Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations

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Abstract

This article proves some properties of the probability density function concerning maxima of a solution to one-dimensional stochastic differential equations. We first obtain lower and upper bounds on the density function of the discrete time maximum of the solution. We then prove that the density function of the discrete time maximum converges to that of the continuous time maximum of the solution. Finally, we prove the positivity of the density function of the continuous time maximum and a relationship between the density functions of the continuous time maximum and the solution itself.

Original languageEnglish
JournalJournal of Theoretical Probability
DOIs
Publication statusPublished - 2019 Jan 1

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Density function
Stochastic differential equations
Continuous time
Discrete-time

Keywords

  • Continuous/discrete time maximum
  • Malliavin calculus
  • Probability density function
  • Stochastic differential equation

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

Cite this

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title = "Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations",
abstract = "This article proves some properties of the probability density function concerning maxima of a solution to one-dimensional stochastic differential equations. We first obtain lower and upper bounds on the density function of the discrete time maximum of the solution. We then prove that the density function of the discrete time maximum converges to that of the continuous time maximum of the solution. Finally, we prove the positivity of the density function of the continuous time maximum and a relationship between the density functions of the continuous time maximum and the solution itself.",
keywords = "Continuous/discrete time maximum, Malliavin calculus, Probability density function, Stochastic differential equation",
author = "Tomonori Nakatsu",
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doi = "10.1007/s10959-019-00885-1",
language = "English",
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AB - This article proves some properties of the probability density function concerning maxima of a solution to one-dimensional stochastic differential equations. We first obtain lower and upper bounds on the density function of the discrete time maximum of the solution. We then prove that the density function of the discrete time maximum converges to that of the continuous time maximum of the solution. Finally, we prove the positivity of the density function of the continuous time maximum and a relationship between the density functions of the continuous time maximum and the solution itself.

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KW - Probability density function

KW - Stochastic differential equation

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