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

Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2010

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The Definition of Brownian Motion and the Wiener Process

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Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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QuestionofDay

Question of the Day

Some mathematical objects are defined by a formula or an expression. Some other mathematical objects are defined by their properties, not explicitly by an expression. That is, the objects are defined by how they act, not by what they are. Can you name a mathematical object defined by its properties?

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Key Concepts

We define Brownian motion in terms of the normal distribution of the increments, the independence of the increments, and the value at 0.
The joint density function for the value of Brownian motion at several times is a multivariate normal distribution.

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Vocabulary

Brownian Motion is the physical phenomenon named after the English botanist Robert Brown who discovered it in 1827. Brownian motion is the zig-zagging motion exhibited by a small particle, such as a grain of pollen, immersed in a liquid or a gas. The first explanation of this phenomenon was given by Albert Einstein in 1905. He showed that Brownian motion could be explained by assuming the immersed particle was constantly buffeted by the actions of the molecules of the surrounding medium. Since then the abstracted process has been used beneficially in such areas as analyzing price levels in the stock market and in quantum mechanics.
The Wiener process is the mathematical definition and abstraction of the physical process as a stochastic process. The American mathematician Norbert Wiener gave the definition and properties in a series of papers starting in 1918. Generally, the terms Brownian motion and Wiener process are the same, although Brownian motion emphasizes the physical aspects and Wiener process emphasizes the mathematical aspects.
Bachelier process is an uncommonly applied term meaning the same thing as Brownian motion and Wiener process. In 1900, Louis Bachelier introduced the limit of random walk as a model for the prices on the Paris stock exchange, and so is the originator of the idea of what is now called Brownian motion. This term is occasionally found in financial literature and European usage.

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Mathematical Ideas

Definition of Wiener Process

Previously, we have considered a discrete time random process, that is, at times $n = 1, 2, 3, \dots$ corresponding to coin flips, we have a sequence of random variables $T_{n}$ . We are now going to consider a continuous time random process, that is a function $W_{t}$ which is a random variable at each time $t \geq 0$ . To say $W_{t}$ is a random variable at each time is too general so we must put some additional restrictions on our process to have something interesting to study.

Definition (Wiener Process). The Standard Wiener Process is a stochastic process $W (t)$ , for $t \geq 0$ , with the following properties:

Every increment $W (t) - W (s)$ over an interval of length $t - s$ is normally distributed with mean $0$ and variance $t - s$ , that is $W (t) - W (s) \sim N (0, t - s)$
For every pair of disjoint time intervals $[t_{1}, t_{2}]$ and $[t_{3}, t_{4}]$ , with $t_{1} < t_{2} \leq t_{3} < t_{4}$ , the increments $W (t_{4}) - W (t_{3})$ and $W (t_{2}) - W (t_{1})$ are independent random variables with distributions given as in part 1, and similarly for $n$ disjoint time intervals where $n$ is an arbitrary positive integer.
$W (0) = 0$
$W (t)$ is continuous for all $t$ .

Note that property 2 says that if we know $W (s) = x_{0}$ , then the independence (and $W (0) = 0$ ) tells us that no further knowledge of the values of $W (τ)$ for $τ < s$ has any additional effect on our knowledge of the probability law governing $W (t) - W (s)$ with $t > s$ . More formally, this says that if $0 \leq t_{0} < t_{1} < \dots < t_{n} < t$ , then

ℙ [W (t) \geq x | W (t_{0}) = x_{0}, W (t_{1}) = x_{1}, \dots W (t_{n}) = x_{n}] = ℙ [W (t) \geq x | W (t_{n}) = x_{n}]

This is a statement of the Markov property of the Wiener process.

Recall that the sum of independent random variables which are respectively normally distributed with mean $μ_{1}$ and $μ_{2}$ and variances $σ_{1}^{2}$ and $σ_{2}^{2}$ is a normally distributed random variable with mean $μ_{1} + μ_{2}$ and variance $σ_{1}^{2} + σ_{2}^{2}$ , see Moment Generating Functions. Therefore for increments $W (t_{3}) - W (t_{2})$ and $W (t_{2}) - W (t_{1})$ the sum $W (t_{3}) - W (t_{2}) + W (t_{2}) - W (t_{1}) = W (t_{3}) - W (t_{1})$ is normally distributed with mean $0$ and variance $t_{3} - t_{1}$ as we expect. Property 2 of the definition is consistent with properties of normal random variables.

Let

p (x, t) = \frac{1}{\sqrt{2 π t}} exp (- x^{2} ∕ (2 t))

denote the probability density for a $N (0, t)$ random variable. Then to derive the joint density of the event

W (t_{1}) = x_{1}, W (t_{2}) = x_{2}, \dots W (t_{n}) = x_{n}

with $t_{1} < t_{2} < \dots < t_{n}$ , it is equivalent to know the joint probability density of the equivalent event

W (t_{1}) - W (0) = x_{1}, W (t_{2}) - W (t_{1}) = x_{2} - x_{1}, \dots, W (t_{n}) - W (t_{n - 1}) = x_{n} - x_{n - 1} .

Then by part 2, we immediately get the expression for the joint probability density function:

f (x_{1}, t_{1}; x_{2}, t_{2}; \dots; x_{n}, t_{n}) = p (x_{1}, t) p (x_{2} - x_{1}, t_{2} - t_{1}) \dots p (x_{n} - x_{n - 1}, t_{n} - t_{n - 1})

Comments on Modeling Security Prices with the Wiener Process

A plot of security prices over time and a plot of one-dimensional Brownian motion versus time has least a superficial resemblance.

Figure 1:

Graph of the Dow-Jones Industrial Average from August, 2008 to August 2009 (blue line) and a random walk with normal increments with the same mean and variance (brown line).

If we were to use Brownian motion to model security prices (ignoring for the moment that some security prices are better modeled with the more sophisticated geometric Brownian motion rather than simple Brownian motion) we would need to verify that security prices have the 4 definitional properties of Brownian motion.

Figure 2:

A standardized density histogram of daily close-to-close returns on the Pepsi Bottling Group, symbol NYSE:PBG, from September 16, 2003 to September 15, 2003, up to September 13, 2006 to September 12, 2006

The assumption of normal distribution of stock price changes seems to be a reasonable first assumption. Figure 2 illustrates this reasonable agreement. The Central Limit Theorem provides a reason to believe the agreement, assuming the requirements of the Central Limit Theorem are met, including independence. Unfortunately, although the figure shows what appears to be reasonable agreement a more rigorous statistical analysis shows that the data distribution does not match normality.
Another good reason for still using the assumption of normality for the increments is that the normal distribution is easy to work with. The probability density is easy to work with, the cumulative distribution is tabulated, the moment-generating function is easy to use, and the sum of independent normal distributions is again normal. A substitution of another distribution is possible but makes the resulting stochastic process models very difficult to work with, and beyond the scope of this treatment.
Moreover, this assumption ignores the small possibility that negative stock prices could result from a large negative change. This is not reasonable (and the log normal distribution from geometric Brownian motion which avoids this possibility is a better model).
Moreover, the assumption of a constant variance on different intervals of the same length is not a good assumption since stock volatility itself seems to be volatile. That is, the variance of a stock price changes and need not be proportional to the length of the time interval.
The assumption of independent increments seems to be a reasonable assumption, at least on a long enough term. From second to second, price increments are probably correlated. From day to day, price increments are probably independent. Of course, the assumption of independent increments in stock prices is the essence of what economists call the Efficient Market Hypothesis, or the Random Walk Hypothesis, which we take as a given in order to apply elementary probability theory.
The assumption of $W (0) = 0$ is simply a normalizing assumption and needs no discussion.
The assumption of continuity is a mathematical abstraction, but it makes sense, particularly if securities are traded minute by minute, or hour-by hour where prices could jump discretely, but then examined on a scale of day by day or week by week so the short-time changes are tiny and in comparison prices appear to change continuously.

At least as a first assumption, we will try to use Brownian motion as a model of stock price movements. Remember the mathematical modeling proverb quoted earlier: All mathematical models are wrong, some mathematical models are useful. The Brownian motion model of stock prices is at least moderately useful.

