Electronic Journal of Probability

Classical and Variational Differentiability of BSDEs with Quadratic Growth

Stefan Ankirchner, Peter Imkeller, and Goncalo Dos Reis

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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

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

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

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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

BSDE forward-backward SDE quadratic growth differentiability stochastic calculus of variations Malliavin calculus Feynman-Kac formula BMO martingale reverse Holder inequality

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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

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