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).
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