Exploring Brownian Motion

1
Exploring Brownian Motion

• Thesis Proposal
• Fei Fei CHU
• Winter 2008
• Advised by Professor C.Shilepsky

2
Introduction

• Brownian Motion
• Named to in honor of Robert Brown
• Physical phenomenon
• Particle random movements
• Why It Is Interesting
• Stochastic process
• Mathematical model
• Simulation
• Widely used in Many Fields

3
Historical Background

• Contributions to Brownian Motion
• Three -berts
• Robert Brown in 1827
• Albert Einstein in 1905
• Norbert Weiner in 1923
• Jean Perrin
• Nobel Prize in physics in 1926

4
Robert Brown

• Discovery in 1826
• Observation by Microscope
• Suspended some of the pollen grains in water
• Pollen was alive
• Particles were filled in the water
• Very obvious movement
• Not the currents in the fluid
• Not evaporating
• belonged to the particle itself?

5
Brown Motion in Physics

• Einstein applied Brownian Motion in
  thermodynamics
• The main physical principle of Brownian motion
  ltEgt mltv2gt/ 2 3kT/2
• mean kinetic energy of Brownian Motion is
  proportional to the temperature
• The mean-square displacement r2 x2 y2 z2 of
  a Brownian particle is described by the equation
  ltr2gt 6kTBt

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1-D Random Walk

• Binomial distribution
• Variance Npq which corresponds to time
• Graph
• Distance n1 n2
• Time increment goes
• to 0, step size is very
• small, continuous
• Normal distribution

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Wiener Process

• Named in honor of Norbert Wiener
• Continuous-time stochastic process
• Mathematics from Brownian Motion
• Wiener process Wt
• W0 0
• Wt is continuous
• Wt Ws has normal distribution
• with expected value 0 and
• variance t-s (0 s lt t )

8
Stock Market

• Brownian Motion Vs. Stock Market
• Wiener Process is a continuous process (step by
  step) in which every step is random in direction
• The price of a stock changing is a continuous
  time phenomenon because prices are changing
  continuously during the exchange time period and
  the value of price change is irregular
• Take natural logarithm on the stock price

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Case Study 1

• The Unknown Gas Leakage
• September 15, 2008, Shanghai
• People were sick and sent to hospital
• My Plan
• Use Weiner Process to model gas diffusion
• Be able to identify high density area
• Computer simulation and see if the result match
  the real case happened in Shanghai

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Case Study 2

• Brownian Motion in the Stock Market
• Logarithms of common-stock prices
• My Plan
• January internship at a company which deal with
  stock market
• Get familiar with the simulation of stock market
  in the way of Brownian Motion
• Get data and make my own simulation

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Tasks

• Continue study in Brownian Motion, especially 3-D
  Brownian Motion
• Internship at the company deal with stock market
  in January
• Get data, apply Brownian Motion and make
  conclusion for both of the cases
• Writing the final paper

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Thank you!

• QA