Strong solutions of stochastic differential equations for multiparameter processes

Abstract

We consider the stochastic differential equation (SDE) $X_t = V_t \int_{[0,t]}f(s,\omega, X.(\omega)),dZ_5(\omega)$, where $V$ and $Z$ are vector valued process indexed by $t\varepsilon\Re^p_+$. The assumptions we make on $Z$ and on the increasing process controlling $Z$ are satisfied by certain classes of square integrable martingales, by processes of finite variation and by mixtures of these types of processes. Existence, uniqueness and the possibility of explosions of a strong solution $X$ are investigated under Lipschitz conditions on $f$. A well-known sufficient condition for non-explosion is shown to work also in the multiparameter case and stability of $X$ under perturbation of $V$, $f$ and $Z$ is proved. Finally more special SDE without Lipschitz conditions are considered, including a class of SDE of the Tsirel'son type.