Search

We formulate a notion of doubly reflected BSDE in the case where the barriers $\xi$ and $\zeta$ do not satisfy any regularity assumption and with a general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $\xi$ is right upper-semicontinuous and $\zeta$ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding $\mathcal{E}^f$-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 \textit{a priori} estimates with universal constants.

International audience; We study (nonlinear) Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale attached to a default jump with intensity process λ = (λ t). The driver of the BSDEs can be of a generalized form involving a singular optional finite variation process. In particular, we provide a comparison theorem and a strict comparison theorem. In the special case of a generalized λ-linear driver, we show an explicit representation of the solution, involving conditional expectation and an adjoint exponential semimartingale; for this representation, we distinguish the case where the singular component of the driver is predictable and the case where it is only optional. We apply our results to the problem of (nonlinear) pricing of European contingent claims in an imperfect market with default. We also study the case of claims generating intermediate cashflows, in particular at the default time, which are modeled by a singular optional process. We give an illustrating example when the seller of the European option is a large investor whose portfolio strategy can influence the probability of default.

This paper studies the superhedging prices and the associated superhedging strategies for European and American options in a non-linear incomplete market with default. We present the seller's and the buyer's point of view. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio process follows non-linear dynamics with a non-linear driver ƒ. By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum over a suitable set of equivalent probability measures *Q* ∈ $\mathcal{Q}$ of the ƒ-evaluation/expectation under *Q* of the payoff. We also provide an infinitesimal characterization of this price as the minimal supersolution of a constrained BSDE with default. By a form of symmetry, we derive corresponding results for the buyer. We also give a dual representation of the seller's (superhedging) price for the American option associated with an irregular payoff (ξ*t*) (not necessarily càdlàg) in terms of the value of a non-linear mixed control/stopping problem. We also provide an infinitesimal characterization of this price in terms of a constrained reflected BSDE. When ξ is càdlàg, we show a duality result for the buyer's price. These results rely on first establishing a non-linear optional decomposition for processes which are $\mathcal{E}$ƒ -strong supermartingales under *Q*, for all *Q* ∈ $\mathcal{Q}$ .

The proof of the second point of Proposition B11 given in the Appendix of Kobylanski and Quenez (2012) ([1]) is only valid in the case where the reward process is right-continuous. In this Erratum, we give the proof in the case where the reward is only right-upper-semicontinuous.

We study double barrier reflected BSDEs (DBBSDEs) with jumps and RCLL barriers, and their links with generalized Dynkin games. We provide existence and uniqueness results and prove that for any Lipschitz driver, the solution of the DBBSDE coincides with the value function of a game problem, which can be seen as a generalization of the classical Dynkin problem to the case of $g$-conditional expectations. Using this characterization, we prove some new results on DBBSDEs with jumps, such as comparison theorems and a priori estimates. We then study DBBSDEs with jumps and RCLL obstacles in the Markovian case and their links with parabolic partial integro-differential variational inequalities (PIDVI) with two obstacles; Nous étudions des équations différentielles stochastiques rétrogrades réfléchies (EDSR) à deux barrières avec sauts dans le cas où les barrières sont modélisées par des processus stochastiques càdlàg. Nous prouvons l'existence et l'unicité de la solution et nous démontrons que dans le cas où le driver est lipschitzien, la solution coincide avec la fonction valeur d'un jeu différentiel stochastique, qui peut s'écrire comme un jeu de Dynkin généralisé avec g- espérance. Grace à cette caractérisation, nous démontrons des théorèmes de comparaison et des estimations a priori. Dans le cas Markovien, nous étudions le lien avec des inéquations variationnelles intégro-différentielles à 2 obstacles

We study the links between reflected backward stochastic differential equations (reflected BSDEs) with jumps and partial integro-differential variational inequalities (PIDVIs). In a Markovian framework, we show that the solution of the reflected BSDE corresponds to the unique viscosity solution of the PIDVI. We apply these results to an optimal stopping problem for dynamic risk measures induced by BSDEs with jumps.; On étudie le lien entre les équations différentielles stochastiques rétrogrades réfléchies avec sauts (EDSR réfléchies) et les inéquations variationnelles integro-différentielles (IVID). Dans un cadre markovien, on montre que la solution de l'EDSR réfléchie avec sauts correspond à l'unique solution de viscosité de l'IVID. On applique ces résultats à l'étude d'un problème d'arrêt optimal pour les mesures de risque dynamiques induites par des EDSR avec sauts.

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping problems related to the case with model ambiguity are investigated.; On étudie le problème d'arrêt optimal de mesures de risque dynamiques représentées par des équations différentielles stochastiques (EDSR) avec sauts et sa relation avec des EDSR réfléchies. On prouve des théorèmes généraux d'existence, d'unicité et de comparaison pour ces équations dans le cas d'un obstacle càdlàg adapté. On montre que la fonction valeur de notre problème d'arrêt optimal est solution d'une EDSR réfléchie. L'existence d'un temps d'arrêt optimal est obtenu sous des conditions de régularité à gauche de l'obstacle. On étudie ensuite le problème d'arrêt optimal dans le cas d'ambiguité de modèle pour la mesure de risque.

http://ejp.ejpecp.org/; International audience; We study the optimal stopping time problem $v(S)={\rm ess}\sup_{\theta \geq S} E[\phi(\theta)|\mathcal {F}_S]$, for any stopping time $S$, where the reward is given by a family $(\phi(\theta),\theta\in\mathcal{T}_0)$ \emph{of non negative random variables} indexed by stopping times. We solve the problem under weak assumptions in terms of integrability and regularity of the reward family. More precisely, we only suppose $v(0) < + \infty$ and $ (\phi(\theta),\theta\in \mathcal{T}_0)$ upper semicontinuous along stopping times in expectation. We show the existence of an optimal stopping time and obtain a characterization of the minimal and the maximal optimal stopping times. We also provide some local properties of the value function family. All the results are written in terms of families of random variables and are proven by only using classical results of the Probability Theory.