Approximate Euler Method for Parabolic Stochastic Partial Differential

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SIAM J. NUMER. ANAL. Vol. 50, No. 6, pp. 2873–2896

APPROXIMATE EULER METHOD FOR PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY SPACE-TIME ´ LEVY NOISE∗ THOMAS DUNST† , ERIKA HAUSENBLAS‡ , AND ANDREAS PROHL† Abstract. We consider parabolic stochastic partial differential equations driven by space-time L´ evy noise. Different discretization methods to accurately simulate jumps are proposed and analyzed in the context of an implicit time-discretization. Computational studies based on a finite element discretization are provided to illustrate combined truncation and time-discretization effects. Key words. stochastic evolution equations, stochastic partial differential equations, numerical approximation, time-discretization, finite elements, Poisson random measure AMS subject classifications. 60H15, 60H35, 65C30, 35R30 DOI. 10.1137/100818297

1. Introduction. In recent years, stochastic partial differential equations (SPDEs) driven by a space-time Lévy noise have received increasing interest; see, e.g., Applebaum and Wu [3], Knoche [13], Mueller [15], and Saint Loubert Bié [19]. One motivation for this work is the model of river pollution proposed by Kwakernaak [14] and studied by Curtain [7]. Let O := (0, 1) × (0, 3) ⊂ R2 and A = DΔ − γ ∂ξ∂ 2 − a; here D > 0 is the dispersion rate, γ > 0 scales convection in the ξ2 -direction, and a ≥ 0 is the leakage rate. Then u : (0, T ) × O → R describes the concentration of a chemical substance which solves the following problem (see Figure 1): ∂u = Au ∂t ∂u =0 ∂n

in OT := (0, T ) × O , on (0, T ) × ΓN ,

and

u=0

on (0, T ) × ΓD .

A chemical substance is now deposited in the region G ⊂ O. The deposits leak, and substance is dripping out. This dripping Lévy noise, out is modeled by a space-time and the related concentration process u(t, ξ); t ∈ [0, T ], ξ ∈ O solves the SPDE ∂u(t, ξ) ˙ ξ) ∀ (t, ξ) ∈ OT , = Au(t, ξ) + L(t, ∂t (1) u(0, ξ) = 0 ∀ ξ ∈ O , ∂u(t, ξ) = 0 ∀ (t, ξ) ∈ (0, T ) × ΓN , u(t, ξ) = 0 ∀ (t, ξ) ∈ (0, T ) × ΓD , ∂ξ2 where L˙ represents space-time Lévy noise. Problem (1) is an example for a more general setting, which we consider in this work. Figures 1 and 2 show simulations ∗ Received by the editors December 15, 2010; accepted for publication (in revised form) July 27, 2012; published electronically November 6, 2012. http://www.siam.org/journals/sinum/50-6/81829.html † Mathematisches Institut, Universit¨ at T¨ ubingen, Auf der Morgenstelle 10, D-72076 T¨ ubingen, Germany ([email protected], [email protected]). ‡ Department Mathematik und Informationstechnologie, Montanuniversit¨ at Leoben, Franz-JosefStraße 18, A-8700 Leoben, Austria ([email protected]).

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T. DUNST, E. HAUSENBLAS, AND A. PROHL

(a) Domain O ΓN


(0, 3)

(1, 3)

(b) t → u(t, A)

A G

B C

ΓD

5

4

ΓD

3

2

1

0

ΓD

(0, 0)

−1 0

(1, 0)

0.5

(c) t → u(t, B)

1 Time

1.5

2

(d) t → u(t, C)

2.5

0.8 0.7

2

0.6 0.5

1.5

0.4 1

0.3 0.2

0.5

0.1 0

0

−0.1 −0.5 0

0.5

1

1.5

Time

−0.2 0

2

0.5

1

1.5

Time

2

Fig. 1. (a) Domain O ⊂ R2 ; in the shaded area G jumps might occur. (b)–(d) Single paths t → u(t, ξ) of the concentration at fixed positions ξ ∈ {A, B, C} ⊂ O.

(a) t = 0.80

(b) t = 0.85

(c) t = 0.90

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

0 3

0 3

0 3

2 1 0

0

0.2

0.4

0.6

0.8

1

2 1 0

0

0.2

0.4

0.6

0.8

1

2 1 0

0

0.2

0.4

0.6

0.8

1

Fig. 2. The solution u(t, ·) to (1) for one trajectory plotted at subsequent times.

for one path, which are based on the approximate Euler scheme (Scheme B) that is proposed and studied below. A first numerical analysis of such SPDEs is [10]. However, in that work increments of the driving Lévy process are exactly simulated. It is only for a small number of Lévy processes that the exact distribution function of the increments is available, while in general it is only the Lévy measure of the driving noise that is given. Therefore, the goal of this paper is to propose different strategies to approximate a given Lévy process and an analysis of errors inherent to a corresponding time-discretization of (15). Several approaches exist to simulate SDEs driven by Lévy processes. Rubenthaler [18] studied the approximate Euler scheme, which combines (explicit) timediscretization, and simulatable approximations of a given Lévy process (with unknown distribution) by truncating jumps below a certain threshold > 0: the numerical strategy then uses “interlacing,” i.e., it simulates subsequent stopping times where jumps occur, corresponding jump heights, as well as an intermediate Brownian motion. We



´ APPROXIMATE EULER METHOD FOR PARABOLIC LEVY SPDE

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remark that this strategy is less efficient in high dimensions, and hence also for SPDEs (after discretization in space). For related numerical works that solve SDEs without approximating the driving Lévy noise, we refer the reader to the works cited in [18]. Such a truncation strategy ignores small jumps, which may cause a rough approximation of Lévy measures of infinite variation, and a reduced precision of iterates of the related approximate Euler scheme; Asmussen and Rosi´ nski [4] have shown that small jumps may be represented by a Wiener process, leading to an improved approximation of the given Lévy process. In [11], weak rates of convergence for iterates of the approximate Euler scheme for SDEs that employs this strategy are shown; strong rates of convergence are obtained in [8], where the proof relies on a central limit theorem when using the quadratic Wasserstein distance. In this work, we give a different proof (cf. Proposition 3.4) that is based on Fourier transforms and uses tail estimates to show a corresponding Lp estimate (1 ≤ p ≤ 2), which is the proper setting of (1), resp., (15), where mild solutions are Lp -valued. The main goal of this work is to obtain accurate, simulatable increments according to the Lévy measure restricted to the set {|x| ≥ κ }, i.e., ν {|x| ≥ κ }, κ ≥ 1, by truncating small (and large) jumps, and approximating the remaining jump heights by a sum of O(−1 ) atoms, such that > 0 controls both truncation and discretization effects. For this purpose, we consider different meshing strategies to assemble a related compound Poisson process to simultaneously represent all small, medium, and large jumps accurately: −1 • local sizes of an equally weighted mesh {Dj }j=1 covering R \ (−, ) (cf. [20]) are determined by the requirement ν(Dj ) = ν(Di ) for 1 ≤ i, j ≤ 1 . Meshes depend on the given Lévy measure ν, will be shown to only roughly approximate small jumps, and only work reliably for certain Lévy measures; cf. Proposition 3.3, Theorem 4.1, and the computational experiments summarized in Figure 8. • Equally spaced meshes allow for an accurate simulation of bounded jumps of size [, 1], while small and large jumps suffer a rough approximation. • The newly introduced stretched meshes accurately resolve small jumps up to size O(2 ), and (relatively less frequent) large jumps up to size O(−γ ) for 0 ≤ γ < 12 ; cf. Proposition 3.3, Theorem 4.1, and the computational evidence in Figure 8. A related analysis shows that stretched meshes lead to the smallest approximation errors of the jump term, which is also confirmed by computational studies in section 5. Thus a proper construction of (discrete) approximate Lévy measures complements the strategy of an additional Gaussian variable to model small jumps, and may even make it dispensable; see Proposition 3.4 and section 5.1 for computational evidence. In section 4, we derive Lp -error estimates for the (implicit) approximate Euler scheme (Scheme B) to numerically solve problem (15). The corresponding analysis accounts for both time-discretization and approximation of the space-time Lévy white noise. The error estimate in Theorem 4.1 quantifies how truncation and discretization effects (using the different mesh strategies above) of the space-time Lévy white noise affect accuracy of the time-discrete approximates of (15). In section 5, we use a space-time discretization based on finite elements to study the effects of truncation and different mesh strategies to approximate the Lévy measure, as well as time-discretization; computational studies are reported that evidence improved approximation properties of the (implicit) approximate Euler method using stretched meshes. The paper is organized as follows. Necessary background material on the space-



