APPENDIX C - Brownian Motion

Summary

The object of this appendix is to explain the property of Brownian motion mentioned in Section 2.5. We will start with a simple model situation (whose relevance may not be immediately apparent). Two people toss a coin for money. If a head comes up, he pays her one pound. If a tail comes up, she pays him one pound. On average, if the game is played many times, each player's winnings and losses will tend to cancel out.

Now suppose that each day the players play a sequence of a certain fixed number of tosses. Call this number n. At the end of the day, they note how many pounds he has paid her. Call this amount the winnings, W. Note that if he has won, W is a negative number. Now square this: W². (Never mind what a “square pound” means.) Note that the square of a number is always positive, whether the number itself is positive or negative. Now average over lots of days. The average value of W will be zero (if it were not, we would suspect that the coin was biased). But the average value of W² is certainly not zero, because each value of W² is positive (or perhaps zero). In fact, it can be shown that the average value of W² is just the number n (the number of games each day).

I will illustrate this with two examples. First suppose n = 2. The possible outcomes of the games are HH, HT, TH, TT, where H stands for head and T for tail. For each of these outcomes, the values of W (the winnings) are 2, 0, 0, −2.