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

Watkins, Joseph C.

Ann. Probab. 13, 531-557 (1985).

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

Keywords:

martingale problem; quadratic variation process; random evolutions; existence and uniqueness

Citations:

Zbl 0547.60040