random_walks - Winter 2010 Pstat160a Handout Random Walks...

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Winter 2010 Pstat160a Handout Random Walks Random walks I. Definition and basic properties. 1. Definition A Random walk is a sequence X 0 , X 1 , . . . , X n , . . . of random variables (a stochastic process , think of n as time) with the property that at each step n , X n +1 = X n ± 1. The probability to go up (by 1) is denoted by p , down (by 1) is 1 p = q . For simplicity, we will assume that the initial position X 0 is a fixed number, but it could be random in general. A Matlab simulation of random walks with p = . 4 and p = . 6 is shown in Figure 1 0 5 10 15 20 25 30 35 40 -10 -5 0 5 10 p=.4 p=.6 Figure 1. Two random walks simulations with p = . 4 and p = . 6 This means: P ( X n +1 = j | X n = i ) = p if j = i + 1 1 p if j = i 1 0 otherwise 2. Random walks as a model for stock prices Random walks arise in many area of science and engineering. Random walks are also used to model stock prices, as we will see later in this class. In that context, suppose that at each step the price S n of a stock can increase by a fraction r > 1 of the previous price, with probability p , or can decrease by a same fraction with probability 1 p (i.e can increase or decrease by (1 r ) × 100%). Then S n = S 0 r #up r #down = S 0 r #up# - down = S 0 r X n where X n is a random walk. Since the function x mapsto→ r x is increasing and one to one, anything we want to know about S n can be inferred from X n . A simulation of this model is shown in Figure 2. 3. Random walks as sum of i.i.d. random variables Is is easy to see that a random walk { X n } n 0 can be represented as follows, X n = X 0 + n summationdisplay i =1 e i , n = 1 , . . . where e i , i = 1 , . . . , are independent random variables, such that e i = 1 with probability p 1 with probability q = 1 p
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2 0 50 100 150 200 250 300 0 10 20 30 40 50 60 Figure 2. 2 simulations of a stock price evolution using the model descibed in this section p = . 51 and r = 1 . 1 We can easily get that E ( e i ) = 2 p 1 = μ E ( e 2 i ) = 1 Var( e i ) = 4 p (1 p ) = σ 2 4. Moments From the representation above we get, E ( X n ) = X 0 + n summationdisplay i =1 E ( e i ) = X 0 + n (2 p 1) = X 0 + Var( X n ) = Var( X 0 ) + Var parenleftBig n summationdisplay i =1 e i parenrightBig = n Var( e i ) = 2 Cov( X n , X n + k ) = Cov ( n summationdisplay i =1 e i , n + k summationdisplay i =1 e i ) = Var ( n summationdisplay i =1 e i ) + Cov ( n summationdisplay i =1 e i , n + k summationdisplay i = n +1 e i ) = σ 2 n (indep. of k 0) or Cov( X n , X m ) = σ 2 min( n, m ) Remark : In the case of a random initial position X 0 , we would need to assume that the e i are independent of X 0 , and we would get instead E ( X n ) = E ( X 0 ) + and Var( X n ) = Var( X 0 ) + . II. Path counting and Applications A path is the set of points that describe the trajectory, or successive values of the random walk. The purpose of this section is to compute the probability of some complicated events for the random walk, which may involve the whole path, and not just its value at a single time n .
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