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2007 Classical and Variational Differentiability of BSDEs with Quadratic Growth
Stefan Ankirchner, Peter Imkeller, Goncalo Dos Reis
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Electron. J. Probab. 12: 1418-1453 (2007). DOI: 10.1214/EJP.v12-462


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).


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Stefan Ankirchner. Peter Imkeller. Goncalo Dos Reis. "Classical and Variational Differentiability of BSDEs with Quadratic Growth." Electron. J. Probab. 12 1418 - 1453, 2007.


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

zbMATH: 1138.60042
MathSciNet: MR2354164
Digital Object Identifier: 10.1214/EJP.v12-462

Primary: 60H10
Secondary: 60H07 , 65C30

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

Vol.12 • 2007
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