Numerical Simulations of Numerical Simulations of Stochastic Differential EquationsStochastic Differential Equations

Presented by:Presented by:

Mikal GrantMikal Grant

OverviewOverview

ProbabilityProbability StochasticStochastic Brownian MotionBrownian Motion ApplicationsApplications Wisdom for the old (“wise”) peopleWisdom for the old (“wise”) people

ProbabilityProbability

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

StochasticStochasticStochastic is synonymous with "random." The word is Stochastic is synonymous with "random." The word is of Greek origin and means "pertaining to chance" of Greek origin and means "pertaining to chance" (Parzen 1962, p. 7). It is used to indicate that a (Parzen 1962, p. 7). It is used to indicate that a particular subject is seen from point of view of particular subject is seen from point of view of randomness. Stochastic is often used as counterpart randomness. Stochastic is often used as counterpart of the word "deterministic," which means that random of the word "deterministic," which means that random phenomena are not involved. Therefore, stochastic phenomena are not involved. Therefore, stochastic models are based on random trials, while models are based on random trials, while deterministic models always produce the same deterministic models always produce the same

output fro a given starting condition.output fro a given starting condition.

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The Path…The Path…

http://www.stat.umn.edu/~charlie/Stoch/brown.html

The randomness…The randomness…

Random WalkRandom Walk

http://www.ms.uky.edu/~mai/java/stat/brmo.html

Essentially a Brownian motion where the Essentially a Brownian motion where the previous change in the value of a variable is previous change in the value of a variable is

unrelated to future or past changes.unrelated to future or past changes.

What’s it all good for?What’s it all good for?

Medical imagingMedical imaging RoboticsRobotics Estimation of extreme floods and droughtsEstimation of extreme floods and droughts Market analysisMarket analysis Decision makingDecision making Aerosol particlesAerosol particles

Medical ImagingMedical Imaging

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

RoboticsRobotics

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

Estimation of Extreme Floods Estimation of Extreme Floods and Droughtsand Droughts

Brownian walks Brownian walks introduced the introduced the idea that floods idea that floods and droughts and droughts could be fractal.could be fractal.

Market AnalysisMarket Analysis

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

Decision MakingDecision Making

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

Motion of Aerosol ParticlesMotion of Aerosol Particles

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

Example…

CombustionCombustion

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

ReviewReview

ProbabilityProbability

StochasticStochastic

Brownian MotionBrownian Motion

ApplicationsApplications

Wisdom for the WiseWisdom for the Wise

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