Optimal stopping of expected profit and cost yields in an investment under uncertainty

TL;DR

This paper studies an optimal stopping problem balancing expected profit and cost in investments under uncertainty, using advanced stochastic methods and variational inequalities to characterize solutions and explore their uniqueness.

Contribution

It introduces a novel approach linking reflected backward SDEs with viscosity solutions for a complex investment decision problem under uncertainty.

Findings

Constructed minimal and maximal solutions via approximation schemes.

Connected solutions to viscosity solutions of variational inequalities.

Showed that uniqueness of solutions generally does not hold.

Abstract

We consider a finite horizon optimal stopping problem related to trade-off strategies between expected profit and cost cash-flows of an investment under uncertainty. The optimal problem is first formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We then construct both a minimal and a maximal solutions using an approximation scheme of the associated system of reflected backward SDEs. When the dependence of the cash-flows on the sources of uncertainty, such as fluctuation market prices, assumed to evolve according to a diffusion process, is made explicit, we also obtain a connection between these solutions and viscosity solutions of a system of variational inequalities (VI) with interconnected obstacles. We also provide two counter-examples showing that uniqueness of solutions of (VI) does not hold in general.