### 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 language | English |
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Journal | Journal of Theoretical Probability |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### Fingerprint

### 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

**Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations.** / Nakatsu, Tomonori.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations

AU - Nakatsu, Tomonori

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

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.

KW - Continuous/discrete time maximum

KW - Malliavin calculus

KW - Probability density function

KW - Stochastic differential equation

UR - http://www.scopus.com/inward/record.url?scp=85061805762&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061805762&partnerID=8YFLogxK

U2 - 10.1007/s10959-019-00885-1

DO - 10.1007/s10959-019-00885-1

M3 - Article

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

ER -