15:45
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.