  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.