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time Lévy noise is provided in the preliminaries. Strong rates of convergence for the approximate Lévy random walk for different meshes, and the gain in accuracy by using a compensatory Wiener process, are studied in section 3. The approximate Euler method to numerically solve problem (15) is analyzed in section 4. Computational studies are reported in section 5. 2. Preliminaries. In this section, we recall the notion of space-time Lévy processes. 2.1. L´ evy processes and L´ evy measures. Let L(R) denote the set of Lévy measures supported on R, i.e., the set of positive measures ν on R such that ν({0}) = 0, ν is σ-finite, and (|x|2 ∧ 1) ν(dx) < ∞ . R

The characteristic function of a Lévy process L = L(t) : t ≥ 0 is uniquely decomposable by the Lévy–Khintchine formula: for any real-valued Lévy process L = {L(t) : t ≥ 0} there exist a nonnegative number Q, a nonnegative measure ν ∈ L(R), and an element m ∈ R such that

1 E eiL(1),x = exp imx − Qx2 − 1 − eixy + ½{|y| 0, and ΔL(0) = 0. The intensity of the Poisson random measure μL is the Lévy measure ν = E μL ([0, 1] × ·) . Conversely, if μ is a given (time-homogeneous) Poisson random measure on R \ {0} over a complete filtered probability space (Ω, F , F, P) and with finite Lévy measure ν ∈ L(R), then the stochastic process L defined by t ζ μ(ds, dζ) [0, ∞) t → L(t) = 0

R

is a compound Poisson process; if the Lévy measure is only σ-finite, μ has to be replaced by the compensated Poisson random measure μ

(ds, dζ) = μ(ds, dζ) − ν(dζ)ds; see, e.g., [2]. A finite αth variation, resp., a finite βth moment, of the Lévy measure ν ∈ L(R) characterizes small, resp., large, jumps; both are relevant quantities to describe its approximation. Definition 2.1. We call a Lévy measure ν of type (α, β) for α ∈ [0, 2] and β ∈ [0, ∞] if α = inf α

≥ 0 : lim ν (x, ∞) |x|α < ∞ and lim ν (−∞, −x) |x|α < ∞ , x→0 x→0 β = sup β ≥ 0 : |x|β ν(dx) < ∞ , {|x|>1}



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and there exist constants C, C, C1 , C2 > 0, such that (2)

ν(dξ) ≤ C1 |ξ|−(1+α) dξ

∀|ξ| ≤ C ;

(3)

ν(dξ) ≤ C2 |ξ|−(1+β) dξ

∀|ξ| ≥ C .

The index α is known as the Blumenthal–Getoor index. 2.2. The space-time L´ evy noise or the space-time Poisson noise. The space-time Lévy white noise is a generalization of the space-time Gaussian white noise. Let M (S) be the set of all R-valued measures on the measurable space (S, S), and let

∈ R for M(S) be the σ-field on M (S) generated by functions iB : M (S) μ → μ(B) d

B ∈ S. In the following, we put S = [0, ∞) × R and S = B [0, ∞) × B(Rd ). Definition 2.2. Let (Ω, F , F, P) be a complete filtered probability space, let O ⊂ Rd be a bounded domain with smooth boundary, and let ν ∈ L(R). The space-time Lévy white noise on O with jump size intensity ν ∈ L(R) is a measurable mapping

L : (Ω, F ) → M [0, ∞) × O , M [0, ∞) × O , such that (i) for all I × B ∈ B [0, ∞) × B(O), the random variable L(I × B) is real-valued, is infinitely divisible, and has the characteristic exponent

iθζ 1 − e − i sin(θζ) ν(dζ) exp iθL(I × B) = exp λ(I) λd (B) R

whenever λ(I) λd (B) < ∞, where λ(·), resp., λd (·), denote the Lebesgue measure; (ii) if the sets I1 × B1 , I2 × B2 ∈ B [0, ∞) × B(O) are disjoint, then the random variables L(I1 ×B1 ) and L(I2 ×B2 ) are independent and L(I1 ×B1 ∪I2 ×B2 ) = L(I1 × B1 ) + L(I2 × B2 ); (iii) the measure-valued process t → L [0, t) × · is F-adapted. Remark 2.1. Fix a nonnegative function ϕ ∈ C0 (O). The real-valued process L(ϕ) = {L(t, ϕ) : 0 ≤ t < ∞} defined by t ϕ(ξ) L(ds, dξ) [0, ∞) t → L(t, ϕ) := 0

O

is an R-valued Lévy process with Lévy measure νϕ : B R B → ( O ϕ(ξ) dξ) ν(B). For more details we refer the reader to [16] and [6, Chapter 7]. 3. Simulation of a L´ evy walk. Let μ be a time-homogeneous Poisson random measure on R with intensity ν ∈ L(R). The R-valued mapping t L : [0, ∞) t → L(t) := (4) ζμ

(ds, dζ) 0

R

is well-defined, and L = {L(t) : 0 ≤ t < ∞} defines a time-homogeneous R-valued Lévy process. The Lévy process may be approximated in different ways, where one is to generate a Lévy random walk: For a given time-step size τ > 0, it is a sequence of random variables

(5) Δ0τ L, Δ1τ L, Δ2τ L, . . . , Δm τ L, . . . ,


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where Δm τ L

tm+1

:= L(tm+1 ) − L(tm ) =

R

tm

ζμ

(ds, dζ)

∀m ∈ N.

m In general, the distribution of Δm τ L is not known, such that Δτ L cannot be simulated directly. In the following, we study two strategies to simulate approximations of the Lévy random walk {Δm τ L : m ∈ N}: truncation, and discretization of the measure ν using (i) stretched meshes to accurately resolve small, medium, and large jumps, and (ii) more common meshes covering (compact subsets of) R, combined with a proper Gaussian variable to account for small jumps. The goal of this work is to simulate space-time white Lévy noise. This is done by a (finite) sum of independent one-dimensional Lévy processes, each having the Lévy measure ν. To conclude, two parameters will be involved in the discretization below: the time-step size τ and the truncation parameter . We start with a Lévy process L defined in (4) having characteristic measure ν and ask for a good approximation of the Lévy random walk given in (5) with time-step size τ .

Truncation of small jumps. Let κ ≥ 1. Fix a truncation parameter 0 < < 1 sufficiently small, and define the approximate Lévy measure

. ν ,κ : B(R) D → ν D ∩ R \ −κ , κ Let μ,κ be the Poisson random measure induced by truncating small jumps, i.e., the Poisson random measure having intensity ν ,κ , μ,κ : B(R+ ) × B(R) I × D → ½D (ζ) μ(ds, dζ) ∈ N ∪ {∞}. R\(−κ ,κ )

I

The approximate Lévy process L,κ = {L,κ(t) : 0 ≤ t < ∞} is defined as t L,κ (t) :=

0

R

ζμ ˜,κ (ds, dζ)

∀t ≥ 0,

where μ ˜,κ denotes the compensated Poisson random measure. Proposition 3.1. Let κ ≥ 1. Assume 1 < p ≤ 2, let τ > 0 be the time-step size, and let 0 < < 1 be the truncation parameter. Suppose that ν ∈ L(R) is of type (α, β) with 0 < α < p. There exists a constant C > 0, such that m p (m ∈ N) . ≤ C τ κ(p−α) E |Δm τ L − Δτ L,κ | Proof. By Burkholder’s inequality, for 1 < p ≤ 2, tm+1 m m p E |Δτ L − Δτ L,κ | = E ζμ ˜(ds, dζ) − R

tm

=E ≤ Cτ

tm+1

tm

κ

−κ

tm+1

tm

p ζμ ˜(ds, dζ)

R

ζμ ˜

,κ

p (ds, dζ)

κ

−κ

|ζ|p ν(dζ) .



