A Rigorous Introduction to Brownian Motion

Transcription

1 A Rigorous Inroducion o Brownian Moion Andy Dahl Augus 19, 1 Absrac In his paper we develop he basic properies of Brownian moion hen go on o answer a few quesions regarding is zero se and is local maxima. Conens 1 The Basics 1 The Relevan Measure Theory 5 3 Markov Properies of Brownian moion 6 4 Furher Properies of Brownian moion 9 1 The Basics The concep of a Brownian moion was discovered when Einsein observed paricles oscillaing in liquid. Since fluid dynamics are so chaoic and rapid a he molecular level, his process can be modeled bes by assuming he paricles move randomly and independenly of heir pas moion. We can also hink of Brownian moion as he limi of a random walk as is ime and space incremens shrink o. In addiion o is physical imporance, Brownian moion is a cenral concep in sochasic calculus which can be used in finance and economics o model sock prices and ineres raes. 1.1 Brownian Moion Defined Since we are rying o capure physical inuiion, we define a Brownian moion by he properies we wan i o have and worry abou proving he exisence of and explicily consrucing such a process laer. 1

2 Definiion 1. An R d -valued sochasic process {B() : } is called a d-dimensional Brownian moion saring a x R d if i has he following four properies: Sar a x: B() x Independen incremens: for all 1... n, he incremens B( n ) B( n 1 ),..., B( ) B( 1 ) are independen random variables Normaliy: for all and h > he incremen B( + h) B() is disribued N(, h) Coninuiy: almos surely, B() is coninuous The firs propery anchors he sochasic process in space. The second capures he coninually random naure of a paricle ha is being consanly buffeed by fluid molecules. The hird is required because he expeced displacemen of a paricle should be proporional o he ime i has been raveling and should be symmerically disribued abou he saring poin. Physical moion is coninuous which explains he fourh requiremen. While a Brownian moion is frequenly denoed {B() } o sress he fac ha i is acually an uncounable family of random variables, we will use B as shorhand, undersanding ha varies over he non-negaive reals. The bulk of he firs and hird secions apply o general Brownian moions and in he fourh we specialize o he linear case. The consrucion of Brownian moion is edious and beyond he scope of his paper. Bu we should remember ha i is he characerisics of Brownian moion, raher han is consrucion, which define i. Indeed, here are even differen consrucions. The deails of he consrucion will no be used in his paper. 1. Nondiffereniabiliy of Brownian moion The mos sriking qualiy of Brownian moion is probably is nowhere differeniabiliy. Theorem 1. Almos surely, Brownian moion is nowhere differeniable The proof consiss primarily of a long compuaion which we do no presen. We will prove laer ha in any small inerval o he righ of some ime s, B aains values greaer han and less han B s. So for all ɛ > and s, we can choose some h (s, s + ɛ) such ha B s+h B s is eiher posiive h or negaive. This suppors he idea ha he upper and lower derivaives

3 of B a every poin are + and, respecively, alhough a good deal of compuaional work goes ino proving ha he upper and lower limis diverge as ɛ. Noneheless, knowing ha hey do diverge does give us insigh ino how rapidly and erraically Brownian moion jumps around. The heorem can also be undersood direcly from he definiion of Brownian moion. If B were differeniable a some poin s, we would know where i was going in some small ime inerval in he fuure, bu he independen incremen propery of B should make us skepical of such a predicion. The nex proposiion is a manifesaion of he combinaion of Brownian moion s nondifferniabiliy and is coninuiy. Proposiion 1. Almos surely, B is no monoonic on any inerval. Proof. Fix a < b. Le P (a, b) be he probabiliy ha B is monoonic on (a, b). Then, by independence of incremens, P (a, b) 1 P (a, a + b ) P (a + b, b) 1 P (a, a + b ) since, even if B is monoonic on (a, a+b a+b ) and (, b), he probabiliy ha i is monoonic in he same direcion on boh inervals is 1. Ieraing he divisions of he inerval ino halves n imes we ge P (a, b) ( 1 )n P (a, a + b a n ) Taking he limi as n, we see P (a, b) ( 1 )n P (a, a + b a ) 1 n, n showing P (a, b), so any fixed inerval is almos surely no monoonic. By aking he counable union over all inervals wih raional endpoins we can see ha B is almos surely no monoonic on any inerval wih raional endpoins. By he densiy of Q R, here is an inerval wih raional endpoins conained wihin every inerval. So every inerval conains a subinerval which is almos surely no monoonic, hus every inerval is almos surely no monoonic. We will use his resul laer when discussing he maxima of B. 1.3 Scaling Properies of Brownian Moion We ofen sudy ransformaions of funcions which leave cerain properies invarian, and i is naural o ask wha ransformaions of B have he same disribuion. 3

