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Exponential moments for numerical approximations of stochastic partial differential equations

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Stochastics and Partial Differential Equations: Analysis and Computations Aims and scope Submit manuscript

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 strong convergence rates to the solutions of such 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 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. 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 a 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 SPDEs. More specifically, the main result of this article proves that these approximation schemes enjoy exponential integrability properties for a large class of semilinear 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.

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Notes

  1. (with \( d = 1 \), \( \mathcal {D} = (0,1) \), \( \eta = 0 \), \( \gamma = \nicefrac {1}{2} \), \( T = T \), \( \varepsilon = \varepsilon - \nicefrac {1}{ \sqrt{3} } \), \( \delta = \delta \), \( U = H \), \( H = H \), \( {\mathbb {H}}= \{ e_n :n \in \mathbb {N}\} \), \( \mathbb {U} = \{ e_n :n \in \mathbb {N}\} \), \( \lambda _{e_N} = - \pi ^2 N^2 \), \( ( \Omega , \mathcal {F}, {\mathbb {P}}, ( \mathcal {F}_t )_{t \in [0,T]} ) = ( \Omega , \mathcal {F}, {\mathbb {P}}, ( \sigma _\Omega ( (W_s )_{s \in [0,t]} ) )_{t \in [0,T]} ) \), \( W = W \), \( Q = Q \), \( A = A \), \( r = ( H_{ \nicefrac {1}{2} } \ni v \mapsto 2 \varepsilon \max \{ 1, \sqrt{ {\text {trace}}_H(Q) } \} + 2 \varepsilon \max \{ 1, \sqrt{ {\text {trace}}_H(Q) } \} \Vert (-A)^{ \nicefrac {1}{2} } v \Vert _H^2 \in [0,\infty ) ) \), \( b = ( (0,1) \times \mathbb {R}\ni (x,y) \mapsto 1 \in \mathbb {R}) \), \( \vartheta = {\text {trace}}_H(Q) \), \( c = 2 \varepsilon \max \{ 1, \sqrt{ {\text {trace}}_H(Q) } \} \), \( R = {\text {Id}}_H \), \( F = F \), \( \xi = ( \Omega \ni \omega \mapsto \xi \in W_0^{1,2}( (0,1), \mathbb {R}) ) \), \( Y^{ \{ 0, T/M,\ldots , T \}, \{ e_1, \ldots , e_N \}, \{ e_1, \ldots , e_N \}} = Y^{N, M} \) for \( N, M \in \mathbb {N}\), \( \varepsilon \in [1, \infty ) \) in the notation of Corollary 4.11).

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Acknowledgements

We gratefully acknowledge Zdzisław Brzeźniak for several useful comments that helped to improve the presentation of the results. This project has been supported through the SNSF-Research Project \( 200021\_156603 \) “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”.

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  1. Department of Mathematics, Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Arnulf Jentzen & Primož Pušnik

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  1. Arnulf Jentzen
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  2. Primož Pušnik
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Correspondence to Primož Pušnik.

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Jentzen, A., Pušnik, P. Exponential moments for numerical approximations of stochastic partial differential equations. Stoch PDE: Anal Comp 6, 565–617 (2018). https://doi.org/10.1007/s40072-018-0116-y

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