On stochastic integrodifferential equations via non-linear integral contractors I

Abstract

The aim of this paper is to study the existence and uniqueness of solutions for a general stochastic integrodifferential equation of the Ito type, by using the concept of non-linear bounded random integral contractors, which includes the Lipschitz condition as a special case. The method applied in this consideration follows partially the basic ideas of the contractor theory introduced earlier by Altman [1, 2] and Kuo [6]. It is also shown that the Lipschitz condition and the condition based on a bounded random integral contractor for the coefficients of the considered equation, in general, cannot be compared.