Brownian Motion

Basic Properties

In 1827 the English botanist Robert Brown observed that microscopic pollen grains suspended in water perform a continual swarming motion.
This phenomenon was first explained by Einstein in 1905 who said the motion comes from the pollen being hit by the molecules in the surrounding water. The mathematical derivation of the Brownian motion process was first done by Wiener in 1918, and in his honor it is often called the Wiener process.

Brownian motion is a continuous-time continuous state-space stochastic process defined as follows:

the process {B(t),t≥0} is a Brownian motion process iff

1. B(0)=0
   
2. {B(t),t≥0} has stationary and independent increments
   
3. for all t>0 B(t)~N(0,σ√t)

Note: in the case of a continuous-time stochastic process we have:

• the process {X(t);t≥0} has stationary increments iff the distribution of X(t+s)-X(t) depends only on s

• the process {X(t);t≥0} has independent increments iff for t₁<t₂<t₃ X(t₃)-X(t₂) is independent of X(t₂)-X(t₁)

Lemma

A Brownian motion process is a Markov Process and a Martingale

proof

by independent and stationary increments.

So Brownian motion is a continuous-time continuous-state space Markov process and a Martingale. In fact BM has just about all the “nice” properties of stochastic processes we encountered earlier.

The Brownian motion process plays a role in the theory of stochastic processes similar to the role of the normal distribution in the theory of random variables.

If σ=1 the process is called standard Brownian motion.

One way to visualize a Brownian motion process is as the limit of symmetric random walks: Let {Z_n,n≥1} be the symmetric random walk on the integers. If we now speed the process up and scale the jumps accordingly we get a Brownian motion process in the limit. More precisely, suppose we jump every δt and make a jump of size δx. if we let Z(t) denote the position of the process at time t then

Here are some sample paths of a standard Brownian motion process.

Covariance Function

Let {B(t);t≥0} be BM, then E[B(t)]=0 and E[B(t)²]=σ²t for all t. Let 0<s<t, then

so in general Cov(B(t),B(s))=σ²min(t,s)

Lemma

say s<t. Then

proof

This generalizes immediately to

and we can see that the joint distribution of a BM is multivariate normal.

From this we can (in principle) compute any probability. For example let’s say we want to find the conditional distribution of B(s)|B(t)=B where s<t:

and so B(s)|B(t)=b ~ N(sb/t,√s(t-s)/t )

Note that the variance of B(s)|B(t)=b does not depend on b!

Definition

A stochastic process {X(t),t≥0} is called a Gaussian process if for any 0<t₁<..<t_n the random vector (X(t₁),..X(t_n)) has a multivariate normal distribution.

Example

as we saw above Brownian motion is a Gaussian process

Example

let {B(t),t≥0} be BM and define

{X(t),0≤t≤1} by X(t)=B(t)|B(1)=0.

This process is called Brownian bridge because it is “tied down” at 0 and 1.

The same argument as above shows that a Brownian bridge is also a Gaussian process. Now

Here is another way to define the Brownian bridge process:

Lemma

let {B(t),t≥0} be BM and define

{X(t),0≤t≤1} by X(t)=B(t)-tB(1)

then {X(t),0≤t≤1} is a Brownian bridge process.

proof
Clearly {X(t),0≤t≤1} is a Gaussian process, so all we need to do is check that it has the right mean and covariance function:

Example: (Empirical Distribution Function)

Say we have observations (X₁,..,X_n). Then in statistics we define the empirical distribution function by

Now N_n(s) = nF_n(s)~Bin(n,F(s))

Now by the strong law of large numbers we have F_n(s)→F(s) and by the Glivenko-Cantelli lemma this convergence is uniform in s, that is

sup|F_n(s)-F(s)|→0

as n→∞

Moreover by the Central Limit Theorem √n[F_n(s)-F(s)] has an asymptotic normal distribution with mean 0 and variance F(s)(1-F(s)).

To continue we will assume that F=U[0,1], so that F(s)=s. This is in fact not much of a restriction because if X is a continuous rv with distribution F then by the probability integral transform Y=F(X)~U[0,1].

Define

α_n(s)=√n[F_n(s)-s]

and let

α(s) = lim_n→∞ α_n(s)

Then {α(s),0<s<1} is a stochastic process. To find out what kind of process it is first note that if s<t

N_n(t)-N_n(s)|N_n(s)=m ~ Bin(n-m,(t-s)/(1-s))

because it is the number of jumps between s and t if we know there were already m jumps at s.

