In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some results from optimal stopping theory, some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales, as well as an appropriate generalization of Itô's formula due to Gal'chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position ξ is assessed by an f -conditional expectation E f (·) (where f is a Lipschitz driver). We characterize the "value function" of the problem in terms of the solution to our RB-SDE. Under an additional assumption of left upper-semicontinuity along stopping times on ξ, we show the existence of an optimal stopping time. We also provide a generalization of Mertens decomposition to the case of strong E f -supermartingales.

We study the properties of nonlinear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale measure associated with a default jump with intensity process (λ t ). We give a priori estimates for these equations and prove comparison and strict comparison theorems. These results are generalized to drivers involving a singular process. The special case of a λ-linear driver is studied, leading to a representation of the solution of the associated BSDE in terms of a conditional expectation and an adjoint exponential semi-martingale. We then apply these results to nonlinear pricing of European contingent claims in an imperfect financial market with a totally defaultable risky asset. The case of claims paying dividends is also studied via a singular process.

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.

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...

By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).

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We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) g-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamadène and J. Zhang, we construct a Nash equilibrium point of this game by a recursive procedure. Our results are obtained in the case of a standard Lipschitz driver g without any additional assumption on the driver besides that ensuring the monotonicity of the corresponding g-expectation.