Utility Maximization of an Indivisible Market with Transaction Costs

TL;DR

This paper addresses the complex problem of utility maximization in markets with indivisible assets and transaction costs, formulating it as an optimal switching problem and establishing conditions for the continuity of the value function.

Contribution

It introduces a novel approach to ensure the continuity of the value function in a degenerate diffusion setting with unbounded domain boundaries.

Findings

Established sufficient conditions for value function continuity.

Characterized the value function using viscosity solutions.

Formulated utility maximization as an optimal switching problem.

Abstract

This work takes up the challenges of utility maximization problem when the market is indivisible and the transaction costs are included. First there is a so-called solvency region given by the minimum margin requirement in the problem formulation. Then the associated utility maximization is formulated as an optimal switching problem. The diffusion turns out to be degenerate and the boundary of domain is an unbounded set. One no longer has the continuity of the value function without posing further conditions due to the degeneracy and the dependence of the random terminal time on the initial data. This paper provides sufficient conditions under which the continuity of the value function is obtained. The essence of our approach is to find a sequence of continuous functions locally uniformly converging to the desired value function. Thanks to continuity, the value function can be characterized by using the notion of viscosity solution of certain quasi-variational inequality.