Lecture 5: Stochastic Processes (II)

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1 Mirada Holmes-Cerfo Applied Stochastic Aalysis, Sprig 2015 Lecture 5: Stochastic Processes (II) Readigs Recommeded: Pavliotis [2014] sectios Pavliotis [2014] sectios , 2.5 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 5.1 Karhue-Loeve decompositio I this sectio we ll cosider a geeral spectral represetatio that works eve for o-statioary processes. Cosider a stochastic process X(t) that has mea zero, ad is cotiuous i the L 2 -sese: EX 2 t <, EX t = 0, lim h 0 E X t+h X t 2 = 0. Its covariace fuctio B(s,t) = EX s X t is symmetric, positive-defiite, ad cotiuous. Suppose the process is defied o a bouded, measurable set D R, e.g. D = [0,1],[a,b], etc. Let s defie a itegral operator K : L 2 (D) L 2 (D) as (K f )(s) B(s,t) f (t)dt (1) It ca be show that this operator is: (i) positive semi-defiite: K f, f 0 for all f, where u,v = D uv dt is the L2 (D) ier product (ii) self-adjoit: K f,g = f,kg D (iii) compact: if { f } =1 is a bouded sequece i L2 (D), the {K f } =1 has a coverget subsequece. Remark. For a fiite-dimesioal aalogy, cosider the matrix K = (B(t i,t j )) i, j=1 ad let u = (u(t i)) be a vector. The Ku is just a matrix (which is symmetric ad oegative) times a vector. Remark. By (i), (ii) it follows that all its eigevalues are real ad oegative. From (iii), the spectral theorem for compact operators implies that K has a coutable sequece of eigevalues tedig to 0. By Mercer s theorem, K has a orthoormal basis of eigefuctios {e i } ad its eigevalues {λ i} are real ad oegative, with 0 as the oly accumulatio poit. Furthermore, we ca write the kerel as B(s,t) = λ i e i (s)e i (t), (2)

2 where the covergece of the sum is absolute ad uiform o D D, ad we ca write every fuctio f L 2 (D) as f = f e (t), where the series coverges i L 2. Our goal is to write a stochastic process also i this form; basically the oly differece will be that the coefficiets f will be radom variables. Theorem (Karhue-Loeve). Let X t be a mea-zero stochastic process defied o t D = [a,b] with X t L 2 (Ω D), i.e. D EX t 2 <. Suppose it has cotiuous covariace fuctio B(s, t), whose correspodig itegral operator has eigevalues {λ i } ad orthoormal eigefuctios {e i}. The X t = ξ i e i (t), with ξ i = X t e i (t) dt. (3) D The above series coverges i mea-square (L 2 (Ω)) ad uiformly i t, i.e. lim sup t D E(X t ξ i e i (t)) 2 0. The coefficiets are radom variables with the followig properties: (i) zero-mea: Eξ i = 0 (ii) ucorrelated: Eξ i ξ j = δ i j λ j Proof. (Mostly from Pavliotis [2014], p.20) First, suppose that X t has represetatio (3). Let s check the properties of the coefficiets. We have Eξ i = (EX t )e i (t) dt = 0 D which shows property (i). We used Fubii s Theorem to iterchage the expectatio ad the itegral, which we ca do because the assumptio that X t L 2 (Ω D) implies that X t L 1 (Ω D). To show property (ii), calculate: Eξ i ξ j = = D D D E(X s X t )e i (s)e j (t) dsdt = B(s,t)e i (s)e j (t) dsdt D D λ j e i (s)e j (s)ds = λ j δ i j, where the secod-last step follows because e j is a eigefuctio, ad the last step because e i, e j are orthogoal if i j. Agai, we used Fubii s Theorem to iterchage the expectatio ad the itegral; this will be a commo calculatio i this course ad we will usually assume it is possible to do so without further remarks. 2

