BSDEs with mean reflection

Abstract

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...

Quadratic BSDEs with mean reflection

“…where the second equation is a running constraint in expectation on the component Y of the solution. The above equation is called BSDE with mean reflection, which was first introduced in [6]. The parameters of the BSDE with mean reflection are the terminal condition ξ, the generator (or driver) f as well as the running loss function ℓ.…”

“…The parameters of the BSDE with mean reflection are the terminal condition ξ, the generator (or driver) f as well as the running loss function ℓ. In [6], the authors have discussed such equation under the standard Lipschitz condition on the generator and the square integrability assumption terminal condition.…”

“…where m is a given threshold, ℓ is a non-decreasing map and can be viewed as a loss function in quantile hedging problems or in stochastic target problems under controlled loss. Recently, motivated by super-hedging of claims under running risk management constraints, Briand, Elie and Hu [6] have formulated a new type of BSDE with constraints, which is called the BSDE with mean reflection. In their framework, the solution Y is required to satisfy the following type of mean reflection constraint:…”

“…In order to establish the well-posedness of BSDEs with mean reflection, in [6] the authors have introduced the notion of deterministic flat solution, i.e., the component K is a deterministic non-decreasing process and satisfies the following type of Skorohod condition,…”

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The present paper is devoted to the study of the well-posedness of BS-DEs with mean reflection whenever the generator has quadratic growth in the z argument. This work is the sequel of [6] in which a notion of BSDEs with mean reflection is developed to tackle the super-hedging problem under running risk management constraints. By the contraction mapping argument, we first prove that the quadratic BSDE with mean reflection admits a unique deterministic flat local solution on a small time interval whenever the terminal value is bounded.

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