Equilibrium Portfolio Selection under Utility-Variance Analysis of Log Returns in Incomplete Markets

TL;DR

This paper develops a framework for equilibrium portfolio strategies in incomplete markets by balancing utility and variance of log returns, addressing time inconsistency through coupled BSDEs and deep learning methods.

Contribution

It introduces a novel equilibrium concept for utility-variance optimization in incomplete markets, including solutions for special cases and an approximate approach for correlated assets.

Findings

Existence of equilibrium strategies in special cases with independent Brownian motions.

Construction of approximate equilibria with error order O(ρ^2) for correlated markets.

Numerical validation and financial insights using deep learning algorithms.

Abstract

This paper investigates a time-inconsistent portfolio selection problem in the incomplete mar ket model, integrating expected utility maximization with risk control. The objective functional balances the expected utility and variance on log returns, giving rise to time inconsistency and motivating the search of a time-consistent equilibrium strategy. We characterize the equilibrium via a coupled quadratic backward stochastic differential equation (BSDE) system and establish the existence theory in two special cases: (i)the two Brownian motions driven the price dynamics and the factor process are independent with $\rho = 0$; (ii) the trading strategy is constrained to be bounded. For the general case with correlation coefficient $\rho \neq 0$, we introduce the notion of an approximate time-consistent equilibrium. Employing the solution structure from the equilibrium in the case $\rho = 0$, we can construct an approximate time-consistent equilibrium in the general case with an error of order $O(\rho^2)$. Numerical examples and financial insights are also presented based on deep learning algorithms.