A note on existence and uniqueness for solutions of multidimensional reflected BSDEs

Chassagneux, Jean François; Élie, Romuald; Kharroubi, Idris

doi:10.1214/ecp.v16-1614

Cited by 48 publications

(77 citation statements)

References 6 publications

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“…The main result of this paper is the existence and uniqueness of deterministic flat solution to the BSDE (5) with mean reflection (6). This is first achieved for the particular case of linear loss function ℓ, see Proposition 4 and Theorem 5 in Section 3.…”

Section: Definitionmentioning

confidence: 93%

“…Lemma 1. Let (Y, Z, K) be a square integrable solution to the BSDE (5) with mean reflection (6). Then Y satisfies the following…”

Section: A Priori Estimatementioning

confidence: 99%

“…Now that the well posedness for constant driver is established, we can focus on the BSDE (5) with mean reflexion (6) in full generality. In order for the solution to be well defined, we will require a Lipschitz property of the operator L, that we present in the following additional Assumption:…”

Section: Existence and Uniqueness For The General Casementioning

confidence: 99%

“…In addition to the running assumptions (H ξ )-(H f )-(H ℓ ), let us moreover assume that (H L ) is satisfied. Then, there exists a unique deterministic flat solution (Y, Z, K) ∈ S 2 × H 2 × A 2 D to the BSDE (5) with mean reflexion (6).…”

Section: Existence and Uniqueness For The General Casementioning

confidence: 99%

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BSDEs with mean reflection

Briand¹,

Élie²,

Hu³

2018

Ann. Appl. Probab.

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In this paper, we study a new type of BSDE, where the distribution of the Y -component of the solution is required to satisfy an additional constraint, written in terms of the expectation of a loss function. This constraint is imposed at any deterministic time t and is typically weaker than the classical pointwise one associated to reflected BSDEs. Focusing on solutions (Y, Z, K) with deterministic K, we obtain the well-posedness of such equation, in the presence of a natural Skorokhod type condition. Such condition indeed ensures the minimality of the enhanced solution, under an additional structural condition on the driver. Our results extend to the more general framework where the constraint is written in terms of a static risk measure on Y . In particular, we provide an application to the super hedging of claims under running risk management constraint. Introduction.Backward Stochastic Differential Equations (BSDEs) have been introduced by Pardoux and Peng [15] and share a strong connection with stochastic control problems. Solving a BSDE typically consists in the obtention of an adapted couple process (Y, Z) with the following dynamics:In their seminal paper, Pardoux and Peng provide the existence of a unique solution (Y, Z) to this equation for a given square integrable terminal condition ξ and a Lipschitz random driver f . Since then, many extensions have been derived in several directions. The regularity of the driver can for example be weakened. The underlying dynamics can be fairly more complex, via for example the addition of jumps. These extensions allow in particular to provide representation of solutions to a large class of stochastic control problems, and to tackle several meaningful applications in mathematical finance.More interestingly, the consideration of additional conditions on the stochastic control problems of interest naturally led to the consideration of constrained BSDEs. In such a case, the solution of a constrained BSDE contains an additional adapted increasing process K, such that (Y, Z, K) satisfiestogether with a chosen constraint on the solution. The process K interprets as the extra cost on the value process Y , due to the additional constraint. In such a framework, this equation admits an infinite ), partially supported by Lebesgue center of mathematics ("Investissements d'avenir" program -ANR-11-LABX-0020-01 and by ANR-15-CE05-0024-02.number of solutions, as the roles of Y and K are too closely connected. The underlying stochastic control problem of interest typically indicates that one should look for the minimal solution (in terms of Y ) of such equation. Motivated by optimal stopping or related obstacle problems, El Karoui et al. [10] introduced the notion of reflected BSDE, where the constraint is of the formThe obstacle process L is a lower bound on the solution Y and interprets as the reward payoff, if one chooses to stop immediately. It is worth noticing that the minimal solution (Y, Z, K) is fully characterized by the following so-called Skorokhod conditionThis conditio...

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Section: Definitionmentioning

confidence: 93%

“…Lemma 1. Let (Y, Z, K) be a square integrable solution to the BSDE (5) with mean reflection (6). Then Y satisfies the following…”

Section: A Priori Estimatementioning

confidence: 99%

Section: Existence and Uniqueness For The General Casementioning

confidence: 99%

Section: Existence and Uniqueness For The General Casementioning

confidence: 99%

See 2 more Smart Citations

BSDEs with mean reflection

Briand¹,

Élie²,

Hu³

2018

Ann. Appl. Probab.

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“…In [8] the existence and uniqueness results of viscosity solutions was extended to the case when the switching costs depend on the state variable. Since then, results have been extended to Knightian uncertainty [14,13,4] and non-Brownian filtration and signed switching costs [19]. For the case when the underlying uncertainty can be modeled by a diffusion process, generalization to the case when the control enters the drift and volatility term was treated in [10].…”

Section: Introductionmentioning

confidence: 99%

A finite horizon optimal switching problem with memory and application to controlled SDDEs

Perninge

2019

Math Meth Oper Res

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We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an optimal switching problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process, when the dynamics of the process depends on the control. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling.We consider a finite horizon problem and thus assume that the terminal time T is fixed with T < ∞.We let (Ω, F, F, P) be a probability space, with F := (F t ) 0≤t≤T a filtration satisfying the usual conditions in addition to being quasi-left continuous.Remark 2.1. Recall here the concepts of quasi-left continuity: A càdlàg process (X t : 0 ≤ t ≤ T ) is quasi-left continuous if for each predictable stopping time γ and every announcing sequence of stopping

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