Numerical Simulations of Stochastic Differential Equations

1
Numerical Simulations of Stochastic Differential
Equations

• Presented by
• Mikal Grant

2
(No Transcript)
3
Overview

• Probability
• Stochastic
• Brownian Motion
• Applications
• Wisdom for the old (wise) people

4
Probability

• Probability is the branch of mathematics that
  studies the possible outcomes of given events
  together with the outcomes' relative likelihoods
  and distributions.

5
(No Transcript)
6
(No Transcript)
7
(No Transcript)
8
(No Transcript)
9
Stochastic
Stochastic is synonymous with "random." The word
is of Greek origin and means "pertaining to
chance" (Parzen 1962, p. 7). It is used to
indicate that a particular subject is seen from
point of view of randomness. Stochastic is often
used as counterpart of the word "deterministic,"
which means that random phenomena are not
involved. Therefore, stochastic models are based
on random trials, while deterministic models
always produce the same output for a given
starting condition.

10
?
11
(No Transcript)
12
The Path
13
The randomness
http//www.stat.umn.edu/charlie/Stoch/brown.html
14
(No Transcript)
15
(No Transcript)
16
Random Walk
Essentially a Brownian motion where the previous
change in the value of a variable is unrelated to
future or past changes.
http//www.ms.uky.edu/mai/java/stat/brmo.html
17
Whats it all good for?

• Medical imaging
• Robotics
• Estimation of extreme floods and droughts
• Market analysis
• Decision making
• Aerosol particles

18
Medical Imaging

• Medical images have a degree of randomness
  (noise on the image) fractional Brownian motion
  models regard natural occurring surfaces as the
  result of random walkshigh degree of pattern
  complexity involved. Only regular lines are
  recognized in human vision as object edges.

19
Robotics

• When a robot moves in a natural environment,
  it is essential to use a terrain modeling
  technology based on observational depth data
  obtained for a range finder. Brownian theory
  makes it possible to affectively move on a rocky
  and sandy terrain by predicting the random
  motions which are Brownian motions implemented in
  discretized random walks.

20
Market Analysis

• Workable hypothesis that takes into account
  uncertainty and randomness thus enabling us to
  make the right investment decisions, or to choose
  the right business strategy.

21
Decision Making
?

• Optimal switching times under economic
  uncertainty and assuming the economic system is a
  stochastic process.

22
Motion of Aerosol Particles

• Understanding how aerosol particles move will
  allow the prediction of their behavior thus
  enabling it be controlled in turn helping control
  their deposition efficiency in the nose and mouth.

Example
23
Combustion

• Predicting the motion of particles during the
  combustion process will allow us to alter the
  composition of the fuel and air to create a
  laminar flame. Applications of this knowledge
  are enormous.

24
Review
Probability Stochastic Brownian
Motion Applications
25
Wisdom for the Wise
The first sign of senility is that a man forgets
his own theorems, the second is that he forgets
to zip up, the third is that he forgets to zip
down.
Paul Erdos
26
???Questions???