Potential Analysis

, Volume 43, Issue 2, pp 289–311 | Cite as

Gaussian Estimates for the Solutions of Some One-dimensional Stochastic Equations

  • Tien Dung Nguyen
  • Nicolas Privault
  • Giovanni Luca Torrisi
Article

Abstract

Using covariance identities based on the Clark-Ocone representation formula we derive Gaussian density bounds and tail estimates for the probability law of the solutions of several types of stochastic differential equations, including Stratonovich equations with boundary condition and irregular drifts, and equations driven by fractional Brownian motion. Our arguments are generally simpler than the existing ones in the literature as our approach avoids the use of the inverse of the Ornstein-Uhlenbeck operator.

Keywords

Malliavin calculus Clark-Ocone formula Probability bounds Fractional Brownian motion 

Mathematical Subject Classification (2010)

60F05 60G57 60H07 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Tien Dung Nguyen
    • 1
  • Nicolas Privault
    • 2
  • Giovanni Luca Torrisi
    • 3
  1. 1.Division of Mathematical SciencesNanyang Technological UniversityNanyang LinkSingapore
  2. 2.Division of Mathematical SciencesNanyang Technological UniversityNanyang LinkSingapore
  3. 3.Istituto per le Applicazioni del Calcolo“Mauro Picone”, CNRRomaItaly

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