Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients
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.
KeywordsEuler scheme Stochastic differential equations Monte Carlo Euler method Local Lipschitz condition
Mathematics Subject Classification (2000)65C05 60H35 65C30
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- 7.M. Hutzenthaler, A. Jentzen, P.E. Kloeden, Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467(2130), 1563–1576 (2011). MathSciNetCrossRefGoogle Scholar
- 15.G.N. Milstein, Numerical Integration of Stochastic Differential Equations (Kluwer Academic, Dordrecht, 1995). Google Scholar
- 19.L. Szpruch, Numerical approximations of nonlinear stochastic systems, Ph.D. thesis, University of Strathclyde, Glasgow, 2010. Google Scholar