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May 2013 Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs
Pauline Barrieu, Nicole El Karoui
Ann. Probab. 41(3B): 1831-1863 (May 2013). DOI: 10.1214/12-AOP743


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


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Pauline Barrieu. Nicole El Karoui. "Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs." Ann. Probab. 41 (3B) 1831 - 1863, May 2013.


Published: May 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1312.60052
MathSciNet: MR3098060
Digital Object Identifier: 10.1214/12-AOP743

Primary: 60G07 , 60G44 , 60H99
Secondary: 91B16 , 91B26

Keywords: BSDE-like semimartingale , entropic inequalities , exponential transformation , monotone stability , quadratic BSDE , Quadratic semimartingale , strong convergence

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 3B • May 2013
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