3 To show covergece of the series to X t, cosider the partial sums S N = N =1 ξ e (t). The E X t S N 2 = EX 2 t + ES 2 N 2E(X t S N ) = B(t,t) + E N k,l=1 ξ k ξ l e k (t)e l (t) 2E N = B(t,t) + k e k=1λ 2 N k (t) 2E k=1 = B(t,t) N k=1 λ k e 2 k (t) ( D B(s,t) X t N =1ξ e (t) ) (just expad) X t X }{{} s e k (s) e k (t) ds (subs. for ξ ) } {{ } λ k e k (t) The fial lie goes to 0 uiformly i t, by Mercer s theorem. Example (Gaussia process). If X t is Gaussia the we ca completely specify the radom coefficiets ξ i. Ideed, ξ i N(0,λ i ), ad ξ i,ξ j are idepedet whe i j. To show that ξ i is Gaussia, we must show X t e i (t) dt is Gaussia. It is clear that if we use a Riema sum approximatio for the itegral this will be Gaussia, sice a fiite sum of Gaussia radom variables is Gaussia. The limit of a sequece of Gaussia radom variables which coverges i probability (amely the Riema sums) is also Gaussia; this is somethig that ca ca be show by cosiderig the characteristic fuctios ad usig Levy s cotiuity theorem. To show that ξ i,ξ j are idepedet for i j, it is eough to kow they are ucorrelated ad Gaussia, sice ucorrelated Gaussia radom variables are idepedet. Note this will ot be true i geeral. This provides a simple way to simulate a Gaussia process, by trucatig ad discretizig its KL expasio. Example (Browia motio). Oe way to defie Browia motio (the Wieer process) W t is by the followig properties: (i) it is Gaussia; (ii) it has mea m(t) = 0, covariace B(s, t) = mi(s, t). (iii) with probability 1, t W t is cotiuous, ad W 0 = 0. Let s calculate the KL expasio o [0,1] usig these properties. First, we fid the eigefuctios. We must solve 1 (s t)e k (t) dt = λ k e k (s) = Takig d ds, d 2 ds 2 gives 0 s 0 1 te k (t)dt + se k (t)dt = λ k e k (s). (4) s 1 λ k e k (s) = se k(s) + s e k (t)dt se k (s), λ k e k = e k. Settig s = 0,1 i (4) to fid e k (0) = 0, e k (1) = 0. Solvig this Sturm-Liouville problem gives ( λ k = π(k 1 ) 2 2 ), e k (s) = 2si (π(k 12 ) )s, k = 1,2,... 3

4 Therefore W t = k=1 2 ξ k (π(k π(k 1 2 ) si 12 ) )t, ξ k N(0,1), i.i.d. Note that this is ot the oly way to expad W t i a set of basis fuctios, for example the Haar basis is also used i some applicatios. 5.2 Browia motio 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 moved about at radom, eve whe the water appeared to be very still. He repeated the experimet with dust particles, ad foud the same behaviour, so he argued the motio was urelated to the fact that the polle came from livig matter. Several mathematicias tried to explai this behaviour (Theile, 1880, Bachelier, 1900). However the pheomeo really took off as a model i physics after a paper by Eistei i 1905, which 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. Here is a movie showig 1µm glass beads sittig i water uder a microscope: Here is aother oe, with actual polle grais (but you have to put up with the soudtrack): 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. The most commo way to defie a Browia Motio is by the followig properties: Defiitio. A Browia motio or Wieer process W t, t 0 is a real-valued stochastic process such that 4

