Lecture 7: Brownian motion

Transcription

1 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Lecture 7: Browia motio Readigs Recommeded: Pavliotis (2014) sectio 1.3, Optioal: Grimmett ad Stirzaker (2001) 8.5, 8.6, 9.6, Koralov ad Siai (2010) Ch. 18, Karatzas ad Shreve (1991), 2.9 (ad other bits of Chapter 2), for detailed results about Browia motio 7.1 Itroductio Browia motio is perhaps the most importat stochastic process we will see i this course. It was first brought to popular attetio i 1827 by the Scottish botaist Robert Brow, who oticed that polle grais suspeded i water move about at radom eve whe the water appears to be very still. Oe explaatio for this observatio could be that the grais come from livig matter ad are movig about of their ow accord. I order to rule out this explaatio, Brow repeated the experimet with iaimate objects like dust particles, ad foud the same behaviour. He was ot the first perso to otice this peculiar motio for example, it is described by a Roma Lucretius i a poem i 60 BC, ad poited out by Dutch scietist Ja Igehousz i 1785 but he was the first to ivestigate it so systematically. Here is a movie showig polle grais diffusig i a still liquid (third movie): Ad here is a movie showig 1µm glass beads sittig i water uder a microscope: Brow s work ispired both physicists to ivestigate the pheomeo. It was fially uderstood by Eistei i 1905, who showed how the radom motio could arise if water were made of may discrete compoets, rather tha formig a cotiuum. He argued that this idirectly cofirmed that matter was made of atoms. Smoluchowski costructed a related model i The equatios were verified experimetally by Jea Baptiste Perri i 1908, who received the Nobel prize for his work. Meawhile, mathematicias realized that a fuctio describig a particle movig accordig to Browia motio would have some bizarre properties, ad that it was ot kow how to actually costruct such a fuctio mathematically. Several mathematicias worked o this problem (e.g.theile, 1880, Bachelier, 1900), but it was ot resolved util Wieer gave a rigorous costructio of a Browia motio i For this reaso i mathematics Browia motio is ofte called a Wieer process.

2 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Figure 1: Some approximate realizatios of Browia motio. These were costructed by simulatig a radom walk with i.i.d. steps with distributio N(0, t), at times t = The total time of each realizatio is 10 uits. 7.2 Defiitios We ll start by lookig at how to costruct a stochastic process that could possibly model the erratic motio of a dust particle or other processes that are very radom. Ideally we d like the process to be as radom as possible i some sese, because if there is ay determiistic or predictable part of the process, the we could model that i a more covetioal way. We could do this by askig for a cotiuous process whose derivative is idepedet at each poit i time the idea beig that if the derivative has some correlatios, the we could predict the value of the process at least some time ito the future, ad the it would t be as radom as possible. It turs out that the best approximatio for such a process is a Browia motio. We ll first study the path properties of Browia motio, ad the we ll look at what we ca say about its statistics. Browia motio is our first example of a diffusio process, which we ll study a lot i the comig lectures, so we ll use this lecture as a opportuity for itroducig some of the tools to thik about more geeral Markov processes. The most commo way to defie a Browia Motio is by the followig properties: Defiitio (#1.). A Browia motio or Wieer process (W t ) t 0 is a real-valued stochastic process such that (i) W 0 = 0; (ii) Idepedet icremets: the radom variables W v W u, W t W s are idepedet wheever u v s t (so the itervals (u,v), (s,t) are disjoit.) (iii) Normal icremets: W s+t W s N(0,t) for all s,t 0. (iv) Cotiuous sample paths: with probability 1, the fuctio t W t is cotiuous. Property (iii) says the icremets are statioary. Therefore a Browia motio has statioary, idepedet icremets,, just like the Poisso process; the differece is that Browia motio has ormal icremets. It is also cotiuous. With these properties i had we ca say a lot about the trajectories ad statistics of the process. Therefore 2

