Numerical Approximations of Non-linear Stochastic Systems.

Numerical Approximations of Non-linear Stochastic Systems. Abstract:  The explicit solution of stochastic differential equations (SDEs can be found only in a few cases. Therefore, there is a need fo accurate numerical approximations that could, for example, enabl  Monte Carlo Simulations. Convergence and stability of these methods are well understood for SDEs with Lipschit  continuous coefficients. Our research focuses on those situations wher  the coefficients of the underlying SDEs are non-Lipschitzian  It was demonstrated in the literature,  that in this case using the classical methods we may fail t  obtain numerically computed paths that are accurate for small step-sizes, or to obtain qualitative information about the behaviour of numerical methods over long time intervals. Our work addresses both of these issues, giving a customized analysis of the most widely used numerical methods.