Electronic Journal of Probability
- Electron. J. Probab.
- Volume 12 (2007), paper no. 53, 1418-1453.
Classical and Variational Differentiability of BSDEs with Quadratic Growth
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).
Electron. J. Probab., Volume 12 (2007), paper no. 53, 1418-1453.
Accepted: 9 November 2007
First available in Project Euclid: 1 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 65C30: Stochastic differential and integral equations
This work is licensed under aCreative Commons Attribution 3.0 License.
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