3 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 it is atural to woder does such a process actually exist? Ad if so, is it uiquely defied by the four properties above? The aswer is yes to both questios, ad is proved i may probability books, e.g. Durrett (2005), p.373, Karatzas ad Shreve (1991), Breima (1992). The major difficulty is i showig property (iv): that there exists a versio of Browia motio that is cotiuous everywhere, almost surely. Remark. It is possible to replace (iii) with the coditio that the icremets W s+t W t do ot deped o t, P( W plus a cotiuity coditio that lim s+t W t δ) s 0 s = 0 for all δ > 0. Oe ca the show the icremets must be Gaussia. See Breima (1992), Ch. 12, p.248. We have already see a Browia motio, i Lecture 5, where we defied it as a particular Gaussia process. Recall that we defied a Browia motio as Defiitio (#2.). A Browia motio or Wieer process is a stochastic process W = (W t ) t 0 with the followig properties: (i) W 0 = 0; (ii) It is a Gaussia process; (iii) It has mea m(t) = 0 ad covariace B(s, t) = mi(s, t). (iv) Cotiuous sample paths: with probability 1, the fuctio t W t is cotiuous. It turs out that Defiitios 1&2 are equivalet (see e.g. Durrett (2005), p.373.), ad you ca work with whichever oe is more coveiet for the problem at had. Proof. Let s show that Defiitio #1 Defiitio #2. Give a BM (#1), we ca calculate the covariace of B s, B t as Cov(B s,b t ) = EB s B t (EB s )(EB t ) = EB s B t. For s < t, we write B t = (B t B s ) + B s ad so EB s B t = EB s (B t B s + B s ) = (EB s )E(B t B s ) + EB 2 s = s. The secod step follows sice B t B s is idepedet of B s (property (ii)), ad the third because the icremets have a give distributio (property (iii).) Therefore the BM satisfies property (iii) of Defiitio #2. To show that a BM (#1) is a Gaussia process, we must calculate its fiite dimesioal distributios. Cosider the two-poit distributios. Let p t,s (x,y) be the joit probability desity of (W t,w s ), i.e. the desity correspodig to the probability P(W s = x,w t = x). The we have p t,s (x,y) = P(W s = y,w t W s = x y) = P(W s = y)p(w t W s = x y) = p t s (x y)p s (y), where p s (y) is the probability desity of W s. The secod step follows by idepedet icremets ad the third by statioary icremets. Substitutig the kow Gaussia distributio of the icremets shows p t,s (x,y) = 1 e (x y)2 2(t s) 2π(t s) 1 2πs e y2 2s. You ca show this is a multivariate Gaussia by maipulatig the quadratic terms i the expoetial. To obtai ay fiite-dimesioal distributio we defie the trasitio desity to be p(x,t y,s) = 1 2π(t s) e (x y)2 2(t s) (1) 3

4 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 ad proceed i a similar fashio to write p t1,t 2,...,t (x 1,x 2,...,x ) = 1 2πt1 e x2 1 2t 1 p(x 2,t 2 x 1,t 1 ) p(x,t x 1,t 1 ). This is agai a multivariate Gaussia. Therefore, we have show that Defiitio #1 Defiitio #2. Exercise Show that Defiitios #2 Defiitio # Browia motio as a limit of radom walks A ituitive way of costructig about Browia motio is as a limit of radom walks. Let X 1,X 2,... be i.i.d. radom variables with mea 0 ad variace 1. For the sake of illustratio let s suppose that X i = ±1 with equal probability; though the argumet below will hold i the more geeral case. Cosider the sum S = j=1 X j, with S 0 = 0. This is a simple symmetric radom walk o the itegers. It is a discrete-time process, but we ca make a cotiuous-time process by liearly iterpolatig betwee values of S. Cosider the properties of (S t ) t N : (i) ES t = 0 (ii) Var(S t ) = t (iii) (S t ) t N has statioary icremets. To see why, ote that S t S s S t s sice both are a fuctio of t s distict X i. (iv) (S t ) t N has idepedet icremets. To see why, let 0 < q < r < s < t, ad ote that S t S s = X s X t, S r S q = X q X r. But the two sets of sequeces ivolve distict X i, so they are idepedet. (v) For t large, S t N(0,t), by the Cetral Limit Theorem. Therefore (S t ) t N has a lot of the properties of a Browia motio. We might woder if there is a way to scale it so it approaches a Browia motio i some limit. We will costruct such a limit by scalig space ad time i a particular way. Suppose we scale space by some factor x, ad time by some factor t. That is, we let the jumps have size x, ad we let the time betwee jumps be t. The rescaled process is S t, x t = x S t/ t = x ( X X t/ t ). (2) We wat to cosider the limit of the process (St t, x ) t N/ t as t, x 0, but how should these small parameters be related? If the limit is to approach somethig fiite, the we eed the variace to be fiite to. Sice Var(S t, x ) = ( x) 2 t t, we should choose ( x) 2 = costat. (3) t Let s suppose the costat equals 1 so the limitig process has the same variace at a poit as a Browia motio. This is a importat poit for a diffusio process, space scales as the square root of time. We will call this diffusive scalig. It will come up agai ad agai throughout the course. 4

