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Generalized Dynkin games and doubly reflected BSDEs with jumps

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Date
2016
Dewey
Probabilités et mathématiques appliquées
Sujet
Dynkin game; mixed game problem; nonlinear expectation; g-evaluation; doubly reflected BSDEs; partial integro-differential variational inequalities; game option
Journal issue
Electronic Journal of Probability
Volume
21
Number
2016
Publication date
2016
Publisher
Electronic Journal of Probability and Electronic Communications in Probability
DOI
http://dx.doi.org/10.1214/16-EJP4568
URI
https://basepub.dauphine.fr/handle/123456789/17266
Collections
  • CEREMADE : Publications
Metadata
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Author
Dumitrescu, Roxana
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Quenez, Marie-Claire
102 Laboratoire de Probabilités et Modèles Aléatoires [LPMA]
Sulem, Agnès
34587 INRIA Rocquencourt
Type
Article accepté pour publication ou publié
Abstract (EN)
We introduce a game problem which can be seen as a generalization of the classical Dynkin game problem to the case of a nonlinear expectation Eg, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driver g. Let ξ,ζ be two RCLL adapted processes with ξ≤ζ. The criterium is given by Jτ,σ=Eg0,τ∧σ(ξτ1{τ≤σ}+ζσ1{σ<τ}) where τ and σ are stopping times valued in [0,T]. Under Mokobodzki's condition, we establish the existence of a value function for this game, i.e. infσsupτJτ,σ=supτinfσJτ,σ. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When ξ and ζ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

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