On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions

Freddy Delbaen; Ying Hu; Adrien Richou

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 2, page 559-574
  • ISSN: 0246-0203

Abstract

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In [Probab. Theory Related Fields141 (2008) 543–567], the authors proved the uniqueness among the solutions of quadratic BSDEs with convex generators and unbounded terminal conditions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman–Kac formula proved in [Probab. Theory Related Fields141 (2008) 543–567].

How to cite

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Delbaen, Freddy, Hu, Ying, and Richou, Adrien. "On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions." Annales de l'I.H.P. Probabilités et statistiques 47.2 (2011): 559-574. <http://eudml.org/doc/239481>.

@article{Delbaen2011,
abstract = {In [Probab. Theory Related Fields141 (2008) 543–567], the authors proved the uniqueness among the solutions of quadratic BSDEs with convex generators and unbounded terminal conditions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman–Kac formula proved in [Probab. Theory Related Fields141 (2008) 543–567].},
author = {Delbaen, Freddy, Hu, Ying, Richou, Adrien},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {backward stochastic differential equations; generator of quadratic growth; unbounded terminal condition; uniqueness result; nonlinear Feynman–Kac formula; backward stochastic differential equations (BSDEs); nonlinear expectation; time consistency},
language = {eng},
number = {2},
pages = {559-574},
publisher = {Gauthier-Villars},
title = {On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions},
url = {http://eudml.org/doc/239481},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Delbaen, Freddy
AU - Hu, Ying
AU - Richou, Adrien
TI - On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 2
SP - 559
EP - 574
AB - In [Probab. Theory Related Fields141 (2008) 543–567], the authors proved the uniqueness among the solutions of quadratic BSDEs with convex generators and unbounded terminal conditions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman–Kac formula proved in [Probab. Theory Related Fields141 (2008) 543–567].
LA - eng
KW - backward stochastic differential equations; generator of quadratic growth; unbounded terminal condition; uniqueness result; nonlinear Feynman–Kac formula; backward stochastic differential equations (BSDEs); nonlinear expectation; time consistency
UR - http://eudml.org/doc/239481
ER -

References

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  1. [1] M. Bardi and I. Capuzzo-Dolcetta. Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston, MA, 1997. Zbl0890.49011MR1484411
  2. [2] P. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica. Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108 (2003) 109–129. Zbl1075.65503MR2008603
  3. [3] P. Briand and Y. Hu. BSDE with quadratic growth and unbounded terminal value. Probab. Theory Related Fields 136 (2006) 604–618. Zbl1109.60052MR2257138
  4. [4] P. Briand and Y. Hu. Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 (2008) 543–567. Zbl1141.60037MR2391164
  5. [5] F. Da Lio and O. Ley. Uniqueness results for convex Hamilton–Jacobi equations under p&gt;1 growth conditions on data. Appl. Math. Optim. To appear. Zbl1223.35115MR2784834
  6. [6] F. Da Lio and O. Ley. Uniqueness results for second-order Bellman–Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45 (2006) 74–106. Zbl1116.49017MR2225298
  7. [7] N. El Karoui, S. Peng and M. C. Quenez. Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71. Zbl0884.90035MR1434407
  8. [8] M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 (2000) 558–602. Zbl1044.60045MR1782267
  9. [9] E. Pardoux and S. Peng. Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991) 200–217. Lecture Notes in Control and Inform. Sci. 176. Springer, Berlin, 1992. Zbl0766.60079MR1176785

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