5 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Let s write t = 1/, x = 1/, ad defie a sequece of processes i terms of parameter. Sice our origial process was a discrete-time process, it is coveiet to make a cotiuous-time process by liearly iterpolatig betwee the discrete values of t. The iterpolated, rescaled process is S () t = S [t] + (t [t])s [t]+1, (4) where [t] meas the largest iteger less tha or equal to t. The secod term o the RHS is oly eeded for the iterpolatio. The Dosker s Theorem or Dosker s Ivariace Priciple says that (S () t ) t 0 coverges i distributio to a Browia motio (W t ) t 0. Essetially, this meas that all the fiite-dimesioal distributios of (S () t coverge to the fiite-dimesioal distributios of a Browia motio. (For a rigorous defiitio of weak covergece, see the Appedix.) Dosker s Theorem is a powerful theorem, which is a geeralizatio of the Cetral Limit Theorem to path space. Provig Dosker s Theorem is a fair amout of work; see e.g. Durrett (1996) p.287 or Karatzas ad Shreve (1991) p.70. However, it is ot hard to see why the fiite-dimesioal distributios should coverge, at least heuristically. For the oe-poit distributios, the Cetral Limit theorem gives that S () t ) t 0 d N(0,t) as (with a little bit of care with the iterpolated parts.) Therefore the oe-poit distributios of coverge to the oe-poit distributios of Browia motio. For the two-poit distributios, we eed to cosider the joit distributio of pairs of radom variables of ( the form St, S s (agai igorig the iterpolated parts here.) Usig a similar techique as i the proof of the CLT (cosiderig (( ) the ( characteristic fuctios) )) shows that such a radom vector has a distributio which 0 mi(s,t) 0 coverges to N,. 0 0 mi(s, t) ) Oe ca treat the k-poit distributios similarly, see e.g. Karatzas ad Shreve (1991), p. 67. The fial part is to show the distributio of the etire process coverges i some sese, for all values of t at oce. We wo t do this here. 7.4 Properties of Browia motio Scalig properties (i) ( W t ) t 0 is a Browia motio (symmetry) (ii) (W t+s W s ) t 0 for fixed s is a Browia motio (traslatio property) 1 (iii) c W ct, with c > 0 is a fixed costat, is a Browia motio (scalig) (iv) (tw 1/t ) t 0 is a Browia motio (time-iversio) Property (iii) shows that Browia motio is like a fractal: it looks statistically the same at all scales, o matter how much you zoom i, provided that space ad time are scaled i the right way (agai, we see the diffusive scalig space time. ) This property follows aturally from the costructio of Browia motio as a limit of radom walks. 5

6 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Proof. (i), (ii), (iii) follow straightforwardly from Defiitio #1, by checkig the required coditios are satisfied. For example, for (iii): let X t = c 1/2 W ct. The (i) X 0 = c 1/2 W 0 = 0. (ii) X t has idepedet icremets this is straightforward to check. (iii) Normal icremets: for t s, X t X s = c 1/2 (W ct W cs ) c 1/2 N(0,c(t s)) N(0,t s). (iv) Cotiuity this follows from cotiuity of W t. To check (iv), we use Defiitio #2 of BM as a Gaussia process. We have that tw t is Gaussia, with mea 0. It has covariace fuctio EstW 1/s W 1/t = st ( 1 s 1 ) t = s t. It is cotiuous for t > 0. It remais to check W that it is cotiuous at 0. But lim t 0 tw 1/t = lim s s s 0 a.s., by the result below Behaviour as t There are several ways to characterize BM i the limit as t : (i) lim t W t t = 0 a.s.. W (ii) limsup t t t =, W limif t t t = (both a.s.). (iii) (Law of the Iterated Logarithm) lim sup t W t 2t loglogt = 1 a.s., limsup t 0+ W t 2t loglog1/t = 1 a.s.. If limsup is replaced by limif i either of the above, the limits are 1. Proof. (i) (from Breima (1992), p. 265) This follows from the Strog Law of Large Numbers. For N we ca write W = (W 1 W 0 ) + (W 2 W 1 ) (W W 1 ), which is a sum of iid radom variables. By the SLLN, W / 0 a.s.. To obtai behaviour at o-iteger t, let Z k = max B(k +t) B(k). 0 t 1 For t [k,k + 1], W t t W k k 1 k(k + 1) W k + 1 k + 1 Z k. The first term o the RHS 0 a.s., ad Z k has the same distributio as max 0 t 1 W t. It ca be show that EZ k <, ad that this implies Z k /k 0 a.s.. (iii) See e.g. Durrett (2005) sectio 7.9 p.431, or Breima (1992), p. 263, or Karatzas ad Shreve (1991), p Oe oly eeds to show oe limit, sice they are related to each other by the time iversio property (iv) of BM. (ii) This follows from the Law of the Iterated Logarithm. 6

