MATH 56A: STOCHASTIC PROCESSESCHAPTER 88. Brownian motionWe will be spending the rest of the course on Brownian motion andintegration with respect to Brownian motion (stochastic integrals). Thetopics in the book are closely interrelated so we need to go over every-thing in the book plus additional material such as L´evy’s remarkabletheorem which I will explain today. Here is an outline of the chapter.(0) Definition of Brownian motion(1) Martingales and L´evy’s theorem(2) Strong Markov property and reflection principle(3) Fractal dimension of the zero set(4) Brownian motion and the heat equation in several dimensions(5) Recurrence and transience(6) Fractal dimension of the path(7) Scaling and the Cauchy distribution(8) Drift8.0. Definition of Brownian motion. First of all this is a randomprocess in continuous time and continuous space. We will start withdimension one: Brownian motion on the real line.The idea is pretty simple. A particle is bouncing around and itsposition at time t is Xt. The process is(1) memoryless: What happens for time s > t depends only on itsposition Xtand not on how it got there.(2) time and space homogeneous: The behavior of the particle re-mains the same if we reset both time and space coordinates.I.e., the distribution of Yt= Xs+t− Xsdepends only on t andis independent of the time s and position Xs.(3) continuity: The particle moves on a continuous path (withoutjumping from one point to another).These conditions almost guarantee that we have Brownian motion.But we need a little more. Here is the definition.Date: December 9, 2006.12 MATH 56A: STOCHASTIC PROCESSES CHAPTER 88.0.1. definition.Definition 8.1. A random function X : [0, ∞) → R written XtisBrownian motion with variance σ2starting at 0 if:(1) X0= 0(2) For any s < t1≤ s2< t2≤ ··· ≤ sn< tnthe random variablesXt1− Xs1, Xt2− Xs2, ··· , Xtn− Xsnare independent.(3) The path Xtis continuous(4) For s < t,Xt− Xs∼ N(0, (t −s)σ2)i.e., Xt−Xsis normally distributed with mean 0 and variance(t − s)σ2.Theorem 8.2. The last condition is equivalent to the condition:(4’) Xt− Xsand Xt+c− Xs+care identically distributed with mean0 and variance (t −s)σ2.Proof. (Outlined in the book.) Certainly (4) ⇒ (4!). To prove theconverse, assume (4!). Let ∆t = (t −s)/N for large N. ThenXt− Xs= (Xs+∆t− Xs) + (Xs+2∆t− Xs+∆t) + ··· + (Xt− Xt−∆t)This is a sum of N independent identically distributed random variableswith mean 0 and variance (∆t)σ2. By the central limit theorem we getXt− Xs≈ N(0, N∆t σ2) = N(0, (t − s)σ2)Now take the limit as N → ∞. (This is not rigorous because we arenot using the precise statement of the CLT.) !Recall that the variance of a random variable X is defined byV ar(X) := E((X − E(X))2) = E(X2) − E(X)2and it has the property that it is additive for independent randomvariables:V ar(X1+ X2+ ··· + Xn) = V ar(X1) + V ar(X2) + ··· + V ar(Xn)8.0.2. as limit of random walk. Brownian motion can be obtained asa limit of random walks: Take time to be integer multiples of a fixedinterval ∆t and we take points on the line at integer multiples of σ√∆t.For each unit time assume that position changes by ±σ√∆t with equalprobability. This is Bernoulli with mean 0 and variance(±σ√∆t) = σ2∆tIn a time interval N∆t the change of position is given by a sum ofN independent random variables. So, the mean would be 0 and theMATH 56A: STOCHASTIC PROCESSES CHAPTER 8 3variance would be N∆tσ2. The point is that this is σ2times the lengthof the time interval. As ∆t → 0, assuming the sequence of randomwalks converges to a continuous function, the limit gives Brownianmotion with variance σ2by the theorem.8.0.3. nowhere differentiable. Notice that, as ∆t goes to zero, the changein position is approximately σ√∆t which is much bigger than ∆t. Thisimplies that the limitlimt→0Xttdiverges. So, Brownian motion is, almost surely, nowhere differentiable.(Almost surely or a.s. means “with probability one.”)8.1. Martingales and L´evy’s theorem.Theorem 8.3. Suppose that Xtis Brownian motion. Then(1) Xtis a continuous martingale.(2) X2t− tσ2is a martingaleProof. (1) is easy: If t > s thenE(Xt|Fs) = E(Xt− Xs|Fs) + E(Xs|Fs)= 0 + Xswhere Fsis the information contained in Xrfor all r ≤ s.For (2), we need the equation:(Xt− Xs)s= X2t− 2XtXs+ 2X2s− X2s= X2t− 2(Xt− Xs)Xs− X2sTaking E(−|Fs) of both sides gives:E((Xt− Xs)2|Fs) = V ar(Xt− Xs) = (t − s)σ2=E(X2t|Fs) − 2E(Xt− Xs|Fs)Xs− X2s= E(X2t|Fs) − X2sWhich givesE(X2t− tσ2|Fs) = X2s− sσ2!L´evy’s theorem is the converse:Theorem 8.4 (L´evy). Suppose that Xtis a continuous martingaleand X2t− tσ2is also a martingale. Then Xtis Brownian motion withvariance σ2This famous theorem has been proved many times. I will try to findthe proof using stochastic integrals. One amazing consequence is thefollowing.4 MATH 56A: STOCHASTIC PROCESSES CHAPTER 8Corollary 8.5. Any continuous martingale Mtis Brownian motionreparametrized and starting at C = M0. I.e.Mt= Xφ(t)+ Cwhere Xsis standard Brownian motion (with σ = 1).Proof. (When I did this in class I forgot to “center” the martingale bysubtracting M0.) The idea is to let φ(t) = E((Mt− C)2) and applyL´evy’s theorem. (I’ll look for the details). !8.2. Strong Markov property and Reflection principle.Theorem 8.6 (strong Markov property). Let T be a stopping time forBrownian motion Xt. LetYt= Xt+T− XTThen Ytis independent of FT(for t > 0).One consequence of this is the reflection principle which the bookuses over and over.8.2.1. reflection principle.Corollary 8.7 (reflection principle). Suppose that a < b then the prob-ability that you will reach b from a within time t is twice the probabilitythat at time t you will be past b. I.e.:P(Xs= b for some 0 < s < t |X0= a) = 2P(Xt> b |X0= a)Proof. If you reach the point b at some time before time t then half thetime you will end up above b and half the time you will end up belowb since the probability that Xt= b is zero. So,P( Xsreaches b sometime before t and ends up higher |X0= a)=12P(Xs= b for some 0 < s < t |X0= a)But the event “Xsreaches b sometime before t and ends up higher” isthe same as the event “Xt> b” since Xtis continuous and thereforecannot get to a point Xt> b starting at a < b without passing throughb. This proves the reflection principle.Why is the reflection principle a corollary of the strong