Homogenization and asymptotics for small transaction costs

TL;DR

This paper analyzes the impact of small transaction costs on optimal investment strategies, providing explicit asymptotic formulas for a broad class of utility functions and asset dynamics using homogenization and viscosity solutions.

Contribution

It introduces a general framework for asymptotic analysis of the Merton problem with small costs, extending previous work to multidimensional cases and general utility functions.

Findings

Explicit first-order asymptotic expansion derived

Closed-form solution for the singular ergodic control problem in 1D

Framework applicable to multidimensional asset models

Abstract

We consider the classical Merton problem of lifetime consumption-portfolio optimization problem with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one-dimensional case. Unlike the existing literature, we consider a general utility function and general dynamics for the underlying assets. Our arguments are based on ideas from the homogenization theory and use the convergence tools from the theory of viscosity solutions. The multidimensional case is studied in our accompanying paper using the same approach.