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

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. 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 also obtain a connection between these solutions and viscosity solutions of a system of variational inequalities (VI) with interconnected obstacles.In other words, the cost Y 2 − a and the profit Y 1 + b act as obstacles that define the exit strategy.The main result of the paper is to show existence of the pair (Y 1 , Y 2 ) that solves the system of equations (1.1) and (1.4) and also to prove that τ * and σ * given respectively by (1.2) and (1.5) are

A full balance sheet two-mode optimal switching problem

“…we take into account the trade-off strategies between expected profit and expected cost yields. It is a combination of ideas and techniques for the two-modes starting and stopping problem developed in [16] and [8], and optimal stopping involving the full balance sheet motivated by problems occurring in merger and acquisition operations introduced in [9]. This problem is a natural extension of [8] and [9] since it incorporates both the action of switching between modes and the action of terminating a project, if it is found to be unprofitable.…”

“…It is a combination of ideas and techniques for the two-modes starting and stopping problem developed in [16] and [8], and optimal stopping involving the full balance sheet motivated by problems occurring in merger and acquisition operations introduced in [9]. This problem is a natural extension of [8] and [9] since it incorporates both the action of switching between modes and the action of terminating a project, if it is found to be unprofitable. For example, being in mode 1, one may want to switch to mode 2 at time t, if the expected profit yield, Y +,1 , in this mode falls below the maximum of the expected profit yield in mode 2, Y +,2 , minus a switching cost ℓ 1 from mode 1 to mode 2, and the expected cost yield in mode 1, Y −,1 minus the cost a 1 incurred when exiting/terminating the production while in mode 1, i.e.…”

“…However, as it is noted in the article, the addition of the risk of abandonment makes it very difficult and highly nonlinear to find this strategy, though there are numerical methods of finding appropriate approximations. A further extension related to this paper is considered in [9]. Instead of only maximizing the expected profit yield with respect to a family of admissible stopping times, the optimal stopping problem is related to trade-off strategies between the expected profit yield (to be maximized) and the expected cost yield (to be minimized).…”

“…This class of switching problems and various extensions to multiple modes has recently attracted a lot of interest. In the following non-exhaustive list of references [2,25,4,3,14,1,10,9,12,18,17,20,24,27,26,30,21,35,31,34] the authors consider various aspects of the multiple modes switching problem, where only switching between the different modes is allowed until the end of the time horizon. Roughly, the setting assumes only that the state process, including the switching costs, are general processes adapted to the Brownian filtration.…”