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

Abstract

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.

Zero-sum Switching Game, Systems of Reflected Backward SDEs and Parabolic PDEs with bilateral interconnected obstacles

“…Here we should point out that since the switching of the system is made from i to i + 1 and the players do not have their proper sets of switching modes, then the method used e.g. in [10] cannot be applied in our framework. As a consequence of this fact, the question of a solution of (1.1) outside the Markovian framework still open.…”

“…It means that the decision makers do not have their proper modes to which they can switch the system when they decide to switch (see e.g. [10] for more details on this model). Therefore a switching strategy for the players are sequences of stopping times u = (σ n ) n≥0 for C 1 and v = (τ n ) n≥0 for C 2 such that σ n ≤ σ n+1 and τ n ≤ τ n+1 for any n ≥ 0.…”

“…From the probabilistic point of view, this zero-sum switching game problem turns into looking for a solution of its associated system of reflected BSDEs with interconnected bilateral obstacles (see e.g. [10] for the case of proper modes of players). A solution for such a system are adapted processes (Y i , Z i , K i,± ) i∈Γ such that for any i ∈ Γ, and s ≤ T ,          Y i and K i,± continuous; K i,± increasing; (Z i (ω) t ) t≤T is dt − square integrable;…”

“…In zero-sum switching games this issue of admissibility of those controls, defined implicitely through (Y i ) i∈Γ , is crucial (see e.g. [10]). Note also that there is another paper by N.Yamada [17] where the solution of system (1.3) is considered in viscosity sense.…”

“…In Section 2, we introduce the zero-sum switching game and especially the notion of coupling which is already used in several papers including [10,15]. In Section 3, we show that the solution of (1.4) is the value of the zero-sum switching game over square integrable controls when Z i , i ∈ Γ, are dt ⊗ dP-square integrable.…”