A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications

Abstract

A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDE) which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. Some numerical tests are presented.

28 pages

References in corpus (2)

• Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
• Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations

Cited by in corpus (20)

• Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients
• Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes
• On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients
• Order-preserving strong schemes for SDEs with locally Lipschitz coefficients
• Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition
• Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
• On Explicit Milstein-type Scheme for Mckean-Vlasov Stochastic Differential Equations with Super-linear Drift Coefficient
• Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
• Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities
• On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions
• Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations
• Parameter Estimation in Nonlinear Multivariate Stochastic Differential Equations Based on Splitting Schemes
• Sampling from mixture distributions based on regime-switching diffusions
• Strongly Asymptotically Optimal Schemes for the Strong Approximation of Stochastic Differential Equations with respect to the Supremum Error
• Perturbation estimates for order-one strong approximations of SDEs without globally monotone coefficients
• Strang Splitting for Parametric Inference in Second-order Stochastic Differential Equations
• Existence, uniqueness, and numerical approximations for stochastic Burgers equations
• A dynamic domain semi-Lagrangian method for stochastic Vlasov equations
• Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions