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Journal of Financial Econometrics
, Volume Advance Article – Aug 2, 2017

44 pages

/lp/ou_press/limit-of-random-measures-associated-with-the-increments-of-a-brownian-XvVM15xFMs

- Publisher
- Oxford University Press
- Copyright
- © The Author, 2017. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com
- ISSN
- 1479-8409
- eISSN
- 1479-8417
- D.O.I.
- 10.1093/jjfinec/nbx021
- Publisher site
- See Article on Publisher Site

Abstract We consider a Brownian semimartingale X (the sum of a stochastic integral w.r.t. a Brownian motion and an integral w.r.t. Lebesgue measure), and for each n an increasing sequence T(n, i) of stopping times and a sequence of positive ℱT(n,i)-measurable variables Δ(n,i) such that S(n,i):=T(n,i)+Δ(n,i)≤T(n,i+1). We are interested in the limiting behavior of processes of the form Utn(g)=δn∑i:S(n,i)≤t[g(T(n,i),ξin)−αin(g)], where δn is a normalizing sequence tending to 0 and ξin=Δ(n,i)−1/2(XS(n,i)−XT(n,i)) and αin(g) are suitable centering terms and g is some predictable function of (ω,t,x). Under rather weak assumptions on the sequences T(n, i) as n goes to infinity, we prove that these processes converge (stably) in law to the stochastic integral of g w.r.t. a random measure B which is, conditionally on the path of X, a Gaussian random measure. We give some applications to rates of convergence in discrete approximations for the p-variation processes and local times. Consider a triangular array (ξin)1≤i≤n of IRd-valued variables and, with any function g on IRd, associate the processes Utn(g)=n−1/2∑1≤i≤[nt][g(ξin)−αin(g)], (1.1) where αin(g) are suitable centering terms. Finding limit theorems for Un(g) is an old problem, solved in many special cases: for example, the ξin s are rowwise i.i.d., or rowwise mixing, or are the increments of martingales. In a series of recent papers (Fujiwara and Kunita, 1990; Kunita, 1991a, 1991b), Fujiwara and Kunita have investigated the properties of the limit Un(g) as a function of g: indeed for suitably chosen centering terms, g↦Utn(g) is linear; then in the simplest case of rowwise i.i.d. the limit appears to be of the form U(g)t=∫[0,t]×I Rdg(x)B(ds,dx), (1.2) where B is a Gaussian random measure, and more precisely a white noise conditioned on the fact that B([0,t]×IRd)=0 for all t (this is just a somewhat sophisticated version of the usual Donsker’s Theorem). 2. In this article, we consider a richer situation. We start with a standard d-dimensional Brownian motion W=(Wi)1≤i≤d on the standard Wiener space (Ω,ℱ,(ℱt)t≥0,P) and the (ξin)1≤i≤d are increments of W. More precisely, for each n we have a strictly increasing sequence of stopping times (T(n,i),i≥1), and associated positive variables Δ(n,i), and we set S(n,i)=T(n,i)+Δ(n,i) and ξin=Δ(n,i)−1/2(WS(n,i)−WT(n,i)). (1.3) Denote by ρ the Gaussian measure N(0,Id) on IRd. We also consider functions g: Ω×IR+×IRd→IRq which are “predictable”, and instead of Equation (1.1) we are interested in the asymptotic behavior of the processes Utn(g)=δn ∑i:S(n,i)≤t (g(T(n,i),ξin)−∫ρ(dx) g(T(n,i),x)). (1.4) where δn is a normalizing sequence going to 0 as n→∞. We need a series of hypotheses for Un(g) to converge to a non-trivial limit. First about g: Assumption K g is a function: Ω×IR+×IRd→IRq, with it is predictable, that is, P⊗Rd-measurable, where P is the predictable σ-field on Ω×IR+, t↦g(ω,t,x) is continuous, there is a non-decreasing adapted finite-valued process γ=(γt) having |g(ω,t,x)|≤γt(ω)(1+|x|γt(ω)). (1.5) □ Second, there are assumptions on the times T(n, i) and Δ(n,i): the increments of W should be taken on non-overlapping intervals, that is S(n,i)≤T(n,i+1). Further, for technical reasons we need S(n, i) to be ℱT(n,i)-measurable: this is a serious restriction, but something of this sort cannot be totally avoided (take for instance Δ(n,i) to be such that ξin=0 identically in Equation (1.3), to see that without strong assumptions on Δ(n,i) we cannot hope for non-trivial limits for Equation (1.4)). Hence, we assume the Assumption A1 For each n∈IN⋆ we are given Tn=(T(n,i),Δ(n,i)):i∈IN) with: The sequence T(n, i) is an increasing family of stopping times with T(n,0)=0 and limi↑T(n,i)=∞ . Each Δ(n,i) is a (0,∞)-valued ℱT(n,i)-measurable random variable, such that S(n,i):=T(n,i)+Δ(n,i)≤T(n,i+1). □ We also need some nice asymptotic behavior of the sequence (Tn) in relation with the normalizing constants δn in Equation (1.4). This is expressed through the following random “empirical measures” on IR+, where ɛa denotes the Dirac mass with support {a}: μn=δn∑i≥0,S(n,i)<∞ ɛS(n,i), (1.6) μn⋆=∑i≥0,S(n,i)<∞ Δ(n,i)δn ɛS(n,i). (1.7) Assumption A2 μn and μn⋆ vaguely converge in probability to some random Radon measures μ and μ⋆. □ Both (A1) and (A2) are satisfied in the so-called regular case, where T(n,i)=i/n,Δ(n,i)=1/n and δn=1/n: then μ=μ⋆ is Lebesgue measure. In general the convergence of μn implies the relative compactness of the sequence μn⋆ (in probability, for the vague topology), and also its convergence (in probability) to μ⋆=0 when μ is a.s. singular w.r.t. Lebesgue measure. 3. Our first main result, under Equations (A1) and (A2), is the existence of a random martingale measure B on IR+×IRd, defined on an extension of the original space (Ω,ℱ,P), such that for any g having (K), Utn(g) converges in law to Ut(g)=∫g(s,x)1[0,t](s) B(ds,dx). The measure B is called the tangent measure to W along the sequence (Tn), and its precise description in terms of W, μ, μ⋆ is given later. However, the statement is simple in the regular case, and goes as follows (all unexplained notions below are recalled in Sections 1 and 2): Theorem 1 Assume that we are in the regular case (or more generally that (A1) and (A2) hold with μ=μ⋆= Lebesgue measure). There is a random measure B on IR+×IRd, defined on a very good extension of the Wiener space, which is a white noise with intensity measure dt×ρ(dx) conditioned on having B([0,t]×IRd)=0 for all t, and which satisfies ∫x1[0,t](s) B(ds,dx)=Wt, (1.8) and such that for every g satisfying (K) the processes Un(g) converge stably (in the sense of Renyi) in law to the process Ut(g)=∫g(s,x)1[0,t](s) B(ds,dx). (1.9) That Equation (1.8) should hold comes from the fact that if g(x) = x then Utn(g)=W[nt]/n. Taking g = 1, hence Utn(g)=0, shows that one must have B([0,t]×IRd)=0. Related results have appeared in various guises in the literature: for instance they come naturally when one studies the error term in approximation for stochastic integrals or differential equations: see Rootzen (1980), which contains a discussion of the interest of stable convergence in this context, or Kurtz and Protter (1991). The main applications we have in mind concern statistical problems related to estimation of the variance coefficient with discrete observations for diffusion processes, in the spirit of Dohnal (1987) or Genon-Catalot and Jacod (1993). This is why we have considered schemes Tn based on stopping times rather than deterministic times (see also the applications relating to local time, in Section 8). 4. Our second main results will be obtained as a consequence of the first one, and concerns m-dimensional “Brownian semimartingales” of the form Xt=x0+∫0tasdWs+∫0tbsds, x0∈IRm, (1.10) with the following: Assumption H a and b are predictable locally bounded processes, with values in IRm⊗IRd and IRm, respectively, and t↦at is continuous. □ In this setting we study the limit of processes like Un(g) in Equation (1.4), with different centering terms, and X instead of W in definition (1.3) of ξin. The limit can still be expressed as a suitable integral w.r.t. the tangent measure B to W, and also as ∫g(s,x)10,t](s) BX(ds,dx) with another random measure BX called the random measure tangent to X along (Tn). 