Exponential moments for numerical approximations of stochastic partial differential equations
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.
KeywordsStochastic partial differential equation Numerical analysis Lyapunov function Exponential moments Approximation scheme
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”.
- 1.Andersson, A., Jentzen, A., Kurniawan, R.: Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. arXiv:1512.06899 (2016)
- 3.Becker, S., Jentzen, A.: Strong, convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations. arXiv:1601.05756 (2016)
- 13.Cox, S.G., Hutzenthaler, M., Jentzen, A.: Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations. arXiv:1309.5595v2 (2014)
- 17.Da Prato, G., Jentzen, A., Röckner, M.A.: Mild Ito formula for SPDEs. arXiv:1009.3526 (2012)
- 26.Harms, P., Stefanovits, D., Teichmann, J., Wüthrich, M.V.: Consistent recalibration of yield curve models. Math. Finance. https://onlinelibrary.wiley.com/doi/abs/10.1111/mafi.12159 (2017)
- 27.Hutzenthaler, M., Jentzen, A.: On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients. arXiv:1401.0295 (2014)
- 30.Hutzenthaler, M., Jentzen, A., Salimova, D.: Strong, convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto–Sivashinsky equations. arXiv:1604.02053 (2016)
- 32.Jentzen, A., Pušnik, P.: Strong, convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. arXiv:1504.03523 (2015)
- 33.Jentzen, A., Pušnik, P.: Exponential moments for numerical approximations of stochastic partial differential equations. arXiv:1609.07031 (2016)