Existence and Uniqueness for Non-Markovian Triangular Quadratic BSDEs

Abstract

We prove the existence and uniqueness of solutions to a class of quadratic backward SDE (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 nonuniqueness example, we answer a “crucial open question” raised by Harter and Richou by showing that the stochastic exponential of an $n \times n$ matrix-valued bounded mean oscillation martingale need not satisfy a reverse Hölder inequality.