5. The article is organized as follows. Part I (Sections 1–4) concerns the Brownian case: Section 1 is devoted to some preliminary results on extensions of spaces and random measures; in Section 2, we describe the tangent random measure to W and state the result, which is proved in Sections 3 and 4. Part II is about Brownian semimartingales of form (1.10): results are gathered in Section 5, and proofs are given in Sections 6 and 7. Finally, Section 8 is devoted to some simple applications (rates of convergence for q-variations, approximation of local times, etc.). PART I: THE BROWNIAN CASE 1 EXTENSION OF SPACES AND MARTINGALE MEASURES In this section, we start with some filtered probability space (Ω,ℱ,(ℱt)t≥0,P). We gather a number of results on extensions of this space and martingale measures: some are new, and some are more or less well known but we have been unable to find precise statements for them in the literature. We state them in a general context, but very often we assume the following hypothesis, which is met by the Wiener space: Assumption B All martingales on (Ω,ℱ,(ℱt)t≥0,P) are continuous, and the σ-field ℱ0 is P-trivial. □ 1.1 Extension of Filtered Spaces We call extension of (Ω,ℱ,(ℱt),P) a filtered probability space (Ω¯,ℱ¯,(ℱ¯)t,P¯) constructed as follows: starting with an auxiliary filtered space (Ω′,ℱ′,(ℱ′t)) and a transition probability Qω(dω′) from (Ω,ℱ) into (Ω′,ℱ′), we set (Ω¯,ℱ¯)=(Ω,ℱ)⊗(Ω′,ℱ′), ℱ¯t=∩s>tℱs⊗ℱ′s and P¯(dω,dω′)=P(dω)Qω(dω′). We also assume that each σ-field ℱ′t− is separable (this is an adhoc definition, sufficient for our purposes here). According to Jacod (1979) (see Lemma (2.17)), the extension is called very good if all martingales on (Ω,ℱ,(ℱt),P) are also martingales on (Ω¯,ℱ¯,(ℱ¯t),P¯) or, equivalently, if ω↦Qω(A′) is ℱt-measurable for every A′∈ℱ′t. A process Z on the extension is called an ℱ-conditional martingale (respectively, Gaussian process) iff for P-almost all ω the process Z(ω,.) is a martingale (respectively, a Gaussian process) on the space (Ω′,ℱ′,(ℱ′t),Qω). A locally square-integrable martingale on the extension is called (ℱt)-localizable if there exists a localizing sequence of stopping times (Tn)relative to (ℱt). Lemma 1 Let M be a right-continuous adapted process on a very good extension, each Mt being P¯-integrable. Then M is an ℱ-conditional martingale iff M is an (ℱ¯t)-martingale orthogonal to all bounded (ℱt)-martingales. Proof Let t≤s, and U and U′ be bounded measurable functions on (Ω,ℱt) and (Ω′,ℱ′t) respectively, and Z be a bounded (ℱt)-martingale. We have E¯(UU′ZsMs)=∫P(dω)U(ω)Zs(ω)∫Qω(dω′)U′(ω′)Ms(ω,ω′), (2.1) E¯(UU′ZtMt)=∫P(dω)U(ω)Zt(ω)∫Qω(dω′)U′(ω′)Mt(ω,ω′). (2.2) If M is an ℱ-conditional martingale, for P-almost all ω we have ∫Qω(dω′)U′(ω′)Ms(ω,ω′)=∫Qω(dω′)U′(ω′)Mt(ω,ω′), and the latter is ℱt-measurable as a function of ω because the extension is very good. Using the fact that Z is a martingale on (Ω,ℱ,(ℱt),P) we have E¯(UU′ZsMs)=E¯(UU′ZtMt), hence ZM is a martingale on the extension: then M is a martingale (take Z = 1), orthogonal to all bounded (ℱt)-martingales. Conversely assume that M is a martingale, orthogonal to all bounded (ℱt)-martingales. Take a bounded ℱs-measurable function V, and consider the (ℱt)-martingale Zt=E(V|ℱt), which has Zs = V. By hypothesis, the left-hand sides of Equations (2.1) and (2.2) are equal, and in the right-hand side of Equation (2.2) we can replace Zt by Zs = V because the last integral is ℱt-measurable in ω. Then (taking U = 1) we have for all V as above: ∫P(dω)V(ω)∫Qω(dω′)U′(ω′)Ms(ω,ω′)=∫P(dω)V(ω)∫Qω(dω′)U′(ω′)Mt(ω,ω′). So for P-almost all ω, Qω(U′Mt(ω,.))=Qω(U′Ms(ω,.)). Because of the separability of the σ-fields ℱ′t− and of the right-continuity of M, we have this relation P-almost surely in ω, simultaneously for all t≤s and all ℱ′t−-measurable variable U′: this gives the result. □ Below ⟨M,N⟩ is the usual predictable bracket of the two locally square-integrable martingales M and N, with the convention ⟨M,N⟩0=E¯(M0N0). If M=(Mi)1≤i≤d is d-dimensional its transpose is MT and MMT, respectively, ⟨M,MT⟩, is the d2-dimensional process with components MiMj, respectively, ⟨Mi,Mj⟩. A process Z is called (ℱt)-locally square-integrable if there is a localizing sequence (Tn) of (ℱt)-stopping times such that each ZTn∧t2 is integrable. Lemma 2 Assume (B) and let Z be a continuous q-dimensional ℱ-conditional Gaussian martingale on a very good extension, which moreover is (ℱt)-locally square-integrable (by Lemma 1 it is an (ℱt)-localizable locally square-integrable martingale, and ⟨Z,ZT⟩ exists). There is a version of ⟨Z,ZT⟩ which is (ℱt)-predictable, hence which does not depend on ω′. Z is ℱ-conditionally centered iff E¯(Z0)=0, in which case the ℱ-conditional law of Z is characterized by the process ⟨Z,ZT⟩ (i.e., for P-almost all ω, the law of Z(ω,.) under Qω depends only on the function t↦⟨Z,ZT⟩t(ω). Proof By (ℱt)-localization we may and will assume that Z is square-integrable. Set Ft(ω)=∫Qω(dω′)Zt(ω,ω′) and Gt(ω)=∫Qω(dω′)(ZtZtT)(ω,ω′). There is a P-full set A such that if ω∈A, under Qω, the process Z(ω,.) is both Gaussian and a martingale, hence it is a process with independent and centered increments: so Ft(ω)=F0(ω) and (ZtZtT)(ω)−Gt(ω) is a martingale. By Lemma 1, ZZT−G is an (ℱ¯t)-martingale, while G0=E¯(Z0Z0T|ℱ)=E¯(Z0Z0T|ℱ0)=E¯(Z0Z0T)=⟨Z,ZT⟩0 (use the very good property of the extension and the fact that ℱ0 is P-trivial). Further since G is continuous (ℱt)-adapted it is (ℱt)-predictable, hence is a version of ⟨Z,ZT⟩. Similarly Ft=F0=E¯(Z0), so the necessary and sufficient condition is trivial. Further if ω∈A and Ft(ω)=0 for all t, the law of Z(ω,.) under Qω is characterized by the covariance ∫Qω(dω′)(ZtZsT)(ω,ω′)=Gs∧t(ω), hence the last claim. □ Lemma 3 Assume (B), and let Z be a continuous q-dimensional local martingale on a very good extension, with the following: E¯(Z0)=0, and Z is orthogonal to all (ℱt)-martingales, and ⟨Z,ZT⟩ has an (ℱt)-predictable version. Then, Z is an ℱ-conditional centered Gaussian martingale. Proof Since ⟨Z,ZT⟩ is (ℱt)-predictable, it is (ℱt)-locally integrable, and as in the previous lemma we may and will assume that Z is in fact square-integrable. Since Z is orthogonal to all (ℱt)-martingales, the same is true of M:=ZZT−⟨Z,ZT⟩=2Z·ZT. Lemma 1 applied to Z and to M shows that for P-almost all ω, under Qω the process Z(ω,.) is a continuous martingale with deterministic bracket ⟨Z,ZT⟩(ω), hence it is a Gaussian martingale, centered by Lemma 2-b because E¯(Z0)=0: hence the result. □ 1.2 Martingale Measures First, we recall some facts about martingale measures: see Walsh (1986) for a complete account. Let again (Ω,ℱ,(ℱt),P) be a filtered probability space. A (finite) L2-valued martingale measure B on IRd is a collection (B(A)t:t∈IR+,A∈Rd) of random variables and a sequence (Tn) of stopping times increasing to +∞, such that for all n∈IN: (i) for all A∈Rd, t↦B(A)t is a square-integrable martingale, (ii) for all t∈IR+, A↦B(A)t is a L2-valued random measure.} (2.3) The measure is called continuous if each t↦B(A)t is a.s. continuous. The (random) covariance measure is ν(ω;[0,t]×A×A′)=⟨B(A),B(A′)⟩t(ω). (2.4) In general [0,t]×A×A′↦ν(ω;[0,t]×A×A′) cannot be extended as a (signed) measure ν(ω;.) on IR+×IRd×IRd. However, it has the following: Property P Each process ν([0,.]×A×A′) is càdlàg predictable. A↦ν([0,t]×A×A′) is an L2-valued measure on (IRd,Rd). It is symmetric positive definite, in the sense that ν((s,t]×A×A′)=ν((s,t]×A′×A) and that for all n∈IN, ai∈IR, Ai∈Rd, then t↦∑1≤i,j≤naiajν([0,t]×Ai×Aj) is a.s. increasing. E[ν([0,Tn]×A×A))<∞ for all A∈Rd, for some localizing sequence (Tn) of stopping times. □ Following Walsh (1986), we say that B (or ν) is worthy if there is a positive random measure η(ω,.) on IR+×IRd×IRd which satisfies (P) and such that |ν|≪η (i.e., for all s≤t, A,A′∈Rd, |ν([0,t]×A×A′)−ν([0,s]×A×A′)|≤η((s,t]×A×A′)). In this case, there is a version of ν which extends as a (signed) measure on IR+×IRd×IRd. If B is worthy, we can define a stochastic integral process f⋆Bt=∫f(.,s,x)1[0,t](s)B(ds,dx) for every predictable function f on Ω×IR+×IRd having ∫f(s,x)f(s,x′)1[0,t](s)η(ds,dx,dx′)<∞ a.s. for all t. Stochastic integrals are characterized by the fact that f⋆Bt=B(A)t if f(ω,s,x)=1A(x), that f↦f⋆B is a.s. linear, and that f⋆B is a locally square-integrable martingale with ⟨f⋆B,f′⋆B⟩t=∫f(s,x)f(s,x′)1[0,t](s) ν(ds,dx,dx′). (2.5) Recall also that a white noise on IR+×IRd with intensity measure m (a positive σ-finite measure on IR+×IRd) is a Gaussian family of centered variables φ=(φ(A):A∈R+⊗Rd) with φ(A) and φ(A′) independent when A∩A′=Ø, and such that E[φ(A)2]=m(A). Obviously m characterizes the law of φ, and if m([0,t]×IRd)<∞ for all t, then B(A)t:=φ([0,t]×A) defines an L2-valued martingale measure on IRd for the filtration ℱt=∩s>tσ(B(A)r:r≤s,A∈IRd), with deterministic covariance measure ν([0,t]×A×A′)=m([0,t]×(A∩A′)). In this case ν is worthy. 2. Consider now a very good extension (Ω¯,ℱ¯,(ℱ¯t),P¯) of (Ω,ℱ,(ℱt),P). By definition an ℱ-conditional Gaussian measure is an L2-valued martingale measure on the extension, such that each finite family (B(A1),⋯,B(An))) is an ℱ-conditional Gaussian process. Further, it is an ℱ-conditional centered Gaussian measure if moreover each B(A) is also an ℱ-conditional centered martingale. Proposition 1 Let B be an ℱ-conditional Gaussian measure on a very good extension. There is a unique decomposition B=B′+B″, where B′ is an L2-valued martingale measure on (Ω,ℱ,(ℱt),P) and B″ is an ℱ-conditional centered Gaussian measure. The corresponding covariance measures ν, ν′, ν″ have ν=ν′+ν″. Under (B), there is a version of ν which does not depend on ω′, and the ℱ-conditional law of B is characterized by B′ and ν (or ν″). Proof Using Equation (2.3)-(i), by (ℱt)-localization we may and will assume that each B(A) belongs to the space ℋ¯2 of all square-integrable martingales on the space (Ω¯,ℱ¯,(ℱ¯t),P¯), which we endow with the Hilbert norm ||M||2=E¯(M∞2). Let ℋ2 be the closed subspace of all elements of ℋ¯2 that are martingales on (Ω,ℱ,(ℱt),P). Call B′(A) the orthogonal projection of B(A) in ℋ¯2, on ℋ2. Since M↦Mt is continuous from ℋ¯2 into L2(P¯), the collection B′=(B′(A)t:t≥0,A∈Rd) is an L2-valued measure martingale on (Ω,ℱ,(ℱt),P). Set B″=B−B′, which is an L2-valued measure martingale on (Ω¯,ℱ¯,(ℱ¯t),P¯), and also clearly an ℱ-conditional Gaussian measure. Since B″(A) is orthogonal to ℋ2, Lemma 1 yields that it is