On the value of optimal stopping games

Ekström, Erik; Villeneuve, Stéphane

doi:10.1214/105051606000000204

Cited by 48 publications

(73 citation statements)

References 17 publications

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“…Martingale methods were further advanced in [18] (see also [23]), and Markovian setting was studied in [8] (via Wald-Bellman equations) and [22] (via penalty equations). More recent papers on optimal stopping games include [14], [16], [1], [11], [6], [7] and [15]. These papers study specific problems and often lead to explicit solutions.…”

Section: Introductionmentioning

confidence: 99%

Optimal Stopping Games for Markov Processes

Ekström¹,

Peškir²

2008

SIAM J. Control Optim.

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, 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.

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Section: Introductionmentioning

confidence: 99%

Optimal Stopping Games for Markov Processes

Ekström¹,

Peškir²

2008

SIAM J. Control Optim.

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“…Now P 1 and P 2 coincide, and we denote it by P. In this case, the value function and the optimal strategies are determined in [6] if µ = r and in [19] for µ < r.…”

Section: Homogeneous Beliefsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Dynkin games with heterogeneous beliefs

2017

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Abstract. We study zero-sum optimal stopping games (Dynkin games) between two players who disagree about the underlying model. In a Markovian setting, a verification result is established showing that if a pair of functions can be found that satisfies some natural conditions, then a Nash equilibrium of stopping times is obtained, with the given functions as the corresponding value functions. In general, however, there is no uniqueness of Nash equilibria, and different equilibria give rise to different value functions. As an example, we provide a thorough study of the game version of the American call option under heterogeneous beliefs. Finally, we also study equilibria in randomized stopping times.

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“…This raises the ques tion whether there is a semiharmonic characterization in the general case (when Gi and G 2 are finite valued). A variant of this question was consid ered earlier under conditions which imply the Nash equilibrium at the first entry times (see [30]), and a one-sided version of the same question (where V equals V) was studied more recently when X is a one-dimensional diffusion (see [11] and [13]). …”

Section: U X {Ra) = E X [G I (X T )I(tmentioning

confidence: 99%

“…Martingale methods were further advanced in [25] (see also [31]), and Markovian setting was studied in [14] (via Wald-Bellman equations) and [30] (via penalty equations). More recent papers on optimal stopping games include [20], [23], [1], [17], [11], [13], [21], [22], [2], and [3]. These papers study specific problems and often lead to explicit solutions.…”

mentioning

confidence: 99%

Optimal Stopping Games and Nash Equilibrium

Peškir¹

2008

Teor. Veroyatnost. i Primenen.

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OPTIMAL STOPPING GAMES AND NASH EQUILIBRIUMПоказывается, что функция цены в игровой задаче об оптималь ной остановке для непрерывного справа сильно марковского про цесса может быть найдена из равенства наименьшей супергармони ческой и наибольшей субгармонической функций, лежащих между функциями выигрыша (семигармоническая характеризация) тогда и только тогда, когда имеет место равновесие по Нэшу (т.е. ког да существует седловая пара оптимальных моментов остановки). В применении к задачам оптимальной остановки показано, что нахождение функции цены сводится к классической характериза-ции этой функции в терминах супергармонических и субгармони ческих функций. Отмеченная эквивалентность показывает, что на хождение функции цены «протягиванием веревки» между «двумя препятствиями» равнозначно доказательству равновесия по Нэшу. В ходе доказательства устанавливаются различные свойства функ ции цены и оптимальных моментов остановки.Ключевые слова и фразы: игровая задача об оптимальной оста новке, равновесие по Нэшу, семигармоническая характеризация функции цены, задача со свободными границами, принцип глад кого склеивания, принцип непрерывного склеивания, оптимальная остановка, марковский процесс, семимартингал.

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On the value of optimal stopping games

Cited by 48 publications

References 17 publications

Optimal Stopping Games for Markov Processes

Optimal Stopping Games for Markov Processes

Dynkin games with heterogeneous beliefs

Optimal Stopping Games and Nash Equilibrium

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