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Thus, we obtain by using (2) for 0 < α < p

κ

(6) −κ

|ζ|p ν(dζ) ≤ C

κ

−κ

|ζ|p |ζ|−α−1 dζ ≤ Cκ(p−α) .

Approximating the L´ evy measure by a discrete measure. The second step is to approximate the truncated Poisson random measure μ,κ by a Poisson random measure μ ,κ which has the discrete Lévy measure ν,κ , where (7)

ν,κ :=

J

ν(Dj ) δdj ∈ L R \ (−κ , κ ) .

j=1

J Here D1 , . . . , DJ are disjoint sets such that j=1 Dj ⊂ R \ (−κ , κ ), with dj ∈ Dj , j = 1, . . . , J, and δdj ∈ M (R) denotes the Dirac measure supported at dj ∈ Dj . For simplicity, let dj = arg minx∈Dj |x|. We choose J = O(−1 ) to control both truncation and discretization errors for a given Poisson random measure μ in terms of the parameter > 0; different triangulation strategies of R\ (−κ, κ ) will be proposed to accurately assemble ν,κ , while keeping the same complexity of the discrete measure ν,κ in terms of the number of used atoms. m L,κ which approximates Δm L,κ . Generating a random variable Δ τ τ At each time-step indexed by m ∈ N, we generate a family {q1m , . . . , qJm } of independent random variables, where qjm is Poisson distributed with parameter τ ν(Dj ), j = 1, . . . , J. In particular,

τ l ν(D )l j P qjm = l = exp −τ ν(Dj ) l!

∀l ∈ N.

m L,κ is then given by The approximation Δ τ (8)

m Δ τ L,κ :=

J

dj · qjm − τ ν(Dj )

(m ∈ N) ,

j=1

and the Lévy random walk in (5) is approximated by

1 L,κ , Δ 2 L,κ , . . . , Δ m L,κ , . . . . 0 L,κ , Δ (9) Δ τ τ τ τ Remark 3.1. The random variable Δm τ L,κ takes into account all jumps in the time interval [tm , tm+1 ) larger than |κ |, adds them up according to their size, and m compensates the sum. In comparison, the random variable Δ τ L,κ counts the number of jumps occurring in the time interval [tm , tm+1 ) in each cell Dj , j = 1, . . . , J, then compensates by τ ν(Dj ) and weights them by the corresponding reference jump size dj , and finally adds them up. m The following result bounds the error Δm τ L,κ − Δτ L,κ in terms of the difference ,κ ν − ν ; its simple proof is based on Burkholder’s inequality. Proposition 3.2. Let κ ≥ 1. Assume 1 < p ≤ 2, let τ > 0, and let 0 < < 1 be given. There exists a constant C > 0, such that m m p |ζ|p (ν − ν,κ )(dζ) (m ∈ N) . E |Δτ L,κ − Δτ L,κ | ≤ C τ R\(−κ ,κ )



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As a consequence, an accurate approximation of increments of the Lévy noise J ,κ Δm L is controlled by accurate approximations ν = ν(D )δ ,κ j dj , which heavily τ j=1 J κ κ rests on the capability of the mesh j=1 Dj ⊂ R \ (− , ) to resolve small and large jumps related to a Lévy measure of type (α, β). For the sake of simplicity, suppose J that jumps are just positive, and thus j=1 Dj ⊂ R+ \ [0, κ ). Below, we compare three grid strategies: (i) Equally weighted mesh: Let D1 = [, x2 ), DJ = [xJ , ∞), and Dj = [xj , xj+1 ) such that ν(Di ) = ν(Dj ) for 1 ≤ i, j ≤ J ≡ −1 ; see [20]. (ii) Equally spaced mesh: Let D1 = [, x2 ) and Dj = [xj , xj+1 ) be such that λ(Di ) = λ(Dj ) for 1 ≤ i, j ≤ J ≡ −1 ; here λ(·) denotes the Lebesgue measure. (iii) Stretched mesh: Let D1 = [2 , x2 ) and Dj = [xj , xj+1 ) such that λ(Dj ) = j , where j = j2 J+j = jγ

(for xj < 1) , ∀j ∈ {1, . . . , J}

(for x ≥ 1) ∀j ∈ {1, . . . , J} J +j

for some 0 < γ < 12 ; see [17, Chapter 10]. While strategy (ii) is limited to accurately resolving a bounded range of jumps Δm τ L,1 , strategy (i) covers the whole range of jumps R+ \[0, κ); this mesh is specific to each ν, leading to fine intervals Dj for small jumps, while the meshing in regions supporting large jumps is coarse. The mesh strategy (iii) starts with λ(D1 ) = 2 and geometrically −1 intervals Dj = [xj , xj+1 ), the widens to resolve frequent small jumps; after J = C −1 2 , and local width is O() to resolve jump heights of order O(1), since J = C 2

xJ+1 = +

J

2

j = +

j=1

2

J

j = 2 +

j=1

J + 1) J( 2 = O(1) . 2

Large jumps up to a height O(−γ ) are then resolved on a mesh that coarsens from

−1 intervals. Note that the local mesh size O() to O(1−γ ) after another J = C −1 −1 −1

stretched mesh strategy employs J + J = C + C intervals. = O( )p many ,κ The following proposition collects bounds for R\(−κ ,κ ) |ζ| (ν − ν )(dζ) and mesh strategies (i) through (iii) for ν ∈ L(R) of type (α, β); its proof shows improved approximation properties of mesh strategy (iii) which matches the structural properties of ν and evidences the dependence of the approximation error on > 0 and α, β ≥ 0. Proposition 3.3. Let 1 < p ≤ 2, 0 < < 1, and ν ∈ L(R) be a Lévy measure of type (α, β) with α < p < β. For some D ⊂ R, let ½D (ζ)|ζ|p (ν − ν,κ )(dζ) (κ = 1, 2) Err ,κ (D) := R\(−κ ,κ )

be the error due to mesh strategies (i) through (iii) above. Then for every compact K R, (i) (ii) (iii)

β−p

Err ,1 (R) ≤ C(1−α) β for 0 ≤ α < 1 p−α Err ,1 (K) ≤ C max , 2(p−α) 1−2γ γ(β−p) Err ,2 (R) ≤ C max , ,

(equally weighted mesh) , (equally spaced mesh) , (stretched mesh) .




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Three sources of errors need to be controlled for stretched meshes (iii): those attributed to small jumps, those attributed to large jumps, and those not resolved by the mesh for finite > 0; in particular, choosing γ < 12 small causes larger truncation errors in the presence of large jumps. The convergence estimate for the mesh (i) is limited to cases 0 ≤ α < 1. Equally spaced meshes are only competitive in case the Lévy measure is supported on compact subsets of R. Proof. (iii) We independently study the approximation of small, medium, and large jumps. We first bound the error due to approximating ν by ν,2 of form (7) on J j=1 Dj . By using Definition 2.1, we obtain ν(Dj ) ≤ C|xj |−(1+α) λ(Dj )

xj ≤ 2

and

j

≤ Cj 2 2 = Cjj .

