Numerical Simulations of Stochastic Differential Equations

1. Numerical Simulations of Stochastic Differential Equations Presented by: Mikal Grant

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.

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 fro a given starting condition.

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12. The Path…

13. The randomness… http://www.stat.umn.edu/~charlie/Stoch/brown.html

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. What’s 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 walks…high 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 dept 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. Estimation of Extreme Floods and Droughts Brownian walks introduced the idea that floods and droughts could be fractal.

21. 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.

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

23. 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…

24. 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.

25. Review Probability Stochastic Brownian Motion Applications

26. Wisdom for the Wise

27. ???Questions???