  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).