=1

J First, we bound the error on j=1 Dj = [2 , xJ+1 ). Because of the fundamental theorem of calculus, the identity xj+1 − xj = j , the bound |xj+1 | ≤ 2|xj |, and the above estimates, J p ,2 p p |ζ| (ν − ν )(dζ) ≤ C |ζ| ν(dζ) − |xj | ν(Dj ) [2 ,xJ+1 )

≤C

j=1

J

|xj+1 | − |xj | p

p

ν(Dj ) ≤ C

j=1

≤C

J

Dj J

|xj+1 |p−1 xj+1 − xj ν(Dj )

j=1

|xj |p−1−(1+α) 2j ≤ C

j=1

J

p−α j p−α−2 ≤ C2(p−α) j

j=1

J

j p−α−2+p−α .

j=1

Using induction and a concavity argument for x → x1−z for 0 < z < 1, one can show J −z −1 that ≤ C J1−z and thus for J = C j=1 j J

j 2(p−α−1) ≤ C 1 + J2(p−α)−1 ≤ C max 1, 1−2(p−α) ,

j=1

such that

(10) [2 ,xJ+1 )

|ζ|p (ν − ν,2 )(dζ) ≤ C max 2(p−α) , . J

j for D

j := [ D xj , x

j+1 ) and x

j := xJ+j . Using

(3), we then conclude that for 1 ≤ j ≤ J, Next, we bound this error on

j=1

j ) ,

j ) ≤ C| xj |−(1+β) λ(D ν(D x

j = xJ +

j

γ ≤ C 1 + j 1+γ = C 1 + jj .

=1

Similar to before, |ζ| (ν − ν )(dζ) ≤ C p

[xJ+1 ) ,xJ+ J+1

,2

J

| xj |p−2−β 2j .

j=1


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and J j 2γ 2 ≤ C J 2γ+1 2 , we obtain Noting | xj |p−2−β ≤ C for all 1 ≤ j ≤ J, j=1 [xJ+1 ) ,xJ+ J+1

|ζ|p (ν − ν,2 )(dζ) ≤ C1−2γ .

Finally, using induction and a convexity argument for x → x1+γ , one can show J γ

1+γ , and thus x ≥ C−γ . Therefore, truncating larger jumps j=1 j ≥ C J J+J+1 introduces an error that is bounded by p |ζ| ν(dζ) ≤ C |ζ|p−(1+β) dζ ≤ C−γ(p−β) . [xJ+ ,∞) J+1

[xJ+ ,∞) J+1

(ii) This assertion easily follows from above. the arguments (i) Note that ν(Dj ) = J1 ν [, ∞) = ν [, ∞) since J = −1 . Because of (3), there holds |xJ |β = O −1 ν([, ∞))−1 . Hence, since p > 1, [,∞)

|ζ|p (ν − ν,1 )(dζ) ≤ C

J

|xj |p−1 λ(Dj )ν(Dj )

j=1 J ≤ Cν [, ∞) |xJ |p−1 λ(Dj ) = Cν [, ∞) |xJ |p , j=1

which implies, using ν [, ∞) = ν [, 1) + ν [1, ∞) ≤ C −α + 1 and β > p, [,∞)

1− βp p p p |ζ|p (ν − ν,1 )(dζ) ≤ C1− β ν [, ∞) ≤ C1− β −α(1− β ) .

Remark 3.2. 1. The given strategy based on the stretched grid (iii) may easily be generalized: Let D1 = [a , x2 ) and Dj = [xj , xj+1 ) such that λ(Dj ) = j , where j = j b a d c

J+j =j

∀j ∈ {1, . . . , J}

∀j ∈ {1, . . . , J}

(for xj < 1) , (for xJ+j ≥ 1)

−1 = O(−1 ) −1 + C for some 0 < a, c and 0 ≤ b, d. Since we only allow J + J = C intervals, this implies a = b + 1. The stretched mesh (iii) corresponds to (a, b, c, d) = (2, 1, 1, γ), while the equally spaced mesh results from (a, b, c, d) = (1, 0, 1, 0). The corresponding approximation error may be shown accordingly, Err ,a (R) ≤ C max a(p−α) , 2(c−d)−1 , (d+1−c)(β−p) . Then choices a ≥ 2, and d2 < c < d + 1 may lead to further reduced approximation errors of ν . 2. Equally weighted meshes depend on the given Lévy measure ν. For J = −1 many atoms, the measure approximation error converges to 0 if α < 1; in situations where α ≥ 1, at least J = −α many atoms are needed to guarantee convergence. 3. It is possible to improve the accuracy of equally weighted meshes by choosing 1 a rougher truncation of small jumps at 2 while keeping J = −1 many atoms. This




2883

approach balances the truncation error and the corresponding approximation error Err , 12 . Approximating the small jumps by a Wiener process. Let κ = 1. So far, small jumps are truncated, but may be approximated by means of a Wiener process, which is defined on the same probability space; it is due to the Lévy–Khintchine formula that both the Lévy process and the Wiener process are independent. Hence, at each time-step m ∈ N, we generate a Gaussian random variable Δm τ W , where (11)

2 Δm τ W ∼ N 0, σ ()τ

and

σ() :=

−

12 ζ 2 ν(dζ) .

Then the Lévy random walk in (5) is approximated by

m 0τ L,1 + Δ0τ W , Δ 1τ L,1 + Δ1τ W , Δ 2τ L,1 + Δ2τ W , . . . , Δ m Δ τ L,1 + Δτ W , . . . . A proper balancing of truncation and time-discretization yields improved convergence rates if the small jumps are additionally accounted for in this way. Proposition 3.4. Let 1 < p ≤ 2, 0 < < 1 sufficiently small, and let ν ∈ L(R) be of type (α, β), α ∈ (0, 2]. Let Lc be a Lévy process with characteristic measure ν (−, ), and let W be a Wiener process of variance σ 2 (). Suppose ≤ Cτ 1/α . Then p α p c m ≤ Cτ 3 p(1− 3 ) E |Δm τ L − Δτ W |

(m ∈ N) ,

c m m where Δm τ L := Δτ L − Δτ L,1 . Proof. We calculate the difference by its Fourier transform using tail estimates. First, note that for some δ > 0

∞ p c m m c m = C L − Δ W P |Δ L − Δ W | ≥ x xp−1 dx E Δm τ τ τ τ 0 ∞ c m p−1 ≤ C δp + P |Δm L − Δ W | ≥ x x dx . τ τ δ

In order to estimate the second integral, we use [12, Lemma 5.1, p. 85]: For any we have random variable X with Fourier transform X, (12)

x x2

P |X| ≥ x ≤ 1 − X(λ) dλ . 2 − x2

By the independence of the Wiener process and the Poisson random measure, we infer that

(13)

m c m m c m E eiλ(Δτ L −Δτ W ) = E eiλΔτ L E e−iλΔτ W τ 2 iyλ 2 = exp τ (e − 1 − iyλ) ν (dy) + σ ()λ , 2 R



2884


where ν := ν (−, ). We substitute (13) into (12) and use the identity eiz = cos(z)+ i sin(z),

x 2

2 −x

x = 2 =

2 x

2 x

2 −x

x 2

2 x

2 −x

x = 2

τ 2 iyλ 2 1 − exp τ (e − 1 − iyλ) ν (dy) + σ ()λ dλ 2 R τ 1 − exp τ (cos(yλ) − 1) ν (dy) + σ 2 ()λ2 2 R + iτ (sin(yλ) − yλ) ν (dy) dλ R

2 x

2 −x

x = 2

m c m 1 − E ei(Δτ L −Δτ W )λ dλ

τ 1 − exp τ (cos(yλ) − 1) ν (dy) + σ 2 ()λ2 2 R dλ × cos τ (sin(yλ) − yλ)ν (dy) R

2 x

2 −x

{1 − ez1 cos (z2 )} dλ ,

where z1 and z2 are defined as τ z1 := τ (cos(yλ) − 1) ν (dy) + σ 2 ()λ2 , 2 R z2 := τ (sin(yλ) − yλ) ν (dy) . R

Next, we apply the Taylor expansion around zero to (z1 , z2 ) → 1 − ez1 cos(z2 ): there exists a constant C > 0 such that z1 e cos(z2 ) − 1 − z1 ≤ C|z1 |2 + C|z2 |2 + C|eξ1 cos(ξ2 )| · |z1 | |z2 |2 + |z1 |2 + C|eξ1 sin(ξ2 )| · |z2 | |z1 |2 + |z2 |2 for some |ξ1 | ≤ |z1 | and |ξ2 | ≤ |z2 |. Hence, we obtain (14)

x 2

2 x

2 −x

≤

x 2

x ≤ 2

m c m 1 − E ei(Δτ L −Δτ W )λ dλ

2 x

2 −x 2 x

2 −x

|1 + z1 − ez1 cos (z2 )| + |z1 | dλ

C|z1 |2 + C|z2 |2 + C|eξ1 cos(ξ2 )| · |z1 | |z2 |2 + |z1 |2 + C|eξ1 sin(ξ2 )| · |z2 | |z1 |2 + |z2 |2 + |z1 | dλ .

