all events happened before time s}. SPDES driven by Poisson Random Measures and their numerical Approximation p.10

11 Poisson Random Measure Remark 2 The mapping S A ν(a) := Eη(A [0,1]) R is a measure on (Z,Z). Assume Z is a separable Banach space. Then the integral L(t) := t 0 is well defined, iff ν is a Lévy measure. Z z η(dz,ds) SPDES driven by Poisson Random Measures and their numerical Approximation p.11

12 Poisson Random Measure Definition 4 (Blumenthal - Getoor - index) For α [0,2] and β 0 we call an R valued Lévy process of type (α,β), iff α = inf { α 0 : lim x 0 ν((x, ))x α < and { β = inf 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.12

13 The Numerical Approximation of a Lévy walk SPDES driven by Poisson Random Measures and their numerical Approximation p.13

14 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.14

15 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.14

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 [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.14

17 The Numerical Approximation of a Lévy walk Different approaches: SPDES driven by Poisson Random Measures and their numerical Approximation p.15

18 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.15

19 The Numerical Approximation of a Lévy walk Generation from compound Poisson Process cutting off the small jumps at ǫ > 0, simulating an exponential 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.16

20 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.17

21 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.18

22 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.19

23 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.20

24 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.21

25 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 Example 1 ˆν cǫ := J j=1 ν(d j)δ cj. 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.22

26 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.23

27 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. (Here, one can fit 2 and γ according to α and β) SPDES driven by Poisson Random Measures and their numerical Approximation p.24

28 The Numerical Approximation The Errors: In general, the error in L p -mean is given by Err ǫ := In particular, we have for the R\( ǫ,ǫ) ξ p (ν ˆν ǫ )(dξ). equally weighted mesh: Err ǫ Cǫ 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.25

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.26

30 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.27

31 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 jk = 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) c j, k 0. The Lévy random walk is approximated by (ˆ 0 τl ǫ, ˆ 1 τl ǫ, ˆ 2 τl ǫ,..., ˆ k τl ǫ,...), SPDES driven by Poisson Random Measures and their numerical Approximation p.28

32 Simulation des Lévy walks SPDES driven by Poisson Random Measures and their numerical Approximation p.29

33 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 jk = 0 = exp( τν(d j )) j = 1,...,k and ) P (q jk = 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.30

34 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.31

35 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 τ L 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.32

36 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.33

37 Simulation des Lévy walks Main ingredients of the proof: For any random variable X, we have E X p C 0 P( X x) x p 1 dx; SPDES driven by Poisson Random Measures and their numerical Approximation p.34

38 Simulation des Lévy walks Main ingredients of the proof: For any random variable X, we have E X p C 0 P( X x) x p 1 dx; For any random variable X with Fourier transform ˆX, we have P( X x) x 2 2 x ( )) 1 R( ˆX(λ) dλ. 2 x SPDES driven by Poisson Random Measures and their numerical Approximation p.34

39 Simulation des Lévy walks Main ingredients of the proof: For any random variable X, we have E X p C 0 P( X x) x p 1 dx; For any random variable X with Fourier transform ˆX, we have P( X x) x 2 2 x ( )) 1 R( ˆX(λ) dλ. 2 x Therefore we have to give an estimate of Ee iλ ( k τ Lc ǫ k τ W ) = Ee iλ k τ Lc ǫ Ee iλ k τ ( W ) = exp τ (e iyλ 1 iyλ)ν ǫ (dy) τσ 2 (ǫ)λ 2 R. (-6) SPDES driven by Poisson Random Measures and their numerical Approximation p.34

40 Simulation des Lévy walks Main ingredients of the proof: For any random variable X, we have E X p C 0 P( X x) x p 1 dx; For any random variable X with Fourier transform ˆX, we have P( X x) x 2 2 x ( )) 1 R( ˆX(λ) dλ. 2 x Therefore we have to give an estimate of Ee iλ ( k τ Lc ǫ k τ W ) = Ee iλ k τ Lc ǫ Ee iλ k τ ( W ) = exp τ (e iyλ 1 iyλ)ν ǫ (dy) τσ 2 (ǫ)λ 2 Taylor expansion of the RHS R. (-9) SPDES driven by Poisson Random Measures and their numerical Approximation p.34

41 Space - Time - White - Noise Let us recall the Definition of a Gaussian white noise (Dalang 2002): Definition 4 Let (Ω,F,P) be a complete probability space and (E,E,σ) a measure space. Then a Gaussian white noise on E based on σ is a measurable mapping W : (Ω,F) (M(E),M(E)) for any A E, W(A) is a Gaussian random variable with mean 0 and variance σ(a), provided σ(a) < ; for any A,B E, if A and B are disjoint, then the random variables W(A) and W(B) are independent and W(A B) = W(A)+W(B). M(E) denotes the set of all measures fromb(e) intor, i.e. M(E) := {ν : B(E) R} andm(e) is theσ-field onm(e) generated by functions i B : M(E) η η(b) R,B E. SPDES driven by Poisson Random Measures and their numerical Approximation p.35

42 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+1. The measure valued process t W( [0,t)) defines a space time Gaussian white noise. λ d+1 denotes the Lebesgue measure inr d. SPDES driven by Poisson Random Measures and their numerical Approximation p.36

43 Space - Time - White - Noise SPDES driven by Poisson Random Measures and their numerical Approximation p.37

44 Space - Time - White - Noise Definition 4 Let (Ω,F,P) be a complete probability space and let (E,E,σ) be a measurable space. Then the Lévy white noise on E based on σ with characteristic jump size measure ν L(R) is a measurable mapping such that L : (Ω,F) (M(E),M(E)) for any A E, L(A) is an infinite divisible random variables with characteristic exponent provided σ(a) <. e iθl(a) = exp ( σ(a) R( 1 e iθx isin(θx) ) ν(dx) ), for any A,B E, if A and B are disjoint, then the random variables L(A) and L(B) are independent and L(A B) = L(A)+L(B). M(E) denotes the set of all measures fromb(e) intor, i.e. M(E) := {ν : B(E) R} andm(e) is the σ-field onm(e) generated by functions i B : M(E) η η(b) R,B E. SPDES driven by Poisson Random Measures and their numerical Approximation p.38

45 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), or Brzezniak and Hausenblas (2010). SPDES driven by Poisson Random Measures and their numerical Approximation p.39

46 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 )/h d, i = 1,...,l. The measure valued process L = { L(t) : t 0} given by L(t) : B(O) A N i=1 ρ i A φ i (x)dxl i (t), t 0, is an approximation in space of the space time Lévy noise. SPDES driven by Poisson Random Measures and their numerical Approximation p.40

47 Space - Time - White - Noise SPDES driven by Poisson Random Measures and their numerical Approximation p.41

48 Space - Time - White - Noise SPDES driven by Poisson Random Measures and their numerical Approximation p.42

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

50 Our Example The Equation: ( ) du(t,ξ) dt = 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 a.s. u(t) = S(t) a u 0 + t 0 S(t s)f(u(s))dt+ t S(t s)g(u(s);z) η(ds,dz) 0 Z a {S(t) : t > 0} denotes the semigroup generated by the Laplace operator with the given boundary conditions. SPDES driven by Poisson Random Measures and their numerical Approximation p.44

51 The Numerical Approximation SPDES driven by Poisson Random Measures and their numerical Approximation p.45

52 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.46

53 Numerical Analysis Theorem 1 (E.H. 2008) Let u = {u(t) : 0 t < } be the mild solution of an SPDE with 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. Then, the error is given 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.47

54 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.48

55 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 + Ξ 2 p d+d pc 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.49

56 The Numerical Approximation The stability condition ( ): There exists a γ < 1 p such that h d+2γ+dp 2 dp τ p 2 h σ(ǫ h ) p h d τ h, h (0,1]. (-9) SPDES driven by Poisson Random Measures and their numerical Approximation p.50

57 Numerical Experiments SPDES driven by Poisson Random Measures and their numerical Approximation p.51

58 Numerical Experiments SPDES driven by Poisson Random Measures and their numerical Approximation p.52

59 Numerical Experiments SPDES driven by Poisson Random Measures and their numerical Approximation p.53

60 the End SPDES driven by Poisson Random Measures and their numerical Approximation p.54