Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

By Charles-Edouard Br'ehier, Jianbo Cui, Jialin Hong. IMA Journal of Numerical Analysis, 2018

108 citations recorded by the source corpus.

This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f6af7d8459407490ec678b88b1ae36c8ccdd1dd

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In Computational Theory and Mathematics

1. Liying Sun
2. Jianbo Cui
3. Zhonghua Qiao
4. Jialin Hong

In Finance

1. Liying Sun
2. Jianbo Cui
3. Jialin Hong

Citations in the corpus

• Strong approximation of monotone stochastic partial differential equations driven by white noise (2020, background): “2( (0;1)) = O N + M 2 (1.3) ; for any m2(0;1=2), where u N denotes the numerical solution and N;M are the dimension of spectral Galerkin and the number of temporal steps, respectively. The authors in [5] analyzed strong convergence rates of some temporal splitting scheme of the stochastic Allen{Cahn”
• Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients (2018, background): “We would 155 like to mention that for SPDEs with non-globally Lipschitz coefficients, there already exist a lot of results on the strong convergence and strong convergence rate of numerical approximations, see [2, 3, 4, 6, 8, 14, 15, 16, 23, 24, 27] and references therein.”