Diffusion Processes. Chapter What is a Diffusion Process?

Transcription

1 Chapter 2 Diffuion Procee 2.1 What i a Diffuion Proce? When we want to model a tochatic proce in continuou time it i almot impoible to pecify in ome reaonable manner a conitent et of finite dimenional ditribution. The one exception i the family of Gauian procee with pecified mean and covariance. It i much more natural and profitable to take an evolutionary approach. For implicity let u take the one dimenional cae where we are trying to define a real valued tochatic proce with continuou trajectorie. The pace Ω = C[,T i the pace on which we wih to contruct the meaure P. We have the σ-field B t = σ{x() : t} defined for t T. The total σ-field B = B T. We try to pecify the meaure P by pecifying approximately the conditional ditribution P [x(t+h) x(t) A B t. Thee ditribution are nearly degenerate and and their mean and variance are pecified a and E P [ x(t + h) x(t) B t = hb(t, ω)) + o(h) (2.1) E P [ (x(t + h) x(t)) 2 B t = ha(t, ω)) + o(h) (2.2) a h, where for each t b(t, ω) anda(t, ω) are B t meaurable function. Since we init on continuity of path, thi will force the ditribution to be nearly Gauian and no additional pecification hould be neceary. We will devote the next few lecture to invetigate thi. Equation (2.1)and (2.2) are infiniteimal differential relation and it i bet to tate them in integrated form that are precie mathematical tatement. We need ome definition. Definition 2.1. We ay that a function f :[,T Ω R i progreively meaurable if, for every t [,T the retiction of f to [,t Ω i a meaurable function of t and ω on ([,t Ω, B[,t B t ) where B[,t i the Borel σ-field on [,t. 17

2 18 CHAPTER 2. DIFFUSION PROCESSES The condition i omewhat tronger than jut demanding that for each t, f(t, ω) ib t i meaurable. The following fact are elementary and left a exercie. Exercie 2.1. If f(t, x) i meaurable function of t and x, thenf(t, x(t, ω)) i progreively meaurable. Exercie 2.2. If f(t, ω) i either left continuou (or right continuou) a function of t for every ω and if in addition f(tomega)ib t meaurable for every t, then f i progreively meaurable. Exercie 2.3. There i a ub σ-field Σ = Σ pm B[,T B T ) uch that progreive meaurability i jut meaurability with repect to Σ pm. In particular tandard operation performed on progreeively meaurable function yield progreively meaurable function. We hall alway init that the function b(, ) anda(, ) be progreively meaurable. Let u uppoe in addition that they are bounded function. The boundedne will be relaxed at a later tage. We reformulate condition 2.1 and 2.2 a and M 1 (t) =x(t) x() M 2 (t) =[M 1 (t) 2 b(, ω)d (2.3) a(,ω))d (2.4) are martingale with repect to (Ω), B t,p). We can then define a Diffuion Proce correponding to a, b a a meaure P on (Ω), B) uch that relative to (Ω), B t,p) M 1 (t) andm 2 (t) are martingale. If in addition we are given a probability meaure µ a the initial ditribution, i.e. µ(a) =P [x() A then we can expect P to be determined by a, b and µ. We aw already that if a 1andb, with µ = δ, we get the tandard Brownian Motion. a = a(t, x(t)) and b = b(t, x(t)), we expect P to be a Markov Proce, becaue the infiniteimal parameter depend only on the current poition and not on the pat hitory. If there i no explicit dependence on time, then the Markov Proce can be expected to have tationary tranition probabilitie. Finally if a(t, x) = a(t) i purely a function of t and b(t, ω)) = b 1 (t) + c(t,)x()d i linear in ω), then one expect P to be Gauian, if µ i o. Becaue the pathe are continuou the ame argument that we provided earlier can be ued to etablih that Z λ (t) =exp[λm 1 (t) λ2 2 =exp[λ[x(t) x() a(,ω)d b(,ω)d λ2 2 a(,ω)d (2.5)

