Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients

21 November 2005
Dr M Tretyakov
Stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients possessing unique solutions make up a very important class in applications. For instance, Langevin-type equations and gradient systems with noise belong to this class. At the same time, most numerical methods for SDEs are derived under the global Lipschitz condition. If this condition is violated, the behaviour of many standard numerical methods in the whole space can lead to incorrect conclusions. This situation is very alarming since we are forced to refuse many effective methods and/or to resort to some comparatively complicated numerical procedures. We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of SDEs with nonglobally Lipschitz coefficients. Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. We prove that accuracy of any method of weak order p is estimated by $\varepsilon+O(h^{p})$, where $\varepsilon$ can be made arbitrarily small with increasing the radius of the sphere. The results obtained are supported by numerical experiments. The concept of rejecting exploding trajectories is applied to computing averages with respect to the invariant law for Langevin-type equations. This approach to computing ergodic limits does not require from numerical methods to be ergodic and even convergent in the nonglobal Lipschitz case. The talk is based on joint papers with G.N. Milstein.
  • Stochastic Analysis Seminar