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On uniformly subelliptic operators and stochastic area
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  • Published: 27 November 2007

On uniformly subelliptic operators and stochastic area

  • Peter Friz1 &
  • Nicolas Victoir2 

Probability Theory and Related Fields volume 142, pages 475–523 (2008)Cite this article

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  • 12 Citations

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Abstract

Let X a be a Markov process with generator ∑ i,j ∂ i ( a ij∂ j · ) where a is a uniformly elliptic symmetric matrix. Thanks to the fundamental works of T. Lyons, stochastic differential equations driven by X a can be solved in the “rough path sense”; that is, pathwise by using a suitable stochastic area process. Our construction of the area, which generalizes previous works of Lyons–Stoica and then Lejay, is based on Dirichlet forms associated to subellitpic operators. This enables us in particular to discuss large deviations and support descriptions in suitable rough path topologies. As typical rough path corollary, Freidlin–Wentzell theory and the Stroock–Varadhan support theorem remain valid for stochastic differential equations driven by X a.

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Author information

Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Peter Friz

  2. Hong Kong, Hong Kong

    Nicolas Victoir

Authors
  1. Peter Friz
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  2. Nicolas Victoir
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Correspondence to Peter Friz.

Additional information

Peter Friz is a Leverhulme Fellow.

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Friz, P., Victoir, N. On uniformly subelliptic operators and stochastic area. Probab. Theory Relat. Fields 142, 475–523 (2008). https://doi.org/10.1007/s00440-007-0113-y

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  • Received: 16 March 2007

  • Revised: 17 October 2007

  • Published: 27 November 2007

  • Issue Date: November 2008

  • DOI: https://doi.org/10.1007/s00440-007-0113-y

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Mathematics Subject Classification (2000)

  • 60H99
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