Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients

• Published in: Foundations of computational mathematics Vol. 11; no. 6; pp. 657 - 706
• Main Authors: Hutzenthaler, Martin, Jentzen, Arnulf
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.12.2011; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational mathematics; Computer Science; Computer simulation; Convergence; Differential equations; Economics; Eulers equations; Foundations; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Monte Carlo methods; Numerical Analysis; Polynomials; Stochastic models; Stochasticity; Stochastic differential equations; Monte Carlo Euler method; 65C30; Local Lipschitz condition; 65C05; Euler scheme; 60H35
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-011-9101-9

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. However, the important case of superlinearly growing coefficients 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.