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Numerical Simulations of Stochastic Differential Equations Presented by: Mikal Grant Slide 2 Slide 3 Overview Probability Stochastic Brownian Motion Applications Wisdom for the old (wise) people Slide 4 Probability Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. Slide 5 Slide 6 Slide 7 Slide 8 Slide 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. Slide 10 Slide 11 Slide 12 The Path Slide 13 http://www.stat.umn.edu/~charlie/Stoch/brown.html The randomness Slide 14 Slide 15 Slide 16 Random Walk http://www.ms.uky.edu/~mai/java/stat/brmo.html Essentially a Brownian motion where the previous change in the value of a variable is unrelated to future or past changes. Slide 17 Whats it all good for? Medical imaging Robotics Estimation of extreme floods and droughts Market analysis Decision making Aerosol particles Slide 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. Slide 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. Slide 20 Estimation of Extreme Floods and Droughts Brownian walks introduced the idea that floods and droughts could be fractal. Slide 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. 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. Slide 22 Decision Making Optimal switching times under economic uncertainty and assuming the economic system is a stochastic process. Optimal switching times under economic uncertainty and assuming the economic system is a stochastic process. ? Slide 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. 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 Slide 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. 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. Slide 25 Review ProbabilityStochastic Brownian Motion Applications Slide 26 Wisdom for the Wise Slide 27 ???Questions???