Estimation of the density of hypoelliptic diffusion processes with application to an extended Itô's formula

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Abstract

We prove a uniform bound for the density, pt(x), of the solution at time t ε (0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p t(x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al.
Original languageEnglish
Pages (from-to)223-247
JournalJournal of Theoretical Probability
Volume15
Issue number1
DOIs
Publication statusPublished - 1 Dec 2002

Keywords

  • Estimation of densities
  • Hypoelliptic diffusion processes
  • Itô's formula
  • Malliavin calculus

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