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Numerical Algorithms
Published in: Numerical Algorithms 4/2020

10-12-2020 | Original Paper

Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions

Authors: Arnulf Jentzen, Felix Lindner, Primož Pušnik

Published in: Numerical Algorithms | Issue 4/2020

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Abstract

In this article we establish exponential moment bounds, moment bounds in fractional order smoothness spaces, a uniform Hölder 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.
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Literature
1.
Beccari, M., Hutzenthaler, M., Jentzen, A., Kurniawan, R., Lindner, F., Salimova, D.: Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities. arXiv:1903.​06066, p. 65 (2019)
2.
Becker, S., Gess, B., Jentzen, A., Kloeden, P.E.: Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. arXiv:1711.​02423, p. 104 (2017)
3.
Becker, S., Jentzen, A.: Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations. Stochastic Process. Appl. 129(1), 28–69 (2019)MathSciNetCrossRef
4.
5.
Birnir, B.: The Kolmogorov-Obukhov Theory of Turbulence. Springer Briefs in Mathematics. Springer, New York (2013). A mathematical theory of turbulenceCrossRef
6.
Blömker, D., Romito, M.: Stochastic PDEs and lack of regularity: a surface growth equation with noise: existence, uniqueness, and blow-up. Jahresber. Dtsch. Math.-Ver. 117(4), 233–286 (2015)MathSciNetCrossRef
7.
Brzeźniak, Z., van Neerven, J.M.A.M., Veraar, M.C., Weis, L.: Itô’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differential Equations 245(1), 30–58 (2008)MathSciNetCrossRef
8.
Da Prato, G., Jentzen, A., Röckner, M.: A mild Itô formula for SPDEs. Trans. Amer. Math Soc. 372(6), 3755–3807 (2019)MathSciNetCrossRef
9.
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)CrossRef
10.
Dörsek, P.: Semigroup splitting and cubature approximations for the stochastic Navier-Stokes equations. SIAM J. Numer. Anal. 50(2), 729–746 (2012)MathSciNetCrossRef
11.
Filipović, D., Tappe, S., Teichmann, J.: Term structure models driven by Wiener processes and Poisson measures: existence and positivity. SIAM J. Financial Math. 1(1), 523–554 (2010)MathSciNetCrossRef
12.
13.
14.
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. Ann. Probab. 48(1), 53–93 (2020)MathSciNetCrossRef
15.
Hutzenthaler, M., Jentzen, A.: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Amer. Math. Soc. 236, 1112 (2015). v + 99MathSciNetMATH
16.
Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467(2130), 1563–1576 (2011)MathSciNetMATH
17.
Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl Probab. 22(4), 1611–1641 (2012)MathSciNetCrossRef
18.
Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Ann. Appl Probab. 23(5), 1913–1966 (2013)MathSciNetCrossRef
19.
Hutzenthaler, M., Jentzen, A., Lindner, F., Pušnik, P.: Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations. arXiv:1911.​01870, p. 60 (2019)
20.
Hutzenthaler, M., Jentzen, A., Wang, X.: Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Math Comp. 87(311), 1353–1413 (2018)MathSciNetCrossRef
21.
Jentzen, A., Pušnik, P.: Exponential moments for numerical approximations of stochastic partial differential equations. Stoch. Partial Differ. Equ. Anal Comput. 6(4), 565–617 (2018)MathSciNetMATH
22.
Jentzen, A., Pušnik, P.: Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. IMA Journal of Numerical Analysis. https://​doi.​org/​10.​1093/​imanum/​drz009, in press, published online: 12 April 2019 (Accessed 1 August 2019) (2019)
23.
Mourrat, J.-C., Weber, H.: Convergence of the two-dimensional dynamic Ising-Kac model to \({\Phi }_{2}^{4}\). Comm Pure Appl. Math. 70(4), 717–812 (2017)MathSciNetCrossRef
24.
25.
Sabanis, S.: Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients. Ann. Appl. Probab. 26(4), 2083–2105 (2016)MathSciNetCrossRef
26.
Sell, G.R., You, Y.: Dynamics of Evolutionary Equations, vol. 143 of Applied Mathematical Sciences. Springer, New York (2002)CrossRef
27.
Tretyakov, M.V., Zhang, Z.A.: fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications. SIAM J. Numer. Anal. 51(6), 3135–3162 (2013)MathSciNetCrossRef
28.
Wang, X., Gan, S.: The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients. J. Difference Equ. Appl. 19(3), 466–490 (2013)MathSciNetCrossRef
Metadata
Title
Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions
Authors
Arnulf Jentzen
Felix Lindner
Primož Pušnik
Publication date
10-12-2020
Publisher
Springer US
Published in
Numerical Algorithms / Issue 4/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00871-y

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