an ℱ-conditional martingale. Further B′(A)0=P¯(B(A)0|ℱ0)=E¯(B(A)0|ℱ) since we have a very good extension. Then E[B″(A)0]=0, and it follows from Lemma 2 that B″ is an ℱ-conditional centered Gaussian measure. We have thus a decomposition B=B′+B″. Now, for any such decomposition B″(A) is orthogonal to ℋ2 by Lemma 1, while B′(A)∈ℋ2, hence uniqueness. The orthogonality of any B′(A) with any B″(A′) readily yields ν=ν′+ν″. b. Since ν is (ℱt)-predictable in the sense of P-(i) and since a version of ν″ is given by ν″([0,t]×A×A′)=∫Qω(dω′) (B′′t(A)B′′t(A′))(ω,ω′) (see the proof of Lemma 2), we see that ν does not depend on ω′. The second claim follows from Lemma 2-b. □ Proposition 2 Let ν=(ν(ω;[0,t]×A×A′):t≥0,A,A′∈Rd) satisfy (P) and be worthy. There is an ℱ-conditional centered Gaussian measure on a very good extension of (Ω,ℱ,(ℱt),P), having ν for covariance measure. Proof Let ℰ be a countable algebra generating the Borel σ-field Rd. Set Ω′=IRl Q+×ℰ, with the “canonical process” B′=(B′(A)t:t∈lQ+,A∈ℰ), and ℱ′t=∩s>tσ(B′(A)r:r≤s,A∈ℰ) and ℱ′=⋁t>0ℱ′t. Then ℱ′ and all ℱ′t− are separable. Using (P-iii) we see that there is a unique probability measure Qω on (Ω′,ℱ′) under which B′ is a centered Gaussian process with covariance Qω[B′(A)B′(A′)]=ν(ω;[0,t⋀s]×A×A′). Further, (P-i) implies that Qω(dω′) is a transition probability from (Ω,ℱ) into (Ω′,ℱ′), and also from (Ω,ℱt) into (Ω′,ℱ′t) for all t. Therefore, the extension (Ω¯,ℱ¯,(ℱ¯t),P¯) of (Ω,ℱ,(ℱt),P) based on (Ω′,ℱ′,(ℱ′t),Qω) (see Section 1.1) is very good. Under Qω, the process (B′(A)t)t∈l Q is also a martingale along lQ+; hence, if we set B″(A)t=limsups∈ l Q,s>t,s→tB′(A)s we obtain a process B″(A) indexed by IR+ which is again a centered Gaussian martingale under each Qω. Further (P-iv) yields P¯(B″(A)t2)<∞, hence by Lemma 1, for each A∈ℰ, (B″(A)Tp∧t)t≥0 is a square-integrable martingale on the extension. Now we use the existence of a positive random measure η having (P) and dominating ν: if An∈ℰ decreases to Ø, then E¯(B″(An)Tp∧t2)≤E¯[η([0,Tp]×An×An)]→0 as n→∞. Thus, A↦B″(A)Tp∧t is an L2-valued measure on (IRd,ℰ). At this point, we can repeat the argument of Walsh (1986) to the effect of constructing B(A) for A∈Rd as the stochastic integral of the function 1A w.r.t. the martingale measure B″ on (IRd,ℰ). The family B=(B(A)t:t≥0,A∈Rd) constructed in this way clearly satisfies Equation (2.3), and B(A)=B″(A) if A∈ℰ. Moreover, if A∈Rd there is a sequence An∈ℰ with B″(An)Tp∧t→B(A)Tp∧t in L2(P¯): we deduce first that Equation (2.4) holds if A∈Rd and A′∈ℰ, and repeating the same argument and using the symmetry in (P)-(iii) gives Equation (2.4) for all A,A′∈Rd, that is ν is the covariance measure of B; we deduce next that, since each B″(An) is orthogonal to all (ℱt)-martingales by Lemma 1, the same is true of B(A) and therefore by Lemma 1 again B(A) is an ℱ-conditional martingale. Furthermore by taking a subsequence we can even suppose that the convergence B″(An)t→B(A)t holds P-a.s. for all t≥0, hence Qω-a.s. for P-almost all ω: since (B″(An1),…,B″(Anp)) is a centered Gaussian process under Qω for Ani∈ℰ, it follows that (B(An1),…,B(Anp)) is also a centered Gaussian process under Qω for P-almost all ω, if Ani∈Rd. Hence, (B(A1),…,B(Ap)) is an ℱ-conditional centered Gaussian martingale for all Ai∈Rd, and we are finished. □ Proposition 3 Assume (B), and let B be a worthy ℱ-conditional centered Gaussian measure on a very good extension, with covariance measure ν (not depending on ω′). Let f:Ω×IR+×IRd→IRq be predictable and integrable w.r.t. B. Then f⋆B is an ℱ-conditional centered Gaussian martingale, orthogonal to all (ℱt)-martingales, and its ℱ-conditional law is determined by its bracket which does not depend on ω′): ⟨f⋆B,fT⋆B⟩t=∫f(s,x)fT(s,x′)1[0,t](s) ν(ds,dx,dx′). (2.6) Proof All claims are obvious when f(ω,t,x)=(1A1(x),…,1Aq(x)) (use Lemma 2 for the last property), and follow by linearity for all “simple” functions. In the general case, the bracket is given by Equation (2.6) (see Equation (2.5)) and thus by (ℱt)-localization we can and will assume that f⋆B is square-integrable. There is a sequence (fn) of simple functions such that fn⋆Bt→f⋆Bt P¯-a.s. and in L2(P¯) for all t. Then repeating the final argument of the previous proof, we obtain that f⋆B is an ℱ-conditional centered Gaussian martingale, orthogonal to all (ℱt)-martingales. The last claim again comes from Lemma 2. □ Remark 1 An ℱ-conditional Gaussian measure is not a Gaussian measure, unless its covariance measure ν is deterministic. If B is an ℱ-conditional centered Gaussian measure, it is not true in general that for P-almost all ω, B(ω,.) is a Gaussian martingale measure on (Ω′,ℱ′,(ℱ′t),Qω). However, when this is true, in Proposition 3 f⋆B(ω,.) is also the “Wiener” integral of the deterministic function (s,x)↦f(ω,s,x)1[0,t](s) w.r.t. the Gaussian measure B(ω,.), relative to Qω. □ 2 THE MAIN RESULT In the rest of the article (Ω,ℱ,(ℱt),P) is the d-dimensional standard Wiener space, with the canonical process W. Recall that ρ=N(0,Id). We write ρ(f)=∫f(x)ρ(dx), and ρ(x1A)=∫x1A(x)ρ(dx), and ρ(ft)(ω)=∫f(ω,t,x)ρ(dx), etc. In order to define the tangent martingale measure, we need the following Lemma, which will be proved in Section 3: Lemma 3 Assume (A1) and (A2). Let λ be the Lebesgue measure on IR+, and μac be the absolutely continuous part of μ w.r.t. λ. There are two nonnegative predictable processes θ, θ⋆ such that μac([0,t])=∫0tθsds, μ⋆([0,t])=∫0tθs⋆ds, (3.1) θs⋆2≤θs. (3.2) Definition 1 A tangent measure to W along the sequence (Tn) satisfying (A1) and (A2) is an ℱ-conditional Gaussian measure B on IRd, defined on a very good extension (Ω¯,ℱ¯,(ℱ¯t),P¯) of the filtered space (Ω,ℱ,(ℱt),P), such that E¯(B(A)0)=0 for all A∈Rd, that ⟨W,B(A)⟩t=ρ(x1A) μ⋆([0,t]) (3.3) for all A∈Rd, and having the covariance measure ν([0,t]×A×A′)=(ρ(A∩A′)−ρ(A)ρ(A′)) μ([0,t]). (3.4) □ The following provides an equivalent definition for a tangent measure, and proves that it exists and is “essentially” unique in the sense that its ℱ-conditional law is uniquely determined (by application of Proposition 1). Proposition 4 B is a tangent measure iff it is an ℱ-conditional Gaussian measure whose decomposition B=B′+B″ of Proposition 1 has, with ν″ covariance measure of B″: B′(A)=ρ(xT1A) θ⋆·W, (3.5) ν″([0,t]×A×A′)=(ρ(A∩A′)−ρ(A)ρ(A′))μ([0,t])−ρ(xT1A)ρ(x1A) ∫0tθs⋆2ds. (3.6) b. There exists a tangent measure, and all of them are worthy. Proof Let B=B′+B″ be the decomposition of the tangent measure B. Then B′(A) is a local martingale on (Ω,ℱ,(ℱt),P), hence B′(A)=αT·W for some predictable d-dimensional process α, while B″(A) is orthogonal to W: thus ⟨W,B′(A)⟩=⟨W,B(A)⟩, and Equations (3.1) and (3.3) yield αt=ρ(x1A)θt⋆ for λ-almost all t and Equation (3.5) follows. The covariance measure ν′ of B′ is trivially given by the last term in Equation (3.6) (with the + sign), so ν=ν′+ν″ gives Equation (3.6). Conversely assume Equations (3.5) and (3.6). Again ⟨W,B′(A)⟩=⟨W,B(A)⟩, hence Equation (3.4) holds, and Equation (3.3) follows from Equation (3.6) and ν=ν′+ν″. b. The formula (3.5) clearly defines an L2-valued martingale measure on (Ω,ℱ,(ℱt),P) ( θ⋆ is integrable w.r.t. W by Equations (3.1) and (3.2)). We apply Proposition 2 to obtain an ℱ-conditionally centered Gaussian measure B″ on a very good extension, with covariance measure ν″ given by Equation (3.6): for this we need to show that ν″ satisfies (P) and is worthy. Recalling that every càdlàg adapted process on the Wiener space is predictable, we have (P-i), while (P-ii) and the symmetry in (P-iii) are obvious. We have (P-iv) because the increasing predictable process μ([0,·]) is locally bounded. If Ai∈Rd, ai∈IR, and f=∑1≤i≤nai1Ai and μs=μ−μac, Equation (3.6) yields ∑aiajν″([0,t]×Ai×Aj)=(ρ(f2)−ρ(f)2) μs([0,t])+∫0t(θs(ρ(f2)−ρ(f)2)−θs⋆2ρ(xTf)ρ(xf)) ds. (3.7) Observe that the orthogonal projection in L2(ρ) of the function f on the linear space spanned by the orthogonal vectors (1,x1,…,xd) is g=ρ(f)+∑1≤i≤d xiρ(xif), hence ρ(f2)−ρ(f)2−ρ(xTf)ρ(xf)=ρ(f2)−ρ(g2)≥0. Taking Equation (3.2) into account, we deduce that Equation (3.7) is non-decreasing in t and thus (P-iii) holds. For the worthyness, we observe that |ν″|≤2η, where η is the positive random measure having η([0,t]×A×A′)=(ρ(A∩A′)−ρ(A)ρ(A′)) μ([0,t]). That η satisfies (P) is obvious. At this point we have the existence of B″, and B=B′+B″ has all properties of (a). Then B is a tangent measure, and its covariance measure ν is given by Equation (3.4) and has |ν|≤η, hence it is worthy. □ If g satisfies (K), then ∫gT(s,x)g(s,x′)1[0,t](s)η(ds,dx,dx′)<∞ (with η as in the previous proof), hence g is integrable w.r.t. B and the brackets are: ⟨g⋆B,gT⋆B⟩t=∫(ρ(gsgsT)−ρ(gs)ρ(gsT))1[0,t](s)μ(ds). (3.8) Note also that g⋆B′=(ρ(gxT)θ⋆)·W, ⟨g⋆B′,WT⟩t=∫(ρ(gsxT)θs⋆) ds. (3.9) (approximate g by simple functions, or use the characterization (2.5) of stochastic integrals), and, ⟨g⋆B″,gT⋆B″⟩t=∫(ρ(gsgsT)−ρ(gs)ρ(gsT))1[0,t](s)μ(ds)−∫(ρ(gsxT)ρ(xgsT)θs⋆2) ds. (3.10) In view of Proposition 3, this implies that the ℱ-conditional law of g⋆B is determined by g⋆B′ and either Equation (3.8) or (3.10). 