Rates of convergence to the local time of a diffusion

Ann. hr.

Herwi

Poinctrrb,

Vol. 34, no 4. 1998. p, 505-544.

ProbubilitPs

rt Strttistiques

Rates of convergence to the local time of a diffusion Jean JACOD Laboratoire de ProbabilitCs, Universite P. et M. Curie et CNRS, UMR 7599, 4 Place Jussieu, 752.52-ParisCedex. France.

ABSTRACT. - In this paper we consider the approximation of the local time L, of a l-dimensional diffusion process X at some level :c, say :I’ = 0 by normalized sums, say U;, of functions of the values X, for %< 711as IL --i cc‘. Our main aim is to prove an associated functional central limit theorem fiving a mixed normal limiting value to the sequenceof processes l/F-( Il;’ - L,), for a suitable value of W. 0 Elsevier, Paris RESUME?. - Dans cet article nous considerons l’approximation du temps local Lt d’un processus de diffusion uni-dimensionnel au niveau :r = 0 par des sommes normalisees lJ+” de fonctions des valeurs X, pour :! 2 ,nt quand IL -+ 3~).Notre objectif principal est de montrer un thkbreme central limite fonctionnel, donnant la convergence de la suite YL”(U,” - A,) vers une limite qui est un melange de processus gaussien, pour une valeur convenable de (2. 0 Elsevier, Paris

1. INTRODUCTION

AND MAIN RESULTS

l-l) It is well known that one can approximate the local time of a Brownian motion, and more generally of continuous semimartingales, in many ways by some sorts of discretizations: one may either discretize “in space”, that is use the random times at which the process hits a grid of mesh l//r,, say (this includes counting the number of upcrossings from 0 to l/7/,), or one

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may discretize “in time”, that is use the values of the process at times i//l, and in both cases let 7~ go to DC. When the basic process is the Brownian motion, space-discretization approximations together with their rates are known. Rates for timediscretization seem to be unknown except in some special cases (see below), and a fortiori no rates are known when the basic process is a diffusion process: finding these rates for time-discretizations, in the form of a central limit theorem, is the main aim of this paper. I,et us introduce our basic assumptions. We consider a tiltered probability space (52.3. (F,)t>~~. P) on which is defined a continuous adapted ldimensional process .Y of the form

where T/1’is a standard Brownian motion. The assumptions on h, CTare such: A. - cr is u continuously differentiable positiw (normwnishing) function on R, suc.h that the rquution Crk; = CJ(I;)~W’, with I;, = >Ycjhas a (necessarily unique und strong) non-exploding solution. Further, the procrss Cl h is such that the luws 0f’~3’ and of‘1 are locally equilwlrnt. HYPOTHESIS

Next. we denote by I, the local time of the process S at level 0, given by L, = /X,1 - ISi,/ - .iit sigrl(X,)dX,.

(1.2)

Next we describe the processes which approximate 1,. The simplest way is to count how many times X,,,, is close enough to 0, that is to consider the processes

for a sequence ‘u,, of positive numbers going to infinity (here, [!I] denotes the integer part of 111 2 0; we take (7 - l)/ 7~instead of 1/71 for coherence with further notation). These processes will converge in probability, after normalization, to L. A bit more generally we can consider the processes

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for a function !J which “goes to 0 fast enough” at infinity: clearly ( I .3) is V(U,,, CJ); for S(Y) = l]-r.i]. Even more general, and of interest for statistical applications, are the processes

It is known that under suitable assumptions on {J and assumptions slightly stronger than (A) on the coefficients b, CJ, the processes -$;l!( fiE,!~)” converge in probability to c,L for some constant (’ depending on ~1 and on the function CT.Further when h = 0 and CT= 1 (i.e. S is a standard Brownian motion), the normalized differences u’/~( +C:( Jz. !I);’ - (.I,, ) converge in law to ~‘Il’f-, where II-’ is another Wiener process, independent of X: for example, Azai’s [3] has shown that ( 1.6)

converges in probability (and also in L2) to &, / i Lt. (1.6) counts the number of “crossings” of the level 0 for the discrete-time process, and it is equal to &U( fi, It,);’ for /!.(.I:. in) = l{.U(,,.+,)<,J). The associated central limit theorem mentioned above has been shown by Borodin ([4],[5]) in a more general context, where ?i,,,, is replaced by & (Zi + + Z, ) and (Z,) are centered i.i.d. variables (and in IS] many other related results are exhibited). Here we will tirst show that under quite general conditions, the processes !y( ?1,,. rt)ppconverge in probability to the process CL, where (’ is a number defined below (depending only on /L and c1(0)), as soon as .v,,/?/ -f 0 and u,, 4 CC. Some related results have been proved by Florens-Zmirou [6]. Next. for the rates of convergence, we restrict to the case where ‘ll,, = 71”

for some (Y E (0.1).

(1.7)

Then we prove that if ~5 = (( 1 - CV) A ~ti)/‘J, the processes r1?( -&+i(7P, h)‘l - d) converge in law to a non-trivia1 limit, provided CY > l/X and we do not know what happens when (Y < l/3. When CP= l/2 this essentially reproves the results of Borodin (relative to the Brownian motion). with a different method allowing processes of the form (I .I). Observe that the rate of convergence 71’ is biggest when cv = l/2, in which case it is I,‘/~‘: this is a bit surprising at first glance. but may be

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interpreted as such: if CY> l/2, then U(n”. I,)” is of “order of magnitude” n,lwru, that is ?t,‘-” is the “number” of nonnegligible terms in (I 5). and it is only natural that the normalizing factor in the associated limit theorem be u(~--‘*)/~. When CI < l/2, we still have T/,‘-” nonnegligible terms, but most of them concern values of .XC~-~),~ which are too far away from 0 to give an appropriate information about the local time at level 0 (see Remark 2) after Theorem 1-2 for more explanations on this phenomenon). 1-2) Now we proceed to state the main results. Let us begin with some notation. If !i is a Bore1 function on [w and y > 0 we set

(1.8) Let /I denote the density of the standard normal law M(O, 1). For small enough Bore1 functions /r. on R’. set

IS,,(2:)=

.I

h(.r.y)p(y)dy.

We will assume that h satisfies the following

(1.9)

with some y 2 0:

h on R2 is Bore1 and satis$es HYPOTHESIS B-y. - he function h(.c, II) < iL(X)P”l’~1, where (1 E W and !L is bounded with jj-, (h) < SW. 0 Finally, associate with any ‘~1> 0 and any function h on W’ the function I!,,,(.1:, ,y) = h( ‘7M; ‘Iq) (

and consider the following

condition on the sequence (u,,):

IL,, -+

71

(1.10)

0,

‘(L,, -t

3i.

(1.11)

Recall also that a sequence (2”) ,,>I of processes is said to converge locally uniformly in time, in probability to a limiting process Z if for any t E Iw+ the sequence su~),+~~12: - Z,?l goes to 0 in probability. THEOREM I. I. -- Assume (A) and (I. I I), and let It sati& (B-O). The processes + U (u,, , h) ‘I converge locally uniformly in time, in probability. to &wh”(,,, )JJ. For example h(.c, ;v) = l(,,.C,,.+,)
a slightly

more general situation.

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Remark. - When u,,/TI. does not go to 0. the theorem cannot be valid in general, since the first summand in V(U,~,g)? is ~(u~,x) if X0 = 2, and this quantity does not go to 0 in general. When u,, does not go to infinity, the result is also wrong: for example if ‘UL1,, - ‘u.is a constant, by Riemann approximation ~V(u~g)~ converges to ,d CJ(X,)~.Sas soon as ~1is continuous. Cl 1-3) Let us turn now to the rates of convergence. For this we need first to recall some facts about stable convergence. Let Y,, be a sequence of random variables with values in a Polish space E, all defined on the same probability space (12,3, I’), and let 6 be a sub-a-field of 3. We say that Y,, converges G-stably in law to Y, if Y is an E-valued random variable defined on an extension (fi, i, r) of the original space and if li;;r E(lJ,f(l;,))

(1.12)

= E’(Uf(Y))

for every bounded continuous f : E --+ R and all bounded S-measurable random variables 0’ (and then ( 1.12) holds for all integrable U). This convergence was introduced by Renyi [lo] and studied by Aldous and Eagleson [I], see also [g]. It is obviously stronger than the convergence in law, and below it will be applied to cadlag processesY” with E being the space of cadlag functions endowed with the Skorokhod topology. For the conditional Guussian martingales, we refer to [9]: denote by (Gt) the filtration generated by the process ,T, and G 5 v Glt A (‘possibly multidimensional) process I’ defined on an extension (52.8, (Gt.). P) of the original filtered space(12.G. (&). P) is a G-conditional Gaussian continuous martingale with bracket C if the process C is adapted to the filtration (Gl) and if Y is a continuous (necessarily Gaussian) martingale with bracket C’‘, for a regular version of the conditional probability knowing G. Next, we need a whole set of new notation. If .f’ and 9 are functions on R’ and [w respectively, we put

Next. set

jj(x) =

.I

p(g)(/:c + :I///- Ixc()dy:

which has: iA>(G) < cc VJa> 0, by a simple calculation. Vol. 34. n” J-1998.

