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

• Pauline Barrieu ^[1] ; Nicole El Karoui ^[2]
  1. [1] London School of Economics and Political Science

     London School of Economics and Political Science

     Reino Unido

     
  2. [2] Pierre and Marie Curie University

     Pierre and Marie Curie University

     París, Francia

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 3, 2, 2013, págs. 1831-1863
• Idioma: inglés
• DOI: 10.1214/12-AOP743
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 H1 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 [Ann. Probab. 28 (2010) 558–602] on bounded solutions.