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Chapter 13 Brownian Motion 13.1 Symmetric Random Walk Toss a fair coin infinitely many times. Define X j != 1 if ! j = H; ,1 if ! j = T: Set M 0 =0 M k = k X j=1 X j ; k 1: 13.2 The Law of Large Numbers We will use the method of moment generating functions to derive the Law of Large Numbers: Theorem 2.38 (Law of Large Numbers:) 1 k M k !0 almost surely, as k!1: 139 140 Proof: ' k u=IEexp u k M k = IE exp 8 : k X j =1 u k X j 9 = ; (Def. of M k : ) = k Y j =1 IE exp u k X j (Independence of the X j ’s) = 1 2 e u k + 1 2 e , u k k ; which implies, log ' k u=klog 1 2 e u k + 1 2 e , u k Let x = 1 k .Then lim k! 1 log ' k u = lim x!0 log 1 2 e ux + 1 2 e ,ux x = lim x!0 u 2 e ux , u 2 e ,ux 1 2 e ux + 1 2 e ,ux (L’Hˆopital’s Rule) =0: Therefore, lim k! 1 ' k u=e 0 =1; which is the m.g.f. for the constant 0. 13.3 Central Limit Theorem We use the method of moment generating functions to prove the Central Limit Theorem. Theorem 3.39 (Central Limit Theorem) 1 p k M k ! Standard normal, as k!1: Proof: ' k u=IEexp u p k M k = 1 2 e u p k + 1 2 e , u p k k ; CHAPTER 13. Brownian Motion 141 so that, log ' k u=klog 1 2 e u p k + 1 2 e , u p k : Let x = 1 p k .Then lim k!1 log ' k u = lim x!0 log 1 2 e ux + 1 2 e ,ux x 2 = lim x!0 u 2 e ux , u 2 e ,ux 2x 1 2 e ux + 1 2 e ,ux (L’Hˆopital’s Rule) = lim x!0 1 1 2 e ux + 1 2 e ,ux : lim x!0 u 2 e ux , u 2 e ,ux 2x = lim x!0 u 2 e ux , u 2 e ,ux 2x = lim x!0 u 2 2 e ux , u 2 2 e ,ux 2 (L’Hˆopital’s Rule) = 1 2 u 2 : Therefore, lim k! 1 ' k u=e 1 2 u 2 ; which is the m.g.f. for a standard normal random variable. 13.4 Brownian Motion as a Limit of Random Walks Let n be a positive integer. If t 0 is of the form k n ,thenset B n t= 1 p n M tn = 1 p n M k : If t 0 is not of the form k n , then define B n t by linear interpolation (See Fig. 13.1). Here are some properties of B 100 t : 142 k/n (k+1)/n Figure 13.1: Linear Interpolationto define B n t . Properties of B 100 1 : B 100 1 = 1 10 100 X j =1 X j (Approximately normal) IEB 100 1 = 1 10 100 X j =1 IEX j =0: varB 100 1 = 1 100 100 X j =1 varX j =1 Properties of B 100 2 : B 100 2 = 1 10 200 X j =1 X j (Approximately normal) IEB 100 2 = 0: varB 100 2 = 2: Also note that: B 100 1 and B 100 2 , B 100 1 are independent. B 100 t is a continuous function of t . To get Brownian motion, let n!1 in B n t; t 0 . 13.5 Brownian Motion (Please refer to Oksendal, Chapter 2.) CHAPTER 13. Brownian Motion 143 t (Ω, F,P) ω B(t) = B(t, ω) Figure 13.2: Continuous-time Brownian Motion. A random variable B t (see Fig. 13.2) is called a Brownian Motion if it satisfies the following properties: 1. B 0 = 0 , 2. B t is a continuous function of t ; 3. B has independent, normally distributed increments: If 0=t 0 t 1 t 2 ::: t n and Y 1 = B t 1 , B t 0 ; Y 2 = B t 2 , B t 1 ; ::: Y n = Bt n , Bt n,1 ; then Y 1 ;Y 2 ;::: ;Y n are independent, IEY j =0 8j; varY j =t j ,t j,1 8j: 13.6 Covariance of Brownian Motion Let 0 s t be given. Then B s and B t , B s are independent, so B s and B t= Bt,Bs + B s are jointly normal. Moreover, IEBs=0; varB s = s; IEBt=0; varB t = t; IEBsBt=IEBsB t , B s + B s = IEBsB t , B s | z 0 + IEB 2 s | z s = s: 144 Thus for any s 0 , t 0 (not necessarily s t ), we have IEBsBt=s^t: 13.7 Finite-Dimensional Distributions of Brownian Motion Let 0 t 1 t 2 ::: t n be given. Then B t 1 ;Bt 2 ;::: ;Bt n is jointly normal with covariance matrix C = 2 6 6 6 4 IEB 2 t 1 IEBt 1 Bt 2 ::: IEBt 1 Bt n IEBt 2 Bt 1 IEB 2 t 2 ::: IEBt 2 Bt n :::::::::::::::::::::::::::::::::::::::::::::::: IEBt n Bt 1 IEBt n Bt 2 ::: IEB 2 t n 3 7 7 7 5 = 2 6 6 6 4 t 1 t 1 ::: t 1 t 1 t 2 ::: t 2 :::::: :::::: ::: t 1 t 2 ::: t n 3 7 7 7 5 13.8 Filtration generated by a Brownian Motion fF tg t0 Required properties: For each t , B t is F t -measurable, For each t and for tt 1 t 2 t n , the Brownian motion increments B t 1 , B t; B t 2 , B t 1 ; :::; Bt n , Bt n,1 are independent of F t . Here is one way to construct F t .Firstfix t .Let s 2 0;t and C 2BIR be given. Put the set fB s 2 C g = f! : B s; ! 