Since ν is of type (α, β), there exists by Definition 2.1 some constant C > 0, such that ν(dy) ≤ C |y|−(1+α) dy ,

|y| ≤ .




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Thus, we can simplify each of the summands in the integral. Taking into account a Taylor expansion around 0 of the function x → cos(x), we obtain by using the definition of σ 2 () 1 cos(yλ) − 1 ν (dy) + σ 2 ()λ2 |z1 | = τ 2 R 2

y 1 1 = τ − λ2 + y 4 λ4 ν (dy) + σ 2 ()λ2 2 4! 2 R ≤ Cτ λ4 |y|3−α dy ≤ Cτ λ4 4−α . −

Hence, |z1 |2 ≤ Cτ 2 λ8 8−2α . By Taylor expansion around 0 of the function x → sin(x), 1 1 |yλ|3 + |yλ|5 ν (dy) sin(yλ) − yλ ν (dy) ≤ τ |z2 | = τ 5! R R 3! ≤ Cτ λ3 |y|2−α dy + Cτ λ5 |y|4−α dy ≤ Cτ λ3 3−α + Cτ λ5 5−α , −

−

and hence |z2 |2 ≤ Cτ 2 λ6 6−2α + λ10 10−2α . Using ≤ Cτ 1/α and setting δ := 1 1 τ 3 1− 3 α , we obtain |ξ1 | ≤ C and |ξ2 | ≤ C. We now insert those estimates in (14) and get x2

m c m x 1 − E ei(Δτ L −Δτ W )λ dλ 2 − x2 x2 x Cτ 2 λ8 8−2α + Cτ 2 λ6 6−2α + Cτ 2 λ10 10−2α + Cτ λ4 4−α + Cτ 3 λ11 11−3α ≤ 2 − x2 + Cτ 3 λ9 9−3α + Cτ 3 λ12 12−3α + Cτ 3 λ10 10−3α + Cτ 3 λ14 14−3α + Cτ 3 λ13 13−3α + Cτ 3 λ15 15−3α dλ ≤ Cτ 2 x−8 8−2α + Cτ 2 x−6 6−2α + Cτ 2 x−10 10−2α + Cτ x−4 4−α + Cτ 3 x−12 12−3α + Cτ 3 x−10 10−3α + Cτ 3 x−14 14−3α , and hence p c m E |Δm ≤ Cδ p + Cτ 2 δ p−8 8−2α + Cτ 2 δ p−6 6−2α + Cτ 2 δ p−10 10−2α τ L − Δτ W | + Cτ δ p−4 4−α + Cτ 3 δ p−12 12−3α + Cτ 3 δ p−10 10−3α + Cτ 3 δ p−14 14−3α . 1

1

By setting δ := τ 3 1− 3 α we obtain under the restriction ≤ Cτ 1/α that τ l δ −r r−lα ≤ C for r ≥ 3 · l holds; this yields that Cτ l δ p−r r−lα converges as fast as Cδ p , and thus p p p c m ≤ Cτ 3 p− 3 α . E |Δm L − Δ W | τ τ We may now combine Propositions 3.1 through 3.3 to control both truncation and discretization errors to approximate ν ∈ L(R); an additional compensation of small jumps by a Gaussian random variable (11) for choices ≤ Cτ 1/α (for meshes (i) and (ii)) and 2 ≤ Cτ 1/α (for mesh (iii)) may reduce the truncation error from



2886


p p O(τ p−α ) (for meshes (i) and (ii)), resp., O(τ 2(p−α) ) (for mesh (iii)), to O τ 3 p− 3 α , p 2(p− p α) 3 resp., to O τ 3 ; cf. Proposition 3.4. The additional compensation of small 1 jumps by a Gaussian random variable yields an advantage if z ≤ Cτ α , 0 < z < 1, which restricts admissible choices of τ . To conclude, using a stretched mesh (iii) seems a promising alternative to using an equally spaced mesh (ii) together with additional compensation by the Gaussian variable (11). 4. The implicit Euler scheme with truncated noise. Fix 1 < p ≤ 2. Let f : R × O → R, where O ⊂ Rd , be Lipschitz continuous in the first variable and of linear growth. For T > 0, we consider

(15)

∂u(t, ξ) ˙ ξ) = Δu(t, ξ) + f u(t, ξ), ξ + L(t, ∂t u(t, ξ) = 0

∀(t, ξ) ∈ OT , ∀ (t, ξ) ∈ (0, T ) × ∂O , ∀ξ ∈ O,

u(0, ξ) = u0 (ξ)

˙ ξ) denotes the formal time derivative of the space-time where u0 ∈ Lp (O), and L(t, Lévy noise. Definition 4.1. An analytically weak solution of (15) is a process u = {u(t, ·) : t ≥ 0}, such that (i) u satisfies the following integral equation for any φ ∈ C0∞ (O), any 0 ≤ t ≤ T , and P-almost surely: t u(t, ξ) φ(ξ) dξ − u(0, ξ) φ(ξ) dξ = u(s, ξ) Δφ(ξ) dsdξ 0 O O O t t f u(s, ξ), ξ φ(ξ) dsdξ + φ(ξ) L(ds, dξ); + 0

0

O

O

(ii) for all φ ∈ C0∞ (O), the real-valued process {u(t), φ : t ∈ [0, T ]} is adapted to {Ft }t≥0 and c´ adl´ ag. It is shown in Saint Lubert Bié [19] or Hausenblas [9] that if p < d2 + 1, then there exists a unique analytically weak solution of problem (15) such that u ∈ Lp (ΩT ; Lp (O)). In fact, the semigroup approach is used in [9] to construct a mild solution. Next, we use the approximate Euler method to discretize (15) in time. For this purpose, consider the space-time random walk 0 Δτ L, Δ1τ L, Δ2τ L, . . . , Δm (16) τ L, . . . . For every m ∈ N, the M (O)-valued random variable Δm τ L := L [tm , tm+1 ), · acts on functions φ ∈ C0 (O) via tm+1 m Δτ L, φ M×C0 := (17) φ(ξ) ζ μ

(ds, dξ, dζ) . tm

O

R

Here, μ denotes the space-time Poisson random measure on OT ×R\{0} with jump size intensity ν, and μ

(ds, dξ, dζ) = μ(ds, dξ, dζ) − ν(dζ)dsdξ the compensated Poisson random measure; see, e.g., [2, 16]. Since we will not specify approximation in space, we introduce a space-time random walk where we only approximate the jump size intensity. In particular, let ,κ be a space-time Poisson ranν,κ ∈ L(R) be the measure defined in (7) and let μ dom measure on OT × R \ {0} with jump size intensity ν,κ . Let

0τ L,κ , Δ 1τ L,κ , Δ 2τ L,κ , . . . , Δ m Δ (18) τ L,κ , . . .



2887


be the approximation of the random walk, where m L,κ , φ Δ := τ M×C0

(19)

tm+1

O

tm

R

φ(ξ) ζ μ ,κ (ds, dξ, dζ)

∀ φ ∈ C0 (O) .

For every m ≥ 1, let vτm : O × Ω → R be a solution of the following implicit Euler scheme at time tm to approximate u(tm , ·). Scheme A. Let vτ0 = u0 . For m ∈ N, the random variable vτm+1 solves vτm+1 = (1 − τ Δ)−1 (vτm + τ f (vτm ) + Δm τ L) ,

(20)

where {Δm τ L : m ∈ N} is the family of random variables given in (16). Note that L is an M (O)-valued random variable; by [1] for 1 ≤ q < ∞, the random variable Δm τ q wm := (1 − τ Δ)−1 Δm τ L is an L -valued distributional solution of

w (ξ)(1 − τ Δ)φ(ξ) dξ = m

O

O

φ(ξ)Δm τ L(dξ)

∀ φ ∈ W01,2 (O) ∩ W 2,2 (O) .

