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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.15601 |
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| _version_ | 1866916331857641472 |
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| author | Klimsiak, Tomasz Rzymowski, Maurycy |
| author_facet | Klimsiak, Tomasz Rzymowski, Maurycy |
| contents | We study Dynkin games governed by a nonlinear $\mathbb E^f$-expectation on a finite interval $[0,T]$, with payoff càdlàg processes $L,U$ of class (D) which are not imposed to satisfy (weak) Mokobodzki's condition - the existence of a càdlàg semimartingale between the barriers. For that purpose we introduce the notion of Mokobodzki's stochastic intervals $\mathscr M(θ)$ (roughly speaking, maximal stochastic interval on which Mokobodzki's condition is satisfied when starting from the stopping time $θ$) and the notion of reflected BSDEs without Mokobodzki's condition (this is a generalization and modification of the notion introduced by Hamadéne and Hassani (2005)). We prove an existence and uniqueness result for RBSDEs with driver $f$ that is non-increasing with respect to the value variable (no restrictions on the growth) and Lipschitz continuous with respect to the control variable, and with data in $L^1$ spaces. Next, by using RBSDEs, we show numerous results on Dynkin games: existence of the value process, saddle points, and convergence of the penalty scheme. We also show that the game is not played beyond $\mathscr M(θ)$, when starting from $θ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15601 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mokobodzki's intervals: an approach to Dynkin games when value process is not a semimartingale Klimsiak, Tomasz Rzymowski, Maurycy Probability Optimization and Control We study Dynkin games governed by a nonlinear $\mathbb E^f$-expectation on a finite interval $[0,T]$, with payoff càdlàg processes $L,U$ of class (D) which are not imposed to satisfy (weak) Mokobodzki's condition - the existence of a càdlàg semimartingale between the barriers. For that purpose we introduce the notion of Mokobodzki's stochastic intervals $\mathscr M(θ)$ (roughly speaking, maximal stochastic interval on which Mokobodzki's condition is satisfied when starting from the stopping time $θ$) and the notion of reflected BSDEs without Mokobodzki's condition (this is a generalization and modification of the notion introduced by Hamadéne and Hassani (2005)). We prove an existence and uniqueness result for RBSDEs with driver $f$ that is non-increasing with respect to the value variable (no restrictions on the growth) and Lipschitz continuous with respect to the control variable, and with data in $L^1$ spaces. Next, by using RBSDEs, we show numerous results on Dynkin games: existence of the value process, saddle points, and convergence of the penalty scheme. We also show that the game is not played beyond $\mathscr M(θ)$, when starting from $θ$. |
| title | Mokobodzki's intervals: an approach to Dynkin games when value process is not a semimartingale |
| topic | Probability Optimization and Control |
| url | https://arxiv.org/abs/2407.15601 |