Strong and weak divergence of exponential and linear-implicit Euler\n approximations for stochastic partial differential equations with\n superlinearly growing nonlinearities

2019/03/14 by Matteo Beccari, Beccari, Matteo, Martin Hutzenthaler +9
Economics, Econometrics and Finance · Engineering · #60H15 #60H35 #65C30 #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Numerical Analysis (math.NA) #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1903.06066

openalex publication_date 2019/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The explicit Euler scheme and similar explicit approximation schemes (such as\nthe Milstein scheme) are known to diverge strongly and numerically weakly in\nthe case of one-dimensional stochastic ordinary differential equations with\nsuperlinearly growing nonlinearities. It remained an open question whether such\na divergence phenomenon also holds in the case of stochastic partial\ndifferential equations with superlinearly growing nonlinearities such as\nstochastic Allen-Cahn equations. In this work we solve this problem by proving\nthat full-discrete exponential Euler and full-discrete linear-implicit Euler\napproximations diverge strongly and numerically weakly in the case of\nstochastic Allen-Cahn equations. This article also contains a short literature\noverview on existing numerical approximation results for stochastic\ndifferential equations with superlinearly growing nonlinearities.\n

Citations

• Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations
• Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations
• The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients
• On the optimal filtering of diffusion processes
• First order strong approximations of scalar SDEs defined in a domain
• Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients
• Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients
• Adapted time steps explicit scheme for monotone BSDEs
• Milstein-type Schemes of SDE Driven by L 'evy Noise with Super-linear\n Diffusion Coefficients
• A note on strong convergence of implicit scheme for SDEs under local one-sided Lipschitz conditions
• Strong convergence rates of semi-discrete splitting approximations for stochastic Allen--Cahn equation
• Multilevel Monte Carlo Method for Ergodic SDEs without Contractivity
• Strong convergence for explicit space-time discrete numerical approximation for 2D stochastic Navier-Stokes equations
• Adaptive time-stepping for Stochastic Partial Differential Equations with non-Lipschitz drift

Related

• Weak approximation of stochastic partial differential equations: the non linear case
• Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients
• Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations
• Strong convergence of the tamed Euler scheme for scalar SDEs with superlinearly growing and discontinuous drift coefficient
• Strong convergence of the exponential Euler scheme for SDEs with superlinear growth coefficients and one-sided Lipschitz drift