Steven R. Dunbar
Department of Mathematics
203 Avery Hall
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and

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Path Properties of Brownian Motion

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_______________________________________________________________________________________________ ### Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

_______________________________________________________________________________________________ ### Section Starter Question

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?

_______________________________________________________________________________________________ ### Key Concepts

1. With probability $1$ a Brownian Motion path is continuous but nowhere diﬀerentiable.
2. Although a Brownian Motion path is continuous, it has many counter-intuitive properties not usually associated with continuous functions.

__________________________________________________________________________ ### Vocabulary

1. Probability theory uses the term almost surely to indicate an event which occurs with probability $1$. The complementary events occurring with probability $0$ 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 an negligible event is not an empty phrase.

__________________________________________________________________________ ### Non-diﬀerentiability of Brownian Motion paths

Probability theory uses the term almost surely to indicate an event which occurs with probability $1$. The complementary events occurring with probability $0$ 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.

Theorem 1. With probability $1$ (i.e. almost surely) Brownian Motion paths are continuous functions.

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 $t=0$. 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 $W\left(t,\omega \right)$ which is a continuous function of $t$ with probability $1$. We need the continuity for much of what we do later, and so this theorem is stated here as a fact without proof.

Theorem 2. With probability $1$ (i.e. almost surely) a Brownian Motion is nowhere (except possibly on set of Lebesgue measure $0$) diﬀerentiable.

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 $f\left(t\right)=|t|$ which is continuous but not diﬀerentiable at $t=0$. Because of the corner at $t=0$, the left and right limits of the diﬀerence quotient exist but are not equal. Even more to the point, the function ${t}^{2∕3}$ is continuous but not diﬀerentiable at $t=0$ because of a sharp “cusp” there. The left and right limits of the diﬀerence quotient do not exist (more precisely, each approaches $±\infty$) at $x=0$. 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.

#### Properties of the Path of Brownian Motion

Theorem 3. With probability $1$ (i.e. almost surely) a Brownian Motion path has no intervals of monotonicity. That is, there is no interval $\left[a,b\right]$ with $W\left({t}_{2}\right)-W\left({t}_{1}\right)>0$ (or $W\left({t}_{2}\right)-W\left({t}_{1}\right)<0$) for all ${t}_{2},{t}_{1}\in \left[a,b\right]$ with ${t}_{2}>{t}_{1}$

Theorem 4. With probability 1 (i.e. almost surely) Brownian Motion $W\left(t\right)$ has

$\begin{array}{llll}\hfill \underset{n\to \infty }{limsup}\frac{W\left(n\right)}{\sqrt{n}}& =+\infty ,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \underset{n\to \infty }{liminf}\frac{W\left(n\right)}{\sqrt{n}}& =-\infty .\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill \end{array}$

From Theorem 4 and the continuity we can deduce that for arbitrarily large ${t}_{1}$, there is a ${t}_{2}>{t}_{1}$ such that $W\left({t}_{2}\right)=0$. That is, Brownian Motion paths always cross the time-axis at some time greater than any arbitrarily large value of $t$. Equivalently, Brownian Motion never eventually stays in the upper half-plane (or lower half-plane).

Theorem 5. With probability 1 (i.e. almost surely), $0$ is an accumulation point of the zeros of $W\left(t\right)$.

From Theorem 4 and the inversion $tW\left(1∕t\right)$ also being a standard Brownian motion, we heuristically deduce that $0$ is an accumulation point of the zeros of $W\left(t\right)$. That is, Standard Brownian Motion crosses the time axis arbitrarily often near $0$.

Theorem 6. With probability 1 (i.e. almost surely) the zero set of Brownian Motion

$\left\{t\in \left[0,\infty \right):W\left(t\right)=0\right\}$

is an uncountable closed set with no isolated points.

Theorem 7. With probability $1$ (i.e. almost surely) the graph of a Brownian Motion path has Hausdorﬀ dimension $3∕2$.

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 $1$. In popular language, this theorem says that Brownian Motion is a fractal.

#### Sources

This section is adapted from Notes on Brownian Motion by Yuval Peres, University of California Berkeley, Department of Statistics.

_______________________________________________________________________________________________ ### Algorithms, Scripts, Simulations

#### Algorithm

For a given value of $p$ and number of steps $N$ on a time interval $\left[0,T\right]$ create a scaled random walk ${Ŵ}_{N}\left(t\right)$ on $\left[0,T\right]$. Then for a given minimum increment ${h}_{0}$ up to a maximum increment ${h}_{1}$ 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

$\sqrt{T∕N}∕\left(T∕N\right)=\sqrt{N∕T}.$

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 ${x}_{0}$. Then using the slider to set the value of $h$, the Geogebra simulation ﬁnds the closest scaled random node greater than ${x}_{0}+h$. The Geogebra simulation draws the secant line through the two nodes to show that the diﬀerence quotients do not appear to converge as $h$ decreases. The ultimate secant line for the smallest value of $h$ is between two adjacent nodes and has a slope of $\sqrt{N∕T}$.

