Probability Theory
Solvability of some quadratic BSDEs without exponential moments
Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 229-233.

We establish the existence and uniqueness of square integrable solutions for a class of one-dimensional quadratic backward stochastic differential equations (QBSDEs). This is done with a merely square integrable terminal condition, and in some cases with a measurable generator. This shows, in particular, that neither the existence of exponential moments for the terminal condition nor the continuity of the generator are needed for the existence and/or uniqueness of solutions for quadratic BSDEs. These conditions are used in the previous papers on QBSDEs. To do this, we show that Itôʼs formula remains valid for functions having a merely locally integrable second (generalized) derivative. A comparison theorem is also established.

Nous établissons lʼexistence et lʼunicité de solutions de carré intégrables pour une classe dʼéquations différentielles stochastiques rétrogrades (EDSR) quadratiques ayant une condition terminale de carré intégrable, et, dans certains cas, un générateur uniquement mesurable. Le présent travail montre, en particulier, que ni lʼexistence des moments exponentiels de la donnée terminale, ni la continuité du générateur ne sont nécessaires à lʼexistence et lʼunicité des EDSR quadratiques. Pour ce faire, nous établissons dʼabord que, pour les solutions dʼEDSR unidimensionnelles de croissance quadratique, la formule dʼItô reste valable pour des fonctions dont la dérivée seconde (au sens des distributions) est seulement localement integrable. Un théorème de comparaison est également établi pour une classe dʼEDSR quadratiques ayant un générateur mesurable.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.04.003
Bahlali, Khaled 1; Eddahbi, Mʼhamed 2; Ouknine, Youssef 3

1 Université de Toulon, IMATH, EA 2134, 83957 La Garde cedex, France
2 UCA, FST, département de mathématiques, B.P. 549, Marrakech, Morocco
3 UCA, FSS, département de mathématiques, B.P. 2390, Marrakech, Morocco
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Bahlali, Khaled; Eddahbi, Mʼhamed; Ouknine, Youssef. Solvability of some quadratic BSDEs without exponential moments. Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 229-233. doi : 10.1016/j.crma.2013.04.003. http://www.numdam.org/articles/10.1016/j.crma.2013.04.003/

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Cited by Sources:

Partially supported by FP 7 PITN-GA-2008-213881, Marie Curie ITN “Deterministic and Stochastic Control Systems”, PHC Volubilis MA/10/224 and PHC Tassili 13MDU887.