Exploring Brownian Motion

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Exploring Brownian Motion

• MPS Senior Thesis
• Fei Fei CHU
• Spring 2009
• Advised by Professor C.Shilepsky

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Introduction
Useful Fields
Robert Brown
Brownian Motion
Stochastic Process
Phenomenon
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Historical Background
C O N T I B U T I O N S
Jean Perrin--Nobel Prize in physics in 1926
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Brownian 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
• k is the Boltzman constant
• The mean-square displacement r2 x2 y2 z2 of
  a Brownian particle is described by the equation
  ltr2gt 6kTBt
• -- B is the mobility of the particle, which is
  inversely proportional to the medium viscosity
  and the size of the particle

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1-D Random Walk

• N total steps
• p is the probability of a step to the right at
• each step
• q is the probability of a step to the left at
• each step
• n1 is the number of steps to the right
• n2 is the number of steps to the left
• P(n1) is the probability of taking n1
• steps to the right out of N steps
• -- Distance traveled n1 n2
• Variance Npq which
• corresponds to time (number of steps)
• Graphgt Binomial distribution
• 
  
  http//en.wikipedia.org/wiki/Random_wa
  lk

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Wiener Process

• As time increment goes to 0, step size is very
  small, number of total steps is getting large
  gtcontinuousgtNormal distribution
• Continuous-time stochastic process
• Wiener process Wt
• Wt is the distribution of the
• particle at time t
• W0 0
• Wt is continuous
• Wt Ws has normal distribution
• with expected value 0 and
• variance t-s (0 s lt t )
• http//en.wikipedia.org/wiki/Wiener_pr
  ocess

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Brownian Motion In Financial Markets
Brownian motion is in stock and future
market Price changes in every time period just
like Brownian particle moves step by step Price
goes up or down as the Brownian particle goes to
right or left Price change is a random value
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Case Study
Price(t)drifttvolatilityWt
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Gas Leakage and Ideal Simulation
Last September in Shanghai, there was a gas
leakage which made some residents around gas
leakage factory feel sick If perfume is sprayed
in a room, people can smell it less or more, soon
or later Gas leakage is a typical Brownian
motion I built an ideal simulation in two
dimensions to use for further analysis.
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My Simulation
Brownian particles move in two dimensions In each
equal period of time, each particle takes one
step with fixed step size 1 and a random
direction from 0 degree to 360 degree
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Statistical Results2 D
1000 particles at 300 step µ1 12
1000 particles at 1500 step µ2 28
1000 particles at 3000 step µ3 38
X-axis is the displacement Y-axis is the number
of particles
µ2 / µ1 2.3 v1500/300 2.23 gtcorresponds to
the Brownian motion theory
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Boundary Simulation
Now add one block as the boundary to the same
simulation When the Brownian particle hit the
block, it goes back to last position, that is, it
stays at the same position when hit the block
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Boundary or no boundary, that is a question
300 steps
1500 steps
3000 steps
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Future Study
More Applications in financial market The detail
Difference between 2-D Brownian Motion with
boundary and without boundary Up to 3-D Brownian
motion
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Acknowledgement
Wells College MPS Faculties Special Thanks to
Professor C. Shilepsky Professor S.
Heinekamp Professor S. Sievers Professor X.
Zhu My Families My Friend--C. Chen
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Thank you!

• QA