N.4) = 1

(1.14)

510

0, JAWD

Next. (E’t)t>O denotes the Brownian semi-group, given by P,li(.r) = ,j’ k(:r + ;r/&)&:,,)&. The following two estimates, where X: denotes a Lebesgue-integrable function. are well known (and for the convenience of the reader they will be reproved, along with other related estimates. in Lemma 3-l below): (1.15) ) f h(k)(J). where h' denotes a universal constant. Hence if /J’,(AZ)< x and /‘jz(li) < x and ,[ X:(.r)tl.~, = 0. the series

(1.16)

F’(k)(.r) = c P,X:(.r) tCN

is absolutely convergent and /E‘(/c)(J)/ 5 ~(ijl(X:)+lj.‘(X:)I.l.l). Iff’has (B2) the function X: = EIf -A( Hf)!/ satisfies the above-mentionned conditions, hence F(k:) exists; further, if /, denotes the law of the pair ( .!I1>I’l) on W x R,, where tr3 is a standard Brownian motion starting at 0, with its local time C at level 0. the following expression is well defined:

When

,f

and ,f’ have (B-Z), we set

ri(f> .f’) = x(iIff;

+ Hf.~(r-l,, --x(H,,~~,) + ~f’.F(ff,--Xm~)r/l

+ HXiH,f)W’) :j&

- Wfbvf’)

- 4H,fOWj’).

(1.18) Using (1.15) again, if i&(X:) < x and X(1X:/) < x, the integral

absolutely converges and IG(A,)(.r)I 5 Kk.(l + IX/). So if .f’ and ,f’ have (B-2). we can set

‘T,‘(.f..f’)= X(HfG(IIff) + HffC=(Hf)) - A(Hf)G(Hj,)(O) - X(H,~)G(H,)(O).

(1.20)

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Clearly */I(.~.) and Y/‘(.~.) are bilinear. That ~/(.f. .f) > 0 and r/j f’. f’) 2 0 is not obvious from ( 1.18) and ( 1.20), but it follows from the fact that these quantities are limits of nonnegative numbers. Below we state the results for a d-dimensional function h. = (II’ ) l 8, urld set n = (( 1 - (Y) A tr)/2. Under eifher. one of the ,fi)llawing:

b) If’ l/Z < CP< 1 (hence h = (1 - tv)/2), wilh

Remurk 1. - There is another, equivalent, way to characterize the limit I” above, when (‘t”. 1-J) = tr;,/,. Namely one can construct on an extension of the space a Wiener process 14’ = (IV’)l<,<,l having E(LVjFV/) = a;,t and independent of S, and we set 1; = ITi,. This formulation is closer to the formulation of Borodin 141when CY= l/2. In this case, the expression ( I. 1X) which is used to compute u,, seems quite different from the corresponding expression in [4], but of course the two agree. Remcrrk 2. - Suppose that we are in the Brownian case, i.e. L = 0 and (T = 1, and also that I/,(.r:. :I/) = !/(.I.). Denote by L” the local time at level .I:, and set (1.2-l)

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We have

If {j is twice differentiable, with ,q.9’. q” satisfying (B-l ), one can prove that when ‘u,, = 71,” and CP< l/2, then T,,“/~A; goes to 0 in probability, while because L is Hiilder in time with any index smaller than l/2 we also have ~“/~(Li,~~l,,, - I,,) - 0: then the leading term in Statement (c) of Theorem l-2 is the second term in the right side of (I .25). When IV > l/2 on the contrary. this second term is of order c”/’ (because of the Hiilder properties of 1,; in :I:) and the leading term in Statement (b) is AI’. When o = l/2, both the first and the second terms in (1.25) have an influence on the limit in Statement (a). In a sense, it is more natural to look at the processes (1.24) rather than ?V(%I,,. 9)” - X(!j)L. 0 When .f(.~..;q) = !/(.I.) the expression for A(j) becomes (see the end of Section 6):

If Hf = A( H,f),ij and [I fj = X(Hf~),ij. all the terms in (1.18) in which F(.) shows up disappear. In particular if f(:r? ?/) = G(z), hence A( F1.f) = 1, (1.18) becomes (1.27) The remainder of the paper is organised as follows: in Section 2 we show how to reduce the proof to the case where ?LIis a standard Brownian motion. Section 3 is devoted to some preliminary estimates on the semi-group and on the local time of the Brownian motion. In Section 4 we prove Theorem l-l. For Theorem l-2. in Section 5 the problem is reduced to a central limit theorem for a suitable martingale (pretty much as one classically proves a central limit theorem for mixing sequences by reduction to a similar result for martingales), and we consider separately the cases o := 1,‘:‘. II > I /2 and I) < l/2 in Sections 6, 7 and 8.

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2. REDUCTION TO THE BROWNIAN CASE In this section, and throughout a number of steps, we show how our results for the process S having (I .I) can be reduced to the case where X is the standard Brownian motion. 2-1) We will use several times the following method for constructing the limiting processes Y and Z with a bracket of the form r:IJ in Theorem I-2 (C is a nonnegative symmetric tl x d matrix): according to [91. the process Y itself may be taken as the canonical process on the canonical space ((2, y-. (j;)) of all continuous @-valued functions on R+; then ((1. +?. (i,)) is the product of ((1.5’. (&)) by (ii. .?. (i,)); finally, the measure on the extension if p(ciw. &J) = P(rlw)l)(ti. dij), where C)((l’w, .) is the unique measure under which Y is a Gaussian martingale with deterministic bracket ~:L(ti)~and Yt, = 0 (observe that C) is a transition ^ . probability from ($1. G) into (12. ;F), and also from (52. Gt) into (lI.Z=,) for all f, because I, is (Go-adapted by (1.2)). 2-2) Here we prove that one may assume F+ = Gt (recall that (4,) is the filtration generated by S). First, if X satisfies (I. 1) and (A) relative to (3,). it also satisfes ( I .I ) and (A) relative to ( Gt), with the same function IT and a drift process b’ and a Brownian motion CI” adapted to (Gt): indeed, X is a continuous semimartingale w.r.t. (Ft), hence w.r.t. (Gt) as well, with the (Gt)-canonical decomposition S = ,I?o + B, + ill, (where A4 is the martingale part), and the quadratic variation of /II is ,I;; (marls. Thus r/r:: = .I;:( l/rr( s,))tLri,, is a (&)-Brownian motion. Further, that I3, = ,I;: b’,rls for some 1): follows for example from Girsanov’s Theorem and Assumption (A). and clearly the pair (h’, CJ) satisties (A) as well. Next, the local time I; as defined by (I 2) is the same for (F,) and for (6,). The processes I;(,IL,,. /r )” do not depend on the filtration. and Step 2-I above yields that the limits in Theorem l-2 depend only on 1; and on the functions II. and n. Therefore one may always replace Ff by Gt, or in other words we can and will assume that I=, = so that the law of .‘i under I’ is I” = P o cp--l. Then standard arguments on changes of space yield that if S satisfies (1.1) and (A) with (6. (r) the process X’ [under 1)‘) satisfies

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(1.1) and (A) with a 6’ having 1)= b’ o 9 and the same (r. and if L’ is the local time of X’ under I”. then L’ o ‘p is a version of L. Further the process U’(u,, ( h)” associated with X’ by (I .5) has U(U,,, I),)” = lj’(~,,) II)” o cp. Finally if we define the extension of (SZ’,3’, (&‘t): I”) as in Step 2-l with L’, we observe that Q(w. .) = Q’(p(w). .). Therefore. with obvious notation, we have ,!?‘(r!‘,f(Y)) = E’jU o yf(k’)), while any random variable on (II, G) is as. of the form II’ o ‘p: all these facts imply that if our theorems hold for X’ under I’, they also hold for ,S. Henceforth, we may assume in the sequel that (si, 3. (3t)) is the canonical space, endowed with the canonical process .I*. 2-4) Here prove the following property: if our results hold for a given pair ((T, b) satisfying (A). they hold for any other such pair (CJ.h’) with the same function 0. Indeed, denote by I” the law of the solution of ( 1. I ) with h’. The two measures P and P’ are equivalent on each n-field 3t (recall that by Step 2-3 we are on the canonical space). By ( I .2), the process I, is also a version of the local time of .‘i under 13’. Further if a sequence A,, of 3f-measurable variables converge in [‘,.-measure to a limit ‘4 (necessarily 3+-measurable as well). we also have A,, + .4 in P:-measure. So our claim is true for Theorem 1-1. Now we define the extension for P’ as in Step 2- 1, except that the measure is now P(&I. 11;) = I”(dw)Q(w, G). If E:(lif(l”‘)) f I;:(rif(J’)) for all integrable variable f r and bounded continuous function .f’ on the space of cjdlag functions. we need to prove the same thing for I” when Ii is in addition bounded. Since Skorokhod convergence is “local” in time. it is enough to prove it when II is 3t-measurable and f depends only on the function up to time I, for any finite f. But if 2, denotes the density of P’ w.r.t. P on the n-field 3$, and since Cj(.. iiz) is 3t-measurable when .4 E 3,. we have E;‘(r’f(I”“)) = E(%,I;f(I’“)), and &‘(lif(‘I’)) = fi;:(Z,!rj(17)). Since VZ, is PP.-integrable when li is bounded, we deduce the result: hence our claim is true also for Theorem l-2. 2-5) Suppose that our results hold for (I. I ), when (T. l/n and (T’ are bounded. We prove here. via a well-known localization procedure, that they also hold without the boundedness of CT,l/n and CT’. In view of 2-4). it is enough to prove this result when h = 0. For each y 2 1 choose a continuously differentiable function IT,, which is bounded, as well as l/g,, and c-$, and such that P,,(I) = O(J) whenever 1~1< ;o. Observe that since b = 0. Equation t 1. I ) may be “inverted” to give IV, = ,l;:(~/~(.~,,)),ls,. Now let LU(;l,) be the (strong) solution to (1.1).

