The Taylor expansion is a powerful tool in the analysis of deterministic functions. A stochastic Taylor expansion together with some general existence results have been developed for diffusion processes and some other classes of processes. We explicitly calculate a stochastic Taylor expansion for multivariate Poisson processes. An extension to diffusion processes with Poisson jumps is straightforward. The expansion is used for two financial applications in the context of risk management. (C) 2003 Elsevier B.V. All rights reserved.