Finite-Horizon Optimal Multiple Switching with Signed Switching Costs

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.