Mean-field type quadratic BSDEs

Hélène Hibon; Ying Hu; Shanjian Tang

doi:10.3934/naco.2022009

Article Contents

2023, Volume 13, Issue 3&4: 392-412. Doi: 10.3934/naco.2022009

This issue Previous Article Forward-backward stochastic differential equations: Initiation, development and beyond Next Article On path-dependent multidimensional forward-backward SDEs

Mean-field type quadratic BSDEs

1.
Univ. Rennes, CNRS, IRMAR-UMR6625, F-35000, Rennes, France
2.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
3.
Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

^*Corresponding author: Ying Hu
^*Corresponding author: Ying Hu

Received: November 2021

Revised: April 2022

Early access: May 2022

Published: September 2023

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Abstract

In this paper, we give several new results on solvability of a quadratic BSDE whose generator depends also on the mean of both variables. First, we consider such a BSDE using John-Nirenberg's inequality for BMO martingales to estimate its contribution to the evolution of the first unknown variable. Then we consider the BSDE having an additive expected value of a quadratic generator in addition to the usual quadratic one. In this case, we use a deterministic shift transformation to the first unknown variable, when the usual quadratic generator depends neither on the first variable nor its mean. The general case can be treated by a fixed point argument.

Keywords:

Backward stochastic differential equation,
quadratic growth,
mean-field type.

Mathematics Subject Classification: Primary: 60H10.

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References

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