On a Chen–Fliess approximation for diffusion functionals

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• 作者：Christian Litterer (1)
  Harald Oberhauser (2)
  
• 关键词：Chen–Fliess series ; Stochastic Taylor expansion ; Control theory ; Primary 60G05 ; 47A60 ; 60Hxx ; Secondary 34H99 ; 46T20
• 刊名：Monatshefte f篓鹿r Mathematik
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：175
• 期：4
• 页码：577-593
• 全文大小：211 KB
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• 作者单位：Christian Litterer (1)
  Harald Oberhauser (2)
  
  1. Centre de Mathématiques Appliquées, école Polytechnique, Route de Saclay, 91128, Palaiseau, France
  2. Oxford-Man Institute, Eagle House, University of Oxford, Walton Well Rd, OX2 6ED, Oxford, UK
  
• ISSN：1436-5081

文摘

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.