The McKean stochastic game driven by a spectrally negative Lévy process

“…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]. They study specific problems and often lead to explicit solutions.…”

Section: Introductionmentioning

confidence: 99%

Optimal Stopping Games and Nash Equilibrium

Peškir¹

2009

Theory Probab. Appl.

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, Probab. Statist. Group Manchester (15 pp) We show that the value function of the optimal stopping game for a rightcontinuous strong Markov process can be identified via equality between the smallest superharmonic and the largest subharmonic function lying between the gain and the loss function (semiharmonic characterisation) if and only if the Nash equilibrium holds (i.e. there exists a saddle point of optimal stopping times). When specialised to optimal stopping problems it is seen that the former identification reduces to the classic characterisation of the value function in terms of superharmonic or subharmonic functions. The equivalence itself shows that finding the value function by 'pulling a rope' between 'two obstacles' is the same as establishing a Nash equilibrium. Further properties of the value function and the optimal stopping times are exhibited in the proof.

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“…Then as above since both V and G 2 are finely continuous, it is easily seen using (2.7) that VpC^J = G 2 (X aD J, when a D2 < oo, and (2.12) im plies that the previous identity also holds when cr D2 -oo. Hence taking any F G Sup[Gi,G 2 ), recalling that F is superharmonic in E\D 2 and F ^ V ^ Gi, it follows as in (2.32) …”

Section: Combining (229) and (230) We See That E X V(x Pn ) -> E X mentioning

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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The McKean stochastic game driven by a spectrally negative Lévy process

Cited by 35 publications

References 13 publications

The Shepp–Shiryaev Stochastic Game Driven by a Spectrally Negative Lévy Process

The Shepp–Shiryaev Stochastic Game Driven by a Spectrally Negative Lévy Process

Optimal Stopping Games and Nash Equilibrium

Optimal Stopping Games and Nash Equilibrium

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