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

TL;DR

This paper tackles time-inconsistent portfolio optimization by marrying expected utility with a variance penalty on log-returns within an incomplete market driven by a stochastic factor. It characterizes time-consistent equilibria via a coupled quadratic BSDE system and proves existence in two tractable settings: zero correlation (ρ=0) and trading constraints, while constructing an approximate equilibrium for small correlation (ρ≠0) with an error of order O(ρ^2). A deep learning–based numerical scheme solves the resulting BSDEs, enabling practical computation and providing financial insights into how risk aversion and variance weighting shape the optimal trajectory. The work extends the literature by addressing incomplete markets and time-inconsistency through a rigorous equilibrium framework and by offering tractable approximate strategies when exact solutions are intractable. These results have potential implications for dynamic risk management and portfolio design under realistic market frictions.

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 $ρ= 0$, we can construct an approximate time-consistent equilibrium in the general case with an error of order $O(ρ^2)$. Numerical examples and financial insights are also presented based on deep learning algorithms.

Figures (5)

• Figure 1: The training loss function of an experiment with $\rho=0$.
• Figure 2: A sample trajectory of $Y$, $\hat{\pi}_t$ and $X_t$ with $\rho=0$
• Figure 3: Plots of the mean of the equilibrium strategy $\hat{\pi}$ of $1000$ trajectories under different $\gamma$ (left panel) and different $\zeta$ (right panel) with $\rho=0$.
• Figure 4: Comparison of conditional variance and conditional expected utility: our model with different $\gamma$ vs. model without variance term
• Figure 5: The mean of $\hat{\pi}_t$ and the factor $X_t$ under trading constraint with different $\rho$.

Theorems & Definitions (49)

• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Lemma 2.6
• proof
• Remark 2.7
• Corollary 2.8
• proof