2. Before stating the main result, we should recall what stable convergence means. This notion was introduced by Renyi (1963); see also Aldous and Eagleson (1978), or jacod and Shiryaev (1987, §VIII-5-c) for a complete account. Let Yn be a sequence of random variables on (Ω,ℱ,P), taking values in a metric space E, and let Y be an E-valued variable defined on an extension (Ω¯,ℱ¯,P¯). We say that Yn converges stably in law to Y if E(Zf(Yn)) → E¯(Zf(Y)) (3.11) for every continuous bounded function f on E and every bounded measurable function Z on (Ω,ℱ). This implies the convergence in law of Yn to Y. Consider also the following subset I of IR+, whose complement is at most countable: I={t≥0:μ({t})=0 P-a.s.}. (3.12) Theorem 2 Assume (A1) and (A2), and let B be a tangent measure to W along the sequence (Tn). Let g satisfy (K), and Un(g) be given by Equation (1.4). If μ has a.s. no atom, the processes Un(g) converge stably in law (for the Skorokhod topology) to g⋆B. For all t1,…,tp in I, the variables Ut1n(g),…,Utpn(g)) converge stably in law to (g⋆Bt1,…,g⋆Btp). Remark 2 When μ has no atom, Lemma 2 applied to Z=g⋆B″ shows that for P-almost all ω, the process Z(ω,.) is Qω-a.s. continuous. Then g⋆B is a.s. continuous (since g⋆B′ is clearly so; in fact B is a continuous martingale measure). If μ has atoms, then g⋆B jumps at each time the bracket (3.8) jumps; now, by Equation (1.4) and since δn→0, the jumps of Un(g) tend uniformly to 0, so we cannot have convergence in law of Un(g) to g⋆B in the Skorokhod sense. □ Remark 3 In case μ=μ⋆=λ, Theorem 1 is a part of Theorem 2 (a part only, because the statement in Theorem 1 does not completely characterizes the random measure B). In this case ν is “continuous in time” and deterministic, so B is a centered Gaussian measure, whose law is determined by ν. Now, if one starts with a white noise B˜ on IR+×IRd with intensity measure dt⊗ρ(dx), a simple conditioning on Gaussian random vectors shows that, conditionally on having B˜([0,t]×IRd)=0 for all t, the covariance of B˜ is given by Equation (3.3) with μ=λ. Further the bracket (3.10) for g(ω,t,x)=x is null because μ=μ⋆=λ, and Equation (3.9) gives x⋆B′=W, hence x⋆B=W: that is, B satisfies all requirements of Theorem 1. More generally, when θ⋆≡1 then B″(ω,.) is under Qω a white noise with intensity measure μ(ω,dt)⊗ρ(dx), conditioned on 1⋆B″(ω)=0 and x⋆B″(ω)=0. When θ⋆≡0 it is a white noise with the same intensity measure, conditioned on 1⋆B″(ω)=0. 3 DISCRETIZATION SCHEMES In all this section we are given a sequence (Tn) satisfying (A1) and (A2). Proof of Lemma 3 We will first prove that a.s., for all s < t: μ⋆((s,t])≤t−s μ((s,t]). (4.1) To this effect, up to taking a subsequence, we may assume that for all fixed ω outside a null set we have μn→μ and μn⋆→μ⋆ weakly. Since Δ(n,i)≤t if S(n,i)≤t, we have μn⋆((s,t))=∑i:s<S(n,i)<t Δ(n,i)δn≤δnt+∑i:s<S(n,i−1),S(n,i)<t Δ(n,i)δn ≤δnt+(∑i:s<S(n,i−1)<t δn)1/2 (∑i:s<S(n,i−1),S(n,i)<t Δ(n,i))1/2≤δnt+μn((s,t)) t−s. Since δn→0, and μ⋆((s,t))≤liminfnμn⋆((s,t)) and limsupnμn([s,t])≤μ([s,t]), we get μ⋆((s,t))≤t−s μ([s,t]). This implies first that μ⋆ has no atom, and secondly that Equation (4.1) holds. b. Let λ′=λ+μ, so that λ=α·λ′ and μ=β·λ′ for two nonnegative predictable processes α, β with α+β=1 (recall that all adapted cadlag processes on the Wiener space are predictable). By applying the martingale construction of Radon–Nikodym derivatives, we deduce from Equation (4.1) that μ⋆ has the form μ⋆=γ·λ′ for some γ satisfying γ≤αβ. First 1{α>0}·λ′=((1/α)1{α>0})·λ. Then μac=((β/α)1{α>0})·λ, that is a version of θ in Equation (3.1) is θ=(β/α)1{α>0}. Next, since α=0 implies γ=0 we get μ⋆=((γ/α)1{α>0})·λ, hence a version of θ⋆ is θ⋆=(γ/α)1{α>0}. Since γ2≤αβ, Equation (3.2) readily follows. □ (2) Next, we show that it is not a restriction to suppose, in addition to Equations (A1) and (A2), the following: Assumption A3 All stopping times of the schemes Tn are finite-valued, and the total mass of μ is infinite. □ Indeed, we wish to prove results of the form (3.11) with Yn=Un(g) and f being continuous for the Skorokhod topology. As is well know, for this it is enough to consider functions f that depend only on the restriction of the path of the process to any finite interval. That is, we really have to consider the processes Un(g) on (arbitrary) finite intervals. So fix T∈I (see Equation (3.12)) and define a new scheme T′n as follows: Replace the times T(n,i)≥T by the times T+jδn for j∈IN, and re-order so as to obtain a new strictly increasing sequence T′(n,i) of stopping times, then set Δ′(n,i)={Δ(n,i)⋀(T−T(n,i)) if T(n,i−1)<Tδnotherwise. This defines new schemes T′n=(T′(n,i),Δ′(n,i):i≥1) which satisfy (A1). The measures μ′n and μ′n⋆ associated with T′n by Equations (1.6) ad (1.7) coincide with μn and μn⋆ on [0,T), and the three measures μ′n, μ′n⋆, and δn ∑i≥1 ɛT+iδn coincide on (T,∞). Since T∈I, the sequence (T′n) satisfies (A2) with μ′=1[0,T)·μ+1[T,∞)·λ and μ′⋆=1[0,T)·μ⋆+1[T,∞)·λ, hence (A3) as well. Further, it follows that the ℱ-conditional distributions of the restriction to [0,T]×IRd of the tangent measures along (Tn) and (T′n) coincide. Now the processes U′n(g) associated with T′n by Equation (1.4) have U′n(g)=Un(g) on [0,T). Then, if we can prove Theorem 2 for (T′n), and since T is arbitrary large, we deduce Theorem 2 for (Tn). Thus, it is no restriction to assume (A3), in addition to (A1) and (A2). (3) As stated in Remark 2, we do not have functional convergence of the Un(g)'s when μ has atoms. And even if μ has no atom we have problems in proving the stable convergence if the support of μ has “holes”. To solve these problems, we add fictitious point to fill in the holes, and also change time to “smooth” out the atoms of μ. This amounts to modify the limiting measures μ and μ⋆ according to the following. For any right-continuous non-decreasing function F: IR+→IR¯+ we call F−1 its right-continuous inverse (taking values in IR¯+ again). We write F(∞)=limt→∞F(t). Let D be the (random) topological support of μ, and set F(t)=μ([0,t]),F⋆(t)=μ⋆([0,t])F′(t)=F(t)+∫0t1Dc(s)ds, F″(t)=inf(s>0: s+F′−1(s)>t).} (4.2) Φ(t)=t−F″(t),A={t: Φ(t+ɛ)>Φ(t) ∀ɛ>0},R(t)=F(Φ(t))+t−ut, where ut=inf(s≥t: s∈A),R⋆(t)=F⋆(Φ(t)).} (4.3) Lemma 4 Each Φ(t) is an (ℱt)-stopping time, and the processes Φ,R, R⋆ are continuous, non-decreasing, adapted to the filtration (ℱΦ(t))t≥0, and R(∞)=Φ(∞)=∞. There are (ℱΦ(t))-predictable processes φ,ψ, ψ⋆ such that a.s. Φ(t)=∫0tφ(s)ds, R(t)=∫0tψ(s)ds, R⋆(t)=∫0tψ⋆(s)ds, (4.4) 0≤φ≤1A, 1Ac≤ψ≤1, 0≤ψ⋆≤φψ. (4.5) Proof As said before, F″ and F′′−1 are continuous and strictly increasing, and F″(t)−F″(s)≤t−s is obvious when s≤t, hence 0≤Φ(t)−Φ(s)≤t−s: therefore, Φ has the form (4.4), with 0≤φ≤1A. Further {Φ(t)≤s}={F″(t)≥t−s}={t≥t−s+F′−1(t−s)}={F′−1(t−s)≤s}={F′(s)≤t−s}∈ℱs because F′ is (ℱt)-adapted. This yields that Φ(∞)=∞ and that Φ(t) is an (ℱt)-stopping time for each t, hence Φ is (ℱΦ(t))-predictable (recall that Φ is continuous) and there is an (ℱΦ(t))-predictable version of φ as well. ii. The following chain of equivalences is obvious: F′(r−)≤v≤F′(r) ⇔ r=F′−1(v) ⇔ F′′−1(v)=v+r ⇔ F″(v+r)=v ⇔ Φ(v+r)=r. Further F′ is strictly increasing, and F′(r)−F′(r−)=F(r)−F(r−). Therefore, if ut′=sup(s≤t:s∈A) (with sup(Ø)=0), we readily deduce from Equation (4.3) that R(t)=F(Φ(t)−)+t−u′t, F(Φ(t)−)≤R(t)≤F(Φ(t)),ut=Φ(t)+F′(Φ(t)),u′t=Φ(t)+F′(Φ(t)−).} (4.6) Therefore, R is non-decreasing, and R(∞)=∞ because Φ(∞)=∞, and F(∞)=∞) by (A3), and R is linear with slope 1 on each interval [u′t,ut]. If s < t and us≤u′t, we also have by Equation (4.6): R(u′t)−R(us)=F(Φ(t)−)−F(Φ(s))≤F′(Φ(t)−)−F′(Φ(s)) = u′t−Φ(t)−us+Φ(s) ≤ u′t−us and it follows that R(t)−R(s)≤t−s, whereas R(t)−R(s)=t−s is obvious when (s,t)⊂Ac. Hence, R has the form (4.4) with ψ satisfying 1Ac≤ψ≤1. Further {ut≥s}={Φ(s)=Φ(t)}∈ℱΦ(t), hence ut is ℱΦ(t)-measurable. Since F and F⋆ are (ℱt)-adapted and right-continuous, F(Φ(t)) and F⋆(Φ(t)) are ℱΦ(t)-measurable, and thus R and R⋆ are (ℱΦ(t))-adapted. Therefore, we can choose a version of ψ that is (ℱΦ(t))-predictable. iii. By definition of R⋆ we deduce from Equations (4.1) and (4.6) that a.s. 0 ≤ R⋆(t)−R⋆(s)≤Φ(t)−Φ(s) F(Φ(t)−)−F(Φ(s))≤Φ(t)−Φ(s) R(t)−R(s). Exactly as in the proof of Lemma 3, we get Equation (4.4) for R⋆ with ψ⋆≤φψ, and ψ⋆ can be chosen (ℱΦ(t))-predictable because R⋆ is (ℱΦ(t))-adapted. □ Lemma 5 There exists an (ℱt)-predictable set B such that the processes θ and θ⋆ in Equation (3.1) have for λ-almost all t: ψ(t)1B(Φ(t))=φ(t)θΦ(t), ψ⋆(t)=φ(t)θΦ(t)⋆. (4.7) Proof Equations (4.3) and (4.4) give ∫0Φ(t)θs⋆ds=∫0tψ⋆(s)ds, and Lebesgue derivation Theorem yields the second property (4.7). Observe that ∫0ut h○Φ(r) ψ(r) dr=∫[0,Φ(t)] h(r) μ(dr) (4.8) is true for h=1[0,v] (it then reduces to R(ut⋀Φ−1(v))=F(Φ(t)⋀v), which holds by Equation (4.3) because ut⋀Φ−1(v) belongs to A), hence for all bounded Borel function h. Recall that μs=μ−μac. Since F is predictable, there is a predictable set B which supports μac and is not charged by μs. In particular, 1B·μ=μac=θ·λ. Further μs(B)=0 implies that 1B○Φ(r)=1B○Φ(t)=0 if t≤r≤ut and t<ut, because then F[Φ(t)−)<F(Φ(t)) by Equation (4.6). Then, applying Equation (4.8) with h=1B gives ∫0t1B(Φ(s)) ψ(s) ds=∫[0,Φ(t)] 1B(r) μ(dr)=∫0Φ(t)θsds, and Lebesgue derivation Theorem again implies the first part of Equation (4.7). □ (4) Now we introduce a time-change. Set S(t)=R−1(t), τ(t)=S○F(t). (4.9) Each S(t) is a finite-valued (ℱΦ(t))-stopping time, because R(∞)=∞ and R is adapted to the filtration (ℱΦ(t)). Further, Lemma 6 Each τ(t) is a finite-valued (ℱΦ(t))-stopping time given by the following formula, where t+=inf(v>t: F(v)>F(t)). τ(t)={Φ−1(t+) if F(t)=F(t+)Φ−1(t+−) if F(t)<F(t+). (4.10) Proof Set s=τ(t). First R(s)=F(t), hence F(Φ(s)−)≤F(t) by Equation (4.6), hence Φ(s)≤t+. Second, for ɛ>0 we have R(s+ɛ)>F(t), hence F(t)<F(Φ(s+ɛ)) by Equation (4.6), hence t+≤Φ(s+ɛ) and by continuity of Φ we get t+≤Φ(s). Now, this and Equation (4.6) imply F(t)=R(s)=F(t+)+s−us; if F(t)=F(t+) this yields s=us=Φ−1(t); otherwise F(t)=F(t+−), hence s=u′s=Φ−1(t+−). Thus, Equation (4.10) is proved. For every (ℱt)-stopping time T, we have {Φ−1(T)<r}={T<Φ(r)}∈ℱΦ(r) and {Φ−1(T−)≤r}={T≤Φ(r)}∈ℱΦ(r), hence both Φ−1(T) and Φ−1(T−) are (ℱΦ(t))-stopping times. The stopping time property of τ(t) follows, because by Equation (4.10) τ(t)=Φ−1(T)⋀Φ−1(T′−) if T=t+ (respectively, ∞) and T′=∞ (respectively, t+) if F(t)=F(t+) (respectively, F(t)<F(t+)). □ Lemma 7 Let k be a locally bounded (ℱt)-predictable process and W′t=WΦ(t). Then, ∫0τ(t)k○Φ(r) ψ(r)dr=∫[0,t]k(r) μ(dr), (4.11) ∫0τ(t)(k1{θ>0})○Φ(r) ψ(r) dW′r=∫[0,t](k1{θ>0})(r) dWr, (4.12)The process (k1{θ>0})○Φ is (ℱΦ(t))-predictable and τ(t) is an (ℱΦ(t))-stopping time, hence the first stochastic integral in Equation (4.12) is meaningful. Proof We use Equation (4.10): if F(t)=F(t+) then τ(t)=uτ(t) and Φ(τ(t))=t+ hence Equation (4.11) follows from Equation (4.8) because μ((t,t+])=0. Suppose now F(t)<F(t+). Then, τ(t)=u′τ(t) and Φ(τ(t))=t+ again, and ψ = 1 on (u′τ(t),uτ(t)) by Equation (4.5), so by Equation (4.8): ∫0τ(t)k○Φ(r) ψ(r) dr=∫0uτ(t)k○Φ(r) ψ(r) dr−k○Φ(τ(t))(uτ(t)−u′τ(t))=∫[0,t+]k(r) μ(dr)−k(t+)μ({t+})=∫[0,t]k(r) μ(dr). b. Set M′t=∫0t(k1{θ>0})○Φ(r) ψ(r) dW′r and Mt=∫0t(k1{θ>0})(r) dWr. The process Φ is a continuous time-change, hence M′t=MΦ(t) a.s. for all t (see e.g., Chapter 10 of jacod (1979)). In particular, M′τ(t)=Mt+ because Φ(τ(t))=t+. If t+=t this gives Equation (4.12). If t<t+ we have θ=0 λ-a.s. on [t,t+], hence Mt+=Mt and Equation (4.12) holds also in this case. □ (5) In fact, Φ, R, and R⋆ appear in the limiting behavior of some denser discretization schemes that are associated to the original ones as follows. We still assume (A1), (A2), and (A3). First set Dtɛ(ω)={x∈[0,t]: d(x,D(ω))≥ɛ} (recall that D is the topological support of μ). Since μ(Dtɛ)=0 and Dtɛ is closed, (A2) yields μn(Dtɛ)→0 for all t. There is an increasing sequence np↑∞ with n≥np ⇒ P(μn(Dp1/p)>1/p)≤1/p, and thus pn=sup(p:np≤n) has: pn↑∞, P(μn(Dpn1/pn)>1/pn)≤1pn. (4.13) Next, we set αn=(δnpn)⋀δn, which is a sequence satisfying αn→0, δn/αn→0, αn/δnpn→0. (4.14) The idea of what follows is such: we first suppress the points T(n, i) for which Δ(n,i)≥αn, and Equation (4.14) ensures that we still keep (A2). Next, we add subdivision points in the complement Dc of D, spaced by δn (so the corresponding “empirical” measure goes to Lebesgue measure on Dc) and distant from the initial subdivision points by αn (which is small, yet “much bigger” than δn by Equation (4.14)). Then, we change time by substituting T′(n,i) with iδn for the ith new subdivision point T′(n,i). Since we must preserve some “stopping time” properties and keep track of the S(n, i)’s as well, things are a bit complicated. We do this step by step. Step 1: Deleting points. We set Jn={i∈IN: Δ(n,i)<αn}, J′n=IN∖Jn, C(n)={T(n,i): i∈Jn}, (4.15) νn=δn∑i∈JnɛT(n,i), νn⋆=∑i∈JnΔ(n,i)δn ɛT(n,i), (4.16) Σ(n,t)={i∈IN: S(n,i)≤t}. (4.17) Lemma 8 We have δn card(J′n∩Σ(n,t))≤tδn/αn→0, (4.18) νn →P μ, νn⋆ →P μ⋆. (4.19) Proof Since ∑i∈Σ(n,t)Δ(n,i)≤t we have card (J′n∩Σ(n,t))≤t/αn and Equation (4.18) follows from Equation (4.14). Next, set ν^n=δn∑i∈JnɛS(n,i) and ν^n⋆=∑i∈JnΔ(n,i)δn ɛS(n,i). We have ν^n≤μn and ν^n⋆≤μn⋆. Also, (μn−ν^n)([0,t])=δncard (J′n∩Σ(n,t)) and (μn⋆−ν^n)⋆([0,t])=δn card(J′n∩Σ(n,t)) by Cauchy–Schwarz inequality. Thus, (A2) and Equation (4.18) give us ν^n→P μ and ν^n⋆→Pμ⋆. Now for all i∈Jn we have Δ(n,i)<αn, hence 0≤S(n,i)−T(n,i)≤αn, thus νn(f)−ν^n(f) and νn⋆(f)−ν^n⋆(f) tend to 0 in probability for every continuous function f with compact support, and Equation (4.19) follows. □ Step 2: Adding points. Now we set C(n,i)={T(n,i)+αn+jδn: j∈IN}∩[0,T(n,i+1))∩Dc. For n fixed, these sets are pairwise disjoint (some or even all may be empty), and also disjoint from C(n). Set also C″(n)=⋃i∈IN C(n,i), C′(n)=C(n)∪C″(n). C′(n) is an optional locally finite random set. We define a strictly increasing sequence of stopping times and a random measure by T′(n,0)=0, T′(n,i+1)=inf(t∈C′(n): t>T′(n,i))μ′n=δn∑i≥0 ɛT′(n,i).} (4.20) Lemma 9 We have μ′n→Pμ′, where the measure μ′ is such that μ′([0,t])=F′(t), as given by Equation (4.2). Proof Up to taking a subsequence we may assume that ∑1/pn<∞ and that outside a P-null set (recall (4.13), (A2), and (4.19)): νn→μ, μn→μ, μn(Dpn1/pn)≤1/pn for n large enough. (4.21) We set μ¯n=δn∑s∈C′(n)ɛs and μ¯=1Dc·λ. Then μ′=μ+μ¯ and, since C(n)∩C″(n)=Ø, we have μ′n=νn+μ¯n, so if we prove μ¯n→μ¯ for all ω having Equation (4.21) then μ′n→μ′ for those ω, and the result will obtain. Hence, below we fix an ω having Equation (4.21). Intervals between successive points in C″(n) have length not smaller than δn, so μ¯n([s,t])≤t−s+δn. Since δn→0 we deduce that the sequence (μ¯n) is relatively compact for the vague topology and all limit points are smaller than λ. Remembering that ω is fixed, it is then enough to show that if a subsequence still denoted by (μ¯n) converges to a limit μ¯′, then μ¯′=μ¯. Let (U, V) be an interval contiguous to D and fix t∈IR+ and ɛ<(V−U)/2. The set C″(n)∩(U,V)∩[0,t] is a finite set whose points are equally spaced by δn, except for gaps of length smaller than δn+αn around all points T(n, i) in (U,V)∩[0,t]. Hence, if Nn denotes the number of points T(n, i) within (U+ɛ,V−ɛ)∩[0,t], the number of points in C″(n)∩(U+ɛ,V−ɛ)∩[0,t] is bigger than (VΛt−UΛt−2ɛ−Nn(δn+αn))/δn. Finally since S(n,i)≤T(n,i+1)<S(n,i+1), the last statement in Equation (4.21) shows that for n large enough we have pn≥t∨(1/ɛ) and Nn≤1+1/δnpn, hence μ¯n([U+ɛ,V−ɛ]∩[0,t])≥V⋀t−U⋀t−2ɛ−(1+1/δnpn)(δn+αn). Since μ¯′([U+ɛ,V−ɛ]∩[0,t])≥limsupnμ¯n([U+ɛ,V−ɛ]∩[0,t]) we deduce from Equation (4.14) and the above that μ¯′([U+ɛ,V−ɛ]∩[0,t])≥V⋀t−U⋀t, which equals μ¯((U,V)∩[0,t]). Since μ¯ is supported by Dc, it follows that μ¯′≥μ¯. Finally μ¯n(D)=0 by construction, hence if D0 is the (possibly empty) interior of D we have μ¯′(D0)=0 because μ¯n→μ¯′. Since the Lebesgue measure of a closed set with empty interior is null and μ¯′≤λ, we deduce that μ¯′(D\D0)=0, hence μ¯′(D)=0, hence μ¯′≤μ¯ because μ¯′≤λ and μ¯=λ on the complement of D. Therefore, μ¯′=μ¯ and the proof is finished. □ Step 3: Changing time. Set A′n={i∈IN: ∃j∈Jn such that T′(n,i)=T(n,j)}Δ′(n,i)=Δ(n,j), S′(n,i)=S(n,j) if i∈A′n and T′(n,i)=T(n,j),} (4.22) T″(n,i)=T′(n,i)+iδn, and S″(n,i)=S′(n,i)+(i+1)δn if i∈A′n. (4.23) If j∈Jn we have C′(n)∩(T(n,j),S(n,j)]=Ø. Therefore, if i∈A′n then S′(n,i)≤T′(n,i+1) and S″(n,i)≤T″(n,i+1), if further S′(n,i)=T′(n,i+1) then S″(n,i)=T″(n,i+1).} (4.24) The locally finite set U(n)={T′(n,i):i∈IN}∪{S′(n,i):i∈A′n} is re-ordered through the following strictly increasing sequence of stopping times: R′(n,0)=0, R′(n,i+1)=inf(t>R′(n,i): t∈U(n)). (4.25) Then, we set R″(n,i)={T″(n,j) if R′(n,i)=T′(n,j)S″(n,j) if R′(n,i)=S′(n,j) and j∈A′n. (4.26) (it is possible that R′(n,i)=S′(n,j)=T′(n,j+1), but by Equation (4.24) there is no ambiguity above), and {An=i∈IN: there is a (unique) j∈A′n with R′(n,i)=T′(n,j),∇(n,i)=R′(n,i+1)−R′(n,i),} (4.27) Σ″(n,t)={i∈IN: R″(n,i+1)≤t}σ(n,t)={i∈A′n: R″(n,i+1)≤t}Φn(t)=R′(n,i+1) if R″(n,i)≤t<R″(n,i+1).