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w.r.t. the same I&‘. If Y-J8= inf(l : IXtJ 2 1)). we clearly have Xt = ,‘i(~,)~ B.S. for all 1: L: TP. Hence all the processes showing up in our results coincide a.~. for X and X(I)) on the interval [O, TP]. Since I;, T #x, the claim is thus obvious. Hence all what precedes shows that it is enough to prove the results when 3; = G, and IT, l/n and n’ are bounded, and ,Y is the solution to the equation

2-6) In our last step, we consider Equation (2.1) with (T, (T’ and l/g bounded, and we show how to reduce the two theorems to the case where S is a standard Brownian motion starting at a fixed point :I’. Consider an arbitrary twice continuously R such that for some ;‘ > 0: T(0) = 0.

T’(0) = 1.

c 5 7”(J)

differentiable

< 5 f

function 7’ : R --+

/T”(X~)I 5 1.

(a.?)

With any function h on R”, and with the function T having (2.2) and the sequence (,IL,,) of positive numbers, we associate the following functions:

Next, if (12.3. (3,)) is the canonical space endowed with the canonical process ,Y, we denote by f’,. the unique measure under which .Y is a standard Brownian motion starting at :I’. The following auxiliary result will be proved in Section 4:

Suppose also that Theormt 1-2 ho1d.v in the abo\v crrnoniml satting, ,fbr e&z measure P,.. We will presently see how to deduce the results in the general case.

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First there is a version of the local time L which works under each P,.. Let 11,be any probability measure on W, and set P = J’ /opt (so under P, X is a standard Brownian motion starting at X0, and the law of XC) is 1~). Since L is the same under each I’,, it is also a version of the local time under I’, and it is obvious that Proposition 2-1 hold also for the measure P. Since in Step 2-l the transition measure Q depends only on u/ through L.(w). if pz is the extension of pr. then with the extension of I’ given by p = ,[ /~(d.r:) p,. it is also obvious that Theorem 1-2 holds for .I’. By Step 2-3, Theorem l-2 and Proposition 2-l hold also when S is a standard Brownian motion on an arbitrary space (!‘I, .7=.(F,), r), with an arbitrary (possibly random) initial value, provided (F,) is the filtration generated by S. Let us now consider the case of Equation (2.1) with CT, CT’ and l/m bounded. Set

(2.3)

This function is of class CJ”, so by Ito’s formula, we immediately deduce from (2.1) that the process X: = S(X,) is a standard Brownian motion, starting at the random point Xh = S( SO). From the above, and since (Ft) is also the filtration generated by X’ because S is invertible, Theorem 1-2 and Proposition 2-l hold for X’. Moreover, we have the

LEMMA 2.2. - The process L’ = L/(T(O) is

LI version of the local rim

of .Y’ nt level 0. Proof: - Let *Y+ and X - (resp. .Y’+ and S’-) be the positive and negative parts of A’ and X’, and let L’ be the local time of X’. Not only do we have (1.2). but also

Now, sign(S(x)) = s&r(x). hence S’+ = S(X+) and AY’-m= -S(--.Y-), so the property S’(0) = l/rr((l). the fact that I, charges only the set where

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X,- = 0 and Ito’s formula yield

I0

S’(-X,-)lpy

<())dX,* - ;

.t S”(-X,,-)l~~~,,<0)“(~i,)2rl.s. J0

Adding these two equations yields

On the other hand, (I .2) applied to S’, together with the fact that d/Y: = S’(X,)dXt + $S”(X,)a(zYt)‘cO (by Ito’s formula again) and that sign(S(x)) = sign(x) again yields

p;1 = ls;l+L;+

.I ‘+ S”(X,)sign(X,)f7(X,)‘&i. .I 0

J(I

Comparing the last two equalities gives L’ = L/n(O). The function S is invertible,

and if S-l

q

denotes its inverse we set

T(x) = -Ll s-y:,+ d(J)

(2.5)

This function is twice differentiable and satisfies (2.2). Let /I be an R”valued function, with which we associate h’ = IL,(~~Jby (1.10). while /I:, is associated with h’ (and T as above and a given sequence (u,,)) by (2.3). Denote by U’(U,, , /I,)~’ the process defined by (1.5) with S’ in place of S. Since X = (T(O)T(~Y’) a simple calculation shows: 1;(1/,,,. I+)‘, = U’(‘ll~,,, l(,)“.

(,2.(j)

If h satisfies (B-r), so does !h’. Thus (2.6) and Lemma 2-2 yield that Theorem l-l is exactly Proposition 2-l (a) for X’. It is also clear that (2.6). Vol.

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Lemma 2-2 and Proposition 2-l(b) for S’ and Theorem I-2 for .Y’ yield Theorem l-2 for AY, since if /J,,~= U” with 0: > l/3 implies I//U: -+ 0 and since J?- > II,~+“-’ (when D is a constant. we have ‘X’(I) = .I’ and thus Proposition 2-l(b) is trivial without any regularity assumption on Ir because A,, = h 1. To summarize, at this point we are left to prove Theorem l-2 and Proposition 2-1, on the canonical space, with the canonical process .Y and the Wiener measures F’, .

3. SOME ESTIMATES We give here some estimates on the semigroup (I’,) of the standard Brownian motion. These are more or less known, but simple to prove. Below, K denotes a constant which may change from line to line; when it depends on an additional parameter 11, we write it h’,,. LEMMA 3.1. - !/‘ t > .L;> 0 ad

T 2 0 we hmr (3.1)

(X2)

(3.3)

(3.4) (3.5) Proc!f. - The density of the law ,%“((I. t) is /)t(,/l) = 5,‘: 2). Since ~,X:(:I,) = ,/‘f~~(g)k(:r: + ?/)d:r/ and /I is bounded, we have (3.1 ). Next,

ap,(‘11)= -+ -. i)Pl

ap,(Pr)= ;(F ,” - l)p{(‘“). i)t

(3.7)

Taylor’s formula yields, with ,$(,I/.) = i)/),(zl)/&:

1 I+l.,r/Jij-’

Since furthermore ~.“‘/~‘j~ < - h- 1

. (3.2) readily follows from (3.8).

Next, (3.6) and (3.7) yield I/+ (:I: - U) - 0, (:r)I < &Iu/( (3.3) follows from (3.X).

1~61 + IJI), hence

Next, we have

Again (3.6) and (3.7) yield //J~(:J.- U) -~~,(~/- IL)/ < Kl.c~-;~~l//. hence (3.4). Finally, P,X:(x) - I’,k(.r:) =

.i

k(u)(pt(l.

- 11,)- /‘+(:I: - v))th.

A further application of (3.6) and (3.7) yields Ic~~(J.- ,!I) - /‘.,(.I’ - cr)l < K(t - .s)/s3/‘, hence (3.5). 0

Proof.

-

1 l+l(.l~+!/Ji)/h

obvious.

Since i

t

I~‘tM:~:)l I

I\‘. (ltl!,JlL+~.Y/hl~

)

.I’ l+,IJ.ijv/iI,h,-.P(Y)&

by an easy calculation.

and

the result is

520

J. JACOD

if flrrthermore

A( X:) = 0, then

Proof. - We have E,,.(I;(fiX+)) C!“i-’

(Pik(.l:fii)I.

= I’;-lJ~(:r:fi),

hence ~(X,.C):

<

Th e estimates (3.1), (3.2). (3.31, CiLf/ -$ 2 20,

XI”=, & < x1 and xi:‘/ f < 1 + log+(r,t) give the results. cl We end this section with some simple calculations on the local time L. Put Cl(f, q. .c) = &(L’:f(>Y,)),

(3.12)

q E N>

where ,f satisties If’(x) / < Kr. ~‘1’ .’ 1 and q > 0. First, according to Revuz and Yor [ 1 I]. if L(a) denotes the local time of X at level CL,under I’” the processes (A’,. L(U),) and ( ;Str.:. kL(c~r),,.l) have the same law. Hence ‘,,“~‘E,./J;;( L’:,,,f( JliA-,,,I)) == d”l;;)(

L( --:r/fi);,J(:r

= &(L(-.r:)‘ff(:r: Similarly,

+ fix,,,,

+ A-,)) = G(f. ‘1,:I.).