5 (i) if t 0 < t 1 < < t, the W(t 0 ),W(t 1 ) W(t 0 ),...,W(t ) W(t 1 ) are idepedet. (ii) For all s,t 0, the radom variable W(s +t) W(s) is Gaussia, with mea zero ad variace s. (iii) With probability 1, t W t is cotiuous, ad W(0) = 0. Remark. It is possible to replace (ii) with the coditio that the icremets W(s +t) W(t) do ot deped P( W(s+t) W(t) δ) o t, plus a cotiuity coditio that lim s 0 s = 0 for all δ > 0. Oe ca the show the icremets must be Gaussia. See Briema [1992], Ch. 12, p.248. Property (i) says the icremets are idepedet. Property (ii) says the icremets are statioary (they have the same distributio for fixed icremet size), ad Gaussia. The above defiitio implies that i the example i sectio 5.1, ad vice-versa. ELFS (or see Durrett [2005], p.373.) A importat questio is whether such a process exists. The aswer is yes, ad is a subtle, techical questio that is dealt with i may probability books, e.g. Durrett [2005], p.373, Karatzas ad Shreve [1991], Briema [1992]. The major difficulty is i showig property (iii): that there exists a versio of Browia motio that is cotiuous everywhere, almost surely Browia motio as a limit of radom walks A ituitive way of thikig about Browia motio is as a limit of radom walks. Let X 1,X 2,... be iid radom variables with mea 0 ad variace 1. Cosider the sum S = j=1 X j, with S 0 = 0. This is a discrete-time process, but we ca make a cotiuous-time process by liearly iterpolatig betwee values of S. We wat the iterpolated process to approach Browia motio as the umber of steps goes to ifiity, so we will eed to scale time ad space appropriately. To figure out how, suppose time of the approximate Browia motio is t = t for some t, ad suppose we scale the jumps to have size x. We eed t, x 0, but how should they be related? Note that ES 2 =, so E( xs t/ t ) 2 ( x) 2 t/ t. Sice EWt 2 ad x = 1/. Let the iterpolated, rescaled process be W t = t, we should choose x = t. Let t = 1/, = S [t] + (t [t])s [t]+1, (5) where [t] meas the largest iteger less tha or equal to t. We would like to show that Wt of covergece. coverges i some way to Browia motio. For this, we eed a defiitio 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. 5

6 If X,X are real-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. (The former defiitio works for radom variables takig values i a path space, while the latter works also for real-valued radom variables.) We kow by the Cetral Limit Theorem that Wt d N(0, t) as. Therefore the oe-poit distributios of Wt coverge to those of Browia motio. To show that the d-poit distributios coverge uses a similar techique as i the proof of the CLT; oe cosiders the characteristic fuctios ad shows these coverge. See Karatzas ad Shreve [1991], p. 67. Showig the etire process Wt coverges i distributio to W t is sigificatly harder ad is the statemet of Doker s Ivariace Priciple; see Karatzas ad Shreve [1991] p.70. This is a geeralizatio of the CLT to path space Properties of Browia motio Scalig properties (i) W t is a Browia motio (symmetry) (ii) W t+s W s for fixed s is a Browia motio (time-homogeeity) 1 (iii) c W ct, with c > 0 is a fixed costat, is a Browia motio (scalig) (iv) tw 1/t is a Browia motio (time-iversio) Proof. (i), (ii), (iii) follow straightforwardly from the defiitio. To check (iv), we use the defiitio 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 If limsup is replaced by limif i either of the above, the limits are 1. a.s.. Proof. (i) (from Briema [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 6

7 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) For the secod part, see e.g. Briema [1992], p. 263, or Karatzas ad Shreve [1991], p The first part follows from the secod usig the time iversio property (iv) of BM. (ii) This follows from the Law of the Iterated Logarithm. Differetiability Theorem. With probability oe, Browia paths are ot Lipschitz cotiuous (ad hece ot differetiable) at ay poit. Remark. We ca see this heuristically as follows. First let s argue why it should ot be differetiable at t = 0. This implies it s ot differetiable aywhere, by the time-homogeeity property (ii) (though be careful: we should really show it s ot differetiable all poits at oce.) If it were differetiable at 0, the we would have W t W = lim t t 0 t = lim s sw 1/s = lim t W t, where W t is aother BM, by the scalig property (iv). But this limit does t exist. Remark. Here is aother heuristic argumet. We have EWt 2 so we would ot expect the derivative to exist at 0. = t, so Var( W t t ) = t = 1 t 2 t. This as t 0, Proof. (From Briema [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/. (6) Let A = {ω : there is a s [0,1] s.t. W t W s 2β t s whe t s 2/}. 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 (6) 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 C = 2 k=1 { B( ) : y k 6β }, 7