4 Example 1. B is a Brownian moion. Coninuiy and independence are clearly mainained by negaive muliplicaion and, since he normal disribuion is symmeric abou zero, all he incremens have he proper means and variances. We now move on o more ineresing and useful ransformaions. Proposiion. Rescaling: If B is a sandard Brownian moion, hen so is he process X ab, for all a >. a Proof. Coninuiy and independence of incremens sill hold. For all > s, he normal random variable X() X(s) a(b( ) B( s )) is disribued a a an(, s ) d N(, s), so X() X(s) N(, s) as desired. a This proposiion ells us ha B is a Brownian moions on all ime scales as long as we compensae for he change in variance of he incremens by aking a scalar muliple of he process. More surprisingly, we can inver he domain of B and sill have a Brownian moion. Proposiion 3. Time-inversion: Le B be a sandard Brownian moion. Then he process { : X : is also a sandard Brownian moion Proof. For Brownian moions, B 1 Cov(B, B +s ) Cov(B, B +s B ) + Cov(B, B ) for all, s. For our process X, Cov(X, X +s ) Cov(B 1, ( + s)b 1 ) +s ( + s)cov(b 1, B 1 ) ( + s) +s 1 + s So Cov(X, X +s X ) Cov(X, X +s ) Var(X ). Because he random variables X +s and X are normal, Cov(X, X +s X ) implies ha X +s X and X are independen. And Var(X +s X ) Var(X +s ) + Var(X ) Cov(X +s, X ) ( + s) + s, so our incremens are independen and have he righ variances. Coninuiy is clear for >. We know ha X has he disribuion of a Brownian moion on Q, so lim n X( 1 n ) lim X() 4

5 and we conclude ha X is coninuous a, so X saisfies he properies of a sandard Brownian moion. We end wih secion wih an example which demonsraes he compuaional usefulness of hese alernaive expressions for Brownian moion. Example. Le B be a sandard Brownian moion and X B 1. X is a sandard Brownian moion, so X lim lim B 1 B The Relevan Measure Theory We assume he reader is familiar wih he elemens of basic probabiliy heory such as expecaion, covariance, normal random variables, ec. Bu we do add rigor o hese noions by developing he underlying measure heory, which will be necessary for our discussion of he Markov properies. Definiion. A σ-algebra Σ on a se S is a subse of S, where S is he power se of S, saisfying: {Ø} Σ for all A Σ, A c Σ for all sequences A, A 1,... Σ, i A i Σ By de Morgan s laws we can see ha σ-algebras are closed under counable inersecions as well. The σ-algebra will be our objec of measuremen, so now we need o develop our mehod of measuremen. Definiion 3. A measure is a counably addiive map µ : Σ [, ], where Σ S is our σ-algebra on some se S. A counably addiive map is one such ha for any sequence A 1, A,... Σ of disjoin evens, µ( i1 A i) i1 µ(a i). A probabiliy measure is a measure µ such ha µ(s) 1. Our definiion implies ha µ({ø}) because µ({ø}) µ({ø} {Ø}) µ({ø}) + µ({ø}). Our nex definiion collecs hese noions. Definiion 4. The probabiliy riple is he riple (Ω, F, P ) where F is a σ-algebra on he se Ω and P : Σ [, 1] is a probabiliy measure. We call Ω he sample space and F he collecion of (P -measurable) evens. This riple provides he background for he sudy probabiliy. foreground are random variables. In he 5