} (4.28) Step 4: Measurability properties. We have the following: Lemma 10 We have {i∈An}∈ℱR′(n,i) and, in restriction to the set {i∈An}, the variables R′(n,i+1) and R″(n,i+1) are ℱR′(n,i)-measurable. Each Φn(t) is an (ℱt)-stopping time; we set ℱtn=ℱΦn(t). Each R″(n,i) is an (ℱtn)-stopping time, and ℱR″(n,i)n=ℱR′(n,i+1) and ℱR″(n,i)−n=ℱR′(n,i) ( ℱ0−n is the trivial σ-field, by convention). Proof It is enough to use (A1) and to observe that {i∈An}∩{R′(n,i+1)≥t}=∪j∈IN{R′(n,i)=T(n,j), t−T(n,j)≤Δ(n,j)<αn},{i∈An}∩{R″(n,i+1)≥t}=∪j∈IN{R′(n,i)=T(n,j), t−T(n,j)−(j+1)δn≤Δ(n,j)<αn}. By definition of Φn(t), {Φn(t)≤s}=∪i∈INDin, Din={R′(n,i+1)≤s, R″(n,i)≤t<R″(n,i+1)}. The sets Din∩{i∈An} and Din∩{i+1∈An} are in ℱs by (a). The set Din∩{i∉An}∩{i+1∉An} is the union for all k∈IN of the sets {R′(n,i+1)=S(n,k+1)≤s, R′(n,i)=S(n,k), Δ(n,k)<αn, Δ(n,k)≤t−T(n,k)−(k+1)δn, t−T(n,k+1)−(k+2)δn<Δ(n,k+1)<αn}, also in ℱs by (A1) and the fact that R′(n,i+1) is a stopping time, hence the claim. c. By definition of Φn again, A∩{R″(n,i)≤t}=A∩{R′(n,i+1)≤Φn(t)}. Then, if A∈ℱR′(n,i) we get A∩{R″(n,i)≤t}∈ℱΦn(t)=ℱtn: hence, R″(n,i) is an (ℱtn)-stopping time (take A=Ω) and ℱR′(n,i+1)⊂ℱR″(n,i)n. The opposite inclusion ℱR″(n,i)n⊂ℱR′(n,i+1) follows from Φn(R″(n,i))=R′(n,i+1) and from Lemma (10.5) of Jacod (1979). Hence, ℱR″(n,i)n=ℱR′(n,i+1). The last claim is obvious if i = 0, so let i≥1. Since R″(n,i−1)<R″(n,i) and ℱR″(n,i−1)n=ℱR′(n,i) we get ℱR′(n,i)n⊂ℱR″(n,i)−. Conversely, ℱR″(n,i)−n is generated by the sets A∩{t<R″(n,i)} for t≥0 and A∈ℱtn; then, A∩{t<R″(n,i)}=A∩{Φn(t)≤R′(n,i)} is ℱR′(n,i)-measurable, hence ℱR″(n,i)−n⊂ℱR′(n,i). □ Step 5: Limiting results. The following (with Φ, ψ, ψ⋆, as in Equation (4.4)) will be crucial for the proof of the main theorems: Lemma 11 The following convergences, where f denotes a bounded continuous function, hold in probability uniformly on compact subsets of IR+: Φn(t) → Φ(t), (4.29) δn ∑i∈σ(n,t) f(R′(n,i)) → ∫0tψ(s) f○Φ(s) ds, (4.30) ∑i∈σ(n,t)V(n,i)δn f(R′(n,i)) → ∫0tψ⋆(s) f○Φ(s) ds. (4.31) Proof For Equations (4.30) and (4.31) it suffices to consider nonnegative functions. Hence, all processes above are increasing, and in addition the limiting processes are continuous: it is then enough to prove the convergence in probability for each t≥0. Up to taking subsequences, we can assume that in A2 and in Lemmas 8 and 9 the convergences hold a.s. So we fix ω such that νn→μ, μ′n→μ′ and νn⋆→μ⋆. Consider the following measures on IR+ (recall that Δ′(n,i) is well defined if i∈A′n: see Equation (4.22)): μ′′n=δn ∑i≥0 ɛT″(n,i),rn=δn ∑i∈A′n ɛT″(n,i), rn⋆=∑i∈A′n Δ′(n,i)δn ɛT″(n,i), and denote by Fn, Fn⋆, Fn′, F′′n, Rn, and Rn⋆ the repartition functions of νn, νn⋆, μ′n, μ′′n, rn, and rn⋆, respectively. c. μ′n→μ′ gives F′n(t)→F′(t) for all t having F′(t)=F′(t−). Since F′−1 is continuous, it follows that F′n−1 → F′−1 locally uniformly . (4.32) Next, if tn denotes the integer part of t/δn, we have Fn′′−1(t)=T″(n,tn+1)=T′(n,tn+1)+(tn+1)δn=Fn′−1(t)+(tn+1)δn, hence Fn′′−1(t)→F′−1(t)+t by Equation (4.32). Since F″ and F′′−1 are continuous and strictly increasing, it follows that, Fn″ → F″ locally uniformly (4.33) (i.e. μ′′n→μ″, with μ″ the measure having F″ for repartition function). To obtain Equation (4.29) it is enough to observe that Fn′−1[(F′′n(t)−δn)+]<Φn(t)≤Fn′−1[(F′′n(t)], and to apply Equations (4.32) and (4.33) and the property Φ=F′−1○F″, which comes from the equivalence F′′−1(v)=v+r ⇔ Φ(v+r)=r in (ii) of the proof of Lemma 4. d. Now, we show that Rn → R pointwise. (4.34) Let j∈A′n and i∈An be related by R′(n,i)=T′(n,j) (or equivalently R″(n,i)=T″(n,j): see Equation (4.23)). We have the following sequence of equivalent properties: T″(n,j)≤t ⇔ R″(n,i)≤t ⇔ R′(n,i)<Φn(t) ⇔ T′(n,j)<Φn(t) (recall Equation (4.28)). Further j∈A′n iff there is k∈Jn with T′(n,j)=T(n,k). Then, in view of Equation (4.16) we get Rn(t)=Fn[Φn(t)−]. Then, νn→μ and Equation (4.29) yields F[Φ(t)−]≤liminfnRn(t)≤limsupnRn(t)≤F[Φ(t)]. This and Equation (4.6) imply Rn(t)→F[Φ(t)] if F[Φ(t)]=F[Φ(t)−], and otherwise, limsupnRn(s)≤F[Φ(t)−] if s<u′t,liminfnRn(s)≥F[Φ(t)] if s>ut.} (4.35) On the other hand, rn≤μn″; hence, Rn(β)−Rn(α)≤F′′n(β)−F′′n(α) if α≤β. Then Equation (4.33) and the fact that F″(β)−F″(α)≤β−α yield limsupn[Rn(β)−Rn(α)]≤β−α. (4.36) Putting together Equations (4.35), (4.36), and F[Φ(t)]−F[Φ(t)−]=un−u′t readily yields Rn(t)→F[Φ(t)]−ut+t=R(t): hence, Equation (4.34) holds. Now we can prove Equation (4.30). Denote by Ψn(t) the left-hand side of Equation (4.30), and by Ψ¯n(t) the same quantity with R′(n,i+1) instead of R′(n,i). If i∈An we have R′(n,i+1)−R′(n,i)≤αn (combine Equations (4.15), (4.22), and (4.25)), while δncard (σ(n,t))≤Rn(t)→R(t) by Equation (4.34): since f is uniformly continuous on [0,t], we deduce that Ψ¯n(t)−Ψn(t)→0. Now R′(n,i+1)=Φn(R″(n,i)), and i∈An iff there is a (unique) J∈A′n such that R″(n,i)=T″(n,j), hence Ψ¯n(f)=∫0tf○Φn(s)rn(ds)−δn∑i∈An,R″(n,i)≤t<R″(n,i+1)f(R″(n,i+1)) and the sum above is in fact bounded by δnsup|f|. By Equation (4.29) f○Φn converges uniformly to the bounded continuous function f○Φ on [0,t], and Equation (4.34) means that rn weakly converges to the measure ψ(s)ds, hence Ψ¯n converges to the right-hand side of Equation (4.30), and Equation (4.30) is proved. Exactly as before, Rn⋆(t)=Fn⋆[Φn(t)−]. Then, νn⋆→μ⋆ and Equation (4.29) and the continuity of F⋆ give Rn⋆→R⋆ pointwise, and Equation (4.31) is deduced from this as Equation (4.30) is from Equation (4.34) in (c) above. □ 4 PROOF OF THEOREM 2 Let g satisfy (K). Since the process (γt)t≥0 is IR+-valued predictable increasing and γ0 is a constant, there is a sequence τp of stopping times increasing to ∞, with γt≤p∨γ0 for all t≤τp. Letting gp(ω,t,x)=g(ω,t∧τp(ω),x), we see that gp satisfies (K) with a process γ which is the constant p∨γ0, and obviously Un(g)t=Un(gp)t and g*Bt=gp*Bt for all t≤τp. Since τp→∞, it is obvious that if the sequence Un(gp) enjoys the limiting behavior described in Theorem 2 for any fixed p, the same is true of the sequence Un(g). In other words, it is enough to consider test functions g having (K) with γt(ω) being a constant. We assume this below, as well as (A1), (A2), and (A3) (as seen before, assuming (A3) is not a restriction). We use all notation of Section 3, and add some more. First, for any process Z we set (recall Equation (4.27) for ∇(n,i)): ∇′inZ=∇(n,i)−1/2 (ZR′(n,i+1)−ZR′(n,i)). Then, define the following processes (Id is the d × d identity matrix): ft=ρ(gtgtT)−ρ(gt)ρ(gtT), ht=ρ(gtxT),Ftn=δn ∑i∈σ(n,t) fR′(n,i), Ft=∫0tfΦ(s)ψ(s) ds, (5.1) Htn=∑i∈σ(n,t) ∇(n,i)δn hR′(n,i), Ht=∫0thΦ(s)ψ⋆(s) ds,Ktn=Φn(t) Id, Kt=Φ(t) Id,W′tn=WΦn(t), W′t=WΦ(t), (5.2) U′tn=∑i∈Σ″(n,t) χin, whereχin=δn 1An(i) (g(R′(n,i),∇inW)−ρ(gR′(n,i))).} (5.3) 2. Now we proceed to study the limiting behavior of U′n. Note that t↦ft and t↦ht are continuous. Then Lemma 11 yields the following convergences in probability, locally uniform in time: W′n→W, Fn→F, Hn→H, Kn→K. (5.4) Recalling that {i∈An}∈ℱR′(n,i) and that the restriction to {i∈An} of the variable ∇(n,i) is ℱR′(n,i)-measurable (Lemma 10-a), we easily deduce from Equation (5.3) that, for some constant K, E(χin|ℱR′(n,i))=0E(χin χin,T|ℱR′(n,i))=1An(i) δn fR′(n,i)E(χin (∇inW)T|ℱR′(n,i))=1An(i) δn hR′(n,i)E(|χin|4|ℱR′(n,i))≤Kδn2.} (5.5) Lemma 12 The processes U′n, W′n, U′nU′n,T−Fn, W′nW′n,T−Kn, U′nW′n,T−Hn are (ℱtn)-local martingales (recall that ℱtn=ℱΦn(t): see Lemma 10). Proof In view of Lemma 10-b, of the fact that Φn(t)→∞ as t→∞ and of Theorems (10.9) and (10.10) of Jacod (1979), the process W′n and W′nW′n,T−Kn are (ℱtn)-local martingale. Now consider a process Vtn=∑i∈Σ″(n,t)ηin=∑i≥0ηin 1{R″(n,i+1)≤t} with ℱR″(n,i+1)n-measurable ηin satisfying ηin=0 when i∉An. By virtue of Lemma 10-a,c Vn is an (ℱtn)-local martingale iff E(ηin|ℱR′(n,i))=0. By Equation (5.5) this applies to Vn=U′n with ηin=χin, and to Vn=U′nU′n,T−Fn with ηin=χinχin,T+U′R″(n,i)nχin,T+χin,TU′R′(n,i)n,T−1An(i) δn fR′(n,i). Set αin=∇(n,i) ∇inW. If Ytn=∑i∈σ(n,t)αin, and again due to Equations (5.5), the previous result also applies to Vn=U′nYn,T−Hn, with ηin=χinαin,T+U′R″(n,i)nαin,T+χin,TYR′(n,i)n,T−1An(i) ∇(n,i)δn hR′(n,i). Finally, U′nW′n,T−Hn=U′nYn,T−Hn+U′n(W′n,T−Yn,T). Now U′n and W′n,T−Yn,T are two (ℱtn)-local martingale, purely discontinuous and with no common jump, hence their product is again a local martingale. □ An application of Aldous’ criterion (apply Equation (5.4) and Lemma 11, and combine Theorem 4.18 and Lemma 4.22 of Chapter VI of Jacod and Shiryaev, 1987) shows that the sequence U′n is tight, and even C-tight (the last inequality in Equation (5.5) implies Lindeberg’s condition). Applying again Equation (5.4) yields that the sequence ζn=(W′n,Fn,Hn,Kn,U′n,U′nU′n,T−Fn) is C-tight and that if ζ=(W¯′,F¯,H¯,K¯,U¯′,M¯) is a limiting process for this sequence, (W′,F,H,K) and (W¯′,F¯,H¯,K¯) have the same distribution and M¯=U¯′U¯′T−F¯ a.s. In other words, if Cq=C(IR+,IRq) is endowed with the canonical process U′ and with the canonical filtration (Ctq), we can realize any limit ζ on the product space (Ω˜,ℱ˜,(ℱ˜t))=(Ω,ℱ,(ℱt))⊗(Cq,C1q,(Ctq)t), so that If we consider a converging subsequence, still denoted by ζn, there is a probability measure P˜ on (Ω˜,ℱ˜) whose Ω−marginal is P, and such that the laws of ζnconverge to the law of ζ=(W′,F,H,K,U′,U′U′T−F) under P˜.} (5.6) Lemma 13 Under P˜ the processes U′, W′, U′U′T−F, W′W′T−K, U′W′T−H are (ℱ˜t)-local martingales, continuous, and null at 0. Proof That the processes are continuous and null at 0 is obvious. We show the martingale property for U′U′T−F only; it is the same (or simpler) for the other processes. Set M=U′jU′k−Fjk and Mn=U′n,jU′n,k−Fn,jk, and also L(n,y)=inf(t: |Mtn|+|Ftn|+|U′tn|>y),L(y)=inf(t: |Mt|+|Ft|+|U′t|>y), Observe that |Mtn|≤y if t<L(n,y) and |ML(n,y)n|≤y+2y|χin|+|χin|2+K′ for some constant K′, if L(n,y)=R″(n,i+1). Thus, E(|Mt∧L(n,y)n|2)≤(y+1)K″ for another constant K″ by Equation (5.5), from which we deduce the uniform integrability of the sequences (Mt∧L(n,y)n)n≥1. On the other hand Equations (5.4) and (5.6) imply the convergence in law of (ζn,Mn,Gn) to (ζ,M,0). Then (see e.g. Proposition VI-2.11 of Jacod and Shiryaev, 1987) for all y in a dense subset of IR+, (ζn,M·∧L(n,y)n)n≥1 converge in law to (ζ,M·∧L(y)). From the uniform integrability above and from Lemma 12, we deduce that M·∧L(y) is a P˜-martingale for the filtration generated by (ζ,M·∧L(y)), that is, for (ℱ˜t). Since L(y)→∞ as y→∞, it follows that M is a local martingale. □ Recalling that 0≤ψ⋆≤φψ and that the process hΦ (time-changed of h by Φ) is (ℱΦ(t))-predictable, and setting 0/0=0, we can define the following continuous local martingales on the extension (Ω˜,ℱ˜,(ℱ˜t)),P˜): M′=α·W′ with αs=ψ⋆(s)ψ(s) hΦ(s), M″=U′−M′. (5.7) Next, due to the structure of (Cq,Cq), there is a regular disintegration P˜(dω,dx)=P(dω)Q˜ω(dx). Lemma 14 The space (Ω˜,ℱ˜,(ℱ˜t)),P˜) is a very good extension of the space (Ω,ℱ,(ℱΦ(t)),P). M″ is an (ℱt)-conditional centered Gaussian martingale, (ℱt)-locally square-integrable, with bracket F′′t=∫0tf″sds, wheref′′s=ψ(s)fΦ(s)−φ(s)αsαsT=ψ(s)(fΦ(s)−φ⋆2φψ(s)hΦ(s)hΦ(s)T).