Comparison theorem for neutral stochastic integro-differential equations in Hilbert space

Abstract

In the present paper, we will discuss a comparison result for solutions to the Cauchy problems for two stochastic differential equations with delay. On this subject number of authors have obtained their comparison results. We deal with the Cauchy problems for two neutral stochastic integro-differential equations. Except transient-(or drift-) and diffusioncoefficients, our equations include also one integro-differential term. Basic difference of our case from the case of all earlier investigated problems is presence of this term. We intoduce a concept of solutions to our problems and prove the comparison theorem for them. According to our result, under certain assumptions on coefficients of equations under consideration, their solutions depend on the transient-coefficients in a monotone way.