Analysis of a Degenerate Parabolic Variational Inequality with Gradient Constraints from Optimal Investment and Consumption with Transaction Costs

Abstract

This paper is concerned with a degenerate parabolic variational inequality with gradient constraints arising from an optimal investment and consumption problem with proportional transaction costs. We are most interested in the resulting free boundaries that correspond to the optimal trading strategies. Under a reasonable technical condition, we present a complete analysis of the smoothness of solution and the behaviors of free boundaries.

Keywords: optimal investment and consumption, transaction costs, finite horizon, free boundaries, variational inequality, gradient constraints