Quadratic BSDEs with mean reflection

Hélène Hibon; Ying Hu; Yiqing Lin; Peng Luo; Falei Wang

doi:10.3934/mcrf.2018031

Article Contents

2018, Volume 8, Issue 3&4: 721-738. Doi: 10.3934/mcrf.2018031

This issue Previous Article Null controllability of the Lotka-McKendrick system with spatial diffusion Next Article Nonlinear backward stochastic evolutionary equations driven by a space-time white noise

Quadratic BSDEs with mean reflection

1.
Institut de Recherche Mathématique de Rennes, Université Rennes 1, 35042 Rennes Cedex, France
2.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
3.
School of Mathematical Sciences, Shanghai Jiao Tong University, 200240 Shanghai, China
4.
Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, France
5.
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
6.
Zhongtai Securities Institute for Financial Studies and Institute for Advanced Research, Shandong University, Jinan 250100, China

* Corresponding authorr: Y. Hu
* Corresponding authorr: Y. Hu

Received: May 2017

Revised: October 2017

Published: September 2018

Y. Hu's research is partially supported by Lebesgue Center of Mathematics "Investissements d'avenir" Program (No. ANR-11-LABX-0020-01), by ANR CAESARS (No. ANR-15-CE05-0024) and by ANR MFG (No. ANR-16-CE40-0015-01). Y. Lin's research is partially supported by the European Research Council under grant 321111. P. Luo's research is partially supported by the National Science Foundation of China "Research Fund for International Young Scientists"(No. 11550110184) and by National Natural Science Foundation of China (No. 11671257). F. Wang's research is partially supported by the National Natural Science Foundation of China (No.11601282 and 11526205), by the Shandong Province Natural Science Foundation (No. ZR2016AQ10) and by the China Scholarship Council (No. 201606225002)

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Abstract

The present paper is devoted to the study of the well-posedness of BSDEs 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. Moreover, we build the global solution on the whole time interval by stitching local solutions when the generator is uniformly bounded with respect to the $y$ argument.

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Mathematics Subject Classification: Primary: 60H10, 60H30.

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