Approximation of the space-time Lévy noise leads to the following approximate Euler scheme. Scheme B. Let v¯τ0 = u0 . For every m ∈ N, let v¯τm+1 be an Lp (O)-valued random variable that solves m L,κ ) , vτm + τ f (¯ vτm ) + Δ v¯τm+1 = (1 − τ Δ)−1 (¯ τ

(21)

m where {Δ τ L,κ : m ∈ N} is the family of random variables given in (18). We are ready to formulate our main result. Theorem 4.1. Fix T > 0 and a polyhedral domain O ⊂ Rd for d = 1, 2, 3. Let u0 ∈ Lp (O) for some 1 < p < 2d + 1 be given. Let (Ω, F , F, P) be a complete filtered probability space, where F = {Ft }t≥0 denotes the filtration, and let t → L [0, t) · × be space-time Lévy noise with jump-size intensity ν ∈ L(R) of type (α, β), where α ∈ vτm : m ∈ N} (0, p) and β ∈ (p, ∞]. Then iterates {vτm : m ∈ N} of Scheme A, and {¯ of Scheme B satisfy sup

0≤m≤M

E vτm − v¯τm pLp (O) ≤ C κ(p−α) + Err ,κ (R)

(κ = 1, 2) ,

depending on the mesh strategies (i) through (iii); see Proposition 3.3. The following proposition controls errors due to the approximation of the spacetime Lévy noise L. m Proposition 4.1. Consider Δm τ L : m ∈ N and Δτ L,κ : m ∈ N , which are defined by (17) and (19). Then we have for m ∈ N and values d − dp < γ < 2p (i) (ii)

m L p −γ,p ≤ C τ κ(p−α) + Err ,κ (R) , E Δm L − Δ τ τ ,κ W p m m L,κ p −γ,p ≤ C τ . E Δτ LW −γ,p + E Δ τ W


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Proof of Proposition 4.1. Let p−1 + (p )−1 = 1. Since γ > d −


d p

d p ,

=

we use

W0γ,p

the continuous embedding (O) → C0 (O) to conclude by duality, and by (17) and (19), p m m L,κ p −γ,p ≤ C E sup m L,κ , φ E Δm . Δ L − Δ L − Δ τ τ τ τ W M×C0 φ∈C0 (O), φ C(O) =1

By Burkholder’s inequality for 1 ≤ p ≤ 2, we resume with ! tm+1 " ! p " p ,κ ,κ ≤CE ζ ( μ−μ )(ds, dξ, dζ) ≤ Cτ E |ζ| (ν − ν )(dζ) tm O R R κ ≤Cτ

−κ

|ζ|p ν(dζ) +

R\(−κ ,κ )

|ζ|p (ν − ν,κ )(dζ)

,

and Propositions 3.1 and 3.3 then show assertion (i). Assertion (ii) follows accordingly. The following calculation bounds the error between iterates from Schemes A and B, which implies the assertion of Theorem 4.1. Proof of Theorem 4.1. For m ≥ 1, iterates vτm and v¯τm satisfy m−1 vτm = (1 − τ Δ)−m u0 + (1 − τ Δ)−(m−k) τ f (vτk ) + Δkτ L k=0

and v¯τm

−m

= (1 − τ Δ)

u0 +

m−1

−(m−k)

(1 − τ Δ)

τ f (¯ vτk )

k L,κ +Δ τ

.

k=0

Step 1. Stability of {vτm : m ∈ N} and {¯ vτm : m ∈ N}. We show stability of iterates in Lp (Ω; Lp (O)), which is needed in Step 2 below. # #p E vτm pLp ≤ C #(1 − τ Δ)−m u0 #Lp m−1 #

#p # # −(m−k) k k τ f (vτ ) + Δτ L # p +CE # (1 − τ Δ) L

k=0

# #p ≤ C #(1 − τ Δ)−m u0 #Lp

m−1 p k + CE τ f (vτ )Lp k=0

+C

m−1

# #p E #(1 − τ Δ)−(m−k) Δkτ L#Lp

k=0

≤C

p u0 Lp

+ CE τ

m−1

f (vτk )pLp

k=0 m−1

1 E Δkτ LpW −γ,p pγ 2 k=0 τ (m − k) m−1 m−1

1 + E vτk pLp + C ≤ C u0 pLp + Cτ +C

k=0

k=0

τ pγ , (τ (m − k)) 2




2889

where d− dp < γ < p2 . Here, the lower bound of γ is needed for the Sobolev embedding in Proposition 4.1 and the upper bound guarantees the stability of the sum; see below. In the above calculations, we applied first the martingale type p property of the space Lp (O) (for more details see [5, Appendix A]) and Minkowsky’s inequality. The product of resolvents (1 − τ Δ)−(m−k) inherits the smoothing property of the semigroup and satisfies (see [10, Proposition 4.2] with X = W −γ,p (O), δ = γ2 , and λ = τ1 ) for all γ ≥ 0 # # Γ m − γ2 − γ −m # # τ 2 φW −γ,p (22) (1 − τ Δ) φ Lp ≤ Γ(m) C φW −γ,p ∀m ∈ N, ≤ (τ m)γ/2

∀ φ ∈ W −γ,p (O) .

This bound is used for the third estimate above; the last estimate uses linear growth of f and Proposition 4.1(ii). m−1 τ ≤ By assumption γ < p2 , hence pγ k=0 (τ (m−k)) pγ 2 < 1, such that we can bound 2 T − pγ 2 x dx ≤ C. We obtain 0 m−1 p E vτm pLp ≤ C u0 Lp + C + Cτ E vτk pLp . k=0

Stability then follows by the discrete version of Gronwall’s lemma. Stability of {¯ vτm : m ∈ N} follows along the same lines. Step 2. Bound for E vτm − v¯τm pLp . E vτm − v¯τm pLp m−1 #p # # # kτ L,κ # (1 − τ Δ)−(m−k) τ f (vτk ) − τ f (¯ vτk ) + Δkτ L − Δ . ≤ CE # p L

k=0

Minkowsky’s inequality and the martingale type p property give m−1 # # m # # m p −(m−k) k k vτ ) # τ f (vτ ) − τ f (¯ E vτ − v¯τ Lp ≤ CE #(1 − τ Δ)

p

Lp

k=0

m−1 #p # # # −(m−k) k k + CE Δτ L − Δτ L,κ # . #(1 − τ Δ) p L

k=0

We use again inequality (22) to conclude for d −

d p

0. For every m = 0, . . . , M , let Xτ, be a random variable that solves m m m+1 m L,κ , Xτ, Δ = Xτ, + σ Xτ, τ

m where Δ evy increment, which is given in section 3. τ L,κ denotes the L´ 0,G m,G Scheme D. Let Xτ, = X0 and > 0. For every m = 0, . . . , M , let Xτ, be a random variable that solves

m,G m m+1,G m,G L,κ + Δm W,κ , Δ = Xτ, + σ Xτ, Xτ, τ τ

m L,κ + Δm W,κ denotes the Lévy increment with compensatory Gaussian where Δ τ τ 2 κ increment Δm τ W,κ ∼ N 0, σ ( )τ ; see (11). We use 100000 realizations and a time-step size τ = 0.01. Since we cannot simulate the Lévy increments exactly, we approximate the exact solution using a stretched mesh with ex = 0.001 and γ = 0.1. In order to compare the stretched mesh with the equally spaced mesh, we fix the amount of intervals to J = −1 es = 100. As a consequence, we have to use a different truncation parameter s for the stretched mesh.



