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

By Pauline Barrieu, Nicole El Karoui. 2011

126 citations recorded by the source corpus.

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.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/2fefa7923b5597f1039c7f54661538b0cb12ab84

How researchers see this paper

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In Finance

1. Romuald Elie
2. P. Briand
3. Adrien Richou
4. Bruno Bouchard
5. Jean-François Chassagneux

In Management Science and Operations Research

1. P. Briand
2. Romuald Elie

Citations in the corpus

• A class of globally solvable Markovian quadratic BSDE systems and applications (2016, unknown): “7) can also be established using a localization technique of [12] or from a forward point of view, as in [5], as follows.”
• A simple constructive approach to quadratic BSDEs with or without delay (2013, result): “Barrieu and El Karoui [2] obtained recently similar results using on the contrary a forward approach for this question.”
• Numerical simulation of quadratic BSDEs (2013, unknown): “has been first proved by Kobylanski [34]. Since then, many authors worked on this question. When the terminal condition is bounded, we refer to [7, 34, 35, 42], and, in the unbounded case, we refer to [3, 8, 9, 19, 20]. We will focus here on the numerical approximation of the so-called “quadratic BSD”