Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Stochastic Processes and
Advanced Mathematical Finance
Path Properties of Brownian Motion
Mathematically Mature: may contain mathematics beyond calculus with proofs.
Provide an example of a continuous function which is not diﬀerentiable at some point. Why does the function fail to have a derivative at that point? What are the possible reasons that a derivative could fail to exist at some point?
Probability theory uses the term almost surely to indicate an event which occurs with probability . The complementary events occurring with probability are sometimes called negligible events. In inﬁnite sample spaces, it is possible to have meaningful events with probability zero. So to say an event occurs “almost surely” or is a negligible event is not an empty phrase.
To state this as a theorem may seem strange in view of property 4 of the deﬁnition of Brownian motion. Property 4 requires that Brownian motion is continuous. However, some authors weaken property 4 in the deﬁnition to only require that Brownian motion be continuous at . Then this theorem shows that the weaker deﬁnition implies the stronger deﬁnition used in this text. This theorem is diﬃcult to prove, and well beyond the scope of this course. In fact, even the statement above is imprecise. Speciﬁcally, there is an explicit representation of the deﬁning properties of Brownian Motion as a random variable which is a continuous function of with probability . We need the continuity for much of what we do later, and so this theorem is stated here as a fact without proof.
This property is even deeper and requires more understanding of analysis to prove than does the continuity theorem, so we will not prove it here. Rather, we use this fact as another piece of evidence of the strangeness of Brownian Motion.
In spite of one’s intuition from calculus, Theorem 2 shows that continuous, nowhere diﬀerentiable functions are actually common. Indeed, continuous, nowhere diﬀerentiable functions are useful for stochastic processes. One can imagine non-diﬀerentiability by considering the function which is continuous but not diﬀerentiable at . Because of the corner at , the left and right limits of the diﬀerence quotient exist but are not equal. Even more to the point, the function is continuous but not diﬀerentiable at because of a sharp “cusp” there. The left and right limits of the diﬀerence quotient do not exist (more precisely, each approaches ) at . One can imagine Brownian Motion as being spiky with tiny cusps and corners at every point. This becomes somewhat easier to imagine by thinking of the limiting approximation of Brownian Motion by scaled random walks. The re-scaled coin-ﬂipping fortune graphs look spiky with many corners. The approximating graphs suggest why the theorem is true, although this is not suﬃcient for the proof.
From Theorem 4 and the continuity we can deduce that for arbitrarily large , there is a such that . That is, Brownian Motion paths always cross the time-axis at some time greater than any arbitrarily large value of . Equivalently, Brownian Motion never eventually stays in the upper half-plane (or lower half-plane).
From Theorem 4 and the inversion also being a standard Brownian motion, we heuristically deduce that is an accumulation point of the zeros of . That is, Standard Brownian Motion crosses the time axis arbitrarily often near .
is an uncountable closed set with no isolated points.
Roughly, this means that the graph of a Brownian Motion path is “fuzzier” or “thicker” than the graph of, for example, a continuously diﬀerentiable function which would have Hausdorﬀ dimension . In popular language, this theorem says that Brownian Motion is a fractal.
This section is adapted from Notes on Brownian Motion by Yuval Peres, University of California Berkeley, Department of Statistics.
For a given value of and number of steps on a time interval create a scaled random walk on . Then for a given minimum increment up to a maximum increment create a sequence of equally spaced increments. Then at a ﬁxed base-point, calculate the diﬀerence quotient for each of the increments. When plotting is available, plot the diﬀerence quotients versus the increments on a semi-logarithmic set of axes.
Because the diﬀerence quotients are computed using the scaled random walk approximation of the Wiener process, the largest possible slope is
So the plotted diﬀerence quotients will “max out” once the increment is less than the scaled random walk step size.
The Geoegebra simulation ﬁnds the closest scaled random walk node less than the basepoint . Then using the slider to set the value of , the Geogebra simulation ﬁnds the closest scaled random node greater than . The Geogebra simulation draws the secant line through the two nodes to show that the diﬀerence quotients do not appear to converge as decreases. The ultimate secant line for the smallest value of is between two adjacent nodes and has a slope of .
R script for..
has no isolated points.
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Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1
Email to Steve Dunbar, sdunbar1 at unl dot edu
Last modiﬁed: Processed from LATEX source on August 1, 2016