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

Abstract

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...

Reflected backward stochastic differential equations with two optional barriers

“…In the present paper we generalize the existence and uniqueness results from [11] in several directions. We consider the case of L p -data with p ≥ 1 (in [11] only the case of p = 2 is considered). As for the generator, we assume that it is Lipschitz continuous with respect to z and only continuous and monotone with respect to y (in [11] it is assumed that f is Lipschitz continuous with respect to y and z).…”

“…To our knowledge, RBSDEs with barriers which are not càdlàg and whose solution satisfies (1.2) are treated only in the papers [1,10,11,17]. Among them, only [11] deals with equations with two barriers. In the present paper we generalize the existence and uniqueness results from [11] in several directions.…”

“…Among them, only [11] deals with equations with two barriers. In the present paper we generalize the existence and uniqueness results from [11] in several directions. We consider the case of L p -data with p ≥ 1 (in [11] only the case of p = 2 is considered).…”

“…We consider the case of L p -data with p ≥ 1 (in [11] only the case of p = 2 is considered). As for the generator, we assume that it is Lipschitz continuous with respect to z and only continuous and monotone with respect to y (in [11] it is assumed that f is Lipschitz continuous with respect to y and z). Moreover, we assume that the generator and the barriers satisfy the so-called generalized Mokobodzki condition which says that there exists a semimartingale X ∈ M loc + V p such that L t ≤ X t ≤ U t , t ∈ [0, T ], and E T 0 |f (r, X r , 0)| dr p + |X| p < ∞, (1.4) where |X| p := (E sup t≤T |X t | p ) 1/p for p > 1 and |X| 1 := sup τ ∈Γ E|X τ | (here M loc is the space of local martingales and V p is the space of finite variation processes with p-integrable variation, and Γ denotes the set of all F-stopping times).…”

“…Moreover, we assume that the generator and the barriers satisfy the so-called generalized Mokobodzki condition which says that there exists a semimartingale X ∈ M loc + V p such that L t ≤ X t ≤ U t , t ∈ [0, T ], and E T 0 |f (r, X r , 0)| dr p + |X| p < ∞, (1.4) where |X| p := (E sup t≤T |X t | p ) 1/p for p > 1 and |X| 1 := sup τ ∈Γ E|X τ | (here M loc is the space of local martingales and V p is the space of finite variation processes with p-integrable variation, and Γ denotes the set of all F-stopping times). In [11] the standard Mokobodzki condition is assumed. It says that L ≤ X ≤ U for some semimartingale X ∈ M loc + V 2 such that |X| 2 < ∞.…”