Harmonic Analysis of Stochastic Equations and Backward Stochastic Differential Equations

TL;DR

This paper develops a harmonic analysis framework for stochastic equations and backward stochastic differential equations using BMO martingale theory, introducing new inequalities and linking solution properties to reverse Hölder inequalities.

Contribution

It introduces a probabilistic Fefferman inequality and explores the connection between solution existence, reverse Hölder inequalities, and Kazamaki's quadratic critical exponent.

Findings

Established new results on existence and uniqueness of solutions.

Linked the reverse Hölder property to solution matrices.

Characterized when Kazamaki's exponent is infinite.

Abstract

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations (SEs) in $\cR^p$ ($p\in [1, \infty)$) and backward stochastic differential equations (BSDEs) in $\cR^p\times \cH^p$ ($p\in (1, \infty)$) and in $\cR^\infty\times \bar{\cH^\infty}^{BMO}$, 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 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 H\"older inequality for some suitable exponent $p\ge 1$. 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.