8 so 2 P(C ) k=1 P ( Wk+2 P max{ Wk+1, Wk+1 Wk, W k Wk 1 ( max { } W3/ W 2/, W2/ W 1/, W1/ 6β W1/ = P( 6β ) 3 ( 6β/ = e x2 /2 dx 2π 6β/ ( 1 6β = e x2 /2 dx 2π 6β ) 3 ) 3 } 6β ) ) The fial itegral coverges to 0 as, so P(W ) 0. Corollary. Almost surely, every sample path of W t has ifiite variatio o every fiite iterval. Proof. If a fuctio has bouded variatio o a iterval I, the it has a derivative existig almost everywhere o I. 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 > 1 2. Proof. See Karatzas ad Shreve [1991], Durrett [2005] Quadratic variatio Eve though Browia motio is ot differetiable, hece has ifiite variatio, it actually has fiite quadratic variatio. This will be a very importat property that we will use whe costructig the stochastic itegral. Defiitio. The pth-variatio of a fuctio f o [a,b] give partitio σ = {t 0,t 1,...,t }, with a = t 0 < t 1 < < t = b is V p [a,b] ( f ;σ) = f (t i ) f (t i 1 ) p. Defiitio. The quadratic variatio of a fuctio f o [0,t] is ofte writte as Qt σ ( f ) V[0,t] 2 ( f,σ), or just if the fuctio is clear. Explicitly, we have Q σ t Q σ t ( f ) = i=0 f (t i+1 ) f (t i ) 2. 8

9 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. 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. W t+1 W ti 2 m.s. t. 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. Proof. (From Koralov ad Siai [2010], p. 269.) ( ( ) 2 E Qt σ (W t ) t = E (Wti W ti 1 ) [ 2 (t i t i 1 ) ]) 2 = = 4 E [ (W ti W ti 1 ) 2 (t i t i 1 ) ] 2 E(W ti W ti 1 ) 4 + (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 (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 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 ), (7) where t 0 < t 1 < < t, ad F i are Borel sets. This seems a lot like the Markov property that we studied i Lectures 2&3 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 (7) is actually the defiitio of a geeral Markov process, 1 ad that Browia motio satisfies this coditio (Koralov ad Siai [2010], p.278.) Therefore Browia motio is our first example of a cotiuous-time, cotiuous-space Markov process; there are may more to come. 1 For a formulatio of the Markov property i terms of σ-algebras, see Lecture 3 otes. 9

10 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 a trasitio desity p(x,t y,s) as 1 P(Γ,t y,s) = e (x y)2 2(t s) dx = p(x, t y, s)dx (8) Γ 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. We were able to write is dow explicitly i (8), because we kow the icremets of Browia Motio are Gaussia. A approach to studyig Markov processes that is commo i the Physics literature 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.) We will explore such techiques o the homework. For ow, let s cosider how the trasitio probabilities for Browia motio evolve, both forward ad backward i time. The trasitio probability desity for Browia motio is statioary i both time ad space: p(x, t y, s) = p(x y,t s 0,0). I additio, oe ca check that it satisfies the PDE This is a versio of the Kolmogorov forward equatio. It also satisfies a PDE i s,y: p t = 1 2 p, p(x,0 y,0) = δ(x y). (9) 2 x2 p s = 1 2 p, p(x,t y,t) = δ(x y). (10) 2 y2 This is a versio of the Kolmogorov backward equatio. It should be solved backward i time i s, startig at s = t. 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 very similar: Defiitio. The geerator of a Markov process is the operator A defied o a subset D(A ) of the set of bouded, measurable fuctios f by A f lim t 0 T t f f t E x f (X t ) f (x) = (A f )(x) = lim, (11) t 0 t The set D(A ) o which this limit exists is the domai of A. We have itroduced the operator T t f (x) E x f (X t ) = f (y)p(dy,t x,0), (12) which acts o the set of bouded, measurable fuctios f. Here E x meas the expected value, give X 0 = x. The covergece is uderstood as the orm covergece, i.e. we mea that lim t 0 t 1 (T t f f ) A f = 0, for a appropriate orm o the fuctio space. R 10

