The G-Martingale Approach for G-Utility Maximization

TL;DR

This paper develops a G-martingale approach within G-expectation space to solve G-utility maximization problems, providing explicit solutions for log-utility and stochastic interest models under uncertainty.

Contribution

It introduces a novel G-martingale framework for utility maximization in G-expectation space, overcoming quadratic variation challenges and deriving explicit solutions.

Findings

Explicit optimal strategies for log-utility are obtained.

A sufficient condition for G-utility maximization is established.

Explicit solutions for stochastic interest models are derived.

Abstract

In this paper, we study representative investor's G-utility maximization problem by G-martingale approach in the framework of G-expectation space proposed by Peng \cite{Pe19}. Financial market has only a bond and a stock with uncertainty characterized by G-Brownian motions. The routine idea of \cite{Wxz} fails because that the quadratic variation process of a G-Brownian motion is also a stochastic process. To overcome this difficulty, an extended nonlinear expectation should be pulled in. A sufficient condition of G-utility maximization is presented firstly. In the case of log-utility, an explicit solution of optimal strategy can be obtained by constructing and solving a couple of G-FBSDEs, then verifying the optimal strategy to meet the sufficient condition. As an application, an explicit solution of a stochastic interest model is obtained by the same approach. All economic meanings of optimal strategies are consistent with our intuitions.