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May, 1985 A Stochastic Integral Representation for Random Evolutions
Joseph C. Watkins
Ann. Probab. 13(2): 531-557 (May, 1985). DOI: 10.1214/aop/1176993007


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.


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Joseph C. Watkins. "A Stochastic Integral Representation for Random Evolutions." Ann. Probab. 13 (2) 531 - 557, May, 1985.


Published: May, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0568.60065
MathSciNet: MR781421
Digital Object Identifier: 10.1214/aop/1176993007

Primary: 60H20
Secondary: 60G44 , 60H05

Keywords: existence and uniqueness theorems , martingale measures , Martingale problem , stochastic integral equations

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 2 • May, 1985
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