Existence and uniqueness for non-Markovian triangular quadratic BSDEs

TL;DR

This paper establishes the existence and uniqueness of solutions for a class of non-Markovian triangular quadratic backward stochastic differential equations, extending prior results and addressing open questions in the field.

Contribution

It generalizes existing results on diagonally quadratic BSDEs to a broader non-Markovian setting and provides new insights into linear BSDEs with unbounded coefficients.

Findings

Proved existence and uniqueness of solutions for triangular quadratic BSDEs.

Extended results to non-Markovian frameworks beyond previous work.

Provided a counterexample showing the stochastic exponential of certain BMO martingales does not satisfy reverse H"older inequality.

Abstract

We prove the existence and uniqueness of solutions to a class of quadratic BSDE systems which we call triangular quadratic. Our results generalize several existing results about diagonally quadratic BSDEs in the non-Markovian setting. As part of our analysis, we obtain new results about linear BSDEs with unbounded coefficients, which may be of independent interest. Through a non-uniqueness example, we answer a "crucial open question" raised by Harter and Richou by showing that the stochastic exponential of an n x n matrix-valued BMO martingale need not satisfy a reverse H\"older inequality.