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 Electronic Journal of Probability > Vol. 12 (2007) > Paper 53 open journal systems 


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: PDF

Pages: 1418-1453

Published on: November 9, 2007
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Electronic Journal of Probability. ISSN: 1083-6489