A consumption-investment problem with a diminishing basket of goods

Abstract

We consider the problem faced by an economic agent trying to find the optimal strategies for the joint management of her consumption from a basket of K goods that may become unavailable for consumption from some random time τ onward, and her investment portfolio in a financial market model comprised of one risk-free security and an arbitrary number of risky securities driven by a multi-dimensional Brownian motion. We apply previous abstract results on stochastic optimal control problem with multiple random time horizons to obtain a sequence of dynamic programming principles and the corresponding Hamilton-Jacobi-Bellman equations. We then proceed with a numerical study of the value function and corresponding optimal strategies for the problem under consideration in the case of discounted constant relative risk aversion utility functions (CRRA).