Connections between Singular Control and Optimal Switching

Abstract

Abstract. This paper builds a new theoretical connection between singular control of finite variation and optimal switching problems. This correspondence provides a novel method for solving high-dimensional singular control problems and enables us to extend the theory of reversible investment: Sufficient conditions are derived for the existence of optimal controls and for the regularity of value functions. Consequently, our regularity result links singular controls and Dynkin games through sequential optimal stopping problems.

Solving Singular Control from Optimal Switching

“…The correspondence established in Guo and Tomecek (2008b) is analogous to the well-known correspondence between a non-decreasing, F-adapted, càglàd singular control (ξ t ) t≥0 and a collection of stopping times (τ…”

“…In our recent work (Guo and Tomecek (2008b)), we established a generic theoretical connection between singular control and optimal switching problems: we defined a consistency property for collections of switching controls, and proved that there is an exact correspondence between the set of finite variation càglàd processes and the set of consistent collections of switching controls.…”

Singular Stochastic Control in Option Hedging with Transaction Costs

“…In our subsequent application to option hedging, S represents the price of the asset underlying the option, whereas in other applications such as to the problem of reversible investment by Guo & Tomecek [21], S represents an economic indicator reflecting demand for a certain commodity. A singular control process is given by a pair (ξ + , ξ − ) of adapted, nondecreasing, LCRL processes such that dξ + and dξ − are supported on disjoint subsets.…”

“…Guo & Tomecek [21] studied the infinite-horizon (T = ∞) reversible investment problem and provided an explicit solution to the problem with the so-called Cobb-Douglas production function F(t, S, x)=S λ x μ , where λ ∈ (0, n), n =[−(α − σ 2 /2)+ (α − σ 2 /2) 2 − 2σ 2 r]/σ 2 > 0 and μ ∈ (0, 1]. The optimal strategy is for the company to increase (resp.…”

Irreversible Capital Accumulation with Economic Impact

“…Related models that have been studied in the mathematics literature include Davis, Dempster, Sethi and Vermes [13], Arntzen [4], Øksendal [42], Wang [48], Chiarolla and Haussmann [11], Bank [6], Alvarez [2,3], Løkka and Zervos [35], Steg [45], Chiarolla and Ferrari [9], De Angelis, Federico and Ferrari [15], and references therein. Furthermore, capacity expansion models with costly reversibility were introduced by Abel and Eberly [1], and were further studied by Guo and Pham [22], Merhi and Zervos [40], Guo and Tomecek [23,24], Guo, Kaminsky, Tomecek and Yuen [21], Løkka and Zervos [36], De Angelis and Ferrari [16], and Federico and Pham [19].…”