3 2.1. WHAT IS A DIFFUSION PROCESS? 19 i a martingale with repect to (Ω), B t,p) for every real λ. We can alo take for our definition of a Diffuion Proce correponding to a, b the condition that Z λ (t) be a martingale with repect to (Ω), B t,p) for every λ. If we do that we did not have to aume that the path were almot urely continuou. (Ω, B t,p) could be any pace uppporting a tochatic proce x(t,ω) uch that the martingale property hold for Z λ (t). If C i an upper bound for a, itieay to check with M 1 (t) defined by equation (2.5) [ E P exp[λ[m 1 (t) M 1 ( exp[ λ2 C 2 The lemma of Garia, Rodemich and Rumey will guarantee that the path can be choen to be continuou. Let (Ω, F,P) be a Probability pace. Let T be the interval [,Tforome finite T or the infinite interval [, ). Let F T Fbe ub σ-field uch that F F t for, t T with <t. We can aume with out lo of generality that F = t T F t. Let a tochatic proce x(t,ω) with value in R n be given. Aume that it i progreively meaurable with repect to (Ω, F t ). We can eaily gneralize the idea decribed in the previou ection to diffuion procee with value in R n. Given a poitive emidefinite n n matrix a = a i,j and an n-vector b = b j, we define the operator (L a,b f)(x) = 1 2 n a i,j 2 f x i x j (x)+ i,j=1 n f x j (x) If a(t,ω)=a i,j (t,ω)andb(t,ω)=b j (t,ω) are progreively meaurable function we define (L t,ω f)(x) =(L a(t,ω),b(t,ω) f)(x) Theorem 2.1. The following defintion are equivalent. x(t,ω) i a diffuion proce correponding to bounded progreively meaurable function a(, ), b(, ) with value in the pace of ymmetric poitive emidefinite n n matrice, and n-vector if 1. x(t,ω) ha an almot urely continuou verion and and are (Ω, F t,p) martingale. 2. For every λ R n y i (t,ω)=x i (t,ω) x i (,ω) z i,j (t,ω)=y i (t,ω) y j (t,ω) j=1 b(,ω)d a i,j (,ω)d Z λ (t,ω)=exp [ <λ,y(t,ω) > 1 2 <λ,a(,ω)λ >d i an (Ω, F t,p) martingale.

4 2 CHAPTER 2. DIFFUSION PROCESSES 3. For every λ R n [ λ (t,ω)=exp i<λ,y(t,ω)+ 1 <λ,a(,ω)λ >d 2 i an (Ω, F t,p) martingale. 4. For every mooth bounded function f on R n with atleat two bounded continuou derivative f(x(t,ω)) f((x(,ω)) i an (Ω, F t,p) martingale. (L,ω f)(x(,ω))d 5. For every mooth bounded function f on T R n with atleat two bounded continuou x derivative and one bounded continuou t derivative f(t,x(t,ω)) f(, (x(,ω)) ( f + L,ωf)(,x(,ω))d i an (Ω, F t,p) martingale. 6. For every mooth bounded function f on T R n with atleat two bounded continuou x derivative and one bounded continuou t derivative [ exp f(t,x(t,ω)) f(, (x(,ω)) ( f + L,ωf)(,x(,ω))d 1 < ( f)(,x(,ω)),a(,ω)( f)(,x(,ω)) >d 2 i an (Ω, F t,p) martingale. 7. Same a (6) except that f i replaced by g of the form g(t,x)=< λ,x > +f(t,x) where f i a in (6) and λ R n i arbitrary. Under any one of the above definition, x(t,ω) ha an almot urely continuou verion atifying [ P up y(,ω) y(,ω) l 2n exp[ l2 t Ct for ome contant C depending only on the dimenion n and the upper bound for a. Here y i (t,ω)=x i (t,ω) x i (,ω) b i (,ω)d