6 Definiion 5. A random variable is an F-measurable map X : Ω R, meaning ha he preimage X 1 (B) F for all B B(R). The law of X is P (X 1 ) : B(R) [, 1]. The random variable X is a correspondence beween evens and ses in R, which formalizes he noion ha X akes on cerain values when cerain evens occur. Of course, his correspondence is no ha ineresing in iself; wha ineress us is he probabiliy of X lying in ses in R, which is given by he law of X. For he law of X o be well defined we need X 1 (B(R) F, since F is he domain of P, which is why we require X o be F-measurable. A sochasic process is a family of random variables ha evolves over ime, and up o his poin we have viewed hese random variables from ime. Bu we can also look a he process a some ime s a which he se {X s} is known, and he probabiliy of evens occuring pas s will depend on his informaion. Definiion 6. A filraion on a probabiliy space (Ω, F, P ) is a family {F } of σ-algebras such ha F s F F for all s <. A sochasic process {X } is adaped o he filraion if X is F measurable for all. An adaped filraion capures he inuiion of our informaion which evolves along wih our process: our informaion grows as ime goes on. 3 Markov Properies of Brownian moion The Markov properies ell us a wha imes s a Brownian moion {B +s } referred o as B +s in he fuure has he same disribuion as a Brownian moion sared a B s or, alernaively, when he process B +s B s is a sandard Brownian moion. We will refer o his phenomenon as Brownian moion saring anew a ime s. Our independence of incremens requiremen migh seem o make his propery rivial, and, for deerminisic imes, i does. Theorem. (Markov Propery) Le B be a Brownian moion and fix s. B +s B s is a sandard Brownian moion independen of {B s}. Proof. I is clear ha B +s is a Brownian moion. Subracing a consan only changes he saring poin, and, in paricular, subracing B s makes he process a sandard Brownian moion. Independence of B before ime s follows from he independence of incremens of Brownian moion. 6

7 A far more ineresing and imporan class of imes is random imes, meaning imes defined by some randomly occurring even. Brownian moion does no necessarily sar afresh a such imes. We provide an example, bu firs sae a definiion. Definiion 7. A coninuous funcion f is said o aain a maximum on an inerval I a s I if f(s) f() for all I We say f aains a local maximum a s if here exiss a non-degenerae inerval I conaining s on which f(s) is a maximum. We say he (local) maximum is sric if he above inequaliy can be replaced by a sric inequaliy. Example 3. Le s be a ime ha B aains a sric local maximum and define X B +s B s. Then here exiss some δ such ha for all r (s δ, s + δ), B r < B s. So P (X δ X < ) P (B s+ δ B s > ). So he incremen X δ X < is cerainly no normal, hus X is no a Brownian moion and B does no sar anew a s. This example shows we need o be careful when considering random imes. Noneheless, Brownian moion does sar anew a some random imes. Definiion 8. A random ime T [, ] defined on a probabiliy space wih filraion F is a sopping ime if {T s} F s for every s >. Given our heurisic undersanding of F as he informaion up o ime, a random ime is a sopping ime if we can deermine wheher i has occurred before s based only on knowing he informaion up o s. This explains why B does no sar anew a he ime s where B aains a local maximum because we do no know ha B s is a local maximum unil ime s + δ. Sopping imes ge heir name because if we sop exacly a ime s we can deermine wheher s is our sopping ime; we do no need o go forward in ime o see ha s is in fac he ime we wan. We provide an example of a sopping ime. Definiion 9. For a Brownian moion in R d and a R d, define T a inf{ B() a} as he firs ime B his a. T a is a sopping ime because we can deermine if s T a by observing wheher B s a and wheher B r a for any r < s, boh of which are known a ime s. For he same reason, he second, hird, or nh ime B his a are also sopping imes, bu he las ime B his a is no a sopping ime because we would need o see infiniely far ino he fuure o know he process never reurns o a again. 7