11 Remark. The set of operators {T t } t 0 forms a operator semigroup, i.e. T s T t f = T s+t f. Note the relatio to the Chapma-Kolmogorov equatios P s P t = P s+t i Lecture 3. You ca show this by cosiderig the geeral versio of the Chapma-Kolmogorov equatios: P(B,s x,t) = R P(B,x y,u)p(dy,u x,t), t u s. Let s calculate the geerator of Browia motio. Suppose f C 0 (R). E x f (X t ) f (x) 1 (A f )(x) = lim = lim ( f (y) f (x))p(y,t x,0)dy t 0 t t 0 t R 1 t [ ] = lim s f (y) f (x))p(y,s x,0)dy ds t 0 t 0 R( 1 t = lim f (y) f (x)) t 0 t 0 R( 1 2 p 2 y 2 (y,s x,0)dyds 1 = lim t 0 t 1 = lim t 0 t = 1 2 t 0 t 0 2 f x 2 (x) R 1 2 f 2 y 2 p(y,s x,0)dyds E x 2 f x 2 (X t) (sice ca t put t uder itegral) (usig the forward equatio) (after itegratig by parts twice) Note that i the secod step, it would be ice to differetiate uder the itegral ad replace t p with 1 2 yy p, directly, but we do t kow that we ca sice p t t=0 is ubouded, so we have to get aroud that by first writig p as a time itegral of its time derivative. This shows that the geerator of Browia motio is the Laplacia: A f = f. Note the aalogy to x 2 Lecture 3: 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 A o a Baach space to be the ifiitesimal geerator of a Markov process (alteratively, a strogly cotiuous oe-parameter semigroup.) Why are we iterested i the geerator? This is the fudametal object from which we ca describe the evolutio of probability ad measurable statistics. Let s start with the latter. Let s defie u(x,t) E x f (X t ) = T t f (x). To fid out how u evolves i time, we calculate: We obtai the u t = lim T t+h f T t f h 0 h Backward Kolmogov Equatio. u t T h (T t f ) T 0 (T t f ) = lim = A T t f = A u. (13) h 0 h = A u, u(x,0) = f (x). (14) 11

12 Note that we could have factored the other way i (13), ad foud that u t = lim T t (T h f ) T t (T h f ) = T t A f, h 0 h assumig that we ca iterchage the limit ad expectatio. (See the hadout for a discussio of coditios uder which this is possible; basically we eed to adapt the Bouded/Mootoe/Domiated Covergece theorems to the case of radom variables.) This shows that for a certai class of processes, T t A = A T t, i.e. the geerator ad trasitio operator commute. Now cosider how the probability desity evolves. This derivatio will be formal ad heuristic, but you ca fid more rigorous derivatios elsewhere. 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. Therefore 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 that u t = T ta f = p(y,t x,0)a f (y)dy = (Ay p) f (y)dy. Here A is the adjoit of A with respect to the L 2 ier product, i.e. ua v = va u, ad the subscript is a remider that it acts o the y-variables of p. We also kow that u t = p t (x,t y,0) f (y)dy. (We ca differetiate uder the itegral provided p t exists ad is locally i L 1. As i the case for Browia motio above, this wo t hold at t = 0, but we ca use the same trick to get aroud it so we just assume it holds here.) Sice the two calculatios hold for all test fuctios f, we have that p t = Ay p(y,t x,0) (weakly). Itegratig over the iitial desity ρ 0 (x) gives Forward Kolmogov Equatio. ρ t = A ρ, ρ(0,y) = ρ 0 (y). (15) 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, (15) must be iterpreted i weak form, for example whe the probability measure is restricted to a submaifold so it cotais δ-fuctios. Refereces Leo Briema. Probability. SIAM, Rick Durrett. Probability: Theory ad Examples. Thomso, 3rd editio, G. Grimmett ad D. Stirzaker. Probability ad Radom Processes. Oxford Uiversity Press, I. Karatzas ad S. E. Shreve. Browia Motio ad Stochastic Calculus. Spriger, L. B. Koralov ad Y. G. Siai. Theory of Probability ad Radom Processes. Spriger, G. A. Pavliotis. Stochastic Processes ad Applicatios. Spriger,