Optimal Stopping Games for Markov Processes

Abstract

, Probab. Statist. Group Manchester (21 pp) Let X = (X t ) t≥0 be a strong Markov process, and let G 1 , G 2 and G 3 be continuous functions satisfying G 1 ≤ G 3 ≤ G 2 and E x sup t |G i (X t )| < ∞ for i = 1, 2, 3 . Consider the optimal stopping game where the sup-player chooses a stopping time τ to maximise, and the inf-player chooses a stopping time σ to minimise, the expected payoffwhere X 0 = x under P x . Define the upper value and the lower value of the game bywhere the horizon T (the upper bound for τ and σ above) may be either finite or infinite (it is assumed thatIf X is right-continuous, then the Stackelberg equilibrium holds, in the sense that V * (x) = V * (x) for all x with V := V * = V * defining a measurable function. If X is right-continuous and left-continuous over stopping times (quasi-left-continuous), then the Nash equilibrium holds, in the sense that there exist stopping times τ * and σ * such thatfor all stopping times τ and σ , implying also that V (x) = M x (τ * , σ * ) for all x . Further properties of the value function V and the optimal stopping times τ * and σ * are exhibited in the proof.

Optimal Stopping Games and Nash Equilibrium

“…It was recently proved in [12] that if X is right-continuous, then the Stackelberg equilibrium (1.3) holds with V := V* -V* defining a measur able function, and if X is right-continuous and left-continuous over stopping times, then the Nash equilibrium (1.4) holds with r. -inf {t > 0: X t e A} and a, = inf {t > 0: X t e D 2 ), (1.5) where Di -{V -d} and D 2 = {V = G 2 }. The two sufficient conditions are known to be most general in optimal stopping theory (see, e.g., [27] and [28]).…”

“…Moreover, if X is only right-continuous and not left-continuous over stopping times, then the Nash equilibrium can break down while the Stackelberg equilibrium still holds (cf. [12,Example 3.1]). …”

Dynkin games with heterogeneous beliefs

“…These games have later found applications for example in mathematical finance, see [13], which motivated further studies, see for example [1], [5], [6], [7], [14] and [19]. Other more recent contributions study various modifications of the zero-sum optimal stopping game.…”

Construction of Nash equilibrium based on multiple stopping problem in multi-person game

“…A game with continuous time where players have possibility to stop more than once was presented in Laraki and Solan (2005). An extensive bibliography on games can be found in Ekström and Peskir (2008), Nowak and Szajowski (1999), , Ramsey and Szajowski (2008) and Solan and Vieille (2003).…”