7 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Figure 2: A sketch of φ h (s,t). From (Evas, 2013) Differetiability Theorem. With probability oe, sample paths of a Browia motio are ot Lipschitz cotiuous (ad hece ot differetiable) at ay poit. A rigorous proof is i the appedix. Here is a heuristic explaatio for why the derivative does t exist, at least at a sigle poit. Suppose we try to calculate the derivative as dw t dt = lim h 0 W t+h W t h But W t+h W t N(0,h) so the variace of the term uder the limit is Var. (5) ( ) Wt+h W t h = 1 h. The quatity o the RHS uder the limit as h 0, so we would ot expect the derivative to exist i ay fiite sese at 0. Aother way to see this, which uses the trasformatio properties of BM, is to argue that X h = W t+h W t is a Browia motio, by the traslatio property, ad the that Y s = sx 1/s for s = 1/t is a Browia motio, by the time-iversio property. But the limit above equals lim s Y s, which does t exist. Of course, the theorem makes a much stroger argumet, which is that the derivative does t exist aywhere, with probability 1. That is, for ay give path, there is ot eve a sigle poit at which a sample path of Browia motio is differetiable! Nevertheless, i the Physics literature it is commo to speak of the derivative of Browia motio, where it is called white oise. It turs out that eve though white oise does t exist i a classical sese, we will work with it i a weaker sese whe we defie stochastic itegrals. Let s set aside for a momet our mathematical kowledge ad preted that the derivative of Browia motio exists, ad see what we ca lear about it. Let ξ t dw t dt where dw t dt is defied i (5). If this is a stochastic process the we ca calculate its covariace fuctio, formally at least. From (5) we have that (e.g. Evas, 2013, p.41) Cov(ξ s,ξ t ) = Eξ s ξ t = lim h 0 φ h (s,t) where ( )( ) Wt+h W t Ws+h W s φ h (s,t) = E = 1 [(t + h) (s + h) (t + h) s t (s + h) +t s]. h h h2 7

8 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 See Figure 2 for a sketch of φ h (s,t). If s t, the for h < t s the quatity o the RHS equals 0. For s = t, it equals h/h 2, which as h 0. However, sice φ h (s,t)ds = 1 for fiite h, we expect this itegral should hold i the limit, ad therefore we expect φ h (s,t) δ(s t) i some sese as h 0. Therefore we expect that, formally at least, we should have i.e. the covariace fuctio should be a delta-fuctio. Eξ s ξ t = δ(s t), (6) Why should ξ t be called white oise? Notice that its covariace fuctio depeds oly o s t so ξ t is weakly statioary, so we ca calculate its spectral decompositio from the last lecture. The spectral desity of δ(t) is f (λ) = 1 e iλt δ(t) dt = 1 for all λ. 2π 2π The spectral desity is flat: all frequecies cotribute equally, just as all frequecies of light cotribute to make white light. Here are some other facts about the sample path properties of Browia motio. Theorem. (a) With probability 1, a Browia sample path is locally Hölder cotiuous with expoet γ for every γ (0, 1 2 ). (b) With probability 1, Browia paths are owhere locally Hölder cotiuous for ay expoet γ > 2 1. Proof. See Karatzas ad Shreve (1991), Durrett (2005). 7.5 Quadratic variatio Recall that the cocept of total variatio from aalysis: Defiitio. The total variatio of a fuctio f (t) o a iterval [a,b] is defied by V [a,b] ( f ) = sup σ i=1 f (t i ) f (t i 1 ), (7) where the supremum is over all partitios σ = {t 0,t 1,...,t } of [a,b] with a = t 0 < t 1 < < t = b. If V [a,b] ( f ) < the f is said to be of bouded variatio, ad if V [a,b] ( f ) = the f is said to be of ifiite variatio. If a fuctio is of bouded variatio o [a,b], the a theorem i aalysis says it has a derivative almost everywhere o [a, b] (i.e. except for a set of measure zero.) Coversely, if a fuctio is owhere differetiable, the it must have ifiite variatio o ay iterval. Sice Browia motio is owhere differetiable, it has ifiite variatio o ay iterval. However, it is actually ot all that bad as a fuctio: it actually has fiite quadratic variatio (i a mea-square sese.) This will tur out to be a very importat property whe we costruct the stochastic itegral. 8