)) (3.13)

for 1 > Cl:

E,,(L:f(ATi,)) = t”%()(L’:.f(xJJi)).

(5.14)

Moreover, on knows (see e.g. (1 11) that L1 under PC, has the same law than ISI/, hence

E(,(L,)= d 1. &,(I$ = 1. The hitting 1.1.1 ~r~~-~‘/“‘ln,

(3.15)

time ?ir of {x} for ,Y, under 1’0, has the density 7’ -+ (7’). Then (3.13) and the Markov property yield

RATES OF CONVERGENCE

TO THE LOCAL

4. THEOREM

TIME

521

OF A DIFFUSION

1-l

4-1) Recall once more that we are on the canonical space, with the canonical process X and the Wiener measure E’, . We first prove a version of Theorem 1-I in the case u,, = ,,G, and for the processes (1.4). In this setting, this version is indeed more general than Theorem l-1, and has interest on its own. THEOREM 4. I. - u) /f g,, is a sequence of ,functions on R satisfying ,fbr all :c E R. as n. - ‘x.:

~,firrthermore sup,, A( ],cJ,,1) < ‘cc, the procexws -& V( fi, locally unij~m~l,y in tirw, i,l T<,-probabili~:. to XL. Let us begin with a lemma.

{J,, )” cnmlerge

LEMMA 4.2. - If the,firnction.s {J,~sati.yfi (4.2) and A({/,, ) = 0. w’e hme +v(fi.!J,,):",

--+

0.

(4.4)

Prooj: - With h(,c~)i’ = sup ,qEW,st[O.t;I~(TJ, !/):‘I. we can write (use the Markov property): I!?,.( I\‘( JT;, S)l’I’)

522

J JACOD

If X(g) = 0, (3.1 I) yields n(g);l 5 K,/j,(!/) (3.10):

and (4.4) follows.

log?, for ~1 > 2, hence by

0

Proof of’ Theorrtn 4.1. - We have E,,.(snp,<+ I\-( fi.!/,,)yI) 5 l.q,&$)I + ~(l,v,rl.~.)r, so ( a ) ob v~ously follows from (3.10). Let us prove (b). The function j of (1.14) is ,ij(:~.) = fi,.(ISr/ - iS,l), and by definition of the local time,

(4.5) so by Lemma (2.14) of [7] we have the following

convergence for all :I:: (4.6)

Now let !I,! be a sequence satisfying (4.2) and A(!/,,) - A. We set / Y,, - !/II - A(!/,, ),$. ( I .14) implies that the sequence (I:, satisfies (4.2) as well and A(!/:,) = 0, hence Lemma 4-2 yields -&I/‘( fi, g:,);’ -+I’, 0. Since ‘lY(fi.
is a I’,-martingale bracket given by (nr(h)“.

M(h)“)f

w.r.t. the filtration

I=;;’ = J=L[,,~I,,,.with predictable

= I,‘(?&,, H,,’ - (H/J’)”

5 I.( II,,. H,,?)“.

(4.8)

RATES OF CONVERGENCE

TO THE LOCAL

TIME

OF A DIFFIISION

523

Observe also that for any function CJon R we have C’(.IL,,,y);’ = I’( J;;. y,,);’

with

In (4.9) above, we have X(~,CJ,,~) = $A(i!/l). the following useful estimate:

!/,,(:I.) = !I( z).

(‘4.9)

Hence (3.10) and (4.9) yield

5 K +,,r7&)l + JiX((,cl, > (

(4.10)

Now we prove three lemmas. which are indeed too strong for proving Theorem I- 1 but will be useful for Proposition 2- I.

sup X( il,, ) < ,I

Y ~

suph,,(.r) r, .I

<

x2.

(4.11)

Proof. - Observe that H,,;, 5 Kh ,,,, so a combination of (4.7) and (4.X) yields that the martingale nZ( I/,, )” has a bracket smaller than K1’(7~,, k,, )‘I (use the second part of (4. I I )). Therefore E( ($ ?~l(l/.,,)” Y“f :21(l~,,)“)~) + 0 by (4.10) and (4.1 1) and (1.1 1). and the result follows from Doob’s inequality. 0

liln lilt1 sul) I,$) (.I,) Ir1.r = 0. ‘I I, I I.rl>y

( 4.13)

----) 0, the pr0ws.w.v !f,fkrther X(,q,,) - CYmd (I,,, scltisjes (I. I I) cd * + 1,’( ‘,,,) , g,, )” converge lo~.~ill~~wCfi)rm!\ in time, in I<,, -prohnhilit~~, to tr 1,.

Proo$ - a) Assume first that SII~,, i-11(!/,, ) < x. In view of (4.9). and if X:,,(.I:) = *cl,, (3 ). we need to prove the convergence, locally uniformly in time in F’f),.-probability. of the sequence &V( fi. X,,,)” to ~rf,. Vbl. 34. II

I-IYYX.

524

J. JACOD

We readily check that )I;,, 1 5 Ku,,/,,& and A(lk,,l) < K and Hence by (I. 11) and ? + 0 /h(L) L KJ-/,7) u,, and A( AI: ) < K.lb,, /fi. the sequence (X:,,) satisfies (4.2), while A(&) = X(!l,,) --t (Y by (a): the last statement in Theorem 4-l gives the result. b) Let us now consider the general case. Up to taking a subsequence, we can assume that j;,,r,5C,I !/rl(:~:)
as K( K, by

9 + x,. By (4.10). the expectation above is smaller than % + &A( /,y,! .- k:;l)). Th us the left side of (4.14) is smaller than lim sup,, A( \,c/,,-- ky 1) by ( 1 1I), and this quantity goes to 0 as (1 -+ “c q (4.13): hence (4.14) holds.

b&m4A 4.5. -- Let g he n hounded und integrable ,func.tion of R, and let T be a func’tior~ satisfying (2.2), and set gIL(:r:) = ~~(~IL,,?‘(:~./,~(,,)). !f u,, x2fi.sfie.v (I. I I) and ,I/,:/,I!. + 0 the processes %V(u,, . g,,)” lwzvrrgge locally un~fiwrnly in time ill I’,.-probahilit,v to X(g) L.

Proof: - By the same argument as in the proof of Theorem 4-l. it suffices to prove the result when $1 > I). in which case it is enough to have the convergence for each time t. We can write

where (by the occupation time formula):

[r1+1 u;’ = c :y i=*

.I II I I IJ

Since .(/ is bounded, r~;’ a 0 follows from (I .l I ). Next. the variables L;;/;;j’r, - L, tend to 0 and remain smaller (for all .I: and IL) than a fixed

RATES OF CONVERGENCE

TO THE LOCAL

TIME

OF A DIFFUSION

525

finite variable U. Therefore

The first term in the right side above goes to 0 as ‘~1--i x for all y, and the second term is smaller than ii j;lJ,>,I,EJ Ig(:c)ldx, which goes to 0 as y -+ x8: thus

Furthermore X(CJ,~)= j’,q(.~,)T’( e) I1.‘I’., w here ?’ is the inverse function ot 7’. Since ?“(:r:/u,,) tends to 1 and remains bounded by (2.2). it follows that A(!/,,) -+ X(q). Therefore the first term of the right side of (4. 15) goes (for all i*/) to X(,q)L,. It remains to prove that l3;’ -I’, r,!’ = E,r (,!$’ IF= ). We have

0. Let C’;’ = cl”ztl ?: ‘, where

(4.1(i) and I/j:’ 1 5 h’~E/ra’. The above sum is thus smaller than Ktv~,//~, goes to 0 by hypothesis: so it remains to prove that [lit] F1,. c /$I ( ,::I i Set ~1,~= u~/u.

-

which

0.

A simple calculation shows that 7:’ = %/II,, ( II,, SL1+ ),

where !I,,, = j;f (P *,,,I .IJI, -- .cJ,, )A. Since l,q,,I < K, by (4. IO) yields that the left side of (4.16) is smaller than h’( % + fiX((.f,,l)), and we are left to prove that A( IS,,/) --f 0. We can find for each integer 1) a function h,, which is Lipschitz with compact support and such that A( /!J - /+I) < l/p. The function k:;:(x) = li:l,(,/~,,T(.c/~~~,,)) has A( I!/,, -- h:;:I) < l/p~. Since the action of the kernel I’, is a convolution, we also have A( jr),,.,, {J,, - I’,,.,, I,:,‘: I) 5 l/pr. Therefore .I I0

A( / I’, ,,, h:;; - h:; I )ds.

526

J. JACOI)

So it remains to prove that for each p. the last term above goes to 0 as /I -- Y. By Lebesgue Theorem. it is even enough to prove that for each s, then X( II?, ,,,,X:;: - X:;;1) + 0. But if 1’ denotes an n/(0.1) random variable, this quantity is less than

Since 71,~+ 0 and since XI,, is Lipschitz with compact support. and in view of (3.2). this last expression clearly goes to 0 as I( --t X. and we are tinished. fl Theorem l-1 obviously follows from these results: assume ( 1.1 1) and let I/. satisfy (B-O). When (log X)/U,, + 0, we can apply Lemma 4-3 with II.,, = /I and Lemma 4-4 with {I,, = 111,and (4.7). Otherwise we apply again Lemma 4-3 with II,, == A and Lemma 4-S with {/ = H,, and T’(X) = .I and (4.7) again. 4-3) Proof of Proposition

2-1. This proof will go through three steps.

