Backward Stochastic PDEs related to the utility maximization problem

TL;DR

This paper derives backward stochastic PDEs for utility maximization in incomplete markets, linking optimal strategies to solutions of these equations and illustrating with power, exponential, and logarithmic utilities.

Contribution

It introduces a novel BSPDE framework for utility maximization in incomplete markets, connecting primal problems with forward SDEs for optimal strategies.

Findings

Derived BSPDEs directly related to the primal utility maximization problem

Established the equivalence between optimal strategies and solutions of forward SDEs

Applied the framework to power, exponential, and logarithmic utilities

Abstract

We study utility maximization problem for general utility functions using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an $R^d$-valued continuous semimartingale. Under some regularity assumptions we derive backward stochastic partial differential equation (BSPDE) related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As examples the cases of power, exponential and logarithmic utilities are considered.