20

15

15 Equally Spaced ε = 0.01 Quantiles

Equally Spaced ε = 0.01 Quantiles

with Gaussian increments

20

10 5 0 −5 −10 −15

10 5 0 −5 −10 −15

−20 −20

−15

−10

−5 0 5 stretched ε = 0.001 Quantiles

10

15

−20 −20

20

−15

−10


10

15

20

M (left) and X M,G (right). Fig. 3. Equally spaced mesh: Quantile-quantile plots of the solution Xτ, τ,

without Gaussian increments

with Gaussian increments

20

20

15

15 Stretched J = 100 Quantiles

Stretched J = 100 Quantiles


without Gaussian increments

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10 5 0 −5 −10 −15 −20 −20

10 5 0 −5 −10 −15

−15

−10


10

15

20

−20 −20

−15

−10


10

15

20

M (left) and X M,G (right). Fig. 4. Stretched mesh: Quantile-quantile plots of the solution Xτ, τ,

Figures 3 and 4 display quantile-quantile plots of the solution at final point T = 1 in the restricted range [−20, 20], evidencing improved accuracy of the solution in the presence of Gaussian increments. By using a stretched mesh strategy, small jumps are truncated at κs < es for κ > 1; as a consequence, an additional approximation by a Gaussian variable is less significant if compared to the equally spaced mesh, since 2−α the additional Gaussian variable scales with σ() ≤ C 2 . Hence, the influence of the additional Gaussian variable grows with increasing values α of the Lévy measure and truncation parameter ; performed computational studies evidence a bad behavior of equally weighted meshes for values α > 1, where also assertion (i) of Proposition 3.3 does not apply, and confirm an improved performance of stretched meshes (see also Remark 3.2), where increased stretching reduces the influence of the Gaussian variable. 5.2. SPDE simulation setup. We report on computational studies for the different discretization strategies from section 4 in the case of both finite and infinite driving Lévy measures of type (α, β). Let O := (0, 5), T = 1, and consider

(24)

1 ∂u(t, ξ) ˙ ξ) = · Δu(t, ξ) + L(t, ∂t 10

π ξ u(0, ξ) = sin 5 u(t, ξ) = 0

∀(t, ξ) ∈ OT , ∀ξ ∈ O, ∀(t, ξ) ∈ (0, 1) × ∂O.

We use different driving Lévy processes: the first Lévy measure is να defined in (23) for different α ∈ (0, 2), which cannot be simulated exactly. The second Lévy



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measure corresponds to an inverse Gaussian process IG(0.5, 1), where increments may be simulated exactly (cf. [20, p. 111]): 1

1 B(R) I → νIG (I) := ½{x>0} (x) √ x−3/2 exp − x dx . 2 2 2π I The third Lévy measure describes a Poisson process and can be simulated exactly:

1 B(R) I → νP (I) := δ0.5 (x) + δ1 (x) dx . I 2 Note that only positive jumps occur for all να , νIG , and νP . The following section concerns discretization in space of Scheme B from section 4. 5.3. Discretization in space using finite elements. Let {Th : 0 < h ≤ 1} be d a family of subdivisions of regular triangulations covering O ⊂ R of maximum mesh size 0 < h ≤ 1, i.e., O = Ki ∈Th Ki ⊂ Rd . We define the finite element space Vh := vh ∈ C0 (O) : vh K ∈ P1 (K)

∀K ∈ Th ,

where P1 denotes the space of polynomials of degree one. Let Nh = {Ni : i = 1, . . . , l} denote the set of nodal points and {φi : i = 1, . . . , l} ⊂ Vh related basis functions such that φi (Nj ) = δij (i, j = 1, . . . , l). The L2 (O)-orthogonal projection Ph : L2 (O) → Vh is defined via

Ph v − v, χ = 0 ∀χ ∈ Vh . m Li at nodal points Ni that In the following, we use approximate Lévy increments Δ τ ,κ are defined in (8).

Let v0 = Ph u0 . For every m = 0, . . . , M , let vm be a Vh -valued Scheme B. h h random variable that solves (25)

l

i m Δ vhm − vhm−1 , φ + τ ∇ vhm , ∇φ = τ L,κ φi , φ

∀φ ∈ Vh .

i=1

The results shown in Figure 1 are based on Scheme B. evy walk. Let κ ≥ 1. Approximate Lévy increments k5.4. Simulation of a L´ Li Δ will be generated for every k ≥ 0 by the following algorithm. τ

,κ 1≤i≤l

Algorithm. Let 0 ≤ k ≤ M . (a) For each interval Dj , simulate an independent Poisson distributed random variable qji,k with parameter τ ν(Dj ). (b) Calculate the increment through k Li := Δ τ ,κ

J

dj · qji,k − τ ν(Dj ) .

j=1

(c) Optional: Approximation of small jumps.

i kτ Li,κ . ∼ N 0, σ 2 (κ )τ and add them to Δ Calculate increments Δkτ W,κ




1.2

2893

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 5

0 5 4

1

3

0.75

2 space

4

0

0.75

2

0.5 1

0.25 0

1

3

0.5 1

time

space

0.25 0

0

time

Fig. 5. One possible path of the solution of (24) with α = 0.25 (left) and α = 1.75 (right).

See Figure 5 for possible outcomes for (24) with Lévy measure να , for α = 0.25 and α = 1.75, simulated with a stretched grid with γ = 0.1 and = 0.001. In the following sections, we study the behavior of the error E u(τ m) − vˆhm 2L2 concerning the time-discretization parameter τ , the truncation parameter , and the measure discretization error Err ,κ (R). After different simulations varying between 100 and 6000 paths we have chosen I ≈ 1000 paths to conduct reliable statistics. 1 1 , 100 ) and γ = 0.1. We study the 5.5. Time-step size error τ . Let (h, ) = ( 260

error of solutions from Scheme B for different time-steps τi = 2−i with i = 4, . . . , 8; m −10 }M . the exact solution { vh,ex m=0 is simulated for the smallest time-step size τex = 2 In the case of the inverse Gaussian or the Poisson process, approximations of the exact solution are simulated using exact increments of the driving noise to study m vh,ex }M error effects. We consider the L2 -norm of the difference between { m=0 and m M

{ vh,i }m=0 from Scheme B at the final time T = 1, M M vh,i (ωj ) − vh,ex (ωj )L2 (O) , (τi , ωj ) :=

and compute the mean over all I paths, ⎞ 12 I 1/2 1 2 M M 2 (τi , ωj ) ⎠ ≈ E vh,i − vh,ex L2 (O) . S(τi ) := ⎝ I j=1 ⎛

To get the approximate rate of convergence rs we use the relation rs,i τi S(τi ) ≈ = 2rs,i S(τi+1 ) τi+1 and compute rs as the mean over all rs,i . For d = 1 and p = 2 [10, Theorem 3.1], the theoretical convergence rate for errors is 0.5. Computed rates of convergence with respect to the time-discretization parameter τ are shown in Figure 6, which change slightly for the different Lévy measures να and are contained in the range [0.35, 0.45], evidencing the theoretical result. For the Lévy measure νIG and νP , computed rates of convergence are rs = 0.55 and rs = 0.41, respectively. 5.6. Truncation error . We study the effect of truncation of a given Lévy

measure ν ∈ L(R) on solutions of the implicit approximate Euler scheme (Scheme B). Here, we just focus on the effect of truncating small jumps, while the effect of the


2894

T. DUNST, E. HAUSENBLAS, AND A. PROHL Error of time discretization

0

10 strong error for α strong error for α strong error for α strong error for α strong error for α strong error for α

strong error for IG process strong error for Poisson process 1/2 τ

= 0.5 = 0.75 =1 = 1.25 = 1.5 = 1.75

1/2

−1

τ

−1

10

10

−2

−2

10 −3 10

−2

−1

10

10

time−step size τ

10 −3 10

−2

−1

10

10

time−step size τ

Fig. 6. Rate of convergence of τ for the SPDE given in (24) driven by the L´ evy measures να (left) and the L´ evy measures νIG and νP (right).

