Exponential moments for numerical approximations of stochastic partial differential equations

Exponential moments for numerical approximations of stochastic partial differential equations

Arnulf Jentzen and Primož Pušnik
Abstract

Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it is an important research problem to construct and analyze discrete numerical approximation schemes which converge with positive strong convergence rates to the solutions of such infinite dimensional SPDEs. In the case of finite dimensional stochastic ordinary differential equations (SODEs) with non-globally monotone nonlinearities it has recently been revealed that exponential integrability properties of the discrete numerical approximation scheme are a key instrument to establish positive strong convergence rates for the considered approximation scheme. Exponential integrability properties for appropriate approximation schemes have been established in the literature in the case of a large class of finite dimensional SODEs with non-globally monotone nonlinearities. To the best of our knowledge, there exists no result in the scientific literature which proves exponential integrability properties for a time discrete approximation scheme in the case of an infinite dimensional SPDE. In particular, to the best of our knowledge, there exists no result in the scientific literature which establishes strong convergence rates for a time discrete approximation scheme in the case of a SPDE with a non-globally monotone nonlinearity. In this paper we propose a new class of tamed space-time-noise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of infinite dimensional SPDEs. More specifically, the main result of this article proves that these approximation schemes enjoy exponential integrability properties for a large class of SPDEs with possibly non-globally monotone nonlinearities. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations, stochastic Kuramoto-Sivashinsky equations, and two-dimensional stochastic Navier-Stokes equations.

1 Introduction

Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. For example, SPDEs frequently appear in models for the approximative pricing of interest-rate based financial derivatives (cf., e.g., Theorem 2.5 in Harms et al. [23] and (1.2) in Filipović et al. [19]), for the approximative description of random surfaces in surface growth models (cf., e.g., (1) in Blömker & Romito [6] and (3) in Hairer [21]), for describing the temporal dynamics associated to Euclidean quantum field theories (cf., e.g., (1.1) in Mourrat & Weber [35]), for the approximative description of velocity fields in fully developed turbulent flows (cf., e.g., (7) in Birnir [4] and (1.5) in Birnir [5]), and for the approximative description of the temporal evolution of the concentration of an undesired (chemical or biological) contaminant in water (e.g., in a water basin, the groundwater system, or a river; cf., e.g., (1.1) in Kouritzin & Long [33] and also (1.1) in Kallianpur & Xiong [31]). Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it is an important research problem to construct and analyze discrete numerical approximation schemes which converge with positive strong convergence rates to the solutions of such infinite dimensional SPDEs. In the case of finite dimensional stochastic ordinary differential equations (SODEs) with non-globally monotone nonlinearities it has recently been revealed in the literature that exponential integrability properties of the discrete numerical approximation scheme are a key ingredient to establish positive strong convergence rates for the considered approximation scheme; cf., e.g., Hutzenthaler et al. [27], Hutzenthaler & Jentzen [24], and Cozma & Reisinger [12]. In particular, e.g., Corollary 3.8 in Hutzenthaler et al. [27] and Proposition 3.3 in Cozma & Reisinger [12] (cf. also Lemma 3.6 in Bou-Rabee & Hairer [8]) establish exponential integrability properties for appropriate stopped/tamed/truncated approximation schemes in the case of a large class of finite dimensional SODEs with non-globally monotone nonlinearities. To the best of our knowledge, there exists no result in the scientific literature which proves exponential integrability properties for a time discrete approximation scheme in the case of an infinite dimensional SPDE. In particular, to the best of our knowledge, there exists no result in the scientific literature which establishes strong convergence rates for a time discrete approximation scheme in the case of a SPDE with a non-globally monotone nonlinearity (cf., e.g., Dörsek [18] and Hutzenthaler & Jentzen [24]). In this paper we propose a new class of tamed space-time-noise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of infinite dimensional SPDEs. More specifically, the main result of this article (see Theorem 3.3 in Section 3 below) proves that these approximation schemes enjoy exponential integrability properties for a large class of SPDEs with possibly non-globally monotone nonlinearities. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations (see Corollary 4.11 in Subsection 4.3 below), stochastic Kuramoto-Sivashinsky equations (see Corollary 4.13 in Subsection 4.4 below), and two-dimensional stochastic Navier-Stokes equations (see Corollary 4.15 in Subsection 4.5 below).

