ON A FINITE HORIZON STARTING AND STOPPING PROBLEM WITH RISK OF ABANDONMENT
We address the issue of finding a strategy to sustain structural profitability of an investment project, whose production activity depends on the market price of a number of underlying commodities. Depending on the fluctuating prices of these commodities, the activity will either continue until the project's profitability reaches a critical low level at which it is stopped and starts again when it becomes profitable. But, if the structural nonprofitability remains for a while, the investment project will face the risk to be abandoned or be definitely closed. We suggest a general probabilistic set up to model profitability as a function of the market price of a set of commodities, and find the related optimal strategy to sustain it, under the constraint that the project faces the abandonment risk when being nonprofitable under a fixed finite time interval. When the market price dynamics is described by a diffusion process, we show that the optimal strategy is related to viscosity solutions of a system of two variational inequalities with inter-connected obstacles.
Real options, security design, backward stochastic differential equation, abandonment risk, Snell envelope, stopping time, stopping and starting, optimal switching, viscosity solution of PDEs, variational inequalities, 60G40, 93E20, 62P20, 91B99
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