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Exponential moments for numerical approximations of stochastic partial differential equations

by A. Jentzen and P. Pusnik

(Report number 2016-41)

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.

Keywords: Stochastic partial differential equations; Exponential moments

@Techreport{JP16_678,
  author = {A. Jentzen and P. Pusnik},
  title = {Exponential moments for numerical approximations 
of stochastic partial differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-41},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-41.pdf },
  year = {2016}
}

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