# Singular Perturbation Expansion for Utility Maximization with Order-ε Quadratic Transaction Costs

Forthcoming, International Journal of Theoretical and Applied Finance

16 Pages Posted: 29 Jun 2017 Last revised: 25 Nov 2019

See all articles by Shiva Chandra

## Shiva Chandra

New York University (NYU) - NYU Tandon School of Engineering

## Andrew Papanicolaou

NYU Tandon School of Engineering, Department of Finance and Risk Engineering

Date Written: May 13, 2019

### Abstract

We present an expansion for portfolio optimization in the presence of small, instantaneous, quadratic transaction costs. Specifically, the magnitude of transaction costs has a coefficient that is of the order $\epsilon$ small, which leads to the optimization problem having an asymptotically-singular Hamilton-Jacobi-Bellman equation whose solution can be expanded in powers of $\sqrt\epsilon$. In this paper we derive explicit formulae for the first two terms of this expansion. Analysis and simulation are provided to show the behavior of this approximating solution.

Keywords: Transaction costs; singular perturbation expansion; stochastic control; Merton problem; aim portfolio

JEL Classification: C61; D40

Suggested Citation

Chandra, Shiva and Papanicolaou, Andrew, Singular Perturbation Expansion for Utility Maximization with Order-ε Quadratic Transaction Costs (May 13, 2019). Forthcoming, International Journal of Theoretical and Applied Finance. Available at SSRN: https://ssrn.com/abstract=2994500 or http://dx.doi.org/10.2139/ssrn.2994500

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