8 T a is really only useful for linear Brownian moions, as i urns ou ha Brownian moions almos surely never hi any specific a R \ {} for d, while in he 1-dimensional case T a is almos surely finie for all a. We now sae he srong Markov propery of Brownian moion and jusify our emphasis on sopping imes. Theorem 3. (Srong Markov Propery) For every almos surely finie sopping ime T, he process B T + B T is a sandard Brownian moion independen of F T. Equivalenly, condiional on F T, B T + is a Brownian moion sared a B T, so, for Brownian moions, we can say ha he only useful informaion conained in F T is he value of B a T : Brownian moion sars anew a T. The srong Markov propery allows us o prove he reflecion principle. Theorem 4. (Reflecion Principle)Le T be a sopping ime. The process given by reflecing a Brownian moion abou ime T is a Brownian moion. More precisely, he process defined by { B(T ) B() : T B B() : T is a Brownian moion. Proof. By he srong Markov propery, B( + T ) B(T ) is a sandard Brownian moion independen of {B() T }. So B(T ) B( + T ) mus be as well. If we aach he process {B() T } in fron of {B( + T ) B(T ) } we will ge a new Brownian moion: he properies of Brownian moion all hold independenly wihin he concaenaed processes, independence of incremens holds beween hem because he processes we concaenae are independen by he srong Markov propery and coninuiy holds because he process are equal a T, he poin where we glue hem ogeher. In he same way, we aach {B() T } o he fron of B(T ) B(+T ), which will give us a process idenical o he former concaenaion, which is jus B. Since his laer concaenaion is B, we conclude B is a sandard Brownian moion. The reflecion principle is exremely useful. We will firs use i o compue he densiy of he maximal process. Proposiion 4. Le B be a linear Brownian moion and M be is maximal process, defined as M max s B s. Then P (M > a) P (B > a) for all a >. 8

9 Proof. P (M > a) P (B > a or M > a while B a) P (B > a) + P (M > a and B a) since he wo evens are disjoin. M > a implies T a <, so by reflecing abou T a, we see ha M > a and B a if and only if B > a. So P (M > a) P (B > a) + P (B > a) P (B > a) 4 Furher Properies of Brownian moion Throughou his secion we specialize o linear Brownian moions. 4.1 The Zeroes of Brownian moion We define he Zero se, Z, as he imes B his, or Z { B }. We firs show ha Z is small in he sense of area. Theorem 5. Wih probabiliy one, he Lebesgue measure of Z is. Proof. Le Z be he Lebesgue measure of Z. We compue he expecaion: E Z E χ {} (W ) d P (W ) d (1 P (W ) + P (W )) d We know Z is non-negaive since i is a measure. We now show ha non-negaive random variables wih expecaion are almos surely. Suppose X is a random variable wih EX and fix a >. Then EX X dp X dp a P (X a) Ω {X a} So P (X a) for all a >. Leing a +, we see P (X > ), and X is almos surely. We conclude ha E Z Z almos surely. We should have expeced his resul because we know ha B is almos surely nonzero for all nonzero since he incremen B B B is a normal random variable. Noneheless, B his infiniely many imes in any inerval o he righ of he origin. Proposiion 5. Almos surely, B has infiniely many zeros in every ime inerval (, ɛ), where ɛ >. 9

10 Proof. We induc on he number of zeros in (, ɛ). Firs, we show ha here mus be a zero in his inerval. Le M + and M be he maximal and minimal processes, respecively, and fix ɛ >. We know ha P (M ɛ + > a) P (B ɛ > a). By aking he limi as a +, we see P (M ɛ + > ) P (B ɛ > ) 1 since B is symmeric. By symmery, P (M (ɛ) < ) 1. Since B is almos surely coninuous, we employ he inermediae value heorem and conclude ha B for some (, ɛ). Now ake some finie se S of zeros of B on he inerval (, ɛ). Le T min( S) be he earlies member of S. Since ɛ was arbirary when we esablished ha B had a zero in (, ɛ), by he same argumen we can see ha B almos surely has a zero in (, T ). By he minimaliy of T his zero mus no be in S, hus here is no finie se conaining all he imes B his zero, so Z is almos surely infinie. This allows us o fully characerize Z. Theorem 6. Almos surely, Z is a perfec se, ha is, Z is closed and has no isolaed poins. Proof. Since B is coninuous, Z B 1 ({}) mus be closed because i is he inverse image of he closed se {}. Consider he ime τ q inf{ q B }, where q is a raional number. This is clearly a sopping ime, and i is almos surely finie because B almos surely crosses for arbirarily large. Moreover, he infimum is a minimum because Z is almos surely closed. We apply he srong Markov propery a τ q and ge ha B +τq B τq B +τq is a sandard Brownian moion. Because we already know ha a Brownian moion crosses in every small inerval o he righ of he origin, τ q is no isolaed from he righ in Z. Now suppose we have some z Z ha is no in {τ q q Q}. Take some sequence, q n, of raional numbers ha converges o z. For each q n, here mus exis some n Z such ha q n n < z since z τ qn. Because q n z, n z, so z is no isolaed from he lef in Z. I is a fac from analysis ha perfec ses are uncounable. This shows ha even hough Z has measure, i sill is very big in a sense. 4. Maxima of Brownian moion Because Brownian moions change direcions so frequenly and dramaically i is no surprising ha hey frequenly aains local maxima. Theorem 7. Almos surely, he se of imes where B() aains a local maximum is dense in [, ) 1

