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

• Applicant: Professor Dr. Martin Hutzenthaler
• Subject Area: Mathematics
• Term: from 2012 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 219293315

Stochastic differential equations (SDEs) are used in all areas for modeling dynamics with stochastic noise. As applied SDEs typically admit no explicit solution, it is crucial to solve SDEs numerically. The majority of applied SDEs have superlinearly growing coefficients and, therefore, do not satisfy the assumptions of the bulk of the literature. We have recently shown that algorithms developed for the case of global Lipschitz coefficients do in general not transfer to the non-global Lipschitz case without modifications. For this reason, we investigate the convergence behavior of suitably modified explicit Euler methods. More precisely we develop a thorough theory of numerical methods which are recursively defined as a general function of the previous state, of the time increment and of the increment of the Brownian motion. The convergence theory will apply to most of the stochastic ordinary differential equations with locally Lipschitz continuous coefficients having finite moments. Establishing the order of convergence will require additional assumptions such as local smoothness. Our main approach is to bring forward the successful Lyapunov technique to the theory of numerical approximations. Moreover, we extend our finite-dimensional results to stochastic partial differential equations. In particular we study a modified version of the exponential Euler method which hasrecently been proposed for the case of additive noise and which has a rather good order of convergence. An additional objective of this project is to publish our results as monograph „Numerical approximation of stochastic differential equations with non-globally Lipschitz continuous coefficients".

• DFG Programme: Research Grants