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.

This paper is a quantitative study of a reserve contract for real-time balancing of a power system. Under this contract, the owner of a storage device, such as a battery, helps smooth fluctuations in electricity demand and supply by using the device to increase electricity consumption. The battery owner must be able to provide immediate physical cover, and should therefore have sufficient storage available in the battery before entering the contract. Accordingly, the following problem can be formulated for the battery owner: determine the optimal time to enter the contract and, if necessary, the optimal time to discharge electricity before entering the contract. This problem is formulated as one of optimal stopping, and is solved explicitly in terms of the model parameters and instantaneous values of the power system imbalance. The optimal operational strategies thus obtained ensure that the battery owner has positive expected economic profit from the contract. Furthermore, they provide explicit conditions under which the optimal discharge time is consistent with the overall objective of power system balancing. This paper also carries out a preliminary investigation of the 'lifetime value' aggregated from an infinite sequence of these balancing reserve contracts. This lifetime value, which can be viewed as a single project valuation of the battery, is shown to be positive and bounded. Therefore, in the long run such reserve contracts can be beneficial to commercial operators of electricity storage, while reducing some of the financial and operational risks in power system balancing.This article is part of the themed issue 'Energy management: flexibility, risk and optimization'.

In this paper we study continuous-time two-player zero-sum optimal switching games on a finite horizon. Using the theory of doubly reflected BSDEs with interconnected barriers, we show that this game has a value and an equilibrium in the players' switching controls.MSC2010 Classification: 91A15, 91A55, 91A05, 93E20, 60G40, 49N25.JEL Classification: C61, C72, C73.

This paper studies a discrete-time optimal switching problem on a finite horizon. The underlying model has a running reward, terminal reward and signed (positive and negative) switching costs. Using the martingale approach to optimal stopping problems, we extend a well known explicit dynamic programming method for computing the value function and the optimal strategy to the case of signed switching costs.MSC2010 Classification: 93E20, 60G40, 91B99, 62P20.

This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black-Scholes market.