} (5.8) Proof Let Z be a bounded martingale on (Ω,ℱ,(ℱΦ(t)),P), and set Nt=E(Z∞|ℱt). We know that N=l·W for some (ℱt)-predictable process l. Like in the proof of Lemma 7, we then have Zt=E(Z∞|ℱΦ(t))=NΦ(t)=∫0tlΦ(s) dW′s. Now W′ is a martingale on the extension (Ω˜,ℱ˜,(ℱ˜t)),P˜) and l○Φ is predictable w.r.t. (ℱ˜t): then Z is a martingale on the extension, which is thus very good. b. Lemma 13 implies that the continuous local martingale U′ has ⟨U′,U′T⟩=F and ⟨U′,W′T⟩=H, and simple calculations show that ⟨M″,M″T⟩=F″ given by Equation (5.8) and ⟨M″,W′T⟩=0. We deduce first that ⟨M″,M″T⟩ is (ℱΦ(t))-adapted. Next, since all bounded (ℱΦ(t))-martingales are stochastic integrals w.r.t. W′ (see (a) above) we deduce that M″ is orthogonal to all bounded (ℱΦ(t))-martingales. Finally, M′′0=0, and M″ is continuous. It remains to apply Lemma 3. □ Corollary 1 The measure P˜ is unique, and Equation (5.6) holds for the initial sequence ζn. We can even strengthen the convergence (5.6) as follows: for all bounded continuous functions k on the Skorokhod space lD(IR+,IRq) and all bounded random variables Z on (Ω,ℱ), we have E(Zk(U′n)) → E˜(Zk(U′)). (5.9) Proof By Lemmas 2 and 14 the ℱ-conditional law of M″ is determined by F″, so the ℱ-conditional law of U′=M′+M″, that is Q˜ω, is P-a.s. unique, so P˜ is unique and thus Equation (5.6) holds for the original sequence ζn; Clearly Equations (5.4) and (5.6) imply Equation (5.9) when Z=l(W′), where l is a continuous bounded function on lD(IR+,IRq). Next, let ℱ′ be the σ-field generated by all variables W′t, t≥0. W′ is a continuous (ℱΦ(t))-local martingale with bracket Kt=Φ(t)Id, the process Φ is ℱ′-measurable, as well as its inverse Φ−1. We have Wt=W′Φ−1(t) because Φ○Φ−1(t)=t, hence Wt is ℱ′-measurable: thus, ℱ′=ℱ. Now let Z be bounded and ℱ-measurable. Since ℱ′=ℱ there are Zp=lp(W′) with lp continuous bounded and Zp→Z in L1(P). Equation (5.9) holds for each Zp, and if C=sup|k| we obtain: |E(Zp k(U′n))−E(Z k(U′n))|≤C E(|Z−Zp|),|E(Zp k(U′))−E(Z k(U′))|≤C E(|Z−Zp|), so Equation (5.9) follows. □ Now we state the relations between the process U′ above and the process g⋆B of Theorem 2, defined on the extension (Ω¯,ℱ¯,(ℱ¯t)),P¯) of (Ω,ℱ,(ℱt),P). For this, we set Ut=U′τ(t) (τ(t) is given by Equation (4.9). (5.10) Lemma 15 Both processes U on the (non-filtered) extension (Ω˜,ℱ˜,P˜) of (Ω,ℱ,P) and g⋆B on the (non-filtered) extension (Ω¯,ℱ¯,P¯) of the same space have the same ℱ-conditional law. Proof First, we show that g⋆B′t=M′τ(t) (see Equations (3.9) and (5.7)). By definition the process W′ is constant on the intervals contiguous to A, hence 1{φ=0}·W′=0 by Equation (4.5). Further the bracket of W′ is absolutely continuous w.r.t. Lebesgue measure by Equation (4.4), hence 1C·W′=0 if λ(C)=0. Therefore, M′=[(θ⋆h 1{θ>0})○Φ]·W′ by Equations (4.7) and (3.2), hence M′τ(t)=g⋆B′t follows from Equation (4.12). Since g⋆B′ is ℱ-measurable, it remains at this point to show that both processes g⋆B″ and M˜″t=M′′τ(t) have the same ℱ-conditional law. Now the time-change τ(t) is ℱ-measurable, so it follows from Lemma 14-b that M″ is an ℱ-conditional centered Gaussian martingale with bracket F˜′′t=F′′τ(t), while g⋆B″ is an ℱ-conditional centered Gaussian martingale with bracket given by Equation (3.10). By Lemma 2-b, it remains to show that F˜″ is given by Equation (3.10). Using Equation (4.7) and ψ = 0 ⇒ ψ⋆=0 and θ=0 ⇒ θ⋆=0, we deduce from Equation (5.8): F′′t=∫0t(fΦ(r)−θ⋆2θ○Φ(r) hΦ(r)hΦ(r)T)ψ(r) dr, and Equation (4.11) gives F′′τ(t)=∫[0,t](fr−θ⋆2θ(r) hrhrT)μ(dr). Thus, F′′τ(t) is equal to Equation (3.10), since θ⋆2θ(r) μ(dr)=θ⋆2(r)dr by Lemma 3. □ Proof of Theorem 2 In a first step, we prove that if χ¯in=δn (g(T(n,i),ξin)−ρ(gT(n,i))), U¯tn=∑i∈Σ(n,t)∩Jnχ¯in, (5.11) (recall Equation (1.3) for ξin and Equation (4.15) for Jn and J′n below), then sups≤t |Usn(g)−U¯sn| →P 0. (5.12) Set ζin=χ¯in 1J′n(i), Xin=∑j≤iζjn, Lin=∑j≤iE(ζjnζjn,T|ℱT(n,j)), Then, Ln is the predictable bracket of the (discrete-time) locally square-integrable martingale Xn w.r.t. the filtration (ℱT(n,i+1))i≥0, for which θ(n,t)=card (Σ(n,t)) is a stopping time. Since Lθ(n,t)n=δn∑i∈Σ(n,t)fT(n,i) 1J′n(i) →P 0 by Equation (4.18), it follows from Lenglart’s inequality that supi≤θ(n,t)|Xin| →P 0. It remains to observe that Utn(g)−U¯tn=Xθ(n,t)n, hence Equation (5.12). Therefore, it is enough to prove the claims of Theorem 2 for U¯n instead of Un(g). b. Next, we observe that i belongs to A iff there is a j∈Jn such that R′(n,i)=T(n,j), in which case ∇(n,i)=Δ(n,j) (see Equations (4.22) and (4.27)) and ξin=χ¯jn. Hence, comparing Equations (5.3) and (5.11) gives that U¯tn=U′sn iff there are as many points in Σ(n,t)∩Jn and in σ(n,s). With the notation of the proof of Lemma 11, these numbers are Fn(t)/δn or 1+Fn(t)/δn (respectively, Rn(s)/δn or 1+Rn(s)/δn). Then, there is τn(t) with U¯tn=U′τn(t)n, Rn−1(Fn(t)−δn)≤τn(t)≤Rn−1(Fn(t)). (5.13) c. Set lD=lD(IR+,IRq), with its Borel σ-field D. Set Y=Ω×lD, with the σ-field Y=ℱ⊗D. We endow (Y,Y) with the probability measures χn and χ defined by χn(A×B)=E(1A 1B(U′n)), χ(A×B)=E˜(1A 1B(U′)). (5.14) By Equation (5.9), χn(Z⊗k)→χ(Z⊗k) for all bounded measurable Z on (Ω,ℱ) and all bounded continuous k on the Polish space (lD,D). By Jacod (1979), Theorem (3.4), we deduce that χn(l)→χ(l) for every bounded measurable l on (Y,Y) such that x↦l(ω,x) is continuous at χ-almost all points (ω,x). Applying this to l(ω,x)=Z(ω)k((xτ(ω,t))t≥0), where Z is bounded measurable on (Ω,ℱ) and k is bounded continuous on (lD,D), we get (see Lemma 5) χn(l)=E(Z k(U′τ(.)n))→χ(l)=E¯(Z k(U′τ(.)))=E¯(Z k(g⋆B)). Applying this to l(ω,x)=Z(ω)k((xτ(ω,t1),…,xτ(ω,tr))) with k bounded continuous on (IRq)r and using the fact that U′ is continuous in time (hence, x↦l(ω,x) is again χ-a.s. continuous), we get similarly E(Z k(U′τ(t1)n),…,(U′τ(tr)n))→E¯(Z k(g⋆Bt1,…,g⋆Btr)). Therefore, in view of Equation (5.13), the result will follow if we prove the following two properties: U′τn(t)n−U′τ(t)n →P 0 for all t∈I(recall Equation (3.12) for I), (5.15) supt≤s|U′τn(t)n−U′τ(t)n| →P 0 for all s if μ has a.s. no atom. (5.16) Up to taking subsequences, we may assume that the convergences (4.19) and (4.34) hold a.s. d. Let us prove two auxiliary facts. First, if t∈I then Equation (4.19) gives that outside a null set Fn(tn)→F(t) whenever tn→t, and if μ has a.s. no atom we have Fn→F a.s., locally uniformly. Then, we have a.s.: Fn(t)−δn→F(t), Fn(t)→F(t) if t∈Isupt≤s|Fn(t)−δn−F(t)|→0, supt≤s|Fn(t)−F(t)|→0for all s if μ has no atom.} (5.17) Second, because of Lemma 5.2, U′ is a martingale with bracket F given by Equation (5.1). Hence, U′ is a.s. constant over the intervals where F is constant, hence over those on which R is constant, and we have a.s.: U′s=U′S(t) if S(t−)≤s≤S(t). (5.18) e. Now we prove Equation (5.15). Let t∈I. Then, Equations (5.17) and (4.34) imply that a.s.: S(F(t)−)≤liminfnτn(t)≤limsupnτn(t)≤S(F(t))=τ(t). (5.19) Since U′n converges in law to the continuous process U′ satisfying Equation (5.18), these inequalities imply Equation (5.15). f. Finally, assume that μ has a.s. no atom. Suppose that Equation (5.16) does not hold. There is ɛ>0, s∈IR+ and a subsequence still denoted by n, and a (random) sequence tn in [0,s], such that P(|U′τn(tn)n−U′τ(tn)n|>ɛ)≥ɛ for all n. (5.20) Up to taking a further subsequence, we can even assume that tn→t∈[0,s] a.s. Since F is continuous, we then have a.s. by Equations (5.17) and (4.34): S(F(t)−)≤liminfnτn(tn)≤limsupnτn(tn) as well as Equation (5.19). Then once more because U′n converges in law to the continuous process U′ satisfying Equation (5.18), these relations imply |U′τn(tn)n−U′τ(t)n| →P 0, which contradicts Equation (5.20). Thus, Equation (5.15) holds, and we are finished. □ PART II: BROWNIAN SEMIMARTINGALES 5 THE RESULTS In this section, the setting is the same as in Section 2, but in addition we have an IRm-valued Brownian semimartingale X of the form (1.10), satisfying (H). We set ΔinX=Δ(n,i)−1/2(XS(n,i)−XT(n,i)). (6.1) We also set c=aaT, and call ρtX=ρtX(ω,dx) the centered Gaussian distribution on IRm with covariance matrix ct(ω). Then, we write ρtX(f)=∫ρtX(ω,dx)f(ω,t,x) for any function f on Ω×IR+×IRm. We are interested in the limiting behavior of processes like Un(g) of Equation (1.4), with ξin replaced by ΔinX. Of course we should also modify the centering term in Equation (1.4), and there are several possibilities for this. The most natural one is the following: Ut1,n(g)=δn ∑i∈Σ(n,t)(g(T(n,i),ΔinX)−E(g(T(n,i),ΔinX)|ℱT(n,i))) (6.2) (see Equation (4.17) for Σ(n,t)), provided the conditional expectations above make sense. However, these conditional expectations are difficult to compute, and it may be more useful to consider Ut2,n(g)=δn ∑i∈Σ(n,t)(g(T(n,i),ΔinX)−ρT(n,i)X(g)), (6.3) which is well-defined if g satisfies (K). Finally, the following has also some interest: Ut3,n(g)=δn ∑i∈Σ(n,t)(g(T(n,i),aT(n,i)ξin)−ρT(n,i)X(g)). (6.4) Observe that under (H) and (K), t↦ρtX(g) is continuous, and Lemma 8 yields for t∈I (recall Equation (3.12) for I): δn ∑i∈Σ(n,t)ρT(n,i)X(g) →P ∫[0,t]ρsX(g) μ(ds), (6.5) and this convergence in probability holds locally uniformly in t if μ has a.s. no atom. The behavior of U3,n(g) is very simple. Indeed if g: Ω×IR+×IRm→IRq satisfies (K), and if (H) holds (hence a is locally bounded), the function g′: Ω×IR+×IRd→IRq defined by g′(ω,t,x)=g(ω,t,at(ω)x) also satisfies (K) and ρtX(g)=ρ(g′t). Hence, Theorem 2 yields: Theorem 3 Assume (A1), (A2), (H), and let B be a tangent measure to W along (Tn). Let g satisfy (K). If μ has a.s. no atom, the processes U3,n(g) converge stably in law to U(g) given by U(g)=g′⋆B, with g′(ω,t,x)=g(ω,t,at(ω)x). (6.6) For all (t1,…,tk) in I, the variables (Ut13,n(g),…,Utk3,n(g)) converge stably in law to the variable (Ut1(g),…,Utk(g)). In view of Equation (6.5), we have the Corollary 2 Assume (A1), (A2), (H), and let g satisfy (K). Then the following convergence δn ∑i∈Σ(n,t)g(T(n,i),aT(n,i)ΔinX) → ∫[0,t]ρsX(g) μ(ds) (6.7)holds in probability, for all t∈I, and also locally uniformly in time if μ has a.s. no atom. Now let us consider the following processes, for A∈Rm: BX(A)t=f⋆Bt, where f(ω,t,x)=1A(at(ω)x). (6.8) It is obvious that BX=(BX(A)t:t≥0,A∈Rm) is a worthy martingale measure on IRm, and that U(g) in Equation (6.6) is U(g)=g⋆BX. Further if B′X and B′′X are defined by Equation (6.8) with B′ and B″ instead of B (recall Proposition 4), then B′X is an L2-valued martingale measure on the Wiener space and B′′X is an ℱ-conditional centered Gaussian measure. Therefore, BX=B′X+B′′X is an ℱ-conditional Gaussian measure. An easy computation using Equations (3.8) and (3.9) shows that, with the notation βtX(g)=∫x g(t,atx) ρ(dx), (6.9)BX satisfies all conditions of the following: Definition 2 A tangent measure to X along the sequence (Tn) is an ℱ-conditional Gaussian measure BX on IRm, defined on a very good extension (Ω¯,ℱ¯,(ℱ¯t),P¯) of (Ω,ℱ,(ℱt),P), such that E¯[BX(A)0]=0 and ⟨W,BX(A)⟩t=∫0t βsX(1A) μ⋆(ds) (6.10) for all A∈Rm, and having the covariance measure νX([0,t]×A×A′)=∫[0,t](ρsX(A∩A′)−ρsX(A)ρsX(A′))μ(ds). (6.11) Again BX is “essentially unique” (its ℱ-conditional law is completely determined). In fact we can construct the tangent measures to all Brownian semimartingales having (H) on the same extension (Ω¯,ℱ¯,(ℱ¯t)),P¯), via (6.8). A result similar to Proposition 4, and formulas similar to Equations (3.8), (3.9), and 3.10) hold for BX: we leave this to the reader. (3) In the rest of the section, BX is a tangent measure to X, and all results below are proved in Section 7. For studying U1,n(g) we need additional assumptions: Assumption H-r ( r∈IR+) E(supt≤s(|at|r+|bt|r))<∞ for all s<∞. Assumption K1 The function g: Ω×IR+×IRm→IRq satisfies (K), and for all ω,s the family of functions x↦g(ω,t,x) indexed by t∈[0,s] is uniformly equicontinuous on each compact subset of IRm. Assumption K2-r ( r∈IR+) We have (K1) and, for some nondecreasing adapted finite-valued process γ=(γt), |g(ω,t,x)|≤γt(ω)(1+|x|r). (6.12) Observe that (H-0) is empty, and that if p < r then (K2-p) implies (K2-r), while (H-r) implies (H-p). Theorem 4 Assume (A1), A2), (H) and one of the following: (H-r) for all r<∞, and (K1), (H-r) and (K2-r) for some r∈[1,∞), (K2‐0) (i.e., (K1) and |g(t,x)|≤γt).Then: (a) The processes U1,n(g) are well-defined (i.e., the conditional expectations in Equation (6.2) make sense), and satisfy for all s<∞: supt≤s|Ut1,n(g)−Ut3,n(g)| →P 0. (6.13) b. If μ has a.s. no atom, the processes U1,n(g) converge stably in law to g⋆BX. c. For all t1,…,t) in I, the variables (Ut11,n(g),…,Utk1,n(g)) converge stably in law to (g⋆Bt1X,…,g⋆BtkX). Corollary 3 Assume (A1), (A2), (H), and (K1). Then the following convergence δn ∑i∈Σ(n,t) g(T(n,i),ΔinX) → ∫[0,t]ρsX(g) μ(ds) (6.14)holds in probability, for all t∈I, and also locally uniformly in time if μ has a.s. no atom. (4) Let us turn to the processes U2,n(g). Again, we need new assumptions: Assumption H′ t↦bt is adapted continuous. The process a is a Brownian semimartingale of the form at=a0+∫0ta′sdWs+∫0tb′sds, (6.15) with a′ and b′ predictable locally bounded and t↦a′t continuous. □ Observe that (H′) implies (H). On the other hand, the following implies (K1): Assumption K′ The function g satisfy (K1), and x↦g(ω,t,x) is differentiable, and the function ∇g (gradient in x) also satisfies (K1). □ In order to define the limiting process, we also need some more notation. First, we consider the process, ρ¯tX(∇g)=12 ∫ρ(dx) ∑1≤i≤d,1≤j,k≤m∂g∂xi(t,atx)at′ijk(xjxk−δjk). (6.16) In the above formula δjk is the Kronecker symbol; recall a=(aij)i≤m,j≤d, so a′=(a′ijk)i≤m;j,k≤d and Equation (6.15) reads componentwise as atij=a0ij+∑1≤k≤d∫0ta′sijkdWsk+∫0tb′sijds. Under the above assumptions, ρ¯tX(∇g) is continuous in t. Finally, we define the q-dimensional process: U¯(g)t=g⋆BtX+∫0t(ρsX(∇g)bs+ρ¯sX(∇g)) μ⋆(ds). (6.17) Theorem 5 Assume (A1), (A2), (H′) and (K′). Then If μ has a.s. no atom, the processes U2,n(g) converge stably in law to U¯(g). For all t1,…,tk in I, the variables (Ut12,n(g),…,Utk2,n(g)) converge stably in law to (U¯(g)t1,…,U¯(g)tk). (5) Finally, we could hope for a central limit theorem associated with the convergence (6.14). For this we need rather strong regularity of g as a function of time. To remain simple, we consider the very special case where g(ω,t,x)=g(x) depends on x only. For such g, (K′) amounts to saying that g is continuously differentiable, with ∇g having polynomial growth. Further, this desired central limit theorem is not true in general (see Remark 4), and we consider only the regular case T(n,i)=i/n and Δ(n,i)=1/n. Then, we are led to consider the processes Vtn(g)=1n ∑1≤i≤[nt]g(n (Xi/n−X(i−1)/n))−∫0tρsX(g)ds. (6.18) Corollary 4 Let g be a continuously differentiable function on IRm with ∇g having polynomial growth. Assume (H′). Then supt≤s|n Vtn(g)−Ut2,n(g)| →P 0 for all s. The processes n Vn(g) converge stably in law to the process U¯(g) of Equation (6.17) (with μ = Lebesgue measure). Remark 4 In contrast with the regular case we do not have in general a rate of convergence δn in Equation (6.14), even when δn=1/n and even when the T(n, i)’s are deterministic. Here is a counter-example: take m=d=q=1, and at = t and b = 0, and g(x)=x2: we have (H′) and (K′). Take T(n,i)=i/nα for some α>1 if i≤n and T(n,i)=∞ otherwise, and Δ(n,i)=1/nα. Then (A1) and (A2) are satisfied with δn=1/n and μ=ɛ0 and μ⋆=0. We have ρtX(g)=t, hence if t≤1 the limit in Equation (6.14) is 0. Denote by Vtn the left-hand side of Equation (6.14). Then, nV1n−U12,n(g)=n−1/2∑1≤i≤nρT(n,i−1)X(g)=∑1≤i≤n(i−1)n−α−1/2=12(n−1)n1.2−α, which is equivalent to n3/2−α/2. By Theorem 5 we have non-degenerate convergence of n V1n if α≥3/2 (with a non-centered limit if α=3/2), and if 1<α<3/2 we have convergence of nα−1V1n to 1/2 in probability. □ (6) The case of stochastic differential equations. Here, we explain how the above assumptions on a, b read when the process X of (1.10) is the solution of the following stochastic differential equation: dXt=A(t,Xt)dWt+B(t,Xt)dt, X0=x0 given in IRm. (6.19) Assume that A and B are locally Lipschitz in space (locally uniformly in time) and with at most linear growth (locally uniformly in time). Then Equation (6.19) has a unique strong non-exploding solution X, and sups≤t|Xs|p is integrable for all p<∞, t<∞, and X is of the form (1.10) with at=A(t,Xt), bt=B(t,Xt). If further A is continuous in time, clearly (H) and (H-r) hold for all r: hence, Theorem 4 applies, provided g satisfies (K1). For (H′) to hold, we need further assumptions: for instance, that A is of class C1,2 on IR+×IRm and B is continuous in time. 6 SOME ESTIMATES Below, Kr denotes a constant depending on r and which may change from line to line, but which does not depend on a, b, g. If s > 0 and t≥0, set δ(t,s)=s−1/2(Xt+s−Xt), δ′(t,s)=s−1/2at(Wt+s−Wt). (7.1) Below, increasing process on IR+j means a process, say G, indexed by IR+j, whose paths (t1,…,tj)↦G(t1,…,tj)(ω) are a.s. with values in IR+ and non-decreasing and right-continuous separately in each variable ti. We also denote by S the family of all pairs (T,Δ) where T is a finite stopping time and Δ an ℱT-measurable (0,∞)-valued random variable. Lemma 16 Assume (H) and (H-r) for some r≥2. There exist two increasing processes χr and χ′r on IR+2, with χ′r(u,0)=0 and such that for all (T,Δ)∈S: E(|δ(T,Δ)|r|ℱT)≤χr(T,Δ), E(|δ′(T,Δ)|r|ℱT)≤χr(T,Δ), (7.2) E(|δ(T,Δ)−δ′(T,Δ)|r|ℱT)≤χ′r(T,Δ). (7.3) Proof Since E(|δ′(T,Δ)|r|ℱT)≤|aT|r E(Δ−r/2|Wt+Δ−Wt|r|ℱT) and Δ is ℱT-measurable, the second inequality in Equation (7.2) holds with χr(u,v)=supt≤u|at|r. By Cauchy–Schwarz and Burkholