Locally Lipschitz path dependent FBSDEs with unbounded terminal conditions in Brownian and Lévy settings

This paper is dedicated to the analysis of forward backward stochastic differential equations driven by a Lévy process. We assume that the generator and the terminal condition are path-dependent and satisfy a local Lipschitz condition. We study solvability and Malliavin differentiability of such BSDEs. The proof of the existence and uniqueness is done in three steps. First of all, we truncate and localize the terminal condition and the generator. Then we use an iteration argument to get bounds for the solutions of the truncated BSDE (independent from the level of truncation). Finally, we let the level of truncation tend to infinity. A stability result ends the proof. The Malliavin differentiability result is based on a recent characterisation for the Malliavin Sobolev space D1,2 by S. Geiss and Zhou.