Mean-field type Quadratic BSDEs

TL;DR

This paper investigates solvability of mean-field quadratic backward stochastic differential equations (BSDEs), introducing new methods such as John-Nirenberg's inequality and shift transformations to handle the mean-dependent quadratic generators.

Contribution

It provides novel solvability results for mean-field quadratic BSDEs using inequalities and transformations, extending existing theory to more complex generator dependencies.

Findings

Established solvability conditions for mean-field quadratic BSDEs.

Applied John-Nirenberg's inequality to estimate contributions of BMO martingales.

Used shift transformations and fixed point arguments for more general cases.

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.