5 2.1. WHAT IS A DIFFUSION PROCESS? 21 Proof. (1) implie (2). Thi wa eentially the content of Theorem and the comment of the previou ection. Alo we aw that the exponential inequality i a conequence of Doob inequality. (2) implie (3). The condition that Z λ (t) i a martingale can be rewritten a a whole collecction of identitie Z λ (t,ω)dp = Z λ (,ω)dp (2.6) A that i valid for every t>, A F and λ R n. Both ide of eqation (2.6) are well defined when λ R n i replaced by λ C n, with complex component and define entire function of the n complex variable λ. Since they agree when the value are real, by analytic continuation, they mut agree for all purely imaginary value of λ a well. Thi i jut (3). (3) implie (4). Thi part of the proof require a imple lemma. Lemma 2.2. Let M(t,ω) be a martingale relative to (ΩF t,p) which ha almot urely continuou trajectorie and A(t,ω) be a progreively meaurable proce that i for almot all ω a continuou function of bounded variation in t. Aume that for every t the random variable ξ(t,ω)=up t M(t) Var [,t A(t,ω) ha a finite expectation. Then T η(t) =M(t)A(t) M()A() M()dA() i again a martingale relative to (Ω, F t,p). Proof. (of lemma.) We need to prove that for every <t, E P [M(t)A(t) M()A() A M(u)dA(u) F = a.e. We can ubdivide the interval [, t into ubinterval with end point = t < t 1 < <t N = t, and approximate M(u)dA(u) by N j=1 M(t j)[a(t j ) A(t j 1 ). The fact that A i continuou and ξ(t) i integrable make the approximation work in L 1 (P )othat [ E P M(u)dA(u) N F = lim N EP M(t j )[A(t j ) A(t j 1 ) F = lim N EP = lim N EP j=1 N [M(t j )A(t j ) M(t j )A(t j 1 ) F j=1 N [M(t j )A(t j ) M(t j 1 )A(t j 1 ) F j=1 = E P [M(t)A(t) M()A()

6 22 CHAPTER 2. DIFFUSION PROCESSES and we are done. We ued the martingale property in going from the econd line to the third when we replaced M(t j )A(t j 1 )bym(t j 1 )A(t j 1 ) Now we return to the proof of the theorem. Let u apply the above lemma with M λ (t) = λ (t) and A λ (t) =exp[i <λ,b() >d 1 <λ,a()λ >d. 2 Then a imple computation yield M λ (t)a λ (t) M λ ()A λ () = e λ (x(t) x()) 1 M λ ()da λ () (L,ω e λ )((x() x())d where e λ (x) =exp[i<λ,x>. Multiplying by exp[i <λ,x() >, which i eentially a contant, we conclude that e λ (x(t)) e λ (x()) (L,ω e λ )((x())d i a martingale. The above expreion i jut what we had to prove, except that our f i pecial namely, the exponential e λ (x). But by linear combination and limit we can eaily pa from exponential to arbitray mooth bounded function with two bounded derivative. We firt take care of infinitely diffrentiable function with compact upport by Fourier integral and then approximate twice differentiable function with thoe. (4) implie (3). The tep can be retraced. We tart with the martingale defined by (4) in the pecial cae of f being e λ and chooe A λ (t) =exp[ i <λ,b() >d+ 1 <λ,a()λ >d 2 and do the computation to get back to the martingale of type (3). (4) implie (5). Thi i baically a computation. If f(t,x) can be approximated by mooth function and o we may aume with out lo of generality more

7 2.1. WHAT IS A DIFFUSION PROCESS? 23 derivative. where E P [f(t,x(t)) f(,x()) F = E P [f(t,x(t)) f(t,x()) F +E P [f(t,x()) f(,x()) F = E P [ (L u,ω f(t, ))(x(u))du F +E P f [ u (u,x())du F = E P [ + E P [ = E P [ (L u,ω f(u, ))(x(u))du F + E P [ (L u,ω [f(t, ) f(u, ))(x(u))du F + E P [ f u (u,x(u))du F [ f u (u,x()) f u (u,x(u))du F [ f u +(L u,ωf)(u,x(u))du F +J J = E P [ + E P [ = E P [ (L u,ω [f(t, ) f(u, ))(x(u))du F [ f f (u,x()) u u (u,x(u))du F ( f v L u,ωf)(v,x(u))du dv F u u E P [ [ = E P =. (L v,ω f u )(u,(x(v))du dv F u v t v u t f (L u,ω )(v,(x(u))du dv v f (L v,ω )(u,(x(v))du dv u The two integral are identical, jut the role of u and v have been interchanged. (5) implie (4). Thi i trivial becaue after all in (5) we are allowed to take f to be purely a function of x. (5) implie (6). Thi i again the lemma on multiplying a martingale by a function of bounded variation. We tart with a function of the form exp[f(t,x) and the martingale exp[f(t,x(t)) exp[f(,x()) ( ef + L,ωe f )(,x())d

