Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs

TL;DR

This paper establishes the stability and convergence of quadratic semimartingales, enabling the existence of solutions for unbounded quadratic BSDEs under minimal integrability conditions, with applications in financial mathematics.

Contribution

It introduces a new stability result for quadratic semimartingales that does not rely on boundedness assumptions, extending the existence theory for quadratic BSDEs.

Findings

Strong convergence of martingale parts in various spaces

Existence of solutions for unbounded quadratic BSDEs

Minimal exponential integrability conditions suffice

Abstract

In this paper, we study the stability and convergence of some general quadratic semimartingales. Motivated by financial applications, we study simultaneously the semimartingale and its opposite. Their characterization and integrability properties are obtained through some useful exponential submartingale inequalities. Then, a general stability result, including the strong convergence of the martingale parts in various spaces ranging from $\mathbb{H}^1$ to BMO, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale. This can be applied in particular to BSDE-like semimartingales. This strong convergence result is then used to prove the existence of solutions of general quadratic BSDEs under minimal exponential integrability assumptions, relying on a regularization in both linear-quadratic growth of the quadratic coefficient itself. On the contrary to most of the existing literature, it does not involve the seminal result of Kobylanski (2000) on bounded solutions.