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On a Chen–Fliess approximation for diffusion functionals

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Abstract

We show that the so-called functional derivatives, as recently introduced by Dupire (Functional Ito calculus, SSRN, 2010), can provide intuitive meaning to classic expansions of path dependent functionals that appear in control theory (work of Brockett, Fliess, Sussmann et. al). We then focus on stochastic differential equations and show that vector fields can be lifted to act as derivations on such functionals. This allows to revisit and generalize the classic stochastic Taylor expansion to arrive at a Chen–Fliess approximation for smooth, path dependent functionals of SDEs with a corresponding \(L^{2}\)-error estimate.

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Notes

  1. T. Lyons personal communication (cf. thesis of T. Fawcett [14] and the transfer report of A. Janssen).

  2. \(\rho _{\infty }\left( \left( t,x\right) ,\left( t,y\right) \right) =0\) iff \(x=y\) on \(\left[ 0,t\right] \); we call a progressively measurable \(F\) continuous at \(\left( t,x\right) \) wrt \(\rho _{\infty }\) if \(\forall \epsilon \exists \delta \) s.t \(\rho \left( \left( t,x\right) ,\left( s,y\right) \right) <\delta \) implies \(\left| F\left( t,x\right) -F\left( s,y\right) \right| <\epsilon \), cf. [11].

  3. \(\delta _{i,j}\) is the usual Kronecker delta, \(\delta _{ij}=1\) if \(i=j,\) otherwise equal \(0\).

  4. in the summation index that appears for the remainder term \(R_{st}^{m}\) we use the convention that \(\left( i_{2},\ldots ,i_{N}\right) =\emptyset \) if \(N=1\).

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Acknowledgments

The research of Christian Litterer and Harald Oberhauser was supported by the European Unions Seventh Framework Programme, ERC grant agreement 258237. The research of Christian Litterer was also supported by ERC grant 321111 RoFiRM, the research of Harald Oberhauser was also supported by ERC grant 291244 Esig.

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  1. Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128, Palaiseau, France

    Christian Litterer

  2. Oxford-Man Institute, Eagle House, University of Oxford, Walton Well Rd, OX2 6ED, Oxford, UK

    Harald Oberhauser

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  1. Christian Litterer
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Correspondence to Harald Oberhauser.

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Communicated by P. Friz.

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Litterer, C., Oberhauser, H. On a Chen–Fliess approximation for diffusion functionals. Monatsh Math 175, 577–593 (2014). https://doi.org/10.1007/s00605-014-0677-4

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