Portfolio Choice under General Utility with Transaction Costs and Search Frictions

2026-09-30

AI Summary

This study addresses the finite-horizon combinatorial optimization problem under transaction costs and search frictions, focusing on optimizing non-concave, non-differentiable utility functions with trading opportunities governed by Cox processes. To this end, it formulates a semilinear nonlocal Hamilton–Jacobi–Bellman (HJB) equation that relaxes conventional regularity assumptions. By integrating the contraction mapping principle, interior Schauder estimates, and verification theory, the authors rigorously establish the existence and uniqueness of a bounded continuous solution to the normalized equation. Furthermore, the optimal Markov feedback strategy is derived. Numerical simulations reveal multi-interval trading behaviors and abrupt weight transitions. Collectively, this work provides a novel theoretical framework for investment decision-making in markets characterized by complex frictions.

Abstract

We study finite-horizon portfolio optimization with proportional transaction costs and trading opportunities arriving at the jump times of a Cox process. Borrowing and short-selling are prohibited, while utility functions need not be concave, increasing, or differentiable. The admissible class includes differentiable utilities with asymptotic elasticity greater than or equal to one. The associated Hamilton--Jacobi--Bellman equation is semi-linear and nonlocal, with control entering only through a zeroth-order term. We show that the normalized equation admits a unique bounded solution that is continuous in all state variables and classical in time and log-price. The proof combines a contraction argument with interior Schauder estimates, and verification identifies the value function and yields an optimal Markovian feedback strategy. Numerical examples with non-concave utilities exhibit, at fixed time and pre-trade wealth, multiple intervals associated with the same trading action, departures from the usual buy--no-trade--sell ordering, and abrupt switches in the optimal post-trade risky weight.

Problem

• Portfolio optimization
• Transaction costs
• Search frictions
• General utility
• Cox process

Innovation

• Portfolio optimization
• Transaction costs
• Search frictions
• Non-concave utility
• Hamilton-Jacobi-Bellman equation