Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

Grigorova, Miryana; Imkeller, Peter; Offen, Elias R.; Ouknine, Youssef; Quenez, Marie‐Claire

doi:10.1214/17-aap1278

Cited by 63 publications

(121 citation statements)

References 35 publications

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“…The approach used in the literature to address the non-linear case (where f is not necessarily equal to 0) is an RBSDE-approach, based on the study of a related Reflected BSDE and on linking directly the solution of the Reflected BSDE with the value family (V (S), S ∈ T 0,T ) (and thus avoiding, in particular, more technical aggregation questions). This approach (cf., e.g., [17], [39]) requires at least the uppersemicontinuity of the reward process ξ which we do not have here (cf. also Remark 10.1).…”

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confidence: 86%

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: 86%

Optimal stopping with f-expectations: The irregular case

Grigorova

Imkeller

Ouknine

et al. 2020

Stochastic Processes and their Applications

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“…Ek s (e)Ñ (ds, de) +h t , α t := − t 0f s ds −Ã t +Ã t and γ t := −C t− +C t− (cf., e.g., Theorem A.3. and Corollary A.2 in [23] Proof of the statement of Proposition 3.20 Let (A, C) (resp. (A , C )) be the Mertens process associated with the strong supermartingale X f (resp.…”

Section: Proofsmentioning

confidence: 91%

“…−2 ]t,T ] e βsỸ s− dÃ s ≤ 0) The proof uses property (3.3) of the definition of the DRBSDE and the properties Y ≥ ξ, Y ≥ ξ (resp. Y ≤ ζ,Ȳ ≤ ζ) ; the details are similar to those in the case of RBSDE (with one lower obstacle) (cf., for instance, the proof of Lemma 3.2 in [23]).…”

Section: Application To the Pricing Of Game Options With Irregular Pamentioning

confidence: 99%

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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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“…As+s) U 2 s (e)φ s (de)dA s .By plugging this last estimate into(20), by choosing α, γ such that As) U 2s (e)φ s (de)dA s + E ds + γE sup t e (β+δ)At h 2 t − ,for some constant C independent of n. Now let S = lim n S n and by the last estimate, considering how S n are defined, we have S = T . This implies thatY ∈ L 2,β (A) ∩ L 2,β (W ), Z ∈ L 2,β (W ) and U ∈ L 2,β (p).…”

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Optimal Stopping of Marked Point Processes and Reflected Backward Stochastic Differential Equations

Foresta

2019

Appl Math Optim

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We define a class of reflected backward stochastic differential equation (RBSDE) driven by a marked point process (MPP) and a Brownian motion, where the solution is constrained to stay above a given càdlàg process. The MPP is only required to be non-explosive and to have totally inaccessible jumps. Under suitable assumptions on the coefficients we obtain existence and uniqueness of the solution, using the Snell envelope theory. We use the equation to represent the value function of an optimal stopping problem, and we characterize the optimal strategy.

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Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

Cited by 63 publications

References 35 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

Optimal Stopping of Marked Point Processes and Reflected Backward Stochastic Differential Equations

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