Conditional Probabilities

According to the defining property 1 of Brownian motion, we know that if $s < t$ , then the conditional density of $X (t)$ given $X (s) = B$ is that of a normal random variable with mean B and variance $t - s$ . That is,

ℙ [X (t) \in (x, x + Δ x) | X (s) = B] \approx \frac{1}{\sqrt{2 π (t - s)}} exp (- {(x - B)}^{2} ∕ 2 (t - s)) Δ x

This gives the probability of Brownian motion being in the neighborhood of $x$ at time $t$ , $t - s$ time units into the future, given that Brownian motion is at $B$ at time $s$ , the present.

However the conditional density of $X (s)$ given that $X (t) = B$ , $s < t$ is also of interest. Notice that this is a much different question, since s is “in the middle” between $0$ where $X (0) = 0$ and $t$ where $X (t) = B$ . That is, we seek the probability of being in the neighborhood of $x$ at time $s$ , $t - s$ time units in the past from the present value $X (t) = B$ .

Theorem 1. The conditional distribution of $X (s)$ , given $X (t) = B$ , $s < t$ , is normal with mean $B s ∕ t$ and variance $(s ∕ t) (t - s)$ .

ℙ [X (s) \in (x, x + Δ x) | X (t) = B] \approx \frac{1}{\sqrt{2 π (s ∕ t) (t - s)}} exp (- {(x - B s ∕ t)}^{2} ∕ 2 (t - s)) Δ x

Proof. The conditional density is

\begin{array}{l} f_{s | t} (x | B) & = & (f_{s} (x) f_{t - s} (B - x)) ∕ f_{t} (B) \\ = K_{1} exp (- x^{2} ∕ (2 s) - {(B - x)}^{2} ∕ (2 (t - s))) \\ = K_{2} exp (- x^{2} (1 ∕ (2 s) + 1 ∕ (2 (t - s))) + B x ∕ (t - s)) \\ = K_{2} exp (- t ∕ (2 s (t - s)) (x^{2} - 2 s B x ∕ t)) \\ = K_{3} exp (- (t {(x - B s ∕ t)}^{2} ∕ (2 s (t - s)))) \end{array}

where $K_{1}$ , $K_{2}$ , and $K_{3}$ are constants that do not depend on $x$ . For example, $K_{1}$ is the product of $1 ∕ \sqrt{2 π s}$ from the $f_{s} (x)$ term, and $1 ∕ \sqrt{2 π (t - s)}$ from the $f_{t - s} (B - x)$ term, times the $1 ∕ f_{t} (B)$ term in the denominator. The $K_{2}$ term multiplies in an $exp (- B^{2} ∕ (2 (t - s)))$ term. The $K_{3}$ term comes from the adjustment in the exponential to account for completing the square. We know that the result is a conditional density, so the $K_{3}$ factor must be the correct normalizing factor, and we recognize from the form that the result is a normal distribution with mean $B s ∕ t$ and variance $(s ∕ t) (t - s)$ . □

Corollary 1. The conditional density of $X (t)$ for $t_{1} < t < t_{2}$ given $X (t_{1}) = A$ and $X (t_{2}) = B$ is a normal density with mean

A + ((B - A) ∕ (t_{2} - t_{1})) (t - t_{1})

and variance

(t_{2} - t) (t - t_{1}) ∕ (t_{2} - t_{1})

Proof. $X (t)$ subject to the conditions $X (t_{1}) = A$ and $X (t_{2}) = B$ has the same density as the random variable $A + X (t - t_{1})$ , under the condition $X (t_{2} - t_{1}) = B - A$ by condition 2 of the definition of Brownian motion. Then apply the theorem with $s = t - t_{1}$ and $t = t_{2} - t_{1}$ . □

Sources

The material in this section is drawn from A First Course in Stochastic Processes by S. Karlin, and H. Taylor, Academic Press, 1975, pages 343–345 and Introduction to Probability Models by S. Ross.