8 24 CHAPTER 2. DIFFUSION PROCESSES and ue A(t) =exp [ 1 2 ( f + L,ωf)(,x())d < ( f)(,x()),a()( f)(,x()) >d (6) implie (5). Thi jut again revering the tep. (6) implie (7). The problem here i that the function <λ, x>are unbounded. If we pick a function h(x) of one variable to equal x in the interval [ 1.1 and then level off moothly we get eaily a mooth bounded function with bounded derivative that agree with x in [ 1, 1. Then the equence h ( x) = kh( x k ) clearly converge to x, h k (x) x and more over h k (x) i uniformly bounded in x and k and h k (x) goe to uniformly in k. We approximate <λ,x>by j λ jh k (x j ) and conider the martingale [ exp λ j h k (x j (t)) λ j h k (x j ()) ψk λ ()d j j where ψ λ k () = and converge to ψ λ () = λ j b j (,ω)h k(x j ())d + 1 a j,j (,ω)h 2 k(x j ())d j j + 1 a i,j (,ω)λ i λ j h 2 i(x i ()h j(x j ()d i,j λ j b j (,ω)d j a i,j (,ω)λ i λ j d a k. By Fatou lemma the limit of nonnegative martingale i alway a upermartingale and therefore in the limit [ exp <λ,x(t) x() > ψ λ ()d i a upermartingale. In particular E P [ exp[< λ,x(t) x() > i,j ψ λ ()d 1 If we now ue the bound on ψ it i eay to obtain the etimate E P [exp[< λ,x(t) x() > C λ Thi provide the neceary uniform integrability to conclude that in the limt we have a martingale. Once we have the etimate, it i eay to ee that we can

9 2.2. RANDOM WALKS AND BROWNIAN MOTION 25 approximate f(t,x)+ <λ,x>by f(t,x)+ j λ jh k (x j ) and pa to the limit, thu obtaining (7) from (6). Of coure (7) implie both (2) and (6). Alo all the exponential etimate follow at thi point. Once we have the etimate there i no difficulty in obtainig (1) from (3). We need only take f(x) =x i and x i x j that can be jutified by the etimate. Some minor manipulation i needed to obtain the reult in the form preented. 2.2 Random walk and Brownian Motion Let 1, 2, be a equence of independent identically ditributed random variable with mean and variance 1. The partial um S k are defined by S = and for k 1 S k = k We recale and interpolate to define tochatic procee n (t) : t 1by ( k ) S k n = n n for k n and for 1 k n and t [ k 1 n, k n ( k ) (k 1 ) n (t) =(nt k +1) n +(k nt)n n n Let P n denote the ditribution of the proce n ( ) on = C[, 1 and P the ditribution of Brownian Motion, or the Wiener meaure a it i often called. We want to explore the ene in which lim P n = P n Lemma 2.3. For any finite collection t 1 < t 2 < < t m 1 of m time point the joint ditribution of (x(t 1 ),,x(t m )) under P n converge, a n, to the correponding ditribution under P. Proof. We are dealing here baically with the central limit theorem for um independent random variable. Let u define k i n =[nt i and the increment ξn i = S kn i S kn i 1 n for i =1, 2,,m with the convention kn =. Foreachn, ξn i are m mutually independent random variable and their ditribution converge a n to Gauian with mean and variance t i t i 1 repectively. We take t =. Thi i of coure the ame ditribution for thee increment under Brownian Motion. The interpolation i of no conequence, becaue the difference between the end point i exactly ome i n. So it doe not really matter if in the definition

