Random solution of a stochastic integral equation :almost sure and mean square convergence of successive approximations

Abstract

Abstract A non-linear random integral equation of the Volterra type of the form was considered by Tsokos (1960), where ω∊Ω, the underlying set of the probability measure space (Ω A, P). He was concerned with the existence of a unique random solution to this equation, where a random solution is defined to be a second-order stochastic process, x(t; ω), which satisfies the equation almost surely. The present paper shows that a sequence of successive approximations, xn)tω) converges to the unique random solution at each t≥0 with probability one and in mean square under the conditions of Tsokos' theorem. The rate of convergence of the successive approximations and a bound on the mean square error of approximation are also given.