scholarly journals Nonlinear Brownian motion - mean square displacement

2004 ◽  
Vol 7 (3) ◽  
pp. 539 ◽  
Author(s):  
Ebeling
Keyword(s):  
Brownian Motion ◽  
Mean Square Displacement ◽  
Mean Square

scholarly journals Option Pricing Based on Modified Advection-Dispersion Equation: Stochastic Representation and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Longjin Lv ◽  
Luna Wang
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Dispersion Equation ◽  
Mean Square Displacement ◽  
Mean Square ◽  
European Option ◽  
Stochastic Representation ◽  
Pricing Models ◽  
Advection Dispersion ◽  
Black Scholes

In this paper, we first investigate the stochastic representation of the modified advection-dispersion equation, which is proved to be a subordinated stochastic process. Taking advantage of this result, we get the analytical solution and mean square displacement for the equation. Then, applying the subordinated Brownian motion into the option pricing problem, we obtain the closed-form pricing formula for the European option, when the underlying of the option contract is supposed to be driven by the subordinated geometric Brownian motion. At last, we compare the obtained option pricing models with the classical Black–Scholes ones.

Brownian Motion on Cantor Sets

2020 ◽  
Vol 21 (3-4) ◽  
pp. 275-281
Author(s):  
Ali Khalili Golmankhaneh ◽  
Saleh Ashrafi ◽  
Dumitru Baleanu ◽  
Arran Fernandez
Keyword(s):  
Brownian Motion ◽  
Classical Method ◽  
Mean Square Displacement ◽  
Planck Equation ◽  
Cantor Sets ◽  
Mean Square ◽  
Fokker Planck Equation ◽  
Fractal Time ◽  
The Mean ◽  
Fokker Planck

AbstractIn this paper, we have investigated the Langevin and Brownian equations on fractal time sets using Fα-calculus and shown that the mean square displacement is not varied linearly with time. We have also generalized the classical method of deriving the Fokker–Planck equation in order to obtain the Fokker–Planck equation on fractal time sets.

scholarly journals Modeling Anomalous Diffusion by a Subordinated Integrated Brownian Motion

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Long Shi ◽  
Aiguo Xiao
Keyword(s):  
Brownian Motion ◽  
Continuous Time ◽  
Waiting Times ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Weak Ergodicity ◽  
Normal Diffusion ◽  
Weak Ergodicity Breaking ◽  
The Mean ◽  
Ergodicity Breaking

We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times are correlated. In a continuum limit, the process can be defined by an integrated Brownian motion subordinated by an inverse α-stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when 0<α<1/3, normal diffusion when α=1/3, and superdiffusion when 1/3<α<1. The time-averaged mean square displacement is also employed to show weak ergodicity breaking occurring in the proposed process. An extension to the fractional case is also considered.

scholarly journals Impulse response function for Brownian motion

Soft Matter ◽  
2021 ◽  
Author(s):  
Nicos Makris
Keyword(s):  
Brownian Motion ◽  
Time Derivative ◽  
Mean Square Displacement ◽  
Impulse Response Function ◽  
Velocity Autocorrelation Function ◽  
Mean Square ◽  
Velocity Autocorrelation ◽  
The Mean ◽  
First Time

Motivated from the central role of the mean-square displacement and its second time-derivative – that is the velocity autocorrelation function in the description of Brownian motion, we revisit the physical meaning of its first time-derivative.

scholarly journals On the Fractal Langevin Equation

2019 ◽  
Vol 3 (1) ◽  
pp. 11 ◽  
Author(s):  
Alireza Khalili Golmankhaneh
Keyword(s):  
Mathematical Model ◽  
Brownian Motion ◽  
Random Walks ◽  
Langevin Equation ◽  
Mean Square Displacement ◽  
Cantor Set ◽  
Langevin Equations ◽  
Mean Square

In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle- τ Cantor set. The fractal mean square displacement of different random walks on the middle- τ Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details.

Numerics for the fractional Langevin equation driven by the fractional Brownian motion

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peng Guo ◽  
Caibin Zeng ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Order ◽  
Langevin Equation ◽  
Mean Square Displacement ◽  
Hurst Parameter ◽  
Mean Square ◽  
Fractional Langevin Equation ◽  
Normal Diffusion ◽  
The Mean

AbstractWe study analytically and numerically the fractional Langevin equation driven by the fractional Brownian motion. The fractional derivative is in Caputo’s sense and the fractional order in this paper is α = 2 − 2H, where H ∈ ($\tfrac{1} {2} $, 1) is the Hurst parameter (or, index). We give numerical schemes for the fractional Langevin equation with or without an external force. From the figures we can find that the mean square displacement of the fractional Langevin equation has the property of the anomalous diffusion. When the fractional order tends to an integer, the diffusion reduces to the normal diffusion.

scholarly journals THE SIMPLE NONDEGENERATE RELATIVISTIC GAS: STATISTICAL PROPERTIES AND BROWNIAN MOTION

2010 ◽  
Vol 24 (31) ◽  
pp. 6043-6048
Author(s):  
A. SANDOVAL-VILLALBAZO ◽  
A. ARAGONÉS-MUÑOZ ◽  
A. L. GARCÍA-PERCIANTE
Keyword(s):  
Brownian Motion ◽  
Thermodynamic Properties ◽  
Brownian Particle ◽  
Mean Square Displacement ◽  
Statistical Properties ◽  
Velocity Space ◽  
Thermal Bath ◽  
Mean Square ◽  
The Mean ◽  
Relativistic Gas

This paper shows a novel calculation of the mean square displacement of a classical Brownian particle in a relativistic thermal bath. Also, the thermodynamic properties of a nondegenerate simple relativistic gas are reviewed in terms of a treatment performed in velocity space.

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