A stochastic integral representation for random evolutions.(English)Zbl 0568.60065

Author’s abstract: Previously [ibid. 12, 480-513 (1984; Zbl 0547.60040)] the author established that the martingales $M^{\#}(t)=(\theta,\quad Y(t)-Y(0)-\int^{t}_{0}\int_{\Xi}A^ 2(\xi)Y(s)\mu (d\xi)ds),$ 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 $Y(t)=Y(0)+\int^{t}_{0}\int_{\Xi}A^ 2(\xi)Y(s)\mu (d\xi)ds+\int^{t}_{0}\int_{\Xi}A(\xi)Y(s)W(d\xi,ds).$ The paper proceeds in giving strong existence and uniqueness theorems for this integral equation.
Reviewer: J.A.Goldstein

MSC:

 60H20 Stochastic integral equations 60H05 Stochastic integrals 60G44 Martingales with continuous parameter

Zbl 0547.60040
Full Text: