Exponential moments for numerical approximations of stochastic partial differential equations
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 nonglobally monotone nonlinearities. Solutions of SPDEs with nonglobally 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 nonglobally 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. 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 this paper we propose a new class of tamed spacetimenoise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of infinite dimensional SPDEs. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations, stochastic KuramotoSivashinsky equations, and twodimensional stochastic NavierStokes equations.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.07031
 Bibcode:
 2016arXiv160907031J
 Keywords:

 Mathematics  Probability
 EPrint:
 44 pages