In this introductory section we now illustrate the proposed approximation schemes and our main result (see Theorem 3.3) in the case of a stochastic Burgers equation (cf., e.g., Section 1 in Da Prato et al. [14] and Section 2 in Hairer & Voss [22]). Let , , , let be non-negative and symmetric, let be a probability space, let be an -cylindrical -Wiener process, let be the Laplacian with Dirichlet boundary conditions on , let , , , satisfy for all , , that , , , let , , be stochastic processes with continuous sample paths which satisfy for all , that , and let , , be stochastic processes which satisfy for all , , that and

(1)

(cf., e.g., [26, 25, 37, 38, 27, 34, 20, 30, 2, 29] for related schemes). In Corollary 4.11 in Subsection 4.3 below we demonstrate that the approximation scheme (1) enjoys finite exponential moments. More precisely, Corollary 4.11 in Subsection 4.3 proves111(with , , , , , , , , , , , , , , , , , , , , , , , for , in the notation of Corollary 4.11) that for all it holds that

(2)

Corollary 4.11 follows from an application of Corollary 3.4 below (see Subsection 4.3 below for details). Corollary 3.4, in turn, is a direct consequence of Theorem 3.3, which is the main result of this article. Theorem 3.3 establishes exponential integrability properties for a more general class of SPDEs (such as stochastic Burgers equations with non-additive noise, stochastic Kuramoto-Sivashinsky equations, and two-dimensional stochastic Navier-Stokes equations on a torus) as well as for a more general type of approximation schemes. Exponential integrability properties such as (2) are a key instrument to establish strong convergence rates for SPDEs with non-globally monotone nonlinearities (cf. [24]). In particular we intend to use (2) and Theorem 3.3, respectively, in succeeding articles to establish strong convergence rates for numerical approximations of stochastic Burgers equations and other SPDEs with non-globally monotone nonlinearities.

While polynomial moment bounds for numerical approximations of infinite dimensional SPDEs and exponential moment bounds for numerical approximations of finite dimensional SODEs have been established in the scientific literature, Theorem 3.3 is – to the best of our knowledge – the first result in the literature which establishes exponential moment bounds for time discrete numerical approximations in the case of infinite dimensional SPDEs. In particular, Theorem 3.3 and its consequences in Corollaries 3.4, 4.11, 4.13, and 4.15, respectively, are – to the best of our knowledge – the first results in the literature that establish exponential integrability properties for time discrete numerical approximations of stochastic Burgers equations, stochastic Kuramoto Sivashinsky equations, and two-dimensional stochastic Navier Stokes equations.

1.1 Notation

Throughout this article the following notation is used. For sets and we denote by the set of all mappings from to . For a topological space and a set we denote by the interior of . For a natural number and normed -vector spaces and we denote by the set of all continuous -linear mappings from to , we denote by the mapping which satisfies for all that , we denote by the set given by , and we denote by the mapping which satisfies for all that . For measurable spaces and we denote by the set of all -measurable functions. For a normed -vector space , a measure space , a real number , and a measurable function we denote by and the extended real numbers given by and . For a topological space we denote by the sigma-algebra of all Borel measurable sets in . For a natural number and a Borel measurable set we denote by the Lebesgue-Borel measure on . For -Hilbert spaces and , a set , and functions and we denote by the function which satisfies for all , that

(3)

For sets and we denote by the real number given by

(4)

For sets and we denote by the function which satisfies for all that . For a set we denote by the power set of , we denote by the number of elements of , and we denote by the set given by . For a normed -vector space with , real numbers , , a set , and an open and convex set we denote by the set given by

(5)

(cf., e.g., (1.12) in Hutzenthaler & Jentzen [25]). We denote by the function which satisfies for all that . For a real number we denote by the set given by . For a real number we denote by the mapping which satisfies for all that

(6)

Let us note for every , that is the maximum step size of the partition . We denote by , , and , , the mappings which satisfy for all , that , , and . For a measure space , a measurable space , a set , and a function we denote by the set given by .

2 Exponential moments for time discrete approximation schemes

2.1 Factorization lemma for conditional expectations

In this subsection we recall in Definitions 2.12.3, Lemma 2.4, Theorem 2.5, and Lemmas 2.62.9 some well known concepts and facts from measure and probability theory. In particular, we recall in Lemma 2.9 below a well-known factorization property for conditional expectations. We use this factorization property in the proofs of our later results. Definitions 2.12.3, Lemma 2.4, Theorem 2.5, and Lemma 2.6 can, e.g., in a very similar form be found in Section 1 in Klenke [32] (see Definition 1.1, Definition 1.10, Theorem 1.16, Theorem 1.18, Theorem 1.19, and Theorem 1.96 in Klenke [32]). Lemmas 2.72.9 can, e.g., in a very similar form be found in Chapter 1 in Da Prato & Zabczyk [16] (see Proposition 1.12 in Da Prato & Zabczyk [16]).

Definition 2.1 (-Stability).

Let be a set. Then we say that is -stable if and only if for all it holds that .

Definition 2.2 (Dynkin system).

Let and be sets. Then we say that is a Dynkin system on if and only if

  1. it holds that ,

  2. it holds for all that , and

  3. it holds for all pairwise disjoint sets that .

