Abstract

We consider the stochastic differential equation (SDE) f(s,w, X.(w))dZs((w), where V and Z are vector valued process indexed by . 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.

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