measure discretization is treated in section 5.7 below. For this purpose, a stretched J mesh with = 0.001 = ex and γ = 0.1 is used, such that νex ,2 = j=1 ν(Dj )δdj and 1 m (h, τ ) = ( 260 , 2−8 ) are fixed. The exact solution { vh,ex }M m=0 is simulated using the ex ,2 discrete Lévy measure ν . For different truncation parameters i ∈ {0.001, 0.0025, m M 0.005, 0.0075, 0.01}, solutions { vh,i }m=0 are then obtained by using the discrete Lévy i ,2 ex ,2 m := ν [i , ∞). The L2 -error of the difference between { vh,ex }M measure ν m=0 m M and { vh,i }m=0 at final time T = 1 is studied in Figure 7. Error of truncation 1 0.9 0.8

convergence rate


Error of time discretization

0

10

empirical convergence rate theoretical convergence rate

0.7

−1

10

0.6 0.5 0.4 0.3 strong error for α = 0.25 strong error for α = 1 strong error for α = 1.75

0.2 0.1 0 0

−2

0.25

0.5

0.75

1

1.25

1.5

type α of the Levy measure

1.75 1.9 2

10 −3 10

−2

truncation parameter ε

10

Fig. 7. Comparison of the theoretical rate (Theorem 4.1) with the observed rates (left). Truncation error: Rate of convergence with respect to for L´ evy measures να (right)

Figure 7 (left) shows how computed convergence rates change for different values of α. This evidences that the convergence rate changes with respect to the truncation parameter for different Lévy measures as stated in Theorem 4.1. 5.7. Measure discretization error Err ,κ (R). We further study the effect of different mesh strategies proposed in section 3 on solutions of the implicit approximate

Therefore we fix (h, τ ) = ( 1 , 2−8 ). The exact solution Euler scheme (Scheme B). 260 Jex m { vh,ex }M evy measure νex ,2 = j=1 ν(Dj,ex )δdj,ex m=0 is simulated using the discrete L´ from section 5.6. For different truncation parameters i ∈ {0.01, 0.02, 0.03, 0.04, 0.05} m,ew M m,es M }m=0 , resp., { vh,i }m=0 , are obtained using an equally weighted mesh solutions { vh,i Ji i , to construct the dis(dj,ew , Dj,ew )j=1 , and an equally spaced mesh (dj,es , Dj,es )Jj=1 Ji Ji i ,1 i ,1 es = crete Lévy measures νew = j=1 ν(Dj,ew )δdj,ew and ν j=1 ν(Dj,es )δdj,es , respectively. In order to compare the different resulting convergence rates, we fix the amount of atoms to Ji = −1 i . Since a stretched mesh with truncation parameter i


0.9

2895

stretched γ = 0.05 stretched γ = 0.1 stretched γ = 0.125 equally weighted equally spaced

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

2

L −error at T=1: rates of convergence



0 0

0,25

0,5

0,75

1

1,25

1,5

type α of the Levy measure

1,75 1,9 2

using different mesh strategies. Fig. 8. L2 -error of solutions from Scheme B,

has a higher amount of intervals, we have to use different truncation parameters si s ,2 to ensure a fair comparison of discretizations νsi = Jj=1 ν(Dj,s )δdj,s of the Lévy m vh,ex }M measure in terms of effort. The L2 -error rate of the difference between { m=0 m,· M and for each mesh strategy { vh,i }m=0 at final time T = 1 is depicted in Figure 8. The L2 -rates of convergence given in Figure 8 evidence improved accuracy of the stretched mesh if compared to both the equally weighted and the equally spaced meshes. The convergence rate obtained for the equally spaced mesh coincides with our theoretical results for a large activity of small jumps (α ≥ 1): the rate decreases linearly similar to 2 − α. In contrast, for small values of α, we observe huge truncation errors due to large jumps, since jumps of size larger than O(1) are ignored for the simulation. The convergence rates for an equally weighted mesh decrease linearly for a low activity of small jumps (α < 1), and stay on a relative low level for bigger activity of the small jumps (α > 1); this observation supports our theoretical results in Proposition 3.3(i). For the stretched mesh with different parameters γ, the observed convergence rates also fit with the theoretical results in Proposition 3.3(iii), in the case of a big activity of small jumps (α ≥ 1). There, the rate decreases linearly similar to 2(2 − α) for each γ, which double the rate of convergence of the equally spaced mesh. For smaller values of α, the convergence rate stays almost constant in the case of γ = 0.1 and γ = 0.125. The parameter γ reflects the resolution for large jumps up to height O(−γ ), which improves from γ = 0.05 to γ = 0.125, resulting in improved rates of convergence in the latter case. REFERENCES [1] H. Amann and P. Quittner, Elliptic boundary value problems involving measures: Existence, regularity, and multiplicity, Adv. Differential Equations, 6 (1998), pp. 753–813. [2] D. Applebaum, Levy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, UK, 2004. [3] D. Applebaum and J. L. Wu, Stochastic partial differential equations driven by L´ evy spacetime white noise, Random Oper. Stochastic Equations, 3 (2000), pp. 245–261. ´ ski, Approximations of small jumps of L´ [4] S. Asmussen and J. Rosin evy processes with a view towards simulation, J. Appl. Probab., 38 (2001), pp. 482–493. [5] Z. Brze´ zniak and E. Hausenblas, Maximal regularity of stochastic convolution with L´ evy noise, Probab. Theory Related Fields, 145 (2009), pp. 615–637. [6] Z. Brze´ zniak and E. Hausenblas, Martingale solutions for stochastic equation of reaction diffusion type driven by L´ evy noise or Poisson random measure, submitted.



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[7] R. Curtain, Infinite dimensional estimation theory applied to a water pollution problem, in Proceedings of the 7th IFIP Conference on Optimization Techniques: Modeling and Optimization in the Service of Man, Part 2, Lecture Notes in Comput. Sci. 41, Springer-Verlag, London, 1976, pp. 685–699. [8] N. Fournier, Simulation and approximation of L´ evy-driven stochastic differential equations, ESAIM Probab. Stat., 15 (2011), pp. 233–248. [9] E. Hausenblas, Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure, Electron. J. Probab., 10 (2005), pp. 1496–1546. [10] E. Hausenblas and I. Marchis, A numerical approximation of SPDEs driven by Poisson random measure, BIT, 46 (2006), pp. 773–811. [11] J. Jacod, T.G. Kurtz, S. M´ el´ eard, and P. Protter, The approximate Euler method for L´ evy driven stochastic differential equations, Ann. Inst. H. Poincar´ e Probab. Statist., 41 (2005), pp. 523–558. [12] O. Kallenberg, Foundations of Modern Probability, 2nd ed., Probab. Appl. (N.Y.), SpringerVerlag, New York, 2002. [13] C. Knoche, SPDEs in infinite dimension with Poisson noise, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 647–652. [14] H. Kwakernaak, Filtering for systems excited by Poisson white noise, in Control Theory, Numerical Methods and Computer Systems Modelling (Internat. Sympos., IRIA LABORIA, Rocquencourt, 1974), Lecture Notes in Econom. and Math. Systems 107, SpringerVerlag, Berlin, 1975, pp. 468–492. [15] C. Mueller, The heat equation with L´ evy noise, Stochastic Process. Appl., 74 (1998), pp. 67–82. [16] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with L´ evy Noise, Encyclopedia Math. Appl. 113, Cambridge University Press, Cambridge, UK, 2007. [17] A. Prohl, Projection and Quasi-compressibility Methods for Solving the Incompressible Navier-Stokes Equations, Teubner-Verlag, Stuttgart, 1997. [18] S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a L´ evy process, Stochastic Process. Appl., 103 (2003), pp. 311–349. ´ [19] E. Saint Loubert Bi´ e, Etude d’une EDPS conduite par un bruit poissonnien (“Study of a stochastic partial differential equation driven by a Poisson noise”), Probab. Theory Related Fields, 111 (1998), pp. 287–321. [20] K. Schoutens, L´ evy Processes in Finance: Pricing Financial Derivatives, Wiley Ser. Probab. Stat., Wiley, West Sussex, 2003.


Approximate Euler Method for Parabolic Stochastic Partial Differential

Approximate Euler Method for Parabolic Stochastic Partial Differential

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