Multi-dimensional backward stochastic differential equations of diagonally quadratic generators

Abstract

In this paper, we study a multi-dimensional BSDE with a “diagonally” quadratic generator, the quadratic part of whose ith component depends only on the ith row of the second unknown variable. Local and global solutions are given, which seem to be the first systematic (positive) results on the general solvability of multi-dimensional quadratic BSDEs. In our proofs, it is natural and crucial to apply both John–Nirenberg and reverse Hölder inequalities for BMO martingales. Our results are finally illustrated to solve the system of “diagonally” quadratic BSDEs arising from a nonzero-sum risk-sensitive stochastic differential game, which answers the open problem posed in El Karoui and Hamadène [Stochastic Process. Appl. 107 (2003), page 164].