A Stochastic Integral Representation for Random Evolutions

Abstract

Previously we established that the martingales $M^\theta(t) = \bigg(\theta, Y(t) - Y(0) - \frac{1}{2} \int^t_0 \int_\Xi A^2(\xi) Y(s)\mu (d\xi) ds\bigg),$ with quadratic variation process $V^\theta(t) = \int^t_0 \int_\Xi (\theta, A(\xi) Y(s))^2\mu (d\xi) ds,$ characterize the limit process for a sequence of random evolutions. This paper shows the equivalence of this presentation to the questions of existence and uniqueness of the stochastic integral equation $Y(t) = Y(0) + \frac{1}{2} \int^t_0 \int_\Xi A^2(\xi) Y(s)\mu (d\xi) ds + \int^t_0 \int_\Xi A(\xi) Y(s) W(d\xi) ds).$ The paper proceeds in giving strong existence and uniqueness theorems for this integral equation.