Numerical Approximations of Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients

Download or read book Numerical Approximations of Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients written by Martin Hutzenthaler and published by . This book was released on 2015 with total page 99 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, we establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, we illustrate our results for several SDEs from finance, physics, biology and chemistry.