Definition 2.3.

Let and be sets with . Then we denote by the set given by

(7)
Lemma 2.4.

Let be a set and let be a Dynkin system on . Then it holds that is -stable if and only if is a sigma-algebra on .

Proof of Lemma 2.4.

Throughout this proof assume w.l.o.g. that is a -stable Dynkin system on (otherwise the statement of Lemma 2.4 is clear). Note that the assumption that is a Dynkin system on ensures for all that

(8)

This and the fact that imply that for all it holds that . Hence, we obtain that for all it holds that

(9)

Combining this, the fact that , and (8) proves that is a sigma-algebra on . The proof of Lemma 2.4 is thus completed. ∎

Theorem 2.5.

Let be a set and let be -stable. Then .

Proof of Theorem 2.5.

Throughout this proof let , , be the sets which satisfy for all that . Note that for all it holds that . This proves that for all it holds that

(10)

In the next step we observe that for all , it holds that

(11)

Moreover, note that for all and all pairwise disjoint sets it holds that

(12)

Combining (10), (11), and (12) proves that for every it holds that is a Dynkin system on . Next note that the assumption that is -stable implies that for all it holds that . This and the fact that for every it holds that is a Dynkin system on proves that for all it holds that . This implies that for all , it holds that . This ensures that for all , it holds that . Hence, we obtain that for all it holds that . In particular, we obtain that for all it holds that . Combining this with the fact that for every it holds that is a Dynkin system on assures that for all it holds that . Therefore, we obtain that for all it holds that . Combining this with Lemma 2.4 completes the proof of Theorem 2.5. ∎

Lemma 2.6.

Let be a measurable space and let . Then there exists a sequence , , which satisfies for all , that , , and .

Proof of Lemma 2.6.

Throughout this proof let , , and , , , be the sets which satisfy for all , that and and let , , be the functions which satisfy for all , that

(13)

Note that for every with it holds that there exist , such that . This and (13) imply that for every with it holds that there exists such that . Hence, we obtain that for all with it holds that

(14)

In addition, note that for all , with it holds that . This proves that for all with it holds that . Combining this and (14) completes the proof of Lemma 2.6. ∎

Lemma 2.7.

Let be a probability space, let and be measurable spaces, let be -independent sigma-algebras, let , , , , and assume for all that . Then it holds that and

(15)
Proof of Lemma 2.7.

Throughout this proof let , , be the functions which satisfy for all , that and let be the set given by . Note that Tonelli’s theorem and the fact that is -measurable show that

(16)

Moreover, observe that for all , it holds that

(17)

This ensures for all that

(18)

Next observe that the monotone convergence theorem proves that for all and all pairwise disjoint sets it holds that

(19)

The monotone convergence theorem for conditional expectations hence shows that for all pairwise disjoint sets it holds that

(20)

Combining (18), (20), and the fact that implies that is a Dynkin system on . Moreover, note that for all , it holds that

(21)

This ensures that . Combining this, the fact that the set is -stable, and Theorem 2.5 (with , in the notation of Theorem 2.5) proves that

(22)

The fact that the set is a Dynkin system on hence assures that . Therefore, we obtain that for all it holds that

(23)

This and (16) complete the proof of Lemma 2.7. ∎

Lemma 2.8.

Let be a probability space, let and be measurable spaces, let be -independent sigma-algebras, let , , , , and assume for all that , . Then it holds that and

(24)
Proof of Lemma 2.8.

Throughout this proof let , , be the functions which satisfy for all , that . Note that for all , it holds that

(25)

The assumption that implies that for all it holds that . Combining this and Lemma 2.7 (with , , , , , , , for in the notation of Lemma 2.7) proves that for all it holds that and

(26)

This and (25) show that and

(27)

This completes the proof of Lemma 2.8. ∎

Lemma 2.9.

Let be a probability space, let and be measurable spaces, let be -independent sigma-algebras, let , , , , and assume for all that . Then it holds that and

(28)
Proof of Lemma 2.9.

Throughout this proof let , , be functions which satisfy for all , that , , and and let , , be the functions which satisfy for all , that . Note that the monotone convergence theorem ensures for all that

(29)

Combining the monotone convergence theorem for conditional expectations and Lemma 2.8 (with , , , , , , , , for in the notation of Lemma 2.8) hence shows

  1. that and

  2. that

    (30)

Combining this and (29) proves that and

(31)

The proof of Lemma 2.9 is thus completed. ∎

2.2 From one-step estimates to exponential moments

In this subsection we establish in Corollary 2.10 below exponential integral properties for approximation schemes (see (34) in Corollary 2.10) under a general one-step condition on the considered approximation scheme (see (