Minimal supersolutions of BSDEs with lower semicontinuous generators
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 524-538.

Nous étudions des sur-solutions minimales d'équations stochastiques rétrogrades. Nous montrons l'existence et l'unicité de telles sur-solutions minimales lorsque le générateur est conjointement semi-continu inférieurement, minoré par une fonction affine de la variable de contrôle et satisfait une condition spécifique de normalisation. Le résultat principal est obtenu en utilisant une convergence de semi-martingales.

We study minimal supersolutions of backward stochastic differential equations. We show the existence and uniqueness of the minimal supersolution, if the generator is jointly lower semicontinuous, bounded from below by an affine function of the control variable, and satisfies a specific normalization property. Semimartingale convergence is used to establish the main result.

DOI : 10.1214/12-AIHP523
Classification : 60H20, 60H30
Mots-clés : supersolutions of backward stochastic differential equations, semimartingale convergence 
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     title = {Minimal supersolutions of {BSDEs} with lower semicontinuous generators},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {524--538},
     publisher = {Gauthier-Villars},
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     year = {2014},
     doi = {10.1214/12-AIHP523},
     mrnumber = {3189083},
     zbl = {1296.60173},
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     url = {http://www.numdam.org/articles/10.1214/12-AIHP523/}
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Heyne, Gregor; Kupper, Michael; Mainberger, Christoph. Minimal supersolutions of BSDEs with lower semicontinuous generators. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 524-538. doi : 10.1214/12-AIHP523. http://www.numdam.org/articles/10.1214/12-AIHP523/

[1] K. Bahlali, E. Essaky and M. Hassani. Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs. C. R. Math. Acad. Sci. Paris 348 (2010) 677-682. | MR | Zbl

[2] M. Barlow and P. E. Protter. On convergence of semimartingales. In Séminaire de Probabilités XXIV 188-193. Lect. Notes Math. 1426. Springer, Berlin, 1988/1989. | Numdam | MR | Zbl

[3] P. Cheridito and M. Stadje. Existence, minimality and approximation of solutions to BSDEs with convex drivers. Stochastic Process. Appl. 122 (2012) 1540-1565. | MR | Zbl

[4] F. Delbaen and W. Schachermayer. A compactness principle for bounded sequences of martingales with applications. In Proceedings of the Seminar of Stochastic Analysis, Random Fields and Applications 133-173. Progress in Probability 45. Birkhäuser, Basel, 1996. | MR | Zbl

[5] F. Delbaen, Y. Hu and X. Bao. Backward SDEs with superquadratic growth. Probab. Theory Related Fields 150 (2011) 145-192. | MR | Zbl

[6] C. Dellacherie and P. A. Meyer. Probabilities and Potential. B: Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam, 1982. Translated from the French by J. P. Wilson. | MR | Zbl

[7] S. Drapeau, G. Heyne and M. Kupper. Minimal supersolutions of convex BSDEs. Ann. Probab. 41 (2013) 3973-4001. | MR | Zbl

[8] N. El Karoui, S. Peng and M. C. Quenez. Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1-71. | MR | Zbl

[9] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springer, New York, 1991. | MR | Zbl

[10] M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 (2000) 558-602. | MR | Zbl

[11] E. Pardoux and S. Peng. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990) 55-61. | MR | Zbl

[12] S. Peng. Backward SDE and related g-expectation. In Backward Stochastic Differential Equation 141-159. Pitman Research Notes in Mathematics Series 364. Longman, Harlow, 1997. | MR | Zbl

[13] S. Peng. Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probab. Theory Related Fields 113 (1999) 473-499. | MR | Zbl

[14] S. Peng and M. Xu. The smallest g-supermartingale and reflected BSDE with single and double L 2 obstacles. Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005) 605-630. | Numdam | MR | Zbl

[15] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Springer, Berlin, 2005. Version 2.1, Corrected third printing. | MR | Zbl

[16] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Fundamental Principles of Mathematical Sciences 293. Springer, Berlin, 1999. | MR | Zbl

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