Numerical solution of stochastic differential equations with non-globally Lipschitz continuous coefficients
Final Report Abstract
Stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs) are a powerful tool for modeling time evolving processes with random influence. Many models, e.g., from financial engineering, from population dynamics and from quantum field theory, contain SDEs with superlinearly growing (and hence non-globally Lipschitz continuous) nonlinearities. Explicit solutions of such equations are typically not available and it is an active topic of research in about the last seventeen years to solve SDEs with superlinearly growing nonlinearities approximatively. The standard approximation method for SODEs, the Euler-Maruyama method, has recently been shown to diverge strongly and numerical weakly in the case of one-dimensional SODEs with superlinearly growing nonlinearities. This project proposes several numerical approximation methods which converge in the strong and numerically weak sense in the case of a large class SDEs with superlinearly growing nonlinearities (and which thus overcome the lack of strong and numerically weak convergence of the Euler-Maruyama method) on the one hand and which are explicit and nearly as easy to simulate as the Euler-Maruyama method on the other hand. The convergence analysis of the proposed approximation schemes is based on a general theory for studying integrability properties of discrete-time stochastic processes. Another research topic of this project is the analysis of the convergence speed of numerical approximation methods for SDEs with non-globally Lipschitz continuous nonlinearities. In particular, within this project the existence of SODEs with smooth and globally bounded coefficients to which the Euler-Maruyama method and other methods such as the Milstein scheme converge in the strong and numerically weak sense without any arbitrarily small polynomial rate of convergence has been revealed. This slow convergence phenomena appears in concrete Monte Carlo simulations and is a consequence of a roughing effect for a certain class of Kolmogorov partial differential equations with smooth coefficients revealed within this project. Sufficient general conditions which ensure that numerical approximation methods for SDEs with non-globally Lipschitz continuous nonlinearities converge with suitable strong and numerically weak polynomial rates of convergence are in general still an open problem for future research.
Publications
- A mild Ito formula for SPDEs, 31 pages
Da Prato, G., Jentzen, A. and Röckner, M.
- Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. The Annals of Applied Probability 22 (2012), no. 4, 1611-1641
Hutzenthaler, M., Jentzen, A. and Kloeden, P. E.
(See online at https://doi.org/10.1214/11-AAP803)
