ABSTRACT
We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a local Lipschitz condition in the Z and U variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these results is a boundedness condition on the terminal value ξ and its Malliavin derivative . Furthermore, we extend existence and uniqueness theorems to cases where the generator is not even locally Lipschitz in U. BSDEs of the latter type find use in exponential utility maximization.
Acknowledgments
Christel Geiss would like to thank the Erwin Schrödinger Institute, Vienna, for hospitality and support, where a part of this work was written.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Alexander Steinicke http://orcid.org/0000-0001-6330-0295