2 C g in F t . Do this for all possible numbers s 2 0;t and C 2BIR . Then put in every other set required by the -algebra properties. This F t contains exactly the information learned by observing the Brownian motion upto time t . fF tg t0 is called the filtration generated by the Brownian motion. CHAPTER 13. Brownian Motion 145 13.9 Martingale Property Theorem 9.40 Brownian motion is a martingale. Proof: Let 0 s t be given. Then IE B tjF s=IEB t , B s + B sjF s = IE B t , B s + B s = B s: Theorem 9.41 Let 2 IR be given. Then Z t = exp n ,Bt , 1 2 2 t o is a martingale. Proof: Let 0 s t be given. Then IE Z tjF s = IE expf, B t , B s+Bs , 1 2 2 t , s+sg Fs = IE Zs expf,B t , B s , 1 2 2 t , sg F s = Z sIE h expf,B t , B s , 1 2 2 t , sg i = Z s exp n 1 2 , 2 varB t , B s , 1 2 2 t , s o = Z s: 13.10 The Limit of a Binomial Model Consider the n ’th Binomial model with the following parameters: u n =1+ p n : “Up” factor. ( 0 ). d n =1, p n : “Down” factor. r =0 . ~p n = 1,d n u n ,d n = = p n 2= p n = 1 2 . ~q n = 1 2 . 146 Let k H denote the number of H in the first k tosses, and let k T denote the number of T in the first k tosses. Then k H + k T=k; k H , k T =M k ; which implies, k H = 1 2 k+M k k T= 1 2 k,M k : In the n ’th model, take n steps per unit time. Set S n 0 =1 .Let t = k n for some k ,andlet S n t= 1+ p n 1 2 nt+M nt 1 , p n 1 2 nt,M nt : Under f IP , the price process S n is a martingale. Theorem 10.42 As n!1 , the distribution of S n t converges to the distribution of expfBt , 1 2 2 tg; where B is a Brownian motion. Note that the correction , 1 2 2 t is necessary in order to have a martingale. Proof: Recall that from the Taylor series we have log1 + x=x, 1 2 x 2 +Ox 3 ; so log S n t= 1 2 nt + M nt log1 + p n + 1 2 nt , M nt log1 , p n = nt 1 2 log1 + p n + 1 2 log1 , p n + M nt 1 2 log1 + p n , 1 2 log1 , p n = nt 1 2 p n , 1 4 2 n , 1 2 p n , 1 4 2 n + On ,3=2 ! + M nt 1 2 p n , 1 4 2 n + 1 2 p n + 1 4 2 n + On ,3=2 ! = , 1 2 2 t + On , 1 2 + 1 p n M nt | z !B t + 1 n M nt | z !0 On , 1 2 As n!1 , the distribution of log S n t approaches the distribution of Bt , 1 2 2 t . CHAPTER 13. Brownian Motion 147 B(t) = B(t, ω) t ω x (Ω, F, P ) x Figure 13.3: Continuous-time Brownian Motion, starting at x 6=0 . 13.11 Starting at Points Other Than 0 (The remaining sections in this chapter were taught Dec 7.) For a Brownian motion B t that starts at 0, we have: IP B 0=0=1: For a Brownian motion B t that starts at x , denote the corresponding probability measure by IP x (See Fig. 13.3), and for such a Brownian motion we have: IP x B 0 = x=1: Note that: If x 6=0 ,then IP x puts all its probability on a completely different set from IP. The distribution of B t under IP x is the same as the distribution of x + B t under IP. 13.12 Markov Property for Brownian Motion We prove that Theorem 12.43 Brownian motion has the Markov property. Proof: Let s 0; t 0 be given (See Fig. 13.4). IE hB s + t F s = IE 2 6 6 4 h B s + t , B s | z Independentof F s + B s | z F s -measurable F s 3 7 7 5 148 s s+t restart B(s) Figure 13.4: Markov Property of Brownian Motion. Use the Independence Lemma. Define g x= IEhBs+t,Bs+x = IE 2 6 6 4 h x + B t | z same distribution as B s + t , B s 3 7 7 5 = IE x hB t: Then IE h B s + t Fs = gBs = E B s hB t: In fact Brownian motion has the strong Markov property. Example 13.1 (Strong Markov Property) See Fig. 13.5. Fix x0 and define = min ft 0; B t=xg: Then we have: IE h B + t F = gB = IE x hB t: