Finite-Horizon Optimal Multiple Switching with Signed Switching Costs

Abstract

This paper is concerned with optimal switching over multiple modes in continuous time and on a finite horizon. The performance index includes a running reward, terminal reward, and switching costs that can belong to a large class of stochastic processes. Particularly, the switching costs are modelled by right-continuous with left-limits processes that are quasi-left-continuous and can take positive and negative values. We provide sufficient conditions leading to a well known probabilistic representation of the value function for the switching problem in terms of interconnected Snell envelopes. We also prove the existence of an optimal strategy in a suitable class of admissible controls, defined iteratively in terms of the Snell envelope processes.

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