On Solutions of General Nonlinear Stochastic Integral Equations

Stochastic or random integral equations are extremely important in the study of many physical phenomena in life sciences and engineering [3, 14, 16]. There are currently two basic versions of stochastic integral equations being studied by probabilists and mathematical statisticians, namely, those integral equations involving Ito-Doob type of stochastic integrals and those which can be formed as probabilistic analogues of classical deterministic integral equations whose formulation involves the usual Lebesgue integral. Equations of the later category have been studied extensively. Several papers have appeared on the problem of existence of solutions of nonlinear stochastic integral equations, and the results are established by using various fixed point techniques [1, 6–11]. Further, asymptotic behavior and stability of solutions of stochastic integral equations are discussed in [2, 4, 5, 12, 13]. In this paper we will prove an existence and uniqueness theorem for a general class of nonlinear stochastic integral equations and to investigate the asymptotic behavior of their solutions. The results are based on a construction of the real Banach space of tempered functions, which contains the space D([0,∞)) of real right continuous functions having left-hand limits. The results of this paper generalize the results of Szynal and Wȩdrychowicz [15].