Generalized Dynkin games and doubly reflected BSDEs with jumps

Dumitrescu, Roxana; Sulem, Agnès

doi:10.1214/16-ejp4568

Cited by 35 publications

(83 citation statements)

References 33 publications

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“…We make the following assumption on the driver (cf., e.g., Theorem 4.2 in [38]). [9]). We recall that under Assumption 5.1 on the driver f , the functional E f S,τ (·) is nondecreasing (cf.…”

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confidence: 99%

Optimal stopping with f-expectations: The irregular case

Grigorova

Imkeller

Ouknine

et al. 2020

Stochastic Processes and their Applications

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We consider the optimal stopping problem with nonlinear f -expectation (induced by a BSDE) without making any regularity assumptions on the payoff process ξ and in the case of a general filtration. We show that the value family can be aggregated by an optional process Y . We characterize the process Y as the E f -Snell envelope of ξ. We also establish an infinitesimal characterization of the value process Y in terms of a Reflected BSDE with ξ as the obstacle. To do this, we first establish some useful properties of irregular RBSDEs, in particular an existence and uniqueness result and a comparison theorem.where T S,T denotes the set of stopping times valued a.s. in [S, T ] and E f S,τ (·) denotes the conditional f -expectation/evaluation at time S when the terminal time is τ .The above non-linear problem has been introduced in [14] in the case of a Brownian filtration and a continuous financial position/pay-off process ξ and applied to the (nonlinear) pricing of American options. It has then attracted considerable interest, in particular, 2 Note that in the case of a not necessarily non-negative pay-off process ξ this result holds up to a translation by the martingale XS := E[ess sup τ ∈T 0,T ξ − τ |FS] (cf. e.g. Appendix A in [30]). More precisely, the property holds forṽ := v + X andξ = ξ + X.

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confidence: 99%

Optimal stopping with f-expectations: The irregular case

Grigorova

Imkeller

Ouknine

et al. 2020

Stochastic Processes and their Applications

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“…optional) processes with integrable variation (cf. Proposition A.7 in [13]), there exist two nondecreasing right-continuous predictable (resp. optional) processes B and B (resp.…”

Section: Proofmentioning

confidence: 99%

“…In the case of non-continuous barriers and/or a filtration associated with a Brownian motion and a random Poisson measure, DRBSDEs have been studied by several authors, cf. [2], [28], [30], [31], [29], [8], [21], [33], [13]. In all of the above-mentioned works on DRBSDEs, the barriers are assumed to be at least right-continuous.…”

Section: Introductionmentioning

confidence: 99%

Doubly Reflected BSDEs and $\mathcal{E} ^{{f}}$-Dynkin games: beyond the right-continuous case

Grigorova¹,

Imkeller²,

Ouknine³

2018

Electron. J. Probab.

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We formulate a notion of doubly reflected BSDE in the case where the barriers ξ and ζ do not satisfy any regularity assumption and with general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where ξ is right upper-semicontinuous and ζ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding E f -Dynkin game, i.e. a game problem over stopping times with (non-linear) f -expectation, where f is the driver of the doubly reflected BSDE. In the general case where the barriers do not satisfy any regularity assumptions, the solution of the doubly reflected BSDE is related to the value of "an extension" of the previous non-linear game problem over a larger set of "stopping strategies" than the set of stopping times. This characterization is then used to establish a comparison result and a priori estimates with universal constants.The criterion is defined as the (linear) expectation of the pay-off, that is, E [I(τ, σ)]. It is well-known that, if ξ is right upper-semicontinuous (right u.s.c) and ζ is right lowersemicontinuous (right l.s.c) and satisfy Mokobodzki's condition, this classical Dynkin game has a (common) value, that is, the following equality holds:Moreover, under the additional assumptions that ξ is left-uppersemicontinuous (leftu.s.c), ζ is left-lowersemicontinuous (left-l.s.c), both along stopping times, and ξ t < ζ t , . Page 2/39DRBSDEs and E f -Dynkin games: beyond right-continuity t < T , there exists a saddle point (cf.[1], [39]). 1 Furthermore, when the processes ξ and ζ are right-continuous, the (common) value of the classical Dynkin game is equal to the solution at time 0 of the doubly reflected BSDE with driver equal to 0 and barriers (ξ, ζ) (cf. [9], [31],[41]).In the second part of the present paper, we consider the following generalization of the classical Dynkin game problem: For each pair (τ, σ), where E f 0,τ ∧σ (·) denotes the f -expectation at time 0 when the terminal time is τ ∧ σ. We refer to this generalized game problem as E f -Dynkin game. This non-linear game problem has been introduced in [13] in the case where ξ and ζ are right-continuous under the name of generalized Dynkin game, the term generalized referring to the presence of a (non-linear) f -expectation in place of the "classical" linear expectation.In the second part of the paper, we first generalize the results of [13] beyond the right-continuity assumption on ξ and ζ (and in the case of a general filtration). More precisely, by using results from the first part of the present paper, combined with some arguments from [13], we show that if ξ is right-u.s.c. and ζ is right-l.s.c. , and if they satisfy Mokobodzki's condition, there exists a (common) value function for the E f -Dynkin( 1.3) and this common value is equal to the solution at time 0 of the doubly reflected BSDE with driver f and barriers (ξ, ζ) from the first part of the paper. Moreover, under the additional assumption that ξ is left u.s...

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“…These games have been studied extensively since then and have garnered renewed interest due to the introduction of game contingent claims (also known as Israeli options) in [11]. The particular variant of the Dynkin game which is described below was studied in recent papers such as [2], [7], and [8].…”

Section: The Dynkin Gamementioning

confidence: 99%

“…This paper complements these findings by proving, under appropriate conditions, that the solution to a two-mode optimal switching problem furnishes the existence of a saddle point for the corresponding Dynkin game. This is accomplished by the method of Snell envelopes which appeared in [1] for optimal switching problems on one hand, and in [2] and [13] for Dynkin games on the other hand. In the process, we relate the solution pair for the two-mode optimal switching problem to a pair of supermartingales which lie between the early exit values of the game.…”

Section: Introductionmentioning

confidence: 99%

Solving finite time horizon Dynkin games by optimal switching

Martyr

2016

J. Appl. Probab.

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This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black-Scholes market.

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Generalized Dynkin games and doubly reflected BSDEs with jumps

Cited by 35 publications

References 33 publications

Optimal stopping with f-expectations: The irregular case

Optimal stopping with f-expectations: The irregular case

Doubly Reflected BSDEs and $\mathcal{E} ^{{f}}$-Dynkin games: beyond the right-continuous case

Solving finite time horizon Dynkin games by optimal switching

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