9 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Defiitio. The pth-variatio of a fuctio f o [a,b] is V p [a,b] ( f ) = sup σ i=1 f (t i ) f (t i 1 ) p. (8) where the supremum is over all partitios σ = {t 0,t 1,...,t } of [a,b] with a = t 0 < t 1 < < t = b. Whe p = 2 the V[a,b] 2 ( f ) is called the quadratic variatio of f o [a,b]. We would like to calculate the quadratic variatio of W t. However, sice the fuctio is a stochastic process we will eed to take care with how the supremum over the partitios is calculated. It turs out that we ca calculate it as a particular kid of limit, but ot all types of stochastic covergece will give a result that is fiite. We will use the followig otio of covergece. Defiitio. A sequece of radom variables X 1,X 2,... coverges i mea-square to aother radom variable m.s. X, writte X X, if E X X 2 0 as. Remark. Covergece i mea-square implies covergece i probability, which i tur implies covergece i distributio. Covergece i mea-square does ot imply almost sure covergece or vice versa. (Almost sure covergece does imply covergece i probability ad hece covergece i distributio.) We will use the idea of covergece i mea-square to calculate the supremum i (8). We eed otatio that explicitly idetifies the partitio ivolved i the sum. Defiitio. The quadratic variatio of a fuctio f o [0,t] with respect to a partitio σ is Q σ t ( f ) = i=0 f (t i+1 ) f (t i ) 2. (9) Lemma. The quadratic variatio of Browia motio Qt σ (W t ) coverges i mea-square to t as σ 0, where σ = max i t i+1 t i, i.e. i=1 W t i+1 W ti 2 m.s. t. Proof. (From Koralov ad Siai (2010), p. 269.) ( ( ) 2 E Qt σ (W t ) t = E (Wti W ti 1 ) i=1[ 2 (t i t i 1 ) ]) 2 = = 4 i=1 i=1 i=1 E [ (W ti W ti 1 ) 2 (t i t i 1 ) ] 2 E(W ti W ti 1 ) 4 + i=1 (see explaatio below) (t i t i 1 ) 2 sice (a b) 2 a 2 + b 2 for a,b 0 (t i t i 1 ) 2 sice E(W t W s ) 4 = 3 t s 2 4 max (t i t i 1 ) 1 i = 4t σ i=1 (t i t i 1 ) The secod equality uses the fact that E ( (W ti W ti 1 ) 2 (t i t i 1 ) )( (W t j W t j 1 ) 2 (t j t j 1 ) ) = 0 if i j. 9

10 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Remark. This meas that icremets of W t behave as t, i.e. formally we ca write ( W) 2 = t, or (dw t ) 2 = dt. We see the diffusive scalig arise yet agai. Remark. Whe we say the quadratic variatio of Browia motio is fiite, we mea that the limit of Q σ t (W t ) exists i mea-square or i probability (which is implied by it existig i a mea-square sese.) It is ot true that the limit exists almost surely. I fact, oe ca show if oe uses the classical defiitio of quadratic variatio by takig the supremum of the sum over all partitios, the Browia motio has ifiite quadratic variatio o ay iterval. Remark. A fuctio with o-zero quadratic variatio must have ifiite total variatio. To see why, write i=0 f (t i+1 ) f (t i ) 2 max f (t i+1) f (t i ) i i=0 f (t i+1 ) f (t i ) As σ 0, max i f (t i+1 ) f (t i ) 0 by defiitio, but the quatity o the LHS goes to a fiite umber. Therefore we must have i=0 f (t i+1) f (t i ). But i=0 f (t i+1) f (t i ) V [a,b] ( f ) so the total variatio is ifiite. Of course, this argumet is for a determiistic fuctio, but you could adapt it for a stochastic oe. 7.6 Browia motio as a Markov process Suppose we kow the value of Browia motio for all times up to some time s. What ca we say about W t, for t > s? Sice W t = W s + (W t W s ), ad the icremet W t W s is idepedet of all observatios up to time s, we ca obtai the distributio of W t usig oly our kowledge of W s, ad ot ay earlier observatios. I other words, we ca write P(W t F W t 1 F 1,...,W t0 F 0 ) = P(W t F W t 1 F 1 ), (10) where t 0 < t 1 < < t, ad F i are Borel sets (evets.) This seems a lot like the Markov property that we studied for Markov chais the differece is that previously, we cosidered Markov chais that could take o a discrete set of values, whereas Browia motio ca take o a cotiuum. It turs out that (10) is actually the defiitio of a geeral Markov process, ad that Browia motio satisfies this coditio (Koralov ad Siai (2010), p.278.) 1 Therefore Browia motio is our first example of a cotiuous-time, cotiuous-space Markov process; there are may more to come. Sice Browia motio is Markov, it makes sese to study its trasitio fuctio P(Γ,t y,s) = P(W t Γ W s = y), where s < t ad Γ is a Borel set. This ca be writte i terms of the trasitio desity p(x,t y,s) as P(Γ,t y,s) = Γ 1 e (x y)2 2(t s) dx = p(x, t y, s)dx (11) 2π(t s) Γ The trasitio desity p(x,t y,s) gives the probability desity to be at x at time t, give the process was at y at time s. It is the cotiuous aalog of the trasitio matrix that we itroduced for Markov chais. 1 See Appedix for the rigorous formulatio of the Markov property. 10

