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Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

Marek T. Malinowski (2015)

Open Mathematics

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...

Some results on stochastic convolutions arising in Volterra equations perturbed by noise

Philippe Clément, Giuseppe Da Prato (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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.

Stochastic viability and a comparison theorem

Anna Milian (1995)

Colloquium Mathematicae

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

Strong and weak solutions to stochastic inclusions

Michał Kisielewicz (1995)

Banach Center Publications

Existence of strong and weak solutions to stochastic inclusions x t - x s s t F τ ( x τ ) d τ + s t G τ ( x τ ) d w τ + s t n H τ , z ( x τ ) q ( d τ , d z ) and x t - x s s t F τ ( x τ ) d τ + s t G τ ( x τ ) d w τ + s t | z | 1 H τ , z ( x τ ) q ( d τ , d z ) + s t | z | > 1 H τ , z ( x τ ) p ( d τ , d z ) , where p and q are certain random measures, is considered.

Strong solutions for stochastic differential equations with jumps

Zenghu Li, Leonid Mytnik (2011)

Annales de l'I.H.P. Probabilités et statistiques

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.

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