Numerical approximation of a cash-constrained firm value with investment opportunities

TL;DR

This paper studies a complex control problem for cash-constrained firms with regime switching, proving the value function's uniqueness and regularity, and providing a convergent numerical approximation to determine optimal investment and dividend policies.

Contribution

It establishes the uniqueness and regularity of the value function and introduces a convergent numerical method for the associated HJB variational inequality.

Findings

Proved the value function is the unique viscosity solution.

Provided a numerical scheme that converges to the value function.

Characterized optimal investment and dividend policies.

Abstract

We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Moreover, we give regularity properties of the value function as well as a description of the shape of the control regions. Based on these theoretical results, a numerical deterministic approximation of the related HJB variational inequality is provided. We finally show that this numerical approximation converges to the value function. This allows us to describe the investment and dividend optimal policies.