The Pathological Properties of Brownian Motion

I turn away with fear and horror from the lamentable plague of continuous functions which do not have derivatives – Charles Hermite (1893)

Brownian motion

Despite being of central importance to the theory of stochastic processes and to many applications in areas such as physics and economics, Brownian motion has some nasty properties such as being nowhere differentiable, which are in stark contrast to the usual well-behaved functions studied in elementary differential calculus. As I intend to post entries on stochastic calculus, it seems that a good place to start is by describing some of the properties of Brownian motion which rule out the use of the standard techniques of differential calculus. Strictly speaking, these properties should not really be regarded as pathological although they can seem so to someone not familiar with such processes and would have been regarded as such at the time of Hermite’s statement above.

Historically, the term `Brownian motion’ refers to the experiments performed by Robert Brown in 1827 where pollen and dust particles floating on the surface of water are observed to move about with a jittery motion. This was explained mathematically by Albert Einstein in 1905 and Marian Smoluchowski in 1906, and is caused by the particles being continuously bombarded by water molecules. Louis Bachelier also studied the mathematical properties of Brownian motion in 1900, applying it to the evolution of stock prices.

Mathematically, Brownian motion is a stochastic process whose increments are independent and identically distributed random variables, and which has continuous sample paths. In the case of the random motion of particles due to collisions with water molecules, as in the experiments performed by Robert Brown, each bombardment by a molecule will not produce a sudden change in the position of the particle. Instead, they will produce a sudden change in the particle’s velocity. So mathematical Brownian motion as described here is better used as model of the velocity of the particle rather than its position (even better – the velocity can be modeled by an Ornstein-Uhlenbeck process). More generally, it is used as a source of random noise in many models of physical and economic systems. It is also referred to as a Wiener process after Norbert Wiener and often represented using a capital W.

A stochastic process can be regarded as a collection of random variables, {X_t}, one for each positive time t. This is actually a map from some underlying probability space {\Omega} to the real numbers, {\omega\mapsto X_t(\omega)}, although the dependence on {\omega} is often suppressed in the notation. Alternatively, for each outcome {\omega\in\Omega}, the sample path {t\mapsto X_t(\omega)} is simply a function on the nonnegative real numbers. The precise definition of standard Brownian motion is as follows.

  • {X_0=0}.
  • For times {t>s\ge 0}, the increment {X_{t}-X_{s}} is normally distributed with mean 0 and variance {t-s}, independently of the values of {X_u} for {u\le s}.
  • The sample paths are continuous.

The first two conditions define the finite distributions of the process. That is, they specify the joint distributions of the process at all finite sets of times. At first sight, it might be thought that this is all that is needed. However, according to probability theory (Kolmogorov’s axioms), the continuity of a process cannot be inferred from the finite distributions and it cannot in general even be assigned a probability. By countable additivity of probability measures, we can only infer the joint properties of the random variables {X_t} at countable sets of times and not simultaneously on the uncountable set of nonnegative real numbers. The best that can be done is to say that there is a continuous modification of the process. That is, if {X} satisfies the first two properties above then there is a process {X^\prime} with continuous sample paths and such that at each time, {X^\prime_t=X_t} with probability one. The third property above says that we always use such a continuous modification. Then, the properties of the sample paths, such as differentiability, are measurable events in the probability space with well defined probabilities.

In fact, it can be shown that all continuous processes with independent increments are normally distributed and, therefore, can be expressed as a standard Brownian motion with a rescaling of the time axis plus a deterministic function. Also, as a consequence of the central limit theorem, Brownian motion occurs as an approximation to discrete processes such as random walks in the limit as the step sizes go to zero. So, Brownian motion is a very general process in stochastic process theory.

As Brownian motion is a random process, any property of its sample paths is satisfied with a certain probability. The properties described below are meant in the almost sure sense. That is, with probability one. Depending on how the probability space is constructed there could well be outcomes in which some or all of these properties are not true, but such outcomes will only occur with zero probability.

Nowhere differentiable

Although it is continuous, the sample paths of Brownian motion are nowhere differentiable. Historically, the first published example of a continuous but nowhere differentiable function was the Weierstrass function, in 1872. In fact, at a cursory glance, it does look very similar to Brownian motion. Previously, it was generally assumed that all functions under consideration were differentiable except, possibly, at some small set of nonsingular points. Most mathematicians believed this to be true for all continuous functions (according to here). The Weierstrass function was published as a pathological counterexample, showing that such functions do exist and is classed as a Monster of Real Analysis. In light of statements such as that by Hermite at the top of this post it is interesting that, in the form of Brownian motion, such functions are central to much mathematics developed in the 20th century with many important and practical applications.

Fractal scaling properties

If {X} is a Brownian motion then, for any constant {a>0}, it follows from the definition that the rescaled process {aX_{a^{-2}t}} is also a Brownian motion. This means that if we zoom in on the sample path of {X} then, at every order of magnification, it still looks like a sample path of standard Brownian motion (an applet demonstrating this property is available here). This is in contrast to differentiable functions which become closer to a straight line when zoomed in.

Infinite variation

Brownian motion has infinite variation over all nontrivial intervals {[a,b]}. For finite variation processes, Riemann-Stieltjes integration can be used to make sense of terms like

\displaystyle  \int_0^T f(t)\,dX_t

for arbitrary continuous integrands f, which allows the derivative {dX_t} to be interpreted in a measure theoretic sense. Unfortunately, this does not work for Brownian motion because this integration is only well-defined for finite variation processes. Stochastic calculus as developed by Kiyoshi Itô is required to solve this problem.

In fact, it has been proven by Lebesgue that any function with finite variation is differentiable almost everywhere. The infinite variation property of Brownian motion is thus a consequence of nowhere differentiability.

It intersects any given value infinitely often

For any given real number {a} then, in the neighbourhood of a time {t} at which {X_t=a}, the Brownian motion {X} will be equal to {a} infinitely often. In fact,

\displaystyle  \left\{t\ge 0\colon X_t=a\right\}

is a Cantor set. That is, it is a nonempty closed set with no isolated points and containing no nontrivial intervals. Such sets are always uncountably infinite in the neighbourhood of any element of the set. A standard example is the middle thirds set, and all Cantor sets are locally homeomorphic to this.

The extrema are dense

A sample path of Brownian motion must have a local maximum or minimum in each nontrivial interval {(a,b)} (otherwise it would be monotonic and hence have finite variation). It follows that times at which the process is at a local maximum or minimum is a dense subset of the nonnegative real numbers.

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