Classical and Variational Differentiability of BSDEs with Quadratic Growth

By Stefan Ankirchner, Peter Imkeller, Gonçalo dos Reis. 2007

80 citations recorded by the source corpus.

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).

Subfields: Applied Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/97126d5f519cbe01a925fe6c1949c9c8e05f4772

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Citations in the corpus

A simple constructive approach to quadratic BSDEs with or without delay (2013, methodology): “Using linearization arguments, we exhibit a priori estimates on the S∞ norm and the BMO norm of the (Y,Z) process, solution of a quadratic BSDE in the spirit of [1, 3].”
Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs (2011, background): “is BMO (see for instance the recent papers by Hu, Imkeller & Muller [27], Ankrichner, Imkeller & Reis [1], [40], or the PhD thesis of Dos Reis [16]).”
Numerical simulation of quadratic BSDEs (2013, unknown): “Moreover, the solution is known to have the following properties; see, for example, [1, 6, 34].”
On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures (2010, methodology): “The fine structure and sensitivity properties of solutions of BSDE or systems of forward and backward stochastic differential equations have been approached by means of the stochastic calculus of variations (see [2] and [1]), and applied to provide explicit descriptions of delta hedges of insurance ”