Homogenization and Asymptotics for Small Transaction Costs: The Multidimensional Case

Abstract

In the context of the multi-dimensional infinite horizon optimal consumption investment problem with small proportional transaction costs, we prove an asymptotic expansion. Similar to the one-dimensional derivation in our accompanying paper, the first order term is expressed in terms of a singular ergodic control problem. Our arguments are based on the theory of viscosity solutions and the techniques of homogenization which leads to a system of corrector equations. In contrast with the one-dimensional case, no explicit solution of the first corrector equation is available and we also prove the existence of a corrector and its properties. Finally, we provide some numerical results which illustrate the structure of the first order optimal controls.

Keywords:

• Asymptotic expansions
• Homogenization
• Transaction costs
• Viscosity solutions

2000 AMS Subject Classfication:

• 91B28
• 35K55
• 60H30