The maximum principle in optimal control of systems driven by martingale measures

We study the relaxed optimal stochastic control problem for systems governed by\nstochastic differential equations (SDEs), driven by an orthogonal continuous\nmartingale measure, where the control is allowed to enter both the drift and\ndiffusion coefficient. The set of admissible controls is a set of measure-valued\nprocesses. Necessary conditions for optimality for these systems in the form of\na maximum principle are established by means of spike variation techniques. Our\nresult extends Peng's maximum principle to the class of measure valued controls.