The Annals of Applied Probability

BSDEs with mean reflection

Philippe Briand, Romuald Elie, and Ying Hu

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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.

Article information

Ann. Appl. Probab., Volume 28, Number 1 (2018), 482-510.

Received: May 2016
Revised: January 2017
First available in Project Euclid: 3 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 91G10: Portfolio theory

Backward stochastic differential equations mean reflection Skorokhod type minimal condition super-hedging risk management constraint


Briand, Philippe; Elie, Romuald; Hu, Ying. BSDEs with mean reflection. Ann. Appl. Probab. 28 (2018), no. 1, 482--510. doi:10.1214/17-AAP1310.

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