Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients

We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (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 epsilon + O(h(p)), where epsilon can be made arbitrarily small with increasing radius of the sphere. The results obtained are supported by numerical experiments.