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

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 $ρ= 0$; (ii) the trading strategy is constrained to be bounded. For the general case with correlation coefficient $ρ\neq 0$, we introduce the notion of an approximate time-consistent equilibrium. Employing the solution structure from the equilibrium in the case $ρ= 0$