The reverse H\"older inequality for matrix-valued stochastic exponentials and applications to quadratic BSDE systems

TL;DR

This paper establishes a link between reverse H"older inequalities for matrix martingales and the well-posedness of quadratic BSDE systems, providing new conditions for their solvability in non-Markovian settings.

Contribution

It introduces a novel equivalence between reverse H"older inequalities and the well-posedness of linear BSDEs with unbounded coefficients, and applies this to quadratic BSDE systems.

Findings

Equivalence between reverse H"older inequality and BSDE well-posedness.

Structural conditions ensuring these properties hold.

Global well-posedness for new classes of non-Markovian quadratic BSDEs.

Abstract

In this paper, we study the connections between three concepts - the reverse H\"older inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo (bounded mean oscillation) coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse H\"older inequality. Furthermore, we give structural conditions under which these two equivalent conditions are satisfied. Finally, we apply our results on linear equations to obtain global well-posedness results for two new classes of non-Markovian quadratic BSDE systems with special structure.