A limit theorem for the martingale problem and continuous dependence of the solutions of stochastic differential equations

Abstract

A topology of compact-open type is defined on the set of pairs consisting of drift and diffusion coefficients for which the Stroock-Varadhan martingale problem has at least one solution. A characterisation is given of subcollections of this set such that the restriction to these subcollections of the function taking each pair into the set of solutions of the corresponding martingale problem is upper semi-continuous. Only weak boundedness conditions are imposed on the coefficients, so that the solutions of the martingale problems under consideration may “escape to infinity” over finite intervals with positive probability. This result is used to obtain a continuous dependence theorem for stochastic differential equations of the Markov type.