Existence, uniqueness and Malliavin differentiability of Lévy-driven BSDEs with locally Lipschitz driver

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 $D\xi$. 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.

KEYWORDS:

• BSDEs with jumps
• locally Lipschitz generator
• quadratic BSDEs
• existence and uniqueness of solutions to BSDEs
• Malliavin differentiability of BSDEs

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

Additional information

Funding

Alexander Steinicke is supported by the Austrian Science Fund (FWF): Project F5508-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.