Nonlinear stochastic ordinary and partial differential equations: regularity properties and approximations

Stochastic Analysis Seminar Series

Abstract: Stochastic differential equations (SDEs), by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs), are a key instrument for modeling time-evolving processes with uncertainties. Nonlinear SDEs appear, for example, in fundamental models from financial engineering, neurobiology, chemistry, and quantum field theory. In particular, we illustrate in this talk how nonlinear SDEs are day after day used in the financial engineering industry to estimate prices of financial derivatives. Since explicit solutions of such equations are typically not available, it is a very active topic of research in the last four decades to solve SDEs approximatively. Although approxmation methods for nonlinear SDEs are highly used in the financial engineering industry and although there are a number of research articles and monographs on the approximation of SDEs, the current state of research for approximating nonlinear SDEs is still at the very beginning. In particular, we reveal in this talk that there exist nonlinear SDEs with smooth and bounded coefficient functions which can, roughly speaking, not be solved approximatively by any time-discrete approximation method in a reasonable computational time. The existence of such equations is a consequence of an irregularity phenomena for a certain class of linear deterministic second-order partial differential equations which we detect in this talk. In addition, we propose a class of new suitably ``tamed'' approximation methods and we derive a sufficient criteria on the coefficient functions of the SDE which ensures that the considered nonlinear SDE can be solved approximatively in a reasonable computional time by the proposed approximation methods. We show that some models from the financial engineering industry fulfill this sufficient criteria and can thus be solved approximatively in a reasonable computional time. Finally, we present a few numerical simulations that indicate that some of the considered financial engineering models can not be solved approximatively in a reasonable computational time by each of the approximation methods that we tested. This talk is based on joint works with Sonja Cox, Martin Hairer, Martin Hutzenthaler, Xiaojie Wang, and Marco Noll. Further details on this topic can be found at [].  


Arnulf Jentzen (ETH Zurich)

Monday, June 8, 2015 - 15:45
to 16:45