lecture-17 - Chapter 17 Diusions and the Wiener Pro cess...

Chapter 17
Di ff usions 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 di ff u-
sions, and their description in terms of stochastic calculus.
Section 17.2 collects some useful properties of the most important
di ff usion, the Wiener process.
Section 17.3 shows, first heuristically and then more rigorously,
that almost all sample paths of the Wiener process don’t have deriva-
tives.
17.1 Di ff usions 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 di ff usions .
Definition 177 (Di ff usion) A stochastic process X adapted to a filtration F
is a di ff usion when it is a strong Markov process with respect to F , homogeneous
in time, and has continuous sample paths.
1
Di ff usions 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 don’t insist that di ff usions be ho-
mogeneous, and some even don’t insist that they be strong Markov processes. But this is the
general sense in which the term is used.
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CHAPTER 17. DIFFUSIONS AND THE WIENER PROCESS 93
source of the term “di ff usion”), fluid flows, noise in communication systems,
financial time series, etc. Probabilistic and statistical studies of time-series data
thus need to understand di ff usions. Second, many discrete Markov models have
large-scale limits which are di ff usion 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 finite-size models. (We saw a hint of this in Section 15.3.) Third, many
statistical-inferential problems can be described in terms of di ff usions, 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 di ff usions is to through the theory of stochas-
tic di ff erential equations; the most important di ff usions can be thought of as,
roughly speaking, the result of adding a noise term to the right-hand side of a
di ff erential equation. A more exact statement is that, just as an autonomous
ordinary di ff erential equation
dx
dt
= f ( x ) , x ( t
0
) = x
0
(17.1)
has the solution