11 Proof. We begin by showing ha B almos surely has a local maximum on every fixed inerval. A ime B aains local maximum is a poin preceded by an increasing inerval and followed by a decreasing inerval. So B has no local max on (a, b) if and only if i is monoonic on (a, b) or monoonically decreasing unil some poin c (a, b) hen monoonically increasing. The former even has probabiliy. And for all c (a, b), (a, a+b ) (a, c) or ( a+b, b) (c, b), so B being monoonic on (a, c) and (c, b) implies ha B is monoonic on eiher (a, a+b a+b ) or (, b), and boh hese evens occur wih probabiliy. By aking a counable union, we see ha all inervals wih raional endpoins almos surely have a ime B aains a local maximum. By he densiy of Q R, here exiss such a raional inerval wihin every inerval. So every real inerval conains a raional inerval which conains a ime B aains a local max, hus conains a ime B aains a local max. Coninuing a heme, having shown ha he se of imes Brownian moion aains a maximum is big in he sense of densiy in [, ), we now show ha i is small in he sense of cardinaliy. Firs we mus prove wo lemmas. Lemma 1. On any wo fixed inervals, B almos surely does no aain he same maximum. Proof. Fix wo disjoin closed inervals, le M be he maximal process and le m be he maximum value aained by B on he firs of he wo disjoin inervals. Name he second inerval [a, b]. Consider he ime τ inf{ a B m}. Since m is known sricly before he ime a, τ is a sopping ime. If τ [a, b], hen B never aains m on [a, b] and cerainly does no have m as a maximum on his inerval, so suppose τ [a, b]. Since B b m almos surely, we can suppose τ [a, b). By he srong Markov propery, X B +τ B τ B +τ m is a Brownian moion. Fix < ɛ < b τ so ha (τ, ɛ + τ) [a, b). We know here exiss some s (, ɛ) such ha X s > since X is a sandard Brownian moion. Thus here exiss some s s + τ (τ, ɛ + τ) (a, b] such ha B s m + X s τ m + X s > m, showing ha he maximum of B on [a, b] is sricly greaer han m. I is imporan o noe ha we have no proved ha almos surely no wo disjoin inervals have he same maximum. In fac, his is no rue; almos surely, here exis disjoin inervals wih he same maximum. Le T a,n be he nh ime he process hi a. Since B is sricly above or below a on he inervals (T a,n 1, T a,n ) for all n, and i has equal probabiliy of being above or below, here almos surely exiss some n such ha he maximum of B on [T a,n 1, T a,n ] is a, which is also he maximum of B on [, T a,1 ]. 11

12 Lemma. Almos surely, every local maximum of Brownian moion is a sric local maximum. Proof. Suppose B aains a local maximum a ime s. Then here exiss some δ such ha B s is a maximum on he inerval (s δ, s+δ). If B s is no a sric maximum, hen here exiss some r (s δ, s + δ) such ha B r B s. We can find disjoin closed inervals wih raional endpoins around r and s conained in (s δ, s + δ). Bu, wih probabiliy 1, B does no aain he same maximum on any wo raional inervals, so such an r almos surely does no exis. Thus, almos surely, every ime B aains a local maximum, i aains a sric local maximum. Wih hese wo lemma we can move on o a major resul. Theorem 8. Almos surely, he se of imes B aains a local maximum is counable. Proof. Le S be he se of imes B aains a local maximum. Define f : Q Q S by f(p, q) inf{ p B max p s q (B s )}. This will conain S, and hus show ha S is almos surely counable, if B almos surely aains a sric maximum on some raional inerval a every ime i aains a local maximum. By he above lemma, we can almos surely find neighborhoods N s around each s S on which B s is a sric maximum. And, by he densiy of Q R, we can find raional inervals wihin each N s conaining each s on which B s is a sric maximum. This classificaion of local maxima is very imporan: now, if we prove ha some propery almos surely holds for an arbirary local maximum, we can ake a counable union and show ha he propery almos surely holds for all local maxima. We demonsrae his echnique. Corollary 1. The local maxima of B are almos surely disinc. Proof. Since here is a counable number of imes B aains a local maximum, here is a counable number of disjoin inervals on which hese local maxima are maxima, and hus here is counable number of pairs of hese inervals. We know ha, almos surely, B does no aain he same maximum on any pair of inervals. Thus we ake a counable union over all pairs of disjoin inervals on which a local maximum of B is a max and prove he corollary. We move on o providing a few compuaions. The reflecion principle will be our key ingredien. 1

