Finite-Horizon Optimal Multiple Switching with Signed Switching Costs

This paper is concerned with optimal switching over multiple modes on a finite-horizon in continuous time. 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 both positive and negative values. By defining an appropriate class of admissible controls, we are able to derive the 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 within this class of controls, defined iteratively in terms of the Snell envelope processes.