10 26 CHAPTER 2. DIFFUSION PROCESSES of n (t) ifwetakek n =[nt ork n =[nt + 1 or take the interpolated value. We can tate thi convergence in the form lim EPn[ f(x(t 1 ),x(t 2 ),,x(t m )) = E P [ f(x(t 1 ),x(t 2 ),,x(t m )) n for every m, anym time point (t 1,t 2,,t m ) and any bounded continuou function f on R m. Thee meaure P n are on the pace of bounded continuou function on [, 1. The pace i a metric pace with d(f,g) =up t 1 f(t) g(t) a the ditance between two continuou function. The main theorem i Theorem 2.4. If F ( ) i a bounded continuou function on then F (ω)dp n = F (ω)dp lim n Proof. The main difference i that function depending on a finite number of coordinate have been replaced by function that are bounded and continuou, but otherwie arbitrary. The proof proceed by approximation. Let u aume Lemma 2.5 which aert that for any ɛ>, there i a compact et K ɛ uch that up n P n [ K ɛ ɛ and P [ K ɛ ɛ. From tandard approximation theory (i.e. Stone-Weiertra Theorem) the continuou function F, which we can aume to be bounded by 1, can be approximated by a function f depending on a finite number of coordinate uch that up ω Kɛ F (ω) f(ω) ɛ. Moreover we can aume without lo of generality that f i alo bounded by 1. We can etimate F (ω)dp n f(ω)dp n F (ω) f(ω) dp n +2P n [Kɛ c 3ɛ K ɛ a well a Therefore and we are done. F (ω)dp f(ω)dp F (ω) f(ω) dp +2P [Kɛ c 3ɛ K ɛ F (ω) dp n F (ω) dp 6ɛ + f(ω) dp n f(ω) dp Remark 2.1. We hall prove Lemma 2.5 under the additional auption that the underlying random variable i have a finite 4-th moment. See the exercie at the end to remove thi condition. Lemma 2.5. Let P n,p be a before. Aume that the random variable i have a finite moment of order four. Then for any ɛ> there exit a compact et K ɛ uch that P n [K ɛ 1 ɛ

11 2.2. RANDOM WALKS AND BROWNIAN MOTION 27 for all n and a well. Proof. The et P [K ɛ 1 ɛ K B,α = {f : f() =, f(t) f() B t α } i a compact ubet of for each fixed B and α. Theorem 1.3 can be ued to give u a uniform etimate on P n [KB,α c which can be made mall by taking B large enough. We need only to check that the condition (1.2) hold for P n with ome contant β,α and C that do not depend on n. Such an etimate clearly hold for the Brownian motion P. If { i } are independent identically ditributed random variable with zero mean, an elementary calculation yield E[( k ) 4 =ke[ k(k 1) [ E[ C1 k + C 2 k 2 (2.7) Let u try to etimate E[( n (t) n ()) 4. If t 2 n we can etiamte n (t) n () M t where M i the maximum lope. There are atmot three interval involved and which implie that E[M 4 n 2 E [ [max i, 2, 3 4 Cn 2 E Pn[ x(t) x() 4 t 4 E[M 4 C t 2 (2.8) If t > 2 n we can find t, uch that n,nt are integer, t t 1 n and 1 n. Applying the etimate (2.8) for the end piece that are increment over incomplete interval and the etimate (2.7) for the piece x(t ) x( ), we get E Pn [ x(t) x() 4 Cn 2 + C n t + C t 2 Since both t and t are atleat 1 n we obtain (1.2). Exercie 2.4. To extend the reult to the cae where only the econd moment exit, we do truncation and write i = Y i +Z i. The pair {(Y i,z i ):1 i n} are mutually independent identically ditributed random vector. We can aume that both Y i and Z i have mean. We can fix it o that Y i ha variance 1 and a finite fourth moment. Z i can be forced to have an arbitrarily mall variance σ 2.Wehave n (t) =Y n (t)+z n (t) and by Kolmogorov inequality P [ up Z n (t) δ δ 2 E [ [Z n (1) 2 = δ 2 σ 2 t 1 which can be made mall uniformly in n if σ 2 i mall enough. Complete the proof.

12 28 CHAPTER 2. DIFFUSION PROCESSES