Finite-horizon optimal multiple switching with signed switching costs

TL;DR

This paper studies finite-horizon optimal switching problems with complex, signed switching costs modeled by stochastic processes, providing probabilistic representations and proving the existence of optimal strategies.

Contribution

It introduces a framework for optimal switching with signed costs using interconnected Snell envelopes and establishes conditions for the existence of optimal controls.

Findings

Representation of the value function via interconnected Snell envelopes

Existence of an optimal switching strategy under certain conditions

Extension to a broad class of stochastic 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 both 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 within a suitable class of admissible controls, defined iteratively in terms of the Snell envelope processes.