Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps

A maximum principle is proved for optimal controls of stochastic systems with random jumps. The control is allowed to enter into both diffusion and jump terms. The form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system (the word "pure" here means the absence of the jump term). In calculating the first-order coefficient for the cost variation, only a property for Lebesgue integrals of scalar-valued functions in the real number space ${\Cal R}$ is used. This shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned.