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Unformatted text preview: Chapter 17 Diffusions and the Wiener Process Section 17.1 introduces the ideas which will occupy us for the next few lectures, the continuous Markov processes known as diffu sions, and their description in terms of stochastic calculus. Section 17.2 collects some useful properties of the most important diffusion, the Wiener process. Section 17.3 shows, first heuristically and then more rigorously, that almost all sample paths of the Wiener process dont have deriva tives. 17.1 Diffusions and Stochastic Calculus So far, we have looked at Markov processes in general, and then paid particular attention to Feller processes, because the Feller properties are very natural con tinuity assumptions to make about stochastic models and have very important consequences, especially the strong Markov property and cadlag sample paths. The natural next step is to go to Markov processes with continuous sample paths. The most important case, overwhelmingly dominating the literature, is that of diffusions . Definition 177 (Diffusion) A stochastic process X adapted to a filtration F is a diffusion when it is a strong Markov process with respect to F , homogeneous in time, and has continuous sample paths. 1 Diffusions matter to us for several reasons. First, they are very natural models of many important systems the motion of physical particles (the 1 Having said that, I should confess that some authors dont insist that diffusions be ho mogeneous, and some even dont insist that they be strong Markov processes. But this is the general sense in which the term is used. 92 CHAPTER 17. DIFFUSIONS AND THE WIENER PROCESS 93 source of the term diffusion), fluid flows, noise in communication systems, financial time series, etc. Probabilistic and statistical studies of timeseries data thus need to understand diffusions. Second, many discrete Markov models have largescale limits which are diffusion processes: these are important in physics and chemistry, population genetics, queueing and network theory, certain as pects of learning theory 2 , etc. These limits are often more tractable than more exact finitesize models. (We saw a hint of this in Section 15.3.) Third, many statisticalinferential problems can be described in terms of diffusions, most prominently ones which concern goodness of fit, the convergence of empirical distributions to true probabilities, and nonparametric estimation problems of many kinds. The easiest way to get at diffusions is to through the theory of stochas tic differential equations; the most important diffusions can be thought of as, roughly speaking, the result of adding a noise term to the righthand side of a differential equation. A more exact statement is that, just as an autonomous ordinary differential equation dx dt = f ( x ) , x ( t ) = x (17.1) has the solution x ( t ) = t t f ( x ) ds + x (17.2) a stochastic differential equation...
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This note was uploaded on 12/20/2011 for the course STAT 36754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
 Spring '06
 Schalizi

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