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Stochastic Differential Equations Driven by Loops

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Book cover XI Symposium on Probability and Stochastic Processes

Part of the book series: Progress in Probability ((PRPR,volume 69))

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Abstract

We study stochastic differential equations of the type

$$\displaystyle{ X_{t} = x +\sum _{ i=1}^{d}\int _{ 0}^{t}V _{ i}(X_{s}^{x}) \circ dM_{ s}^{i},\text{ }0 \leq t \leq T, }$$

where (M s )0 ≤ s ≤ T is a semimartingale generating a loop in the free Carnot group of step N and show how the properties of the random variable X T x are closely related to the Lie subalgebra generated by the commutators of the V i ’s with length greater than N + 1. It is furthermore shown that if f is a smooth function, then

$$\displaystyle{\lim _{T\rightarrow 0}\frac{\mathbb{E}(\,f(X_{T}^{x})) - f(x)} {T^{N+1}} = (\Delta _{N}\,f)(x),}$$

where \(\Delta _{N}\) is a second order operator related to the V i ′s.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Fabrice Baudoin

Authors
  1. Fabrice Baudoin
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Correspondence to Fabrice Baudoin .

Editor information

Editors and Affiliations

  1. Probabilidad y Estadística, IIMAS UNAM, Mexico

    Ramsés H. Mena

  2. CIMAT, Guanajuato, Mexico

    Juan Carlos Pardo

  3. CIMAT, Guanajuato, Mexico

    Víctor Rivero

  4. Instituto de Matemáticas, UNAM, Mexico

    Gerónimo Uribe Bravo

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Baudoin, F. (2015). Stochastic Differential Equations Driven by Loops. In: Mena, R., Pardo, J., Rivero, V., Uribe Bravo, G. (eds) XI Symposium on Probability and Stochastic Processes. Progress in Probability, vol 69. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-13984-5_3

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