11 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 We are able to write it dow explicitly i (11), because we kow the icremets of Browia Motio are Gaussia. A geeral Markov process is time-homogeeous if the trasitio fuctio oly depeds o the differece i times: P(Γ,t x,s) = P(Γ,t s x,0) P(Γ,t s x). The trasitio fuctio for a geeral Markov process satisfies the Chapma-Kolmogorov equatios, which ow take this form: Chapma-Kolmogorov Equatio. P(Γ,t x,s) = P(Γ, t y, u)p(dy, u x, s) (12) for all s u t. For a time-homogeeous Markov process, the Chapma-Kolmogorov equatios are P(Γ,t + s x) = P(Γ, t x)p(dz, s x). (13) See e.g. Pavliotis (2014), p. 35 for a (formal) proof. You could also check it holds directly for the case of a Browia motio for which you have a explicit formula for the trasitio probabilities. Oe approach to studyig Markov processes is to start with a set of trasitio fuctios or desities with kow ifiitesimal properties, ad to show they satisfy certai evolutio equatios (ad/or aalyze the cosistecy of the iitial assumptios) (e.g. Gardier, 2009). We will explore such techiques o the homework. Aother approach to studyig Markov processes is through the trasitio semigroup (Durrett, 1996, Chapter 7), (Varadha, 2007). Let (X t ) t 0 be a time-homogeeous Markov process, ad let s defie the operator T t f (x) E x f (X t ) = f (y)p(dy,t x), (14) R where E x meas the expected value, give X 0 = x, ad t 0. The domai of T t is the set of bouded, measurable fuctios f : R R. We calculate (T t+s ) f (x) = f (y)p(dy,t + s x) = f (y)p(dy, x z)p(dz, t x) = (T s f )(z)p(dz,t x) Therefore = (T t T s )( f )(x). T s+t f = (T s T t ) f. (15) This is the semigroup property for the set of operators {T t } t 0. Note the relatio to the Chapma-Kolmogorov equatios P s P t = P s+t for discrete-time Markov chais. We use the orm f = sup x f (x) ad work i the space of bouded measurable fuctios L = { f : f < }. You ca see that T t is a cotractio semigroup o L, i.e. T t f f, ad therefore T t is a bouded liear operator. (Exercise: show this!) 11

12 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 For a cotiuous-time Markov chai, we calculated the ifiitesimal geerator ad derived a umber of properties from this. It turs out this cocept is also very useful whe studyig more geeral Markov processes. The defiitio of the geerator is similar: Defiitio. The ifiitesimal geerator of a Markov process is the operator L defied by L f lim t 0 T t f f t E x f (X t ) f (x) (L f )(x) = lim, (16) t 0 t The set D(L ) L o which this limit exists is the domai of L. The covergece is uderstood as the orm covergece, i.e. that lim t 0 t 1 (T t f f ) g = 0 for some fuctio g, which is idetified as L f. Formally, we ca write 2 Tt = e L t I +tl + t2 2 L 2 +. (17) Example. Let s calculate the geerator of Browia motio (followig Varadha, 2007). We have that (T t f )(x) = The ifiitesimal geerator is 1 f (y) e (y x)2 2t dy = 2πt (L f )(x) = lim t 0 f (x + z t) f (x) t f (x + z 1 t) e z2 2 dz. 2π 1 e z2 2 dz. (18) 2π Suppose f is bouded ad has three bouded derivatives, so we ca expad it usig Taylor s formula as f (x + z t) f (x) = z t f (x) + tz2 2 f (x) +t 3/2 R(t,z) where the remaider term satisfies R(t,z) C z 3 for some costat C. Substitutig this expasio ito (18) ad calculatig the itegral directly shows that The geerator of Browia motio is the Laplacia operator. L f = 1 2 f = 1 d 2 f 2 dx 2. (19) Remark. Note the aalogy to the geerator for a cotiuous-time Markov chai: there, the geerator was a matrix applied to a vector (or, i the case of the Poisso process, a liear fuctio applied to a coutable sequece.) Here, the geerator is still liear, but ow it acts o fuctios, ot vectors: it is a liear operator. This will be true for Markov processes i geeral: their evolutio ca be described by a geerator, which is always a liear fuctioal. The fuctioal wo t always be a partial differetial operator though for a wide class of processes, amely diffusio processes, it is. Aother commo possibility is a itegral operator, which occurs for jump processes. The Hille-Yosida theorem provides the coditios for a closed liear operator L o a Baach space to be the ifiitesimal geerator of a Markov process. 2 This ca be show to be true rigorously wheever L is bouded, or (equivaletly) T t is a uiformly cotiuous semigroup. We will always assume these coditios hold. 12

