## The Annals of Applied Probability

### 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.

#### Article information

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

Dates
Revised: January 2017
First available in Project Euclid: 3 March 2018

https://projecteuclid.org/euclid.aoap/1520046093

Digital Object Identifier
doi:10.1214/17-AAP1310

Mathematical Reviews number (MathSciNet)
MR3770882

Zentralblatt MATH identifier
06873689

#### Citation

Briand, Philippe; Elie, Romuald; Hu, Ying. BSDEs with mean reflection. Ann. Appl. Probab. 28 (2018), no. 1, 482--510. doi:10.1214/17-AAP1310. https://projecteuclid.org/euclid.aoap/1520046093

#### References

• [1] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
• [2] Bouchard, B., Elie, R. and Réveillac, A. (2015). BSDEs with weak terminal condition. Ann. Probab. 43 572–604.
• [3] Buckdahn, R. and Hu, Y. (1998). Pricing of American contingent claims with jump stock price and constrained portfolios. Math. Oper. Res. 23 177–203.
• [4] Buckdahn, R. and Hu, Y. (1998). Hedging contingent claims for a large investor in an incomplete market. Adv. in Appl. Probab. 30 239–255.
• [5] Buckdahn, R., Li, J. and Peng, S. (2009). Mean-field backward stochastic differential equations and related partial differential equations. Stochastic Process. Appl. 119 3133–3154.
• [6] Chassagneux, J.-F., Elie, R. and Kharroubi, I. (2011). A note on existence and uniqueness for solutions of multidimensional reflected BSDEs. Electron. Commun. Probab. 16 120–128.
• [7] Chaudru de Raynal, P. E. and Garcia Trillos, C. A. (2015). A cubature based algorithm to solve decoupled McKean–Vlasov forward–backward stochastic differential equations. Stochastic Process. Appl. 125 2206–2255.
• [8] Cvitanić, J. and Karatzas, I. (1996). Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 2024–2056.
• [9] Cvitanić, J., Karatzas, I. and Soner, H. M. (1998). Backward stochastic differential equations with constraints on the gains-process. Ann. Probab. 26 1522–1551.
• [10] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 702–737.
• [11] Hamadène, S. and Jeanblanc, M. (2007). On the starting and stopping problem: Application in reversible investments. Math. Oper. Res. 32 182–192.
• [12] Hamadène, S. and Zhang, J. (2010). Switching problem and related system of reflected backward SDEs. Stochastic Process. Appl. 120 403–426.
• [13] Hu, Y. and Tang, S. (2010). Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Related Fields 147 89–121.
• [14] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
• [15] Peng, S. and Xu, M. (2010). Reflected BSDE with a constraint and its applications in an incomplete market. Bernoulli 16 614–640.