Stochastic differential equations driven by fractional Brownian motion and Poisson point process

TL;DR

This paper investigates stochastic differential equations driven by fractional Brownian motion and Poisson processes, establishing existence and uniqueness of solutions through advanced fractional calculus and extending prior results in the field.

Contribution

It introduces new fractional calculus on the fractional Wiener-Poisson space and proves existence and uniqueness of solutions for SDEs driven by combined fractional Brownian motion and Poisson noise.

Findings

Established weak solution existence and uniqueness.

Proved strong solution existence using extended Krylov estimates.

Extended previous results by Mishura and Nualart.

Abstract

In this paper, we study a class of stochastic differential equations with additive noise that contains a fractional Brownian motion (fBM) 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 the long term dependence between the claim payment and the past history of liability becomes the main focus. We establish some new fractional calculus on the fractional Wiener-Poisson space, from which we define the weak solution of the SDE and prove its existence and uniqueness. Using an extended form of Krylov-type estimate for the combined noise of fBM and compound Poisson, we prove the existence of the strong solution, along the lines of Gy\"{o}ngy and Pardoux (Probab. Theory Related Fields 94 (1993) 413-425). Our result in particular extends the one by Mishura and Nualart (Statist. Probab. Lett. 70 (2004) 253-261).