On uniqueness and existence of solutions to stochastic set-valued differential equations with fractional Brownian motions

Abstract

This paper is concerned with a class of stochastic set differential equations (SSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition. The solutions of SSDEs with an fBm are set-valued stochastic processes. We first provide some prior inequalities on set valued integrals including the Itô type set-valued one with respect to an fBm. Second, the existence and uniqueness theorem of solutions to SSDEs is proven under some assumptions. Furthermore, their continuous dependence on initial data is studied. Finally, a numerical example with the evolution of interest from the financial market is analysed. The Gronwall inequality is employed.

Keywords

• stochastic set differential equation (ssde)
• fractional brownian motion (fbm)
• uniqueness and existence
• lipschitzian condition
• gronwall's inequality