On some limit theorems for occupation times of one dimensional Brownian motion and its continuous additive functionals locally of zero energy

Abstract We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

Download Full-text

Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

Advances in Applied Probability ◽

10.1239/aap/1427814588 ◽

2015 ◽

Vol 47 (1) ◽

pp. 210-230 ◽

Cited By ~ 18

Author(s):

Hongzhong Zhang

Keyword(s):

Brownian Motion ◽

Diffusion Process ◽

Time Horizon ◽

Three Dimensional ◽

Bessel Processes ◽

Occupation Times ◽

Exponential Time ◽

One Dimensional ◽

Brownian Motion With Drift ◽

Running Maximum

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

Download Full-text

Limit Theorems for Local and Occupation Times of Random Walks and Brownian Motion on a Spider

Journal of Theoretical Probability ◽

10.1007/s10959-017-0788-7 ◽

2017 ◽

Vol 32 (1) ◽

pp. 330-352

Author(s):

Endre Csáki ◽

Miklós Csörgő ◽

Antónia Földes ◽

Pál Révész

Keyword(s):

Brownian Motion ◽

Random Walks ◽

Limit Theorems ◽

Occupation Times ◽

And Brownian Motion

Download Full-text

On limit theorems of some extensions of fractional Brownian motion and their additive functionals

Stochastics and Dynamics ◽

10.1142/s0219493717500228 ◽

2017 ◽

Vol 17 (03) ◽

pp. 1750022

Author(s):

M. Ait Ouahra ◽

S. Moussaten ◽

A. Sghir

Keyword(s):

Brownian Motion ◽

Fractional Brownian Motion ◽

Local Time ◽

Limit Theorems ◽

Fractional Derivatives ◽

Slowly Varying Function ◽

Bifractional Brownian Motion ◽

Additive Functionals ◽

Subfractional Brownian Motion

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function.

Download Full-text

XIX.—A Generalization of the Classical Random-walk problem, and a Simple Model of Brownian Motion Based Thereon

Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences ◽

10.1017/s0080454100007147 ◽

1952 ◽

Vol 63 (3) ◽

pp. 268-279

Author(s):

G. Klein

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Thermal Equilibrium ◽

Diffusion Theory ◽

Initial Kinetic Energy ◽

Time Varying ◽

Zero Energy ◽

Discrete Probability ◽

One Dimensional ◽

Special Cases

SynopsisSuggested by the analogy between the classical one-dimensional random-walk and the approximate (diffusion) theory of Brownian motion, a generalization of the random-walk is proposed to serve as a model for the more accurate description of the phenomenon. Using the methods of the calculus of finite differences, some general results are obtained concerning averages based on a time-varying bivariate discrete probability distribution in which the variates stand in the particular relation of “position” and “velocity.” These are applied to the special cases of Brownian motion from initial thermal equilibrium, and from arbitrary initial kinetic energy. In the latter case the model describes accurately quantized Brownian motion of two energy states, one of zero energy.

Download Full-text