13 Proposiion 6. T a has he disribuion given by Proof. P (T a d) a π 3 exp( a )d P (T a ) P (M a) P (B a) s a πs 3 exp( a s )ds a 1 π exp( x )dx subsiuing x a. Differeniaing wih respec o gives he resul. We will use he reflecion principle again in he nex proposiion. Proposiion 7. The processes M and M B have he join disribuion Proof. P (M da, M B db) (a + b) π 3 + b) exp( (a )dadb P (M a, B b) P (T a, B b) P (T a, a B b) P (a B b) P (B a b) a (y b) exp( )dy π a b 1 π exp( x )dx where B is he Brownian moion B refleced abou T a and we subsiued x y b. Taking he derivaive wih respec o b we ge P (M a, B db) Now differeniaing wih respec o a we ge So P (M a, B db) a (y b) (y b) exp( )dydb π 3 (a b) π 3 exp( P (M a, M B db) P (M da, B a db) (a b) )dadb (a + b) π 3 + b) exp( (a )dadb 13

14 Corollary. We can now show ha M B has he same disribuion as B. Inegraing P (M a, M B db) over a gives P (M B da) (a + b) exp( )da exp( a π π )da This proposiion leads us o anoher calculaion. Proposiion 8. Le θ max{s B(s) M(s)}. The join disribuion of θ and M is given by P (M da, θ ds) a π ( s)s 3 exp( a s )dads and hus he cumulaive disribuion funcion of θ is P {θ s} π arcsin s Proof. Le X be he sandard Brownian moion given by X B +s B s and le N be he maximal process of X. Then N s max{x u u s} max{b u+s B s u s} max{b u s u } B s We know θ s if and only if M s M which happens if and only if M s max{b u s u } N s + B s, and we can almos surely replace he inequaliy by a sric inequaliy. So P (M da, θ s) P (M s da, M s B s > N s ) P (M s da, M s B s > c N s dc) P (N s dc) c [, ) P (M s da, M s B s > c) P (N s dc) c [, ) P (M s da, M s W s db) P (N s dc) c [, ) b>c c da π s( s) da π s( s) (a + b) πs 3 da π exp( a ) exp( (a + b) s exp( (a + c) s s (c + a) exp( a s s c ) exp( π( s) ( s) )dbdcda c ) exp( ( s) )dc s( s) ) exp( a )dc exp( u )du 14

15 where we subsiued u respec o s, we ge s c+ a s( s) P (M da, θ ds) da π exp( a )(exp( (a c + a s. Differeniaing wih s( s) s s s ) da π exp( a )( ( s) a exp( a )ds) s ( s)s 3 a π ( s)s 3 exp( a s )dads To ge o he cumulaive disribuion of θ, we inegrae: P (θ s) P (M da, θ dr) s s r [,s] a [, ) a π ( r)r 3 exp( a r )dadr du π 1 u π arcsin where we subsiued u r. s Acknowledgmens s )ds) d s ds (a ) s dr π ( r)r The help and guidance my menor Yuan Shao provided was invaluable. He suggesed a opic which urned ou o be ineresing and useful, provided ineresing problems and insighful advice along he way, and was even willing o help me chug hrough some ugly inegrals. I would also like o he Universiy of Chicago REU for providing me wih a producive and fun summer doing mah. References [1] Peer Mörers and Yuval Peres, Brownian Moion, hp:// 15