On the Uniqueness of Solutions to Martingale Problems for Diffusion Operators with Progressively Measurable Random Coefficients

The uniqueness of solutions to martingale problems for diffusion operators with progressively measurable coefficients is studied and a uniqueness result is obtained: the uniqueness holds under the conditions of the boundedness and uniform ellipticity for the coefficients of the diffusion operators and under an additional condition for the diffusion coefficients. Construction of appropriate approximation consisting of simple functions to the diffusion coefficients plays a key role; the additional condition is used to ensure the simpleness and then the uniqueness follows from the result in the case of diffusion operators with simple type coefficients, which is due to Stroock and Varadhan.