13 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Why are we iterested i the geerator? This is the fudametal object from which we ca describe the evolutio of probability ad measurable statistics. Notice that we ca approximate E x f (X t ) f (x) +tl f (x), so L describes what happes to the process i a small amout of time, i.e. how the probability moves aroud. Differet kids of terms i L, i.e. first derivatives, secod derivatives, itegrals, etc, have differet physical meaig. Let s derive (formally) a more specific evolutio equatio. Notice that, from differetiaig the formal relatio (17) term-by-term, we obtai dt t dt = L e L t = L T t, Therefore T t L = L T t, ad by itegratig, we have that dt t dt = e L t L = T t L. (20) t T t f = f + T s L f ds. (21) 0 This calculatio was formal, but it ca be made rigorous with a few extra lies of work, see e.g. Durrett (1996), Sectio 7.1 Theorem 1.5 p.247. We use the first relatio i (20) to get a versio of the Kolmogorov backward equatio. Backward Kolmogorov Equatio. Let u(x,t) E x f (X t ) = T t f (x). If f D(L ), the u t = L u, u(x,0) = f (x). (22) Now let s cosider how the probability measure for the process evolves. Let µ t (dx) be the law of the process at time t, i.e. µ t (A) = P(X t A) = P(dy,t x,0)µ 0 (dx). y A x R Let s defie a operator T t to be the pushforward of the iitial probability measure: T t µ(dx,0) µ(dx,t). Tt is a operator that acts o probability measures ad produces a probability measure. The T t, Tt formally adjoits i L 2. To see why, ote that we have ad also Therefore T t f = R R f (x)µ t (dx) = R T t f = T t f (x) µ 0 (dx). R f (x)(t t µ 0 (dx)) T t f (x) µ 0 (dx) = f (x)(tt µ 0 (dx)) (23) R or i other words T t f, µ 0 = f,t t µ 0, where f,g = R f (x)g(x)dx is the L2 ier product. 13 are

14 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Let L be the L 2 -adjoit of the geerator L, i.e. it is such that (L f )gdx = f (L g)dx R for all fuctios f,g such that f is i D(L ), g is a probability measure. The from (23) you ca show that = e tl. A similar argumet that led to the backward equatio gives the forward equatio: T t Forward Kolmogorov Equatio. µ t t R = L µ t, µ 0 = (give iitial desity.) (24) Remark. The forward equatio may be easier to uderstad i the case whe the process has a smooth probability desity for all time. Suppose the probability desity at time t is ρ(y,t) = ρ 0 (x)p(y,t x,0)dx, where ρ 0 (y) is the iitial desity. To compute ρ t, we eed a equatio for t p(y,t x,0). We will do this idirectly, workig with our previous calculatios for the backward equatio. Let f be a suitable test fuctio, e.g. a bouded, measurable fuctio with compact support. If we let u = T t f as before, we kow from the backward equatio that u t = T tl f = p(y,t x,0)l f (y)dy = (Ly p) f (y)dy. Here L is the adjoit of L with respect to the L 2 ier product, i.e. ul v = vl u, ad the subscript o L is a remider that it acts o the y-variables of p. We also kow that u p t = (x,t y,0) f (y)dy. t (We ca differetiate uder the itegral provided p t exists ad is locally i L 1. This wo t ecessarily hold at t = 0, but we ca get aroud this by itegratig over small times ad the takig the derivative; we ll just igore this issue for ow.) Sice the two calculatios hold for all test fuctios f, we have that p t = Ly p(y,t x,0) (weakly). Itegratig over the iitial desity ρ 0 (x) gives This is a versio of the forward equatio formulated i terms of desities. ρ t = L ρ, ρ(0,y) = ρ 0 (y). (25) Remark. The forward equatio exists for all the processes we will ecouter, though i geeral its mathematical existece is trickier to establish tha the backward equatio. Sometimes, (25) must be iterpreted i weak form, for example whe the probability measure is restricted to a submaifold so it cotais δ-fuctios. Example. For Browia motio, the backward equatio for a fuctio u(x,t) is u t = 1 2 u 2 x 2, u(x,0) = f (x) I terms of the trasitio probabilities p(x,t y,t) the backward equatio is p s = 1 2 p 2 y 2, p(x,t y,t) = δ(x y). 14

