A note on the existence of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition
Federica Masiero (Università di Milano Bicocca)
Abstract
In [Stochastc Process. Appl., 122(9):3173-3208], the author proved the existence and the uniqueness of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition when the generator and the terminal condition are locally Lipschitz. In this paper, we prove that the existence result remains true for these BSDEs when the regularity assumption on the terminal condition is weakened.
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Pages: 1-15
Publication Date: April 17, 2013
DOI: 10.1214/EJP.v18-2124
References
- Barrieu-ElKaroui-11 P. Barrieu and N. El~Karoui. Monotone stability of quadratic semimartingales with applications to general quadratic BSDEs and unbounded existence result. to appear in Annals of Probability.
- Briand-Hu-06 P. Briand and Y. Hu. BSDE with quadratic growth and unbounded terminal value. Probab. Theory Related Fields, 136(4):604--618, 2006.
- Briand-Hu-08 P. Briand and Y. Hu. Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields, 141(3-4):543--567, 2008.
- Cheridito-Stadje-12 P. Cheridito and M. Stadje. Existence, minimality and approximation of solutions to bsdes with convex drivers. Stochastic Process. Appl., 122(4):1540 -- 1565, 2012.
- Delbaen-Hu-Bao-09 F. Delbaen, Y. Hu, and X. Bao. Backward SDEs with superquadratic growth. Probab. Theory Related Fields, pages 1--48, 2010.
- Delbaen-Hu-Richou-09 F. Delbaen, Y. Hu, and A. Richou. On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions. Ann. Inst. Henri Poincaré Probab. Stat., 47(2):559--574, 2011.
- ElKaroui-Peng-Quenez-97 N. El~Karoui, S. Peng, and M. C. Quenez. Backward stochastic differential equations in finance. Math. Finance, 7(1):1--71, 1997.
- Fuhrman-Tessitore-02 M. Fuhrman and G. Tessitore. The Bismut-Elworthy formula for backward SDEs and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces. Stoch. Stoch. Rep., 74(1-2):429--464, 2002.
- Gilding-Guedda-Kersner-03 B. H. Gilding, M. Guedda, and R. Kersner. The Cauchy problem for u_t=Δ u+ert nabla uert ^q. J. Math. Anal. Appl., 284(2):733--755, 2003.
- Gladkov-Guedda-Kersner-08 A. Gladkov, M. Guedda, and R. Kersner. A KPZ growth model with possibly unbounded data: correctness and blow-up. Nonlinear Anal., 68(7):2079--2091, 2008.
- Hu-Imkeller-Muller-05 Y. Hu, P. Imkeller, and M. Müller. Utility maximization in incomplete markets. Ann. Appl. Probab., 15(3):1691--1712, 2005.
- Kobylanski-00 M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab., 28(2):558--602, 2000.
- Mania-Schweizer-05 M. Mania and M. Schweizer. Dynamic exponential utility indifference valuation. Ann. Appl. Probab., 15(3):2113--2143, 2005.
- Masiero, Federica. Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian. Discrete Contin. Dyn. Syst. 32 (2012), no. 1, 223--263. MR2837060
- Pardoux, É.; Peng, S. G. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55--61. MR1037747
- Richou, Adrien. Numerical simulation of BSDEs with drivers of quadratic growth. Ann. Appl. Probab. 21 (2011), no. 5, 1933--1964. MR2884055
- Richou, Adrien. Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition. Stochastic Process. Appl. 122 (2012), no. 9, 3173--3208. MR2946439
- A. Richou. Étude théorique et numérique des équation différentielles stochastiques rétrogrades. PhD thesis, Université de Rennes 1, November 2010.
- Rouge, Richard; El Karoui, Nicole. Pricing via utility maximization and entropy. INFORMS Applied Probability Conference (Ulm, 1999). Math. Finance 10 (2000), no. 2, 259--276. MR1802922
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