Harmonic analysis of stochastic equations and backward stochastic differential equations

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in ${mathcal{R}^p}$ ( ${pin [1,infty)}$ ) and backward stochastic differential equations (BSDEs) in ${mathcal{R}^ptimes mathcal{H}^p}$ ( ${pin (1, infty)}$ ) and in ${mathcal{R}^inftytimesoverline{L^infty}^{rm BMO}}$ , 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 cases for stochastic differential equations (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, some relations are established between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite.