{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lec0419 - IEOR 4106 Spring 2011 Professor Whitt Topics for...

This preview shows pages 1–2. Sign up to view the full content.

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.)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}