MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties of a Browia motio. Historical otes 765 Ja Igehousz observatios of carbo dust i alcohol. 828 Robert Brow observed that polle grais suspeded i water perform a cotiual swarmig motio. 900 Bachelier s work The theory of speculatio o usig Browia motio to model stock prices. 905 Eistei ad Smoluchovski. Physical iterpretatio of Browia motio. 920 s Wieer cocise mathematical descriptio. 2 Costructio of a Browia motio from a radom walk The developmets i this lecture follow closely the book by Resick [3]. I this sectio we provide a heuristic costructio of a Browia motio from a radom walk. The derivatio below is ot a proof. We will provide a rigorous costructio of a Browia motio whe we study the weak covergece theory. The large deviatios theory predicts expoetial decay of probabilities P( i X > a) whe a > µ = E[X ] ad E[e θx ] are fiite. Naturally, the decay will be

2 slower the closer a is to µ. We cosidered oly the case whe a was a costat. But what if a is a fuctio of : a = a? The Cetral Limit Theorem tells us that the decay disappears whe a. Recall Theorem (CLT). Give a i.i.d. sequece (X, ) with E[X ] = µ, var[x ] = σ 2. For every costat a X a i µ 2 i lim P( a) = e t 2 dt. σ 2π Now let us look at a sequece of partial sums S = i (X i µ). For simplicity assume µ = 0 so that we look at S = i Xi. Ca we say aythig about S as a fuctio of? I fact, let us make it a fuctio of a real variable t R + ad rescale it by o i tj as follows. Defie B(t) = X i for every t 0. Deote by N(µ, σ 2 ) the distributio fuctio of a ormal r.v. with mea µ ad variace σ 2.. For every fixed 0 s < t, by CLT we have the distributio of X σ lsj<i ltj ltj lsj covergig to the stadard ormal distributio as. Igorig the differece betwee ltj lsj ad (t s) which is at most, ad writig st<i t X i B (t) B (s) = σ t s σ t s we obtai that B 2 (t) B (s) coverges i distributio to N(0, σ (t s)). i o i t Xi 2. Fix t < t 2 ad cosider B (t ) = ad B (t 2 ) B (t o ) = Xi t <i t 2. The two sums cotai differet elemets of the sequece X, X 2,.... Sice the sequece is i.i.d. B (t ) ad B (t 2 ) B (t ) are idepedet. Namely for every x, x 2 P(B (t ) x, B (t 2 ) B (t ) x 2 ) = P(B (t ) x )P(B (t 2 ) B (t ) x 2 ). This geeralizes to ay fiite collectio of icremets B (t ), B (t 2 ) B (t ),..., B (t k ) B (t k ). Thus B (t) has idepedet icremets. 2

3 3. Give a small E, for every t the differece B (t + E) B (t) coverges i distributio to N(0, σ 2 E). Whe E is very small this differece is very close to zero with probability approachig as. Namely, B (t) is icreasigly close to a cotiuous fuctio as. 4. B (0) = 0 by defiitio. 3 Defiitio The Browia motio is the limit B(t) of B (t) as. We will delay aswerig questios about the existece of this limit as well as the sese i which we take this limit (remember we are dealig with processes ad ot values here) till future lectures ad ow simply postulate the existece of a process satisfyig the properties above. I the theorem below, for every cotiuous fuctio ω C[0, ) we let B(t, ω) deote ω(t). This otatio is more cosistet with a stadard covetio of deotig Browia motio by B ad its value at time t by B(t). Defiitio (Wieer measure). Give Ω = C[0, ), Borel σ-field B defied o C[0, ) ad ay value σ > 0, a probability measure P satisfyig the followig properties is called the Wieer measure:. P(B(0) = 0) =. 2. P has the idepedet icremets property. Namely for every 0 t < < t k < ad x,..., x k R, P(ω C[0, ) : B(t 2, ω) B(t, ω) x,..., B(t k, ω) B(t k, ω) x k ) = P(ω C[0, ) : B(t i, ω) B(t i ), ω) x i ) 2 i k 3. For every 0 s < t the distributio of B(t) B(s) is ormal N(0, σ 2 (t s)). I particular, the variace is a liear fuctio of the legth of the time icremet t s ad the icremets are statioary. The stochastic process B described by this probability space (C[0, ), B, P) is called Browia motio. Whe σ =, it is called the stadard Browia motio. 3

4 Theorem 2 (Existece of Wieer measure). For every σ 0 there exists a uique Wieer measure. (what is the Wieer measure whe σ = 0?). As we metioed before, we delay the proof of this fudametal result. For ow just assume that the theorem holds ad study the properties. Remarks : I future we will ot be explicitly writig samples ω whe discussig Browia motio. Also whe we say B(t) is a Browia motio, we uderstad it both as a Wieer measure or simply a sample of it, depedig o the cotext. There should be o cofusio. It turs out that for ay give σ such a probability measure is uique. O the other had ote that if B(t) is a Browia motio, the B(t) is also a Browia motio. Simply check that all of the coditios of the Wieer measure hold. Why is there o cotradictio? Sometimes we will cosider a Browia motio which does ot start at zero: B(0) = x for some value x = 0. We may defie this process as x + B(t), where B is Browia motio. Problem.. Let Ω be the space of all (ot ecessarily cotiuous) fuctios ω : R + R. i Costruct a example of a stochastic process i Ω which satisfies coditios (a)-(c) of the Browia motio, but such that every path is almost surely discotiuous. i Costruct a example of a stochastic process i Ω which satisfies coditios (a)-(c) of the Browia motio, but such that every path is almost surely discotiuous i every poit t [0, ]. HINT: work with the Browia motio. 2. Suppose B(t) is a stochastic process defied o the set of all (ot ecessarily cotiuous) fuctios x : R + R satisfyig properties (a)-(c) of Defiitio. Prove that for every t 0, lim B(t+ ) = B(t) almost surely. 4

5 Problem 2. Let A R be the set of all homogeeous liear fuctios x(t) = at where a varies over all values a R. B(t) deotes the stadard Browia motio. Prove that P(B A R ) = 0. 4 Properties We ow derive several properties of a Browia motio. We assume that B(t) is a stadard Browia motio. Joit distributio. Fix 0 < t < t 2 < < t k. Let us fid the joit distributio of the radom vector (B(t ),..., B(t k )). Give x,..., x k R let us fid the joit desity of (B(t ), B(t 2 ),..., B(t k )) i (x,..., x k ). It is equal to the joit desity of (B(t ), B(t 2 ) B(t ),..., B(t k ) B(t k )) i (x, x 2 x,..., x k x k ), which by idepedet Gaussia icremets property of the Browia motio is equal to k i= 2π(t i+ t i ) (x x ) 2 i. e i+ i 2(t i+ t ) Differetial Property. For ay s > 0, B s (t) = B(t + s) B(s), t 0 is a Browia motio. Ideed B s (0) = 0 ad the process has idepedet icremets B s (t 2 ) B s (t ) = B(t 2 + s) B(t + s) which have a Guassia distributio with variace t 2 t. Scalig. For every c, cb(t) is a Browia motio with variace σ 2 = c 2. Ideed, cotiuity ad the statioary idepedet icremet properties as well as the Gaussia distributio of the icremets, follow immediately. The variace of the icremets cb(t 2 ) cb(t ) is c 2 (t 2 t ). For every positive c > 0, B( t ) is a Browia motio with variace c c. Ideed, the process is cotiuous. The icremets are statioary, idepedet with Gaussia distributio. For every t < t 2, by defiitio of the stadard Browia motio, the variace of B( t 2 ) B( t ) is (t2 t )/c = (t 2 t ) c c c. Combiig these two properties we obtai that cb( t ) c is also a stadard Browia motio. 5

6 Covariace. Fix 0 s t. Let us compute the covariace Cov(B(t), B(s)) = E[B(t)B(s)] E[B(t)]E[B(s)] a = E[B(t)B(s)] = E[(B(s) + B(t) B(s))B(s)] b = E[B 2 (s)] + E[B(t) B(s)]E[B(s)] = s + 0 = s. Here (a) follows sice E[B(t)] = 0 for all t ad (b) follows sice by defiitio of the stadard Browia motio we have E[B 2 (s)] = s ad by idepedet icremets property we have E[(B(t) B(s))B(s)] = E[(B(t) B(s))(B(s) B(0))] = E[(B(t) B(s)]E[(B(s) B(0)] = 0, sice icremets have a zero mea Gaussia distributio. Time reversal. Give a stadard Browia motio B(t) cosider the process () defied by () B (t) B (t) = tb( t ) for all t > 0 ad B () (0) = 0. I other words we reverse time by the trasformatio t t. We claim that B () (t) is also a stadard Browia motio. Proof. We eed to verify properties (a)-(c) of Defiitio plus cotiuity. The cotiuity at ay poit t > 0 follows immediately sice /t is cotiuous fuctio. B is cotiuous by assumptio, therefore tb( t ) is cotiuous for all t > 0. The cotiuity at t = 0 is the most difficult part of the proof ad we delay it till the ed. For ow let us check (a)-(c). (a) follows sice we defied B () (0) to be zero. We delay (b) till we establish ormality i (c) (c) Take ay s < t. Write tb( t ) sb( s ) as tb( ) sb( ) = (t s)b( ) + sb( ) sb( ) t s t t s The distributio of B( ) B( ) is Gaussia with zero mea ad variace t s s t, sice B is stadard Browia motio. By scalig property, the distributio of sb( ) sb( t s ) is zero mea Gaussia with variace s 2 ( s t ). The distributio of (t s)b( t ) is zero mea Gaussia with variace (t s) 2 ( t ) ad also it is idepedet from sb( t ) sb( s ) by idepedet icremets properties of 6

7 the Browia motio. Therefore tb( ) sb( t s ) is zero mea Gaussia with variace s 2 ( ) + (t s) 2 ( ) = t s. s t t This proves (c). We ow retur to (b). Take ay t < t 2 < t 3. We established i (c) that all the differeces B () (t 2 ) B () (t ), B () (t 3 ) B () (t 2 ), B () (t 3 ) B () (t ) = B ()(t 3 ) B () (t 2 ) + B () (t 2 ) B () (t ) are zero mea Gaussia with variaces t 2 t, t 3 t 2 ad t 3 t respectively. I particular the variace of B () (t () () () 3) B (t ) is the sum of the variaces of B (t 3 ) B (t 2 ) ad B () (t 2 ) B () (t ). This implies that the covariace of the summads is zero. Moreover, from part (b) it is ot difficult to establish that B () (t 3 ) B () (t 2 ) ad B () (t 2 ) B () (t ) are joitly Gaussia. Recall, that two joitly Gaussia radom variables are idepedet if ad oly if their covariace is zero. It remais to prove the cotiuity at zero of B () (t). We eed to show the cotiuity almost surely, so that the zero measure set correspodig to the samples ω C[0, ) where the cotiuity does ot hold, ca be throw away. Thus, we eed to show that the probability measure of the set A = {ω C[0, ) : lim tb(, ω) = 0} t 0 t is equal to uity. We will use Strog Law of Large Numbers (SLLN). First set t = / ad cosider tb( t ) = B()/. Because of the idepedet Gaussia icremets property B() = i (B(i) B(i )) is the sum of idepedet i.i.d. stadard ormal radom variables. By SLLN we have the B()/ E[B() B(0)] = 0 a.s. We showed covergece to zero alog the sequece t = / almost surely. Now we eed to take care of the other values of t, or equivaletly, values s [, + ). For ay such s we have B(s) B() B(s) B() B() B() + s s s s B() + sup B(s) B() s s + B() + sup 2 B(s) B(). s + We kow from SLLN that B()/ 0 a.s. Moreover the B()/ 2 0. () 7

8 a.s. Now cosider the secod term ad set Z = sup s + B(s) B(). We claim that for every E > 0, P(Z / > E i.o.) = P(ω C[0, ) : Z (ω)/ > E i.o.) = 0 (2) where i.o. stads for ifiitely ofte. Suppose (2) was ideed the case. The equality meas that for almost all samples ω the iequality Z (ω)/ > E happes for at most fiitely may. This meas exactly that for almost all ω (that is a.s.) Z (ω)/ 0 as. Combiig with () we would coclude that a.s. B(s) B() sup 0, s s + as. Sice we already kow that B()/ 0 we would coclude that a.s. lim s B(s)/s = 0 ad this meas almost sure cotiuity of B () (t) at zero. It remais to show (2). We observe that due to the idepedet statioary icremets property, the distributio of Z is the same as that of Z. This is the distributio of the maximum of the absolute value of a stadard Browia motio durig the iterval [0, ]. I the followig lecture we will show that this maximum has fiite expectatio: E[ Z ] <. O the other had (+)ɛ E[ Z ] = P( Z > x)dx = 0 P( Z > x)dx 0 ɛ =0 0 E =0 P( Z > ( + )E). We coclude that the sum i the right-had side is fiite. By i.i.d. of Z 0 Z 0 Z P( > E) = P( > E) = = Thus the sum o the left-had side is fiite. Now we use the Borel-Catelli Lemma to coclude that (2) ideed holds. 5 Additioal readig materials Sectios 6. ad 6.4 from Chapter 6 of Resick s book Advetures i Stochastic Processes i the course packet. Durrett [2], Sectio 7. Billigsley [], Chapter 8. 8

9 Refereces [] P. Billigsley, Covergece of probability measures, Wiley-Itersciece publicatio, 999. [2] R. Durrett, Probability: theory ad examples, Duxbury Press, secod editio, 996. [3] S. Resick, Advetures i stochastic processes, Birkhuser Bosto, Ic.,

10 MIT OpeCourseWare J / 6.265J Advaced Stochastic Processes Fall 203 For iformatio about citig these materials or our Terms of Use, visit:

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