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

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 solution and a maximal solution using an approximation scheme of the associated system of reflected backward stochastic differential equations (SDEs). We also address the question of uniqueness of solutions of this system of 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 obtain a connection between these solutions and viscosity solutions of a system of variational inequalities with interconnected obstacles.