Martingale measures and stochastic calculus

In this paper, martingale measures, introduced by J.B. Walsh, are investigated. We prove, with techniques of stochastic calculus, that each continuous orthogonal martingale measure is the time-changed image martingale measure of a white noise. We also exhibit a representation theorem for certain vector martingale measures as stochastic integrals of orthogonal martingale measures. Thus we can study the following martingale problem: {Mathematical expression} where L is a second order differential operator and q a predictable random measure-valued process. We prove that this problem is bound to a stochastic differential equation with a term integral with respect to a martingale measure.