Step I. - First, we prove the claim (a). We assume that Ir satisfies (B-O) with (1, and it. Define h,, by (2.3), and set X,,, = HI,, In view of (2.2). it *is clear that i//,,,(.r,.!/)I < i,,,(.~)~,“‘l~~i, where (1’ = C/,/C and ii.,,(L) = ~h(ll,,‘l’(.r:/(/,,)). Obviously (2.2) yields that the sequence i, ,$ satisfies (4. I I ). hence Lemma 4-3 yields that !:LI ‘(u,, ~It.,, -- X:,,)” goes to 0 locally uniformly in time in I’-probability. So it remains to prove that +Ir(~~l,,,, k,,)” tends to X(H,,)L locally uniformly in time in P, -probability. For this, by using exactly the same argument than in the end of the proof of Lemma 4-4, we can and will assume that ,lr(ir) < x;. Observe that

Set ,?,)(,r3 ;(I) = J; (7’( T( k)

where

+- +)

- ;i).

A change of variable yields

RATES OF CONVERGENCE

Now,

(2.2)

implies

that

TO THE LOCAL

jtr,,(.r:.!/)

‘TIME OF A DIFFl:SION

- ;!/I _< A-($$

+ 17’($)1)

Ir’($ + T(($)) - 11 < A?($$ + I?‘($))) and ,r I Thus /)((v,,(:I:, 11)) < Kp(,t/;). while I/~‘(:t/)l < Kp(!//2). Taylor’s formula that //j(c~,~(.r..!/)) - //(!/)I < I<,I(?/E/‘)($ and thus I~,,(:I:,;Y) - /)(!/)I < h.,,(:l,~/L’)(%~ ,f,l(,~,) = H,,(,/L,,‘~(.I./,/I,,)) we have

+ IT(+)/).

527 and

‘--I:“) < i. It follows by + /7’( -,+-)I). Therefore if

Combining (4.10) and the facts that it is bounded and that ,r,(jr) < x, we deduce that I:‘.r( 2 I#‘( II,,, (A,,, - .f,,l)F) 5 A’,(k + $7). and thus ? V( ‘lb,)~A:,, - f,()” tends to 0 in I), -probability locally uniformly in time: therefore it remains to prove that !?I’(‘(/,,, . .f,,)” tends to A( Hi, )I, in P,.-probability locally uniformly in time. Suppose first that (log IL)/Y~,, (1. The sequence J”,, satisfies (4.12) and (4.13) (for the later, observe that I,/,, / < jr,, and that ,/I+, i,, (+h -i ,/I,,,~,,Ejh(.r,)m’:r:).On the other hand, a change of < variable yields

which clearly goes to A(HI,) (since 7” is bounded and goes to 0 at 0): the result then follows from Lemma 4-4. Next, suppose that /I,: /II - 0: the result readily follows from Lemma J-5. Srep 2. - From now on WC assume that h satishes (B-l ). In this step we prove that with notation (4.7). the process AI” = finI(l) - 11)” tends to 0 locally uniformly in time in I’,. probability. In view of (4.8) and of Doob’s inequality, it is enough to prove that R,.( 2 V-(,/L,,.Ii(,,,,Pf,,~);‘) - 0. set {/,,(I) = 11,I,,, ,!)1. As seen at the beginning of Step I, lh.,, (.I:. !/)I 5 &,,(:r)P’~‘~‘, where (I’ = (I/C- and A,,(J) = ~j,,(,,,,,T(.~./1~,,)): it readily follows that T/,,(J) is bounded uniformly in .I’, II. Thus in view of (1.1 I ) and (4. IO), it remains to show that A(!/,,) + 0. We have

528

I. JACOD

Denote below by (( y I/ the quantity (Jy(z. ~)z~(y)cl~~~y)1’5. For any 9 E N one may find a continuous function f, on W* with compact support, such that 11/L - ,9<] 111 l/q. With the notation of Step 1, a change of variables yields

for a constant K depending only on E in (2.2). Now, if o;L denotes the right side of (4.18) with scl instead of IL, it then follows that -A-fl. Furthermore, since 9, is continuous with compact >J5support, it is i’mmediate (by (2.2) again) that CL: -+ 0 as 71,-+ c0, for all q: thus X(9,,) -+ 0 readily follows, and Step 2 is complete. Step 3. - In view of Step 2 and of (4.7). in order to obtain Proposition 2-l(b) it remains to prove that EV(?l,,,. HI,~,-I~)” tends to 0 locally uniformly in time in P,.-probability when n/,r~t -+ 0 and when I/. is differentiable in the first variable with a partial derivative satisfying (B-l ). We use again the notation of Step 1. In particular HI,,,-,, = X:,, - H,,. Using (4.17), we obtain A( /AI,, - f,, I) 5 K( * + 5). while by definitions of k,, and .f,, and the boundedness of i,, we have IX:,, - ,frl I 5 K. Then (4.10) implies that

Since we have (1. I 1) and r)./?bi --f 0, the above goes to 0, and we are left to prove the convergence of ~~V(IA?~, f7< - H,,)‘” = -& V(&. {I,~)” to 0, where !J~~(.I.)= &(J;, - Hf,)(~r/.,,/fi) (use (4.9)). We assume that 11’ = i)lt,/ti.~ exists and has (B-l). We have

:l: ) - J;;;: :I: )pl+L(g a, (:I:) = ,& ( T ( J;r because H,,J is the derivative by (2.2), it foilows that

+?(T(+-)-g)))iir!

of HI,. Since IT(*)

- -$=I 5 K/xl/+

RATES OF CONVERGENCE

TO THE LOCAL

TIME

529

OF A DIFFUSION

Since I),‘, hence H,,!, satisfy (B-2), we easily deduce that X( I:J~~]) < h-vG/up,/‘. while on the other hand lg.,iI 5 K& by construction of g,,: hence the sequence !I,, satisfies (4. I), and the result follows from Theorem 4- 1. 4-4) Finally we give a result which is simple, but will be used later. THEOREM 4.6. - Assume (1.11) and (A), and let 6 = (~‘)L<,+I be LI d-dimensional flmction satisfving (B-O). Therl the proces.~es Y” = p$Y(‘U,,. 11)” - 1:(*/L!I TH,,)“) converge stably in law trr N process I’. dc$ned on an extension of the space, and which is an 3-cotrditianal Gaussian continuous martingale M’ith brackets

(Y’Y)

= A(Nf,‘,,, - H,,~N,,,)L

Proof. - We have that I-‘” = v&&M(/2.)” (notation (4.7)) is a martingale w.r.t. the filtration (3~,,,l,,,)~>a. Further, under P,. any martingale (w.r.t. (3t)) orthogonal to --\; is constant. Hence by Theorem 3-2 of 19.1, the result will follow from the next three properties, where for any process Z we put Ag’Z = Z,,,, - Zi,-lil,,:

[rrtl c

E,,,(~.:‘E’“‘~~‘);““J3~)

,=l

/I

-+{‘J A(N,,i,,, - N,,fH,,., If+.

(‘Al!))

I4 ~E,(/A:‘E’“‘I’l(,~:.,~~.,,>~-r~3~)

4”’

0

VlE > 0.

(4.21)

i=l

By polarization, it is enough to prove these when h is l-dimensional, which we assume in the sequel. First, by (4.8) the left side of (4.19) is %k’(~,,.1f,~z - (H,,)“):. hence (4.19) obtains by Theorem I-l. --Second, by a simple computation the left side of (4.20) is ckT( II,, , !I);. where ,9(1:) = j’ /),(:I:, :y)y/,(~)dy, hence (4.20) obtains by Theorem I- 1 and 11ri - ‘3.2. yields IA:,,“‘1 < Third, another computation that (where (L and 6 are as in (BKir(U,,X(i&l)/,l) JGJrLC “lfiYsl 0)). Hence (A:,Y”I > t- + IfiAyXl > -h;- log(7~,/~,,) for some Vol. 34. IlO 4.lYY8

530

.I. JACOD

h-, > 0. Therefore

(the constant K depends on (I and on 5.). Therefore the left side of (4.21) is smaller than (T,, II.,,/,u)V( u,, . is):: we conclude by Theorem I-l once more and by the property y,, + 0.

5. A BASIC

MARTINGALE

It remains to provje Theorem l-2. for the Brownian measures 9,. on the canonical space; the &dimensional function /I, satisfies (B-r) for some 7‘ > 3, and ‘I!,, = II” with l/:3 < (Y < 1, and ~5= (( 1 - O) A rr)/2. For proving the convergence of the sequence rr”(~l~(?~“. II)” - X( ff,,)L) we will use the same method as for Theorem 4-6. However the above processes are not in general martingales. and our tirst task is to write them as sums of a sequence of martingales to which the previous method applies, and a sequence of processes which go to 0. In this section, we perform this decomposition. The processes of interest may be decomposed in a sum of four terms:

&‘(r,” where M” for ij):

. Ir )” ~- A( If,, )L = M” + N” + OR” + My’.