Again using the CLT is seems clear that the joint distribution of α(s) and α(t) is a bivariate normal, and in fact that {α(s),0<s<1} is a Gaussian process.

To continue first note that N_n(s)~Bin(n,s) and therefore

EN_n(s) = ns

VarN_n(s) = ns(1-s)

E[N_n(s)²] = VarN_n(s) + [EN_n(s)]² = ns(1-s)+n²s² = ns(1-s+ns)

Now

Continuity and Differentiability

When studying a continuous-time stochastic process it is often useful to think of any particular realization of the process as a function. Say S is the sample space of the process, that is the set of all possible paths {X(t),t≥0}, and let w\(\in\) S. Then f(t) = X(t,ω) is a function. (Usually we suppress ω, though).

In the case of BM, what are the properties of a typical realization B(t)? First let’s look at continuity:

Now by the definition we have that B(t+h)-B(t) ~ N(0,√h), therefore E[(B(t+h)-B(t))²] = h and so the size of an increment of |B(t+h)-B(t)| is about √h. So as h→0 √h→0 which implies continuity.

How about differentiability? Now we have

and we see that BM is nowhere differentiable!
(Of course this is rather heuristic but it can be made rigorous).

The idea of functions that are continuous but nowhere differentiable has a very interesting history. It was first discussed in 1806 by André Marie Ampère and trying to show that such a function exists was one of the main open problems during the 19^th century. More than fifty years later it was Karl Theodor Wilhelm Weierstrass who finally succeeded in constructing such a function as follows:

The hard part here was not the construction but to show that the function existed! For the proof he developed what is now known as the Stone-Weierstrass theorem.

Shorty after that a new branch of mathematics called functional analysis was developed. It studies the properties of real-valued functions on function space. Here are some examples of such functionals:

Of course one needs to specify the space of functions for which a certain functional applies. Standard “function spaces” are C, the space of all continuous functions and C¹, the space of all continuous functions with a continuous derivative.

One of the results of functional analysis is that C is much larger than C¹, actually of a higher order of infinity, shown with the Baire category theorem. So consider the following “experiment”: pick any continuous function in C. Then the probability that it has a continuous derivative anywhere is 0! So functions such as Weierstrass (or the paths of BM) are not the exception, they are the rule. Or, all the functions we study in mathematics are completely irrelevant in nature.

The Reflection Principle

Let B(t) be standard BM. Let x>0 and t>0. Consider a path of BM with B(0)=0 and B(t)>x. Then because of the continuity of the paths by the intermediate value theorem there has to be a time τ such that B(τ)=x for the first time. Let’s define a new path B*(u) obtained from B(u) by reflection:

Because of the symmetry of BM after hitting x the process is equally likely to follow the path of B and B*, therefore

Let M(t) be the largest value attained by a BM up to time t. Then we have just shown that

P(M(t)>x) = 2[1-Φ(x/√t)]

where Φ is the cdf f a standard normal rv.

First Hitting Time

Let τ_x be the first time the BM path hits some x>0 starting at 0, or

clearly we have

τ_x≤t iff M(t)≥x

so

P(τ_x≤t) = P(M(t)≥x) = 2[1-Φ(x/√t)]

and so

Zeros of Brownian Motion

say B is a standard BM process with B(0)=0. We want to find the probability that B has at least one zero in the interval (t,t+s]. Let’s denote this by ν(t,t+s)

To start define

H_t(z,x) = P(τ_x≤t|B(0)=z)

so H_t(z,x) is the probability that a standard BM starting at B(0)=z will reach the level x before time t. Above we found an integral for H_t(0,x).

Because of the symmetry and the spatial homogeneity of BM we have

H_t(0,x)=H_t(x,0)=H_t(-x,0)

and so

where the last equality follows from standard trigonometry.

Some other Properties of Brownian Motion

1. BM will eventually hit any and every real value, no matter how large or how negative! It may be a million units above the axis, but it will (wp1) be back down again to 0, by some later time.

2. Once BM hits zero (or any particular value), it immediately hits it again infinitely often, and then again from time to time in the future.

3. Spatial Homogeneity: B(t) + x for any x \(\in\) R is a BM started at x.

4. Symmetry: -B(t) is a Brownian motion

5. Scaling: √c•B(t/c) for any c > 0 is a BM

6. Time inversion:
   
   is a BM.

7. BM is time reversible

8. BM is self-similar (that is its paths are fractals). Brownian Motion is an example of a process that has a fractal dimension of 2. So in moving from a given location in space to any other, the path taken by the particle is almost certain to fill the whole space before it reaches the exact point that is the ‘destination’ (hence the fractal dimension of 2).