Problems to Work for Understanding

Let W(t) be standard Brownian motion.
1. Find the probability that $0 < W (1) < 1$ .
2. Find the probability that $0 < W (1) < 1$ and $1 < W (2) - W (1) < 3$ .
3. Find the probability that $0 < W (1) < 1$ and $1 < W (2) - W (1) < 3$ and $0 < W (3) - W (2) < 1 ∕ 2$ .
Let W(t) be standard Brownian motion.
1. Find the probability that $0 < W (1) < 1$ .
2. Find the probability that $0 < W (1) < 1$ and $1 < W (2) < 3$ .
3. Find the probability that $0 < W (1) < 1$ and $1 < W (2) < 3$ and $0 < W (3) < 1 ∕ 2$ .
4. Explain why this problem is different from the previous problem, and also explain how to numerically evaluate to the proabilities.
Let W(t) be standard Brownian motion.
1. Find the probability that $W (5) \leq 3$ given that $W (1) = 1$ .
2. Find the number $c$ such that $Pr [W (9) > c | W (1) = 1] = 0.10$ .
Suppose that the fluctuations of a share of stock of a certain company are well described by a Standard Brownian Motion process. Suppose that the company is bankrupt if ever the share price drops to zero. If the starting share price is $A (0) = 5$ , what is the probability that the share price is above $10$ at $t = 25$ ?. What is the probability that the company is bankrupt by $t = 25$ ? Explain why these are not the same.
Suppose you own one share of stock whose price changes according to a Standard Brownian Motion Process. Suppose you purchased the stock at a price $b + c$ , $c > 0$ and the present price is $b$ . You have decided to sell the stock either when it reaches the price $b + c$ or when an additional time $t$ goes by, whichever comes first. What is the probability that you do not recover your purchase price?
Let $Z$ be a normally distributed random variable, with mean $0$ and variance $1$ , $Z \sim N (0, 1)$ . Then consider the continuous time stochastic process $X (t) = \sqrt{t} Z$ . Show that the distribution of $X (t)$ is normal with mean $0$ with variance $t$ . Is $X (t)$ a Brownian motion?
Let $W_{1} (t)$ be a Brownian motion and $W_{2} (t)$ be another independent Brownian motion, and $ρ$ is a constant between $- 1$ and $1$ . Then consider the process $X (t) = ρ W_{1} (t) + \sqrt{1 - ρ^{2}} W_{2} (t)$ . Is this $X (t)$ a Brownian motion?
What is the distribution of $W (s) + W (t)$ , for $0 \leq s \leq t$ ? (Hint: Note that $W (s)$ and $W (t)$ are not independent. But you can write $W (s) + W (t)$ as a sum of independent variables. Done properly, this problem requires almost no calculation.)
For two random variables $X$ and $Y$ , statisticians call $Cov (X, Y) = E [(X - E [X]) (Y - E [Y])]$
the covariance of $X$ and $Y$ . If $X$ and $Y$ are independent, then $Cov (X, Y) = 0$ . A positive value of $Cov (X, Y)$ indicates that $Y$ tends to increases as $X$ does, while a negative value indicates that $Y$ tends to decrease when $X$ increases. Thus, $Cov (X, Y)$ is an indication of the mutual dependence of $X$ and $Y$ . Show that
$Cov (W (s), W (t)) = E [W (s) W (t)] = min (t, s)$
Show that the probability density function $p (t; x, y) = \frac{1}{\sqrt{2 π t}} exp (- {(x - y)}^{2} ∕ (2 t))$
satisfies the partial differential equation for heat flow (the heat equation)
$\frac{\partial p}{\partial t} = \frac{1}{2} \frac{\partial^{2} p}{\partial x^{2}}$

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Books

Reading Suggestion:

References

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

[2] Sheldon M. Ross. Introduction to Probability Models. Elsevier, 8th edition edition, 2003.

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Links

Outside Readings and Links:

Copyright 1967 by Princeton University Press, Edward Nelson.. On line book Dynamical Theories of Brownian Motion. It has a great historical review about Brownian Motion.
National Taiwan Normal University, Department of Physics. A simulation of Brownian Motion which also allows you to change certain parameters.
Department of Physics, University of Virginia, Drew Dolgert . Applet is a simple demonstration of Einstein’s explanation for Brownian Motion.
Department of Mathematics,University of Utah, Jim Carlson . A Java applet demonstrates Brownian Paths noticed by Robert Brown.
Department of Mathematics,University of Utah, Jim Carlson. Some applets demonstrate Brownian motion, including Brownian paths and Brownian clouds.
School of Statistics,University of Minnesota, Twin Cities,Charlie Geyer. An applet that draws one-dimensional Brownian motion.

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