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.

A probabilistic verification theorem for the finite horizon two-player zero-sum optimal switching game in continuous time

“…If a sufficiently regular solution of the system (1) exists then v i (s, x) is nothing else but the optimal profit that can be generated by switching, with initial conditions i for the system's mode and x for the process X at time s ∈ [0, T]. Probabilistic solution methods for optimal switching problems have been investigated since the 1970s and 1980s in various degrees of generality (see, for instance, [4], [26], [28], [34], and [35]), and most of the recent research in this area has been a combination of the martingale approach via Snell envelopes [11], [24], and the theory of backward stochastic differential equations (BSDE) [7], [14], [20]. The latter methods lead to the study of the following system of reflected BSDEs with lower interconnected obstacles.…”

“…Since σ * n ≥ ρ n for n ≥ 0, it follows that P({σ * n < T for all n ≥ 0}) = 0. Also, the consistency property (4) ensures that it is not optimal for a single player to switch twice at the same instant, so we have σ * n < σ * n+1 on {σ * n < T} for n ≥ 1 (see [17] or [24]). By the construction of α * , noting that u…”

A finite horizon optimal switching problem with memory and application to controlled SDDEs

“…In [8] the existence and uniqueness results of viscosity solutions was extended to the case when the switching costs depend on the state variable. Since then, results have been extended to Knightian uncertainty [14,13,4] and non-Brownian filtration and signed switching costs [19]. For the case when the underlying uncertainty can be modeled by a diffusion process, generalization to the case when the control enters the drift and volatility term was treated in [10].…”

An investment model with switching costs and the option to abandon

“…Its numerous applications include the optimal scheduling of production in a real asset such as a power plant that can operate in distinct modes, say "open" and "closed", as well as the optimal timing of sequentially investing and disinvesting, e.g., in a given stock. The references Bayraktar and Egami [1], Brekke and Øksendal [2], Carmona and Ludkovski [4], Djehiche, Hamadène and Popier [7], Duckworth and Zervos [8], El Asri [9], El Asri and Hamadène [10], Elie and Kharroubi [11], Gassiat, Kharroubi and Pham [12], Guo and Tomecek [13], Hamadène and Jeanblanc [14], Hamadène and Zhang [15], Johnson and Zervos [17], Korn, Melnyk and Seifried [19], Lumley and Zervos [20], Ly Vath and Pham [21], Martyr [22], Pham [23], Pham, Ly Vath and Zhou [24], René, Campi, Langrené and Pham [25], Song, Yin and Zhang [26], Tang and Yong [27], Tsekrekos and Yannacopoulos [29], Zhang and Zhang [31], and Zhang [32] provide an alphabetically ordered list of important contributions in the area.…”