Mean-field type Quadratic BSDEs

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.

Quadratic mean-field reflected BSDEs

“…However, it gets much more complicated for the quadratic case, which needs further study (even for quadratic mean-field BSDEs, see, e.g. [13]).…”

“…Applying Doob's maximal inequality and Hölder's inequality, we get that for each p ≥ 1 and t ∈ [0, T ], (13) and assumption (H4), we derive that…”

Mean Field Portfolio Games

“…The BSDE (3.6) is a quadratic one of conditional mean field type and it does not satisfy the assumptions in [14]. In particular, the quadratic growth in (3.6) comes from both ( Z, Z 0 ) and the conditional expectation of ( Z, Z 0 ).…”

Mean-field backward stochastic differential equations with mean reflection and nonlinear resistance

“…We take à ≥ Ã0 and choose δ à as (24). For T ∈ (0, δ Ã], define (Y 0 , Z 0 , K 0 ) = (0, 0, 0) and by (8)…”

“…When the mean reflection constraint and resistance term vanish, it reduces to a standard mean field BSDE. For the study of mean-field BSDEs, we refer the reader to [6,8].…”