Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convol
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Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions Arnulf Jentzen1 · Felix Lindner2 · Primoˇz Puˇsnik1 Received: 19 December 2018 / Accepted: 27 December 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this article we establish exponential moment bounds, moment bounds in fractional order smoothness spaces, a uniform H¨older continuity in time, and strong convergence rates for a class of fully discrete exponential Euler-type numerical approximations of infinite dimensional stochastic convolution processes. The considered approximations involve specific taming and truncation terms and are therefore well suited to be used in the context of SPDEs with non-globally Lipschitz continuous nonlinearities. Keywords Stochastic partial differential equation · SPDE · Stochastic convolution · Tamed-truncated numerical approximation · Exponential moment bound · Strong convergence rate
1 Introduction Stochastic partial differential equations (SPDEs) of evolutionary type are important modeling tools in economics and the natural sciences (see, e.g., Birnir [4, Equation (7)], Birnir [5, Equation (1.5)], Bl¨omker & Romito [6, Equation (1)], Filipovi´c Arnulf Jentzen
[email protected] Felix Lindner [email protected] Primoˇz Puˇsnik [email protected] 1
Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, Zurich, Switzerland
2
Institute of Mathematics, Faculty of Mathematics and Natural Sciences, University of Kassel, Kassel, Germany
Numerical Algorithms
et al. [11, Equation (1.2)], Hairer [12, Equation (3)], Harms et al. [13, Theorem 3.5], and Mourrat & Weber [23, Equation (1.1)]). However, exact solutions to SPDEs are usually not known explicitly. Therefore, it has been and still is a very active research area to develop and analyze numerical approximation methods which approximate the exact solutions of SPDEs with a reasonable approximation accuracy in a reasonable computational time. It is known that in order to approximate the exact solutions of stochastic evolution equations appropriately, the numerical methods employed should enjoy similar statistical properties, such as finite uniform moment bounds (see, e.g., Hutzenthaler & Jentzen [15] and the references therein). Unfortunately, moments of the easily realizable Euler-Maruyama and exponential Euler approximation methods are known to diverge for some stochastic differential equations (SDEs) and SPDEs with superlinearly growing nonlinearties (see, e.g., Beccari et al. [1] and Hutzenthaler et al. [16, 18]). This poses the challenge to develop new efficient approximation methods which preserve finite moments (see, e.g., Hutzenthaler & Jentzen [15, Corollary 2.21 and Theorem 3.15], Hutzenthaler et al. [17, Theorem 1.1 and Lemma 3.9], Sabanis [24, Theorem 2.2, Corollary 2.3, and Lemmas 3.2–3.3], Sabanis [25, Theorems 1–3 and Lemmas 1–2], Tretyakov & Zhang [27, Theorem 2.1], and Wang & Gan [28, Theore
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