= ;;&;nf(/r.)”

(5.1)

(see (4.7)). and H = X(N,,), and (recall (1.14)

(5.2) Xl’ = -11.‘( fi. fi

!/);’ - L ,,,,],,, .

()I’ = Lr,,t ,,,, - L,.

As seen before, the process M” is a martingale w.r.t. the tiltration (?‘L~,,J,~,)~>~,,as well as R” by (4.5). It is not true for N” and 0”. but in this section we prove that ,tlhC)” is “negligible”. while ,brN” is a martingale plus a “negligible” term. The term Q” is easy to deal with: the local time 1, has Holder paths with index ,c for any c < l/2 (this is classical result, following for example

RATES OF CONVERGENCE

TO THE LOCAL

TIME

OF A DIFFUSION

531

from Kolmogorov’s criterion combined with (3.14)). Hence. since ~5< t/2. we have for all w E II:

The case of N” is more complicated. &V( fi. A:,,)“. where

By (4.9). we have I/~N”

X,,,(.r) = l,“(li”~‘i’ll,,(rr”--‘!‘.r.)

- Hy(:l.)).

=

(5. I)

In the rest of the paper, the constants K, h--) may depend on the function k via jl and rj. Observe that (B-7.) and (1.14) yield Ik,,(.r)I

5 r
&(A,,,)

for y E [O. 7.1. A( A,,,) = 0.

5 fw(l

+ 7P1)f1”-q (xi)

We also consider /j E (0. 1). to be chosen later, and set TP,,= [u’] and

532

J. JACOD

(5.11)

Observe that

<,‘I= I;,,(fix, ) + F,,(fix; ) - jr’,,(J;lx+ ) = J&/%X,) - i&6lX+l), !I so a simple computation

(5.12)

yields 7,,AN”

= W”

+ A’” + B’” .

(5.13)

First (5.8) and (5.11) yield that supt IAl’ ] is smaller, up to a multiplicative constant, than (Y).‘-~/’ + u’-~” + IL*+~~-~) log rb. Now, the definition of h implies that all the powers of ‘n in the previous bound are negative, and thus (5.14) 5.1. - ai If/-, E (1 - it, I), then I3” goes to 0 in I’, -probabilit): un$ormly over all finite intervals. LEMMA

hi !f /-I E ((I - ‘Ltr)+. I), then F;,(lB;‘)‘)

-+ 0.

Proc$ - Set y,, = I’ ((,,,+I XI,,. Observe that A(P,Az) = 0 as soon as X(X:) = 0, so here (5.5) yields A(!/,,) = 0. Hence (a) (resp. (b)) follows from Theorem 4- 1(a) (resp. Lemma 4-2) if we prove that the sequence Q,, satisfies (4.1) when ij > 1 - rr (resp. (4.2) when ,8 > (1 - ‘AU)+). When (1 2 l/2,

(5.9) yields

Iylr 1 < Kd-“)/2-~‘,

X(lyrtl)

[j1(!/,,)

jq /9n I”) < h’,,,l --cl’-:,.‘/2

5 K71(‘-“)fQ.

2 K7,,(1-fJ- j)/L’.

and all the claims are obvious. When CY< l/2,

(5.9) yields for all E E (0,~ - 31:

I!/,, 1 < KY,,‘/“---“/” /jl(gl,) A(

19,, I’)

jj(l.c/,ll) < K&(1--“)/2 <

jT((&:~;j/2-0

5 h;(,$-“-,f)/2

+ r,E- ‘+1-h/2)+

+ TIE++-j--~+)% +

7,E+3/2-2.1-2r>)

Then again, all the claims are easy to check, since CP> l/3.

0

RATES OF CONVERGENCE

LEMMA

5.2. - VP

TO THE LOCAL

E (( 1 - 2cx)+. I), then

TIMb

533

OF A DIFFUSION

SU~,~~~ ~TL~~V~’- bV:lI

-+*‘,

0

for all t.

Proof. - Let /j be as above, and choose [Y in (1 - CY.:, 1). With i-1’ we associate the processes IV”“, A’“, and B’“. Let C’” := A” + I?” = 7LhN71_ W” and Chl = A’” + B’“, By the previous lemma and (5.14), C”’ tends to 0 in probability, locally uniformly in time, and also CT and Ci7’ tend to 0 in B”(P,.). Then the processes 2” = (7” - P = W’” - W’” are martingales and satisfy ZZJ(/2,“1’) - 0 for all t. Following Aldous [2], we deduce that in fact 2” tends in probability to 0, locally uniformly in time. hence the same holds for C’“. Cl

6. THEOREM

1-2, THE CASE
Here we prove Theorem l-2 when o = l/2, hence 6 = l/4. Let us take /j = l/4 in the definition (5.10) of W”. In view of (5.1), (5.3) and Lemma 5-2, we are left to prove that the sequence

stably converges to the limit Y, as described in Theorem l-2. Note that I’” is a locally square-integrable martingale w.r.t. the filtration (?=I,~,I,,,)~>~,so exactly as for Theorem 4-6 the result will follow from (4.21), (4.22) and the following (which replaces (4.19)): c

E,.(Al’Y”‘Al’Y”“IF?+

iPJ. ,r,(h’. Ir’)L,.

(6.2)

i=l

By polarization, it is enough to prove these when h is l-dimensional, which we assume in the sequel. Combining (6.1), (5.2), (5.10). (5.12), (4.7), plus the facts that I;,, = ‘v,““(HI, - 0.6) and that E;,,= p,, + k:,,, we obtain

Proof

of

4.21. -

Kn,-“/2(e”“lJ;;~:‘sl+(1 Vol. 34. IlO d-1998

By (6.3), (5.8) and (B-r), IA:,Y”l” < ogn,)“+n”(A~LI”). Since E,~(n”~A~L~“I.F~+)) =

5.14

J. JACOD

G(1.6;,fi_YL) by (3.13), while bounded, we “get E,.(IA:‘S”‘i’II=~)) t:,.(I~~“~‘“/‘li~:,l.‘,i>E holds.

I.Fd))

(3.16) yields that CJ(1.6..) is 5 h’(log?r)(‘/~,~‘/~. This yields

< ~-~~I(logll)“/rr,“i’.

and thus (4.21)

Proof’ of 4.20. - In view of (6.3) and (3.13). the left side of (4.20) is &V(fi.!g,,); + ~V(L:;;.~g:,);‘+ *l’(fi.!y::):I. where

g::(.r) = &G(‘.

I../,) - G(.f. 1.x)).

with

j’(x) = .I’.

First. by Cauchy-Schwarz inequality lq,, (,r)l 5 +-i ,I’ lylr(:r.. !/)/p( ;~~)d;~+ + dm. Then (5.5) and (3.2) give Prk~(x) 2 Kfi/( I + I.rI+t).

It follows that I!/,, / and A( /T/,~I) are smaller than A% ‘I ‘. so the sequence q,, satisfies (4.1 ). and +I,-( fi. !I,~); -r, 0. $=(I’~~,~(.r))‘/‘. We have Next, Id, (:{.)I 2 K~~~*(logr~)‘)/( 1 + /Y/J ‘/‘/“‘-‘) by (5.7). Then (PI ?‘i(x))“’ 5 Krr’(log I,)/( 1 + I,IY~~~~I’-‘) and thus

J;,(:K) < (3.9) yields

so the sequence Hence ly,,I and A(l.v,,I) are smaller than K(log?r.)/r?/“. ,q,, satisfies (4.1) and 5 L’( fi. !I:, );’ +“I 0 by Theorem 4-l. Finally. consider &. Since &(LIAri,) = 0 for symmetry reasons, one has G(f$ 1.x) = 0 by (3.16), so (3.15) and (3.16) give

By the change of variable I’ = .I.‘/s we obtain

RATES OF CONVERGENCE

‘TO THE LOCAL

TIME

OF A DlFFUSlON

Then the sequence !& satisfies (4. I ) and h V( 6.9::); 4-l again.

535

--tP, 0 by Theorem

Proof of 6.2. - In this proof, we set X: = HI, - f/g = k,,/r~l/‘~ and 1F’L= F,, /,rr’/’ and $y, = i’? /II ‘/‘. Observe that X(X:) = 0, so fi’ := &‘(A,) may be defined by ( 1.16). and by (5.7) we see that both FA (:c) and p,{ (I) converge to F(x) and stay smaller than A’( 1 + 1.~1). By (6.3) and a simple computation, and if ,f,,.J(?/) = h(:c, ~1-x) + FA (a). the left side of (6.2) is $=( L’( ,/‘G, ~/,~)i’ + H’V(fi, !/‘)I’ - 2HV( &. !I::);’ ). where (recall (1.131, (3.12) and (3.13)):

!j,,(.cj=

b(~:)

+ z), z+::2(.rj - i-;; (#

g’(n) = G( 1.2. :r).

y::(r)

+ 21&:,

lzI.j.

