Stochastic Differential Equations Driven by Multifractional Brownian Motion and Poisson Point Process (Journal of Partial Differential Equations, 2020)
In this paper, we study a class of stochastic differential equations with additive noise that contains a non-stationary multifractional Brownian motion (mBm) witha Hurst parameter as a function of time and a Poisson point process of class (QL). Thedifferential equation of this kind is motivated by the reserve processes in a generalinsurance model, in which between the claim payment and the past history of liability present the long term dependence. By using the variable order fractional calculuson the fractional Wiener-Poisson space and a multifractional derivative operator, andemploying Girsanov theorem for multifractional Brownian motion, we prove the existence of weak solutions to the SDEs under consideration, As a consequence, we deducethe uniqueness in law and the pathwise uniqueness.