Harmonic Analysis of Stochastic Equations and Backward Stochastic Differential Equations

The BMOmartingale theory is extensively used to study nonlinear multi-dimensional stochastic equations (SEs) inRp (p ∈ [1,∞)) and backward stochastic differential equations (BSDEs) in Rp × Hp (p ∈ (1,∞)) and in R∞ × H∞, with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman’s inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear case for SDEs and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Hölder inequality for some suitable exponent p ≥ 1. Finally, we establish some relations between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the solution operator for the M -driven SDE, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite. 2000 Mathematics Subject Classification. Primary 60H10, 60H20, 60H99; Secondary 60G44, 60G46.