Convergence of the stochastic Euler scheme for locally Lipschitz coefficients

Abstract

Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case of superlinearly growing coefficients, however, has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.

Published at http://www.springerlink.com/content/g076w80730811vv3 in the Foundations of Computational Mathematics 2011

References in corpus (1)

• Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients

Cited by in corpus (14)

• Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
• Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients
• Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations
• Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations
• Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations
• Strong Approximation of Monotone Stochastic Partial Differential Equations Driven by Multiplicative Noise
• Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
• Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients
• Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients
• Stochastic Calculus with Jumps Processes : Theory and Numerical Techniques
• Perturbation estimates for order-one strong approximations of SDEs without globally monotone coefficients
• Stability equivalence among stochastic differential equations and stochastic differential equations with piecewise continuous arguments and corresponding Euler-Maruyama methods
• The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients
• Convergence and stability of the semi-tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients