Stochastic Differential Equations Driven by Multifractional Brownian Motion and Poisson Point Process

Abstract

In this paper, we study a class of stochastic differential equations with additive noise that contains a non-stationary multifractional Brownian motion (mBm) with a Hurst parameter as a function of time and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which between the claim payment and the past history of liability present the long term dependence. By using the variable order fractional calculus on the fractional Wiener-Poisson space and a multifractional derivative operator, and employing Girsanov theorem for multifractional Brownian motion, we prove the existence of weak solutions to the SDEs under consideration, As a consequence, we deduce the uniqueness in law and the pathwise uniqueness.