= G(.f;i,x. 1.x)

First we observe that by (3.15) and (3.16).

Hence A(.(/‘) =

fij;:( 5 - &)ck = -L II&G’ and we obtain that +“( fi, y’); converges in P.,.-probability to AL, by Theorem 1-I. So in order to prove (6.2), and in view of Theorem 4-1, it suffices to show that the two sequences !I,, and ~1:: satisfy (4.2) and

X(&L) -

A( Hl,’ + 2&J).

A(!/::) -+ h(6).

(Ll)

The sequence !I,$: First we have

Since IP,,,,,+lk(a)l 5 K/l,,ll’( 1+1x/7/ ‘/lijn-‘) by (5.9) and IFAl 5 Kllog II by (5.8), we see that X((PU.r,.+lk)(21?,‘, -I- P,,,,,+lk)) -+ 0. Since I;;,(x) + E’(Y) and jF,l~(x)j _< Ii( I + i:r]), we see that I?h,f-;,(,~:) + I?,,F(J:) and that II?h,F:,(:~:)I 5 K&(x)( 1 + Ix/): so X(fi,,,~;) ---) X(~?,,,F). hence the first property (6.4). If FL = F,,/~L’/.‘, we have PIEx’ 5 2P1X:’ + 2PI pi, and HI,,E’;, = H,,,k + I?,,>,;, (B-1) implies il.-(k”) < cc for all s E [0, 1.1.so (3.2) yields P,k”(:I.) 5 K(c4’++) 5 $$-, while clearly jZ?l,,k/ < j,,m 5 Kh by Cauchy-Schwarz inequality. On the other hand we have seen in the Vol. 34. n” 4.IYYX

536

J. JACOD

proof of (4.20) that l’Ipi(:r) 5 K,,&(lo~7~)‘)/(1 + /XII --L/8(12r--L,),and by (5.7) the same majoration holds for ~Z(.I:), while (fiI,,I.;V,1 < h-i,,dm by Cauchy-Schwarz inequality again. Putting all these together yields 1!/,,(J~,I i

1

(log I,.)' I + /:rI~. + 1 + I~I:/,,I/K12r~--:!1

(-

A- h+ f-

Then I:/,, 1 5 K(log II.):! and A( I!],, I) < K(log~)‘nl/” and X(,r$) < K(log n)“7W and D1(!/,,) 5 K(log~)% ‘/’ . It readily follows that the sequence !I~, satisfies (4.2). The sequence

!J:::

(3.16) yields for .I’ # 0:

I;;(,(Ll(h(O, X1) +E::(X)))

if .I: = 0

,f/::(:l:)=

.’

I (,

1:I.l -g J1-lJ21F7.“/2 f’

if :I: # 0

x E,,(L,(h(:rLY,J1-r-

- .I:)) + F,;(x,J1-r-)))fl7~ (6.5)

The change of variable .I’ =

:t~!lJ;;

gives

A(!/::) =

Then F:, ( .I,) - F(x) and lF’,:(:r,)J 5 K(1 + 1~1) readily give the last property (6.4) (recall (1.17), and observe that p is the law of (-Yt. L,) under PC,). Further, (H-K) implies that the expectation in the expression (6.5) for :I’ # 0 is smaller than It~e”~~“~.Hence

It is then obvious that the sequence

,cJ::

satisfies (4.2).

Proof of 1.26. - Finally, let us verify that (1.26) holds when /l.(:l.. :I/) = !/(:I:). First we have El,, = ,CJ. Second,

by (3.16) and the change of variable :I: = :Y/,

while G’( 1,l. :r:) = D(X).

RATES OF CONVEKCL:NCE

7. THEOREM

TO THE LOCAL

TIME

OF A lJIFl+JSIOK

537

1-2, THE CASE rv > l/2

Here we prove Theorem l-2 when o E (l/Z. I), hence h = (1 - ck)/‘t. We will choose /j such that 0 < /9 < (1 - O) A (20 - 1): this choice is possible, and yields /r < l/3 and ij < 0. First, observe that I,‘/“( h\‘( fi.jj)” - L) = r,,‘/‘( R” + C)“) (this is (5.1) when I/.(./:. :1/)= !/(I)). and by Theorem l-2 applied with (Y = l/2 and //,(I, 1~)= j(:~,) these processes converge stably in law. In view of (5.3). it follows that IL’/-‘R” stably converge in law as well. Coming back to the case (Y > l/2, hence h < l/-2, we deduce that ,lhI?’ converges in law to 0. This and (5.3) and Lemma S-2 imply that for Theorem l-2 with o > l/2 we are left to prove that the sequence

stably converges to the limit as described in the theorem. Again I-” is a locally square-integrable martingale w.r.t. the filtration (~f,,t~,,l)t~o, so exactly as for Theorem 4-6 the result will follow from the properties (4.20). (4.21) and

By polarization, it is enough to prove these when h is l-dimensional. which we assume in the sequel. By (4.7), (S.lO), (5.12) and (7.1). we have

Prooj #j’ 4.2 1 - (B-V), (5.8) jqTL-h(&6q~SI + ,,‘~-1,/2, ogv), hence ~~~‘l(~-*/~)log I,). This quantity is smaller hence (4.2 I ) follows from Tchebicheff’s

and (7.3) yield I.:‘}-“/ 5 E’(j~j,k”‘I~[~+) < h-q(~rP” $ than l/‘rr” if (I is large enough, inequality,

Proofc!f’4.20. - By (7.?), the left side of (4.20) is -&V( 6.

!/,, )I’. where

In view of Theorem 4-I(a), it suffices to prove that the sequence {I,~ satisfies (4.1). Since IY > l/2. it follows from (5.7) and (3.9) that Vol. 34. rl@ 4.1098

538

J. JACOD

atom 5 Krr~1-“(10g~~)2/(1 + 1:r~-.‘/~1*r-~). On the other hand. (5.5) and (1.9) yield Ptkz(z) < Kn-‘y(c-1.1/2 + l/(1 + I.I:I”-~)) hence 1’1F;f(11.) 5 K,,.t-“(log

,,,)2

(

1 +

By Cauchy-Schwarz inequality -& dm. Thus (7.4) yields

1

1

/3:71-‘v212’.-2

lglL(z:)j

Hence l:g1115 K,/I-(~-“)/’ and A( I~~,,I) < (log we obtain the desired result.

+

<

1 +

l:[.I’.-l

(7.4) >

&~.(:uL”-.~/~)

+

,‘)/‘: since 0 < (Y,

71.)/71(+

Proofof7.2. - In view of (7.3), the left side of (7.2) is u”+‘V(YL”, (III,,)~): + ~V(fi..yn);’ + &ViJli,e,,j;. where

ck-1’2:C))&,(3:

+

H/,2 -

?/){J(?/)fh/.

First, Theorem 1-l gives that 71'1-1V(71'1,H~12 - (Ii,,)");' f (Hh)*)L, in &-probability. Thus it is enough to prove that

X(Hfj2

-

Let us study first g,l: we will prove that this sequence satisfies (4.2) and A(gl,) -+ X(Hz), thus obtaining the first condition in (7.5) by Theorem 4-1. By (5.7) and (7.4) we have h’q,71 (.Yn(~)l

I

1’2--‘r(10g

7,42 (

1 +

):C7,,-/f/2p-2

1

1 + +

13./‘.?1

1

1

Hence lgn / < K(log 7~,)‘/rr.“-~/~, and A( lglLl) 2 K(log 71)‘)/,,1’y-1/2--,j/2, and /711(gR)5 K(log 71.)2/71.‘r-‘/2--1’ and A( 1~ ! ,L 12) 5 K(log ‘71,)~/712’~~ l-“/2: in view of /j < o and /j < 20 - 1, we readily observe that the sequence gII satisfies (4.2).

RATES OF CONVERGENCE

TO THE LOCAL

TIME

Next, we easily obtain X(.r/,,) = X($=(P,T - k:)). $$c

539

OF A DIFFUSION

But we may write

- @rT) = TJ,,+ ;I/,, + z,, , where -1/2H,,(n ‘/I,,(.I.) = ‘71 yjl, (x ) = ‘71w“pjj(,r:)’

<1-l/2

.X)2.

_ ,,1/2--l,9(:1:)7111-1/2~~l(7,,‘t.-1/2:I.).

71r = L(h;,,(k,, fi

+ r;,,) - (I:c,,,+,k,,)(F,, f i-l,),.

Then (Y~~(:I.)I < K7~,1!2-‘1(4(.r) + 71°-1’2h(7~,1~-1/2%)), and it follows that Q,,) + 0. Moreover, (SS), (5.7) (5.8) and (5.9) 71)/(1 4 1:1:71-“/~/“-~)~ hence X( 1~~~l) < give jz,&(:~:)I 5 Ei’n’/“-“(log &-71-(-1/2- ‘f/2) log ,,, 4 0. Finally, we trivially have X(r),,) = X(&t): hence the first part of (7.5) holds. Let us turn now to the sequence P,,. We have If?,,(.I:)/ 5 K,~,-‘~(?)~‘~1/2~~)~~~‘~(:~.). By (7.4) we obtain that /lirl(.~:)I < K(log n)/t~(4Lc’-1/2.r): thus (Y,?1 5 K lof?;II and X( (8,,I) 5 Kr/,r/‘--” log 71.--+ 0, and Theorem 4-l(a) yields the second condition in (7.5).

