Solution of a stochastic integral equation using integral contractors

Abstract

A general stochastic integral equation of the form x ( t; ω ) = h ( t; ω ) + ∫ 0 t ƒ ( t, s, x ( s; ω ); ω ) ds + ∫ 0 t g ( t, s, x ( s; ω ); ω ) dβ ( s; ω ) is studied, where t ⩾ 0, ω ε Ω , ( Ω , A , P ) is a complete probability space, and β ( t; ω ) is a Brownian motion process. The concept of a bounded integral vector contractor is utilized to obtain very general conditions for the existence of solutions to the stochastic integral equation. The existence theorems are then applied to give stability results.