Multi-dimensional quadratic BSDEs

Shanjian Tang (Fudan University)
Thursday, January 29, 2015 - 5:00pm
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

Quadratic BSDEs refer to those BSDEs whose generators grow quadratically in the second unkown variable. In this talk, I will start with recalling J. M. Bismut's Ph. D work on the linear quadratic optimal stochastic control problem and the introduction of backward stochastic Riccati equations, which motivated the study of general quadratic BSDEs. Then I review the theory of one-dimensional quadratic BSDEs and show the difficulty in a general solution of multi-dimensional quadratic BSDEs even when the terminal value is essentially bounded. Finally, I introduce our recent results jointed with Ying HU on adapted solution of a multi-dimensional BSDE with a "diagonall" quadratic generator, the quadratic part of whose $i$th component only depends on the $i$th 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 crucial to apply both John-Nirenberg and reverse Hölder inequalities for BMO martingales, all details of which can be found in our online preprint arXiv:1408.4579.