8. THEOREM

1-2, THE CASE cv < l/2

Here we prove Theorem l-2 when o E (l/3.1/2), will choose /j such that 1 - 20 < ij < f

(*

jj < I -CY,

/-I < 2tt!,

hence b = (r/2. We

3/j < lf2tu

).

(X.1)

This choice is possible, since l/3 < CY< l/2. As seen in Section 7, ,r,l’dR’C stably converge in law. Since 6 < l/4, we deduce that r~‘R” converges in law to 0. On the other hand, r,(1-f’)/2fV” converges in law by Theorem 4-6 and since 6 < (1 - o)/2 it follows that the sequence ?L~&“’ tends in law to (I. These two facts, plus (5.3) and Lemma 5-2, imply that for Theorem l-2 with IY < I/2 we are left to prove that the sequence of martingales E”’ = W” stably converges to the limit as described in the theorem. Hence once again the result will follow from the properties (4.20), (4.21) and

540

J. JACOD

By polarization, it is enough to prove these when h is l-dimensional, which we assume in the sequel. By (5.10) and (5.12). we have

Proof of 4.21. - (5.8) yields lA:‘Y”i obviously holds.

5 K(log7~~)/~~~0/2, and thus (4.21)

In view of (8.3), the left side of (4.20) is q,,); + +L’(t/;;. q:,);‘, where

Proof oj‘ 4.20. -

-&l’(Jrr.

_ ,,,“/2+-,11/“-~l)-1/2 /,%I l.1.) I

-5

h -, ---

1

+]:1j-

for 3 E [l/Z. (1. -. I )/2].

Then clearly /!/,,I 5 K?lW1/’ and X(Jg,,]) 2 K~u-“” for r > 0 small enough: the sequence !I,( satisties (4.1) and hV( 4s. ~1,) );’ -+I’, 0 by Theorem 4- I. Next, observe that A( A:,,) = 0. hence X(F,, ) = 0 and A(!{:, ) = 0. So in view of Theorem 4- 1. it suffices to prove that the sequence !I:, satisfies (4.2). It follows from (5.7) and (3.9) that for 3: E [O.T’- I]:

By Cauchy-Schwarz inequality ]:I:,(:I,)] < $=(Prp;f(.l:) jliz. Thus for ,r > 0 small enough,

:jl(!,:,) ,q I{/;,l')

< <

h-,(log/l,)(7,, h-<(log

',,,)?(,,

'-iL' '/L'+l/L'-c,

+ 7,50/2-

l-+q.

+ ,,,,:,,/L'--1 tf ),

In view of (8.1) it is then obvious that (4.2) is met by !I:,.

RATES OF CONVERGENCE

Prmf’oj’8.2.

TO THE LOCAL

TIME

OF A DIFFUSION

- By (8.3). the left side of (8.2) is -$I’(fi.~j,,);‘.

541 where

!I,, = h(ptFif - iz). In fact F,, = F,, -- X:,,+ P,,.,,+lk,,, so we can write !I,, = yfz + z,, + r::,, with

?,’ = -&b-;; u,,

- F;;‘).

Then (8.3) will follows from the next three properties:

2) By (5.8) we have /z,~/ 2 ~,~/,-“IL’(lo~,~b)lP,,.,,+lli,, 1, while (5.9) gives II:,.,, +111:,,1 5 IGt.‘12- ‘- “/2 and A( IP,,,,,+tk,,i) 5 Krb(‘-“-“)/’ because /j > 1 - 20. It readily follows that the sequence z,, satisfies (4. I ), and the second property in (8.4) is satisfied. 3) In order to prove the last property in (8.4) it is enough by Theorem 4-l to show that the sequence z:, satisfies (4.2), since obviously A(;;:,) = 0. By (3.4) and (5.5) we have lP,klL(.~: + !I) - Z’jX,,,(.L:)I 5 KI:fllp/“‘L)/;j. Then /F,,(:r + ~1)- F),(.r)l <_ K~f”“‘(l + 1~1)log’n. and

“- lqog + 71.

,, l/“-u ,,i- t1)(1/2-cl I ,,)Y -I__ -+ l+l.I:I’ ( I + I.I:rr-Jq1

where the last equality follows from (5.7) and (3.9). It follows (using again A > 1 - 20, and (5.5)) that

pl(z:,)

5 K(log 11.)V.

x(2:;‘)

clearly

< K(log r,)%1’2.

Hence the sequence ,?;I satisfies (4.2) as soon as /l < l/:3, which is met by (8.1): so we have the last property in (8.4).

542

J. JACOD

4) In view of Theorem 4-l. the first property in (8.4) will hold if we prove that the sequence y,, satisfies (4.2) and X(y/,) - 7/1(/~:/L). By (5.8), therefore (5.5) yields I;~J,~ / 5 K logn, we have ly,, 1 I U.-fY/2(10g71)1k,,j, rklog’/l and X(;q,‘,) 2 and A( 1~~~1)5 K log ‘II, and /j~(?/,~) < Kn1/2K(log 745 which clearly imply that (4.2) is satisfied. 5) To simplify the notation we write u,, = 71.0-1/‘. Set &,,(:I:) = = ,~f,~,~b~P,,(,~~,,:r:). A simple calculation H,,(z) - H,cj(.l:/v,,)/1!,,. so II:,, shows P,k,,(.r) = ‘~),,I~.*P,,.;:B,,(~~,,:I.),and thus (X.3) Now, /j;(je,, 1) < K for i = 0, 1, ‘2 (use the fact that ‘II,, < I). So we deduce from (3.1), from (3.3) and A(a,,) = 0, and from (3.51, that

Putting all these together with (8.5) yields

Hence :y:,(.r) = Klli,,(z)l(71(i).--l)/2

27!?k,, (J:) &” F’,a,, (,u,,.I:)& has ~J/~~(.c)- q:, (:[.)I < + 7tl-!j--3~~)/2 + IiI;17,, --(“+a)/z). Since A( I/C,,/) < Kn” we obtain A(/g?, - u:,/) 5 K(I),“-- ‘/2 + and /jl(k,,) 5 l1~71.“~~“+““, 71(1-.G2fk)/‘) + 0 because :I + Ztr > 1 and (1 < l/2. Further if ?J;:= 2P,, .I;,” PsF,,(1.s we have X(:q/:,) = X(qY::). Hence it remains to prove that A($) --+ .r/‘(it,h.). 6) Now we study t’,Y,, = IT,Iih - HP,jj,, , where ij,, ( :I,) = jj( :/./,o,,) /v,, . In fact we compare ,I;;^ 1’,8,, (.I.)& with G( .I:). where C; = G’( H,, ). If we set y,<(x) = I:,H,, (.r) - &P-,~‘/~,’ Set also r:‘(x)

= &r-.“/”

we have G(x) = ,I;;” y,q(:r:)d.s.

- F*jj,,(:r:), so that

RATES OF CONVERGENCE

NOW,

&($,,)

TO THE LOCAL

TIME

OF A DIFFUSION

543

= ,~f~/-I;(i], so (3.1) and (3.3) and oII I 1 yield

Dividing the integral in (8.6) in two pieces, from 0 to E and from E to ‘x. and using (8.7) for the first piece and (8.8) for the second piece, we get

If fil = 2P,,G, and taking E = ‘I!,, above, we thus get I$( I:) - .f,, (x)1 I KIIr,,(.c)(&(l + 1x1). Thus X(I:y:: - f,,I) < KJV,, + 0, and it remains to prove that X(fT1) - v’(h.,h). 7) Apply (3.4) and the majorations /$===(&~” - l)i 5 Klzl/s (easily deduced from (3.6) and (3.7)) to obtain [T,~(:I:) - yo(:r:)J < KIxl/.~i. Thus if we cut the integral G(u,,:I.) - G(0) = ,I~~“(y,~(~,~,:r)- ys(0))tls in three pieces, from 0 to E (use (8.7)), from E to .-1 (use what precedes) and from A to CC (use (8.8)), we get if 0 < E < I < A < x:

Taking E = ‘~~,,and L:I = I /!J: gives

(X.9) Now we can write

= axi H,, G) - 20

.i

g(:r:)G(,fr,,.r,)d:r

Recalling that $(/I,~ I,.) = 2X( H,,G) - 2M~(O) and that X(g) = 1, it follows that Ih(f,,)

- ‘r/‘(h, /I.)[ 5 K

by (8.9). and we are finished. Vol 34, Ilo 4.1998

.I _< Kfi(1

tj(:r:)(G(u,,:r) - G(O)(tl:r +/jr(G))

-4 0

544

J. JACOD

REFERENCES