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Lecture1_Spring2010

Lecture1_Spring2010 - Stochastic Calculus Spring 2010 22...

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Stochastic Calculus, Spring 2010, 22 January, Lecture 1 Construction of the Brownian motion Reading for this lecture (for references see the end of the lecture): [1] pp. 72-108 Scaled Random Walks. Our main object of study in this course will be stochastic processes . Loosely speaking, stochastic process is a collection of random variables indexed by “time” variable t : X ( t ) . Variable t could take values in a discreet set, for instance in the set of positive integers or it could take values in a continuous set, for instance in [0 , ) . An example of stochastic process is a stock price that changes in time. Our main objective for today will be to build a very important stochastic process called Brownian Motion . As you will see, Brownian Motion is a continuous-time stochastic process, it is usually indexed by [0 , ) . To build a Brownian motion we begin with a simpler stochastic process that is called Symmetric Random Walk (SRW). To construct a SRW we repeatedly toss a fair coin: P ( H ) = P ( T ) = 1 / 2 . We define the successive outcomes of the tosses by w 1 , w 2 , w 3 · · · ∈ { H, T } and define w = w 1 w 2 w 3 . . . , in other words, w is the infinite sequence of tosses and w n is the outcome of the n th toss. Next, let X j be a random variable defined as X j = braceleftBigg 1 if w j = H, 1 if w j = T. Define discrete-time stochastic process M ( n ) in the following way: M (0) = 0 , M ( n ) = n summationdisplay j =1 X j . The process M ( n ) is called a symmetric random walk. With each step it either steps up one unit or down one unit, and each of the two possibilities is equally likely. Exercise 1. How many are there sequences of H and T of length n ? For a fixed positive integer n what is the distribution of M ( n ) , in particular for a given integer k what is P ( M ( n ) = k )? Increments of the SRW. A random walk has independent increments. This means that if we choose nonnegative integers 0 = n 0 < n 1 < · · · < n k , then the random variables M ( n 1 ) = M ( n 1 ) M ( n 0 ) , M ( n 2 ) M ( n 1 ) , . . . , M ( n k ) M ( n k - 1 ) are independent. Each of these random variables M ( n i +1 ) M ( n i ) = n i +1 summationdisplay j = n i +1 X j 1 this version January 24, 2010 1
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2 is called an increment of the random walk. It is the change in the position of the random walk between times n i and n i +1 . Increments over non-overlapping time intervals are independent because they depend on different coin tosses. Moreover, E ( M ( n i +1 ) M ( n i )) = n i +1 summationdisplay j = n i +1 E X j = 0 and Var( M ( n i +1 ) M ( n i )) = n i +1 summationdisplay j = n i +1 Var X j = n i +1 n i since Var X j = 1 .
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