## Electronic Journal of Probability

### Classical and Variational Differentiability of BSDEs with Quadratic Growth

#### 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).

#### Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 53, 1418-1453.

Dates
Accepted: 9 November 2007
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464818524

Digital Object Identifier
doi:10.1214/EJP.v12-462

Mathematical Reviews number (MathSciNet)
MR2354164

Zentralblatt MATH identifier
1138.60042

Rights

#### Citation

Ankirchner, Stefan; Imkeller, Peter; Dos Reis, Goncalo. Classical and Variational Differentiability of BSDEs with Quadratic Growth. Electron. J. Probab. 12 (2007), paper no. 53, 1418--1453. doi:10.1214/EJP.v12-462. https://projecteuclid.org/euclid.ejp/1464818524

#### References

• S. Ankirchner, P. Imkeller, A. Popier. Optimal cross hedging of insurance derivatives. Preprint, (2005).
• P. Briand, F. Confortola. BSDEs with stochastic Lipschitz condition and quadratic PDEs in hilbert spaces. To appear in Stochastic Process. Appl. (2007)
• 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
• 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.
• 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.
• N. El Karoui; S. Peng; M. C. Quenez. Backward stochastic differential equations in finance. Math. Finance 7 (1997), no. 1, 1–71.
• U. Horst; M. Müller. On the spanning property of risk bonds priced by equilibrium. To appear in Mathematics of Operations Reasearch (2006).
• N. Kazamaki. Continuous exponential martingales and BMO. Lecture Notes in Mathematics, 1579. Springer-Verlag, Berlin, 1994. viii+91 pp. ISBN: 3-540-58042-5
• M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 (2000), no. 2, 558–602.
• 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.
• M.A. Morlais. Quadratic BSDEs driven by continuous martingale and application to maximization problem. Preprint. (2007). Available at arXiv:math/0610749v2.
• 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.
• 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.