Solvability of some quadratic BSDEs without exponential moments

Khaled Bahlali; Mʼhamed Eddahbi; Youssef Ouknine

doi:10.1016/j.crma.2013.04.003

Probability Theory

Solvability of some quadratic BSDEs without exponential moments

Présenté par : the Editorial Board

Khaled Bahlali ¹ ; Mʼhamed Eddahbi ² ; Youssef Ouknine 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

Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 229-233.

Résumé
Abstract

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.

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.

Reçu le : 2013-02-06
Accepté le : 2013-04-02
Publié le : 2013-03-01

DOI : 10.1016/j.crma.2013.04.003

Affiliations des auteurs :

Khaled Bahlali ¹ ; Mʼhamed Eddahbi ² ; Youssef Ouknine 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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     author = {Khaled Bahlali and M'hamed Eddahbi and Youssef Ouknine},
     title = {Solvability of some quadratic {BSDEs} without exponential moments},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {229--233},
     publisher = {Elsevier},
     volume = {351},
     number = {5-6},
     year = {2013},
     doi = {10.1016/j.crma.2013.04.003},
     language = {en},
}

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Khaled Bahlali; Mʼhamed Eddahbi; Youssef Ouknine. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.04.003/

Bibliographie
Cité par

[1] K. Bahlali Flows of homeomorphisms for stochastic differential equations with measurable drift, Stoch. Stoch. Rep., Volume 67 (1999), pp. 53-82

[2] K. Bahlali; S. Hamadène; B. Mezerdi Backward stochastic differential equations with two reflecting barriers and application, SPA Stoch. Process. Appl., Volume 115 (2005), pp. 1107-1129

[3] P. Barrieu; N. El Karoui Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs, Ann. Probab. (2013) http://www.imstat.org/aop/futurepapers.htm (preprint in press)

[4] P. Briand; Y. Hu BSDE with quadratic growth and unbounded terminal value, Probab. Theory Relat. Fields, Volume 136 (2006) no. 4, pp. 604-618

[5] A. Dermoune; S. Hamadène; Y. Ouknine Backward stochastic differential equation with local time, Stoch. Stoch. Rep., Volume 66 (1999) no. 1–2, pp. 103-119

[6] R.M. Dudley Wiener functionals as Itô integrals, Ann. Probab., Volume 5 (1977) no. 1, pp. 140-141

[7] M. Eddahbi; Y. Ouknine Limit theorems for BSDE with local time applications to non-linear PDE, Stoch. Stoch. Rep., Volume 73 (2002) no. 1–2, pp. 159-179

[8] E. Essaky; Hassani Generalized BSDE with 2-reflecting barriers and stochastic quadratic growth, J. Differential Equations, Volume 254 (2013) no. 3, pp. 1500-1528

[9] M. Kobylanski Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., Volume 28 (2000) no. 2, pp. 558-602

[10] N.V. Krylov Controlled Diffusion Processes, Springer-Verlag, 1980

[11] J.-P. Lepeltier; J. San Martin Backward stochastic differential equations with continuous coefficient, Stat. Probab. Lett., Volume 32 (1997) no. 4, pp. 425-430

[12] E. Pardoux; S. Peng Adapted solution of a backward stochastic differential equation, Syst. Control Lett., Volume 14 (1990), pp. 55-61

[13] R. Tevzadze Solvability of backward stochastic differential equations with quadratic growth, Stoch. Process. Appl., Volume 118 (2008) no. 3, pp. 503-515

Cité par 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.

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