Quadratic BSDEs driven by a continuous martingale and application to utility maximization problem

TL;DR

This paper investigates quadratic BSDEs driven by continuous martingales, establishing their existence and uniqueness, and applies these results to solve utility maximization problems with various utility functions.

Contribution

It introduces new existence and uniqueness results for quadratic BSDEs driven by continuous martingales and applies them to utility maximization with different utility functions.

Findings

Proved existence and uniqueness of solutions for quadratic BSDEs.

Applied BSDE results to utility maximization with exponential, power, and logarithmic utilities.

Demonstrated the relevance of quadratic BSDEs in financial optimization problems.

Abstract

In this paper, we study a class of quadratic Backward Stochastic Differential Equations (BSDEs) which arises naturally when studying the problem of utility maximization with portfolio constraints. We first establish existence and uniqueness results for such BSDEs and then, we give an application to the utility maximization problem. Three cases of utility functions will be discussed: the exponential, power and logarithmic ones.