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

• Applicant: Professor Dr. Arnulf Jentzen
• Subject Area: Mathematics
• Term: from 2011 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 196762502

Stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs) are a fundamental instrument for modeling all kind of dynamics with stochastic influence in nature or in man-made complex systems. Since explicit solutions of such equations are typically not available, it is a very active research topic in the last four decades to solve SDEs approximatively. At the moment there is, however, a big discrepancy between the assumptions used for numerical approximation methods for SDEs and the assumptions fulfilled by such equations in "real world" applications, e.g., in population dynamics, in molecular dynamics or in option pricing. More precisely, in many of such applications, SDEs with non-globally Lipschitz nonlinearities appear while in the vast majority of research articles for approximating SDEs the nonlinear terms in the SDE are assumed to be globally Lipschitz continuous. In particular, it has been an open question whether the standard numerical method for approximating SODEs, i.e. the stochastic Euler scheme, converges strongly to the exact solution of an SODE with a superlinearly growing (and hence not globally Lipschitz continuous) drift coefficient such as a cubic drift of the form x-x^3. This problem is precisely described on page 20 in [Higham, Mao & Stuart; Strong convergence of Euler type methods for nonlinear SDEs, 2002]. In the recent article [Hutzenthaler & Jentzen; Non-globally Lipschitz counterexamples for the stochastic Euler scheme, 2009] this question has been answered to the negative, i.e. Eulers method fails to converge strongly to the exact solution of such an SODE. Starting from this recent development, the goal of this project is to construct and to analyze new algorithms which overcome the lack of strong convergence of Eulers method and which solve SODEs and SPDEs with non-globally Lipschitz continuous nonlinearities approximatively.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Dr. E Weinan