Classical and Variational Differentiability of BSDEs with Quadratic Growth

Stefan Ankirchner (Humboldt Uiversitaet Berlin)
Peter Imkeller (Humboldt Uiversitaet Berlin)
Goncalo JN Dos Reis (Humboldt Uiversitaet Berlin)

Abstract


We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect to the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDEs to be differentiable in the variational sense (Malliavin differentiable).

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1418-1453

Publication Date: November 9, 2007

DOI: 10.1214/EJP.v12-462

References

  1. S. Ankirchner, P. Imkeller, A. Popier. Optimal cross hedging of insurance derivatives. Preprint, (2005).
  2. P. Briand, F. Confortola. BSDEs with stochastic Lipschitz condition and quadratic PDEs in hilbert spaces. To appear in Stochastic Process. Appl. (2007)
  3. S. Chaumont, P. Imkeller, M. Müller, U. Horst. A simple model for trading climate risk. Vierteljahrshefte zur Wirtschaftsforschung / Quarterly Journal of Economic Research, 74 (2005), no. 2, 175--195. Available at RePEc:diw:diwvjh:74-2-6
  4. S. Chaumont; P. Imkeller; M. Müller. Equilibrium trading of climate and weather risk and numerical simulation in a Markovian framework. Stoch. Environ. Res. Risk Assess. 20 (2006), no. 3, 184--205. MR2297447
  5. P. Cheridito; H. Soner; N. Touzi; N. Victoir. Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 (2007), no. 7, 1081--1110. MR2319056
  6. N. El Karoui; S. Peng; M. C. Quenez. Backward stochastic differential equations in finance. Math. Finance 7 (1997), no. 1, 1--71. MR1434407 (98d:90030)
  7. U. Horst; M. Müller. On the spanning property of risk bonds priced by equilibrium. To appear in Mathematics of Operations Reasearch (2006).
  8. N. Kazamaki. Continuous exponential martingales and BMO. Lecture Notes in Mathematics, 1579. Springer-Verlag, Berlin, 1994. viii+91 pp. ISBN: 3-540-58042-5 MR1299529 (95k:60110)
  9. M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 (2000), no. 2, 558--602. MR1782267 (2001h:60110)
  10. H. Kunita. Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1990. xiv+346 pp. ISBN: 0-521-35050-6 MR1070361 (91m:60107)
  11. M.A. Morlais. Quadratic BSDEs driven by continuous martingale and application to maximization problem. Preprint. (2007). Available at arXiv:math/0610749v2.
  12. D. Nualart. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, New York, 1995. xii+266 pp. ISBN: 0-387-94432-X MR1344217 (96k:60130)
  13. P. E. Protter. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4 MR2020294 (2005k:60008)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.