15 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 This equatio should be solved backward i time i s, startig at s = t. The forward equatio for the probability desity ρ(x,t) is ρ t = 1 2 ρ 2 x 2, ρ(x,0) = ρ 0(x). I terms of the trasitio probabilities p(x,t y,s) the backward equatio is p t = 1 2 p 2 x 2, p(x,0 y,0) = δ(x y). You ca check that these equatios hold directly from the explicit form of p(x,t y,s). I geeral, you wo t have a explicit formula for the trasitio probabilities, so you will obtai them by solvig the correspodig PDEs. 7.7 Appedix Covergece i distributio Here is the defiitio of covergece i distributio (see e.g. Durrett (1996), Chapter 8, Theorem 1.1.) Defiitio. Cosider a sequece of radom variables X 1,X 2,... defied o a sequece of probability spaces {(Ω,F,P )} =1 ad takig values i some metric space (S,ρ). Let (Ω,F,P) be aother probability space, o which aother radom variable X is defied, which takes values i (S,ρ). The {X } =1 coverges i d distributio, or coverges weakly to X, writte X X, if E f (X ) E f (X) for all bouded, cotiuous, real-valued fuctios f, where E,E deote expectatios with respect to the measures associated with X,X respectively. If X,X are real-valued (i.e. ot path-valued) the a equivalet defiitio is that F (x) F(x) at each poit of cotiuity of F(x), where F,F are the cumulative distributio fuctios of the radom variables. If the probability space is R (so each X is a stochastic process), the weak covergece is equivalet to covergece of fiite dimesioal distributios; see e.g. Durrett (1996), Chapter 8 Theorem 2.7 p Browia motio is ot differetiable Proof of the o-differetiability of Browia motio. This proof is from Breima (1992), p. 261, i tur from Dvoretsky, Erdös, ad Kakutai (1961. The same proof is preseted i Durrett (2005), p 377. Notice that if a fuctio x(t) has a derivative x (s), with x (s) < β at some poit s [0,1], the there is a 0 such that for > 0, x(t) x(s) 2β t s, if t s 2/. (26) Let A = {ω : there is a s [0,1] s.t. W t W s 2β t s whe t s 2/}. 15

16 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 The A icrease with, ad the limit set A icludes the set of all sample paths o [0,1] havig a derivative at ay poit which is less tha β i absolute value. If (26) holds, the let k be the largest iteger such that k/ s, so that } Therefore, if we let y k = max{ Wk+2 Wk+1, Wk+1 Wk, W k Wk 1 { C = B( ) : at least oe y k 6β }, 6β. the A C. To show P(A) = 0, which implies the theorem, it is sufficiet to get lim P(C ) = 0. But so 2 P(C ) k=1 P C = ( Wk+2 P max{ 2 k=1 Wk+1 { B( ) : y k 6β, Wk+1 }, Wk, W k Wk 1 ( max { } W3/ W 2/, W2/ W 1/, W1/ 6β = P( W 1/ 6β ( = 2π 6β/ 6β/ ) 3 ) 3 e x2 /2 dx ( 1 6β = e x2 /2 dx 2π 6β The fial itegral coverges to 0 as, so P(W ) 0. ) 3 } 6β ) ) Markov property The Markov property i cotiuous time ca be formulated more rigorously i terms of σ-algebras. Let (Ω,F,P) a the probability space ad let {F t } t 0 be a filtratio: a icreasig sequece of σ-algebras such that F t F for each t, ad t 1 t 2 F t1 F t2. We suppose the process X t is adapted to the filtratio {F t } t 0 : each X t is measurable with respect to F t. For example, this will be true automatically if we let F t be the σ-algebra geerated by (X s ) 0 s t, i.e. geerated by the pre-images Xs 1 (B) for Borel sets B R. The X t has the Markov property if E( f (X t ) F s ) = E( f (X t ) σ(x s )) for all 0 s t ad bouded, measurable fuctios f. Aother way to say this is P(X t A F s ) = P(X t A σ(x s )), where P( ) is a regular coditioal probability (see Koralov ad Siai (2010), p.184.) 16

17 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2017 Refereces Breima, L. (1992). Probability. SIAM. Durrett, R. (1996). Stochastic Calculus: A practical itroductio. CRC Press, Taylo & Fracis Group. Durrett, R. (2005). Probability: Theory ad Examples. Thomso, 3rd editio. Evas, L. C. (2013). A Itroductio to Stochastic Differetial Equatios. America Mathematical Society. Gardier, C. (2009). Stochastic methods: A hadbook for the atural scieces. Spriger, 4th editio. Grimmett, G. ad Stirzaker, D. (2001). Probability ad Radom Processes. Oxford Uiversity Press. Karatzas, I. ad Shreve, S. E. (1991). Browia Motio ad Stochastic Calculus. Spriger. Koralov, L. B. ad Siai, Y. G. (2010). Theory of Probability ad Radom Processes. Spriger. Pavliotis, G. A. (2014). Stochastic Processes ad Applicatios. Spriger. Varadha, S. R. S. (2007). Stochastic Processes, volume 16 of Courat Lecture Notes. America Mathematical Society. 17