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Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients

by M. Hutzenthaler and A. Jentzen

(Report number 2013-45)

Abstract
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 discretetime 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.

Keywords: stochastic differential equation, rare event, strong convergence, numerical approximation, local Lipschitz condition, Lyapunov condition

@Techreport{HJ13_543,
  author = {M. Hutzenthaler and A. Jentzen},
  title = {Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-45},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-45.pdf },
  year = {2013}
}

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