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

Abstract

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 to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.

Keywords

• set-valued stochastic integral equation
• set-valued stochastic integrals
• fuzzy stochastic integral equation
• fuzzy stochastic differential equation
• semimartingale
• maruyama approximation
• existence and uniqueness of solution
• osgood’s condition
• bihari’s inequality