We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect...

Regularity of stochastic convolutions corresponding to a Volterra equation, perturbed by a white noise, is studied. Under suitable assumptions, hölderianity of the corresponding trajectories is proved.

We give explicit necessary and sufficient conditions for the viability of polyhedrons with respect to Itô equations. Using the viability criterion we obtain a comparison theorem for multi-dimensional Itô processes

Existence of strong and weak solutions to stochastic inclusions $x_t-x_s\in {\int }_s^t{F}_{\tau }\left(x_{\tau }\right)d\tau +{\int }_s^t{G}_{\tau }\left(x_{\tau }\right)d{w}_{\tau }+{\int }_s^t{\int }_{{ℝ}^n}{H}_{\tau ,z}\left(x_{\tau }\right)q(d\tau ,dz)$ and $x_t-x_s\in {\int }_s^t{F}_{\tau }\left(x_{\tau }\right)d\tau +{\int }_s^t{G}_{\tau }\left(x_{\tau }\right)d{w}_{\tau }+{\int }_s^t{\int }_{\left|z\right|\le 1}{H}_{\tau ,z}\left(x_{\tau }\right)q(d\tau ,dz)+{\int }_s^t{\int }_{\left|z\right|>1}{H}_{\tau ,z}\left(x_{\tau }\right)p(d\tau ,dz)$, where p and q are certain random measures, is considered.

General stochastic equations with jumps are studied. We provide criteria for the uniqueness and existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type. The results are applied to stochastic equations driven by spectrally positive Lévy processes.

By using successive approximation, we prove existence and uniqueness result for a class of neutral functional stochastic differential equations in Hilbert spaces with non-Lipschitzian coefficients