Simple Bounds for Utility Maximization with Small Transaction Costs

TL;DR

This paper derives simple $ ext{L}_p$-error bounds for approximating frictionless wealth in markets with small transaction costs, providing insights into the value function and optimal strategies using elementary and Malliavin calculus techniques.

Contribution

It introduces elementary methods to bound the approximation error and offers new regularity conditions for optimal trading strategies in the presence of small transaction costs.

Findings

Derived $ ext{L}_p$-error bounds for wealth approximation

Established lower bounds for the frictional value function

Provided regularity conditions for optimal strategies

Abstract

Using elementary arguments, we show how to derive $\mathbf{L}_p$-error bounds for the approximation of frictionless wealth process in markets with proportional transaction costs. For utilities with bounded risk aversion, these estimates yield lower bounds for the frictional value function, which pave the way for its asymptotic analysis using stability results for viscosity solutions. Using tools from Malliavin calculus, we also derive simple sufficient conditions for the regularity of frictionless optimal trading strategies, the second main ingredient for the asymptotic analysis of small transaction costs.