An existence result for the Cauchy problem for stochastic systems with heredity

Abstract

In this paper, we consider stochastic differential equations with heredity in a Hilbert space. In [14], the existence and uniqueness of a solution of such a class of systems was studied using linear growth and some less-restrictive conditions (Ousgood or Holder type) than the Lipschitz condition on the nonlinear terms. Our objective here is to study the existence problem by dropping even the linear growth condition and instead replacing it by some less restrictive conditions. We prove our existence result by the method of successive approximations and a comparison principle.