Geogebra
R
1p <- 0.5
2N <- 1000
3
4T <- 1
5
6S <- array(0, c(N+1))
7rw <- cumsum( 2 * ( runif(N) <= p)-1 )
8S[2:(N+1)] <- rw
9
10WcaretN <- function(x) {
11    Delta <- T/N
12
13    # add 1 since arrays are 1-based
14    prior = floor(x/Delta) + 1
15    subsequent = ceiling(x/Delta) + 1
16
17    retval <- sqrt(Delta)*(S[prior] + ((x/Delta+1) - prior)*(S[subsequent] - S[prior]))
18}
19
20h0 <- 1e-7
21h1 <- 1e-2
22m = 30
23basepoint = 0.5
24
25h <- seq(h0, h1, length=m)
26x0 <- array(basepoint, c(m))
27
28diffquotients <- abs(WcaretN( x0 + h) - WcaretN(x0) )/h
29
30plot(h, diffquotients, type = "l", log = "y", xlab = expression(h),
31      ylab = expression(abs(W(x0+h) - W(x0))/h))
32max(diffquotients, na.rm=TRUE)
Octave
1p = 0.5;
2
3global N = 1000;
4global T = 1;
5
6global S
7S = zeros(N+1, 1);
8S(2:N+1) = cumsum( 2 * (rand(N,1)<=p) - 1);
9
10function retval = WcaretN(x)
11  global N;
12  global T;
13  global S;
14  step = T/N;
15
16  # add 1 since arrays are 1-based
17  prior = floor(x/step) + 1;
18  subsequent = ceil(x/step) + 1;
19
20  retval = sqrt(step)*(S(prior) + ((x/step+1) - prior).*(S(subsequent)-S(prior)));
21
22endfunction
23
24h0 = 1e-7;
25h1 = 1e-2;
26m = 30;
27basepoint = 0.5;
28
29h = transpose(linspace(h0,h1, m));
30x0 = basepoint * ones(m,1);
31
32diffquotients = abs( WcaretN( x0 + h) - WcaretN(x0) ) ./ h
33
34semilogy( h, diffquotients)
35xlabel("h")
36ylabel("abs(W(x0+h) - W(x0))/h)")
Perl
1$p = 0.5; 2 3$N = 1000;
4$T = 1; 5 6# the random walk 7$S = zeros( $N + 1 ); 8$S ( 1 : $N ) .= cumusumover( 2 * ( random($N) <= $p ) - 1 ); 9 10# function WcaretN interpolating random walk 11sub WcaretN { 12 my$x = shift @_;
13    $Delta =$T / $N; 14 15$prior      = floor( $x /$Delta );
16    $subsequent = ceil($x / $Delta ); 17 18$retval =
19        sqrt($Delta) 20 * ($S ($prior) 21 + ( ($x / $Delta ) -$prior )
22            * ( $S ($subsequent) - $S ($prior) ) );
23}
24
25$h0 = 1e-7; 26$h1        = 1e-2;
27$m = 30; 28$basepoint = 0.5;
29
30$h = zeroes($m)->xlinvals( $h0,$h1 );
31$x0 =$basepoint * ones($m); 32 33$diffquotients = abs( WcaretN( $x0 +$h ) - WcaretN($x0) ) /$h;
34
35# file output to use with external plotting programming
36# such as gnuplot, R, octave, etc.
37# Start gnuplot, then from gnuplot prompt
38#    set logscale y
39#   plot "pathproperties.dat" with lines
40
41open( F, ">pathproperties.dat" ) || die "cannot write: $! "; 42foreach$j ( 0 .. $m - 1 ) { 43 print F$h->range( [$j] ), " ",$diffquotients->range( [\$j] ), "\n";
44}
45close(F);
SciPy
1
2import scipy
3
4p = 0.5
5
6N = 1000
7T = 1.
8
9# the random walk
10S = scipy.zeros(N+1)
11S[1:N+1] = scipy.cumsum( 2*( scipy.random.random(N) <= p ) - 1 )
12
13def WcaretN(x):
14    Delta = T/N
15    prior = scipy.floor(x/Delta).astype(int)
16    subsequent = scipy.ceil(x/Delta).astype(int)
17    return scipy.sqrt(Delta)*(S[prior] + (x/Delta - prior)*(S[subsequent] - S[prior]))
18
19h0 = 1e-7
20h1 = 1e-2
21m = 30
22basepoint = 0.5
23
24h = scipy.linspace( h0, h1, m)
25x0 = basepoint * scipy.ones(30)
26
27diffquotients = scipy.absolute( WcaretN( x0 + h ) - WcaretN( x0 ) )/h
28
29# optional file output to use with external plotting programming
30# such as gnuplot, R, octave, etc.
31# Start gnuplot, then from gnuplot prompt
32#    set logscale y
33#    plot "pathproperties.dat" with lines
34f = open(pathproperties.dat, w)
35for j in range(0, m-1):
36    f.write( str(h[j])+ +str(diffquotients[j])+\n);
37f.close()

_______________________________________________________________________________________________ ### Problems to Work for Understanding

1. In an inﬁnite sequence of fair coin ﬂips, consider the event that there are only ﬁnitely many tails. What is the probability of this event? Is this event empty? Is this event impossible?
2. Provide a more complete heuristic argument based on Theorem 4 that almost surely there is a sequence ${t}_{n}$ with $\underset{t\to \infty }{lim}{t}_{n}=\infty$ such that $W\left(t\right)=0$
3. Provide a heuristic argument based on Theorem 5 and the shifting property that the zero set of Brownian Motion
$\left\{t\in \left[0,\infty \right):W\left(t\right)=0\right\}$

has no isolated points.

4. Looking in more advanced references, ﬁnd another property of Brownian Motion which illustrates strange path properties.

__________________________________________________________________________ ### References

   David Freedman. Brownian Motion and Diﬀusions. Holden-Day, 1971. QA274.75F74.

   I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. Springer Verlag, second edition, 1997.

   S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.

   Steven E. Shreve. Stochastic Calculus For Finance, volume II of Springer Finance. Springer Verlag, 2004.

   Steven E. Shreve. Stochastic Calculus For Finance, volume I of Springer Finance. Springer Verlag, 2004.

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