lec0419 - IEOR 4106, Spring 2011, Professor Whitt Topics...

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IEOR 4106, Spring 2011, Professor Whitt Topics for Discussion: Tuesday, April 19 Introduction to Brownian Motion, Chapter 10, Sections 10.1-10.3 We follow the Ross text in Sections 10.1-10.3 closely in this opening lecture on Brownian motion 1 Section 10.1 Basic Properties 1. A Random Walk with Frequent Small Steps Brownian motion can be thought of as a random walk with frequent small steps. As usual with random walks, the steps are random, being independent and identically distributed (i.i.d.). The random walk can be a simple random walk that goes either up or down a constant amount, each with probability 1 / 2. However, let the time between steps by Δ t and let the size of the step be Δ x . Then, at each step, the random walk goes up Δ x or down Δ x , each with probability 1 / 2. But the time taken for each step is Δ t . The idea is to let Δ t and Δ x both grow small. If we examine what happens over an interval [0 ,t ], then we will see that the way that Δ t decreases toward 0 should be related to the way that Δ x decreases toward 0 in order for something interesting to happen. We look at the random walk at time t . By time t , the random walk will have taken b t/ Δ t c steps, where b x c is the greatest integer less than x , i.e., the integer part of x . However, each step is only ± Δ x . We can express this special random walk at time t directly in terms of a simple random walk with steps ± 1 as follows. Let the process we are constructing at time t be X ( t ) = Δ x ( Y 1 + Y 2 + ··· + Y b t/ Δ t c ) = Δ xS b t/ Δ t c , (1) where { Y k : k 1 } is the sequence of steps in a simple random walk, i.e., a sequence of i.i.d. random variables with P ( Y k = 1) = P ( Y k = - 1) = 1 / 2 for all k 1 , (2) and { S k : k 1 } is the random walk itself, i.e., the associated sequence of partial sums defined by S k Y 1 + Y 2 + ··· + Y k , k 1 , (3) where we understand S 0 0. (See (10.1) in the book.) It suffices to consider the mean and variance of the random variable X ( t ) defined in (1). First, the mean is
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This note was uploaded on 01/16/2012 for the course IEOR 4106 taught by Professor Whitward during the Spring '11 term at Columbia College.

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lec0419 - IEOR 4106, Spring 2011, Professor Whitt Topics...

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