BSDEs and log-utility maximization for L\'{e}vy processes

TL;DR

This paper proves existence and uniqueness of solutions for a class of backward stochastic differential equations driven by Lévy processes and applies these results to solve a logarithmic utility maximization problem in such stochastic environments.

Contribution

It introduces new existence and uniqueness results for BSDEs with Lévy processes and applies them to utility maximization, extending previous work to more general Lévy-driven models.

Findings

Established existence and uniqueness of BSDE solutions for Lévy processes.

Solved the logarithmic utility maximization problem in Lévy process filtrations.

Extended BSDE theory to include generators with sublinear growth.

Abstract

In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for L\'{e}vy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary L\'{e}vy process.