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.
Publié le : 1985-05-14
Classification:
Martingale problem,
martingale measures,
stochastic integral equations,
existence and uniqueness theorems,
60H20,
60H05,
60G44
@article{1176993007,
author = {Watkins, Joseph C.},
title = {A Stochastic Integral Representation for Random Evolutions},
journal = {Ann. Probab.},
volume = {13},
number = {4},
year = {1985},
pages = { 531-557},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993007}
}
Watkins, Joseph C. A Stochastic Integral Representation for Random Evolutions. Ann. Probab., Tome 13 (1985) no. 4, pp. 531-557. http://gdmltest.u-ga.fr/item/1176993007/