BSDEs with logarithmic growth driven by a Brownian motion and a Poisson random measure and connection to stochastic control problem

TL;DR

This paper investigates one-dimensional BSDEs with logarithmic growth driven by Brownian motion and Poisson jumps, establishing existence, uniqueness, and their link to stochastic control problems with optimal strategies.

Contribution

It introduces new existence and uniqueness results for BSDEs with logarithmic growth and connects these equations to stochastic control problems, including the existence of optimal strategies.

Findings

Proved existence and uniqueness of solutions for BSDEs with logarithmic growth.

Established the connection between BSDEs and stochastic control problems.

Demonstrated the existence of optimal control strategies.

Abstract

In this paper, we study one-dimensional backward stochastic differential equation with jump under logarithmic growth assumption in the z-variable (|z|\sqrt{|\ln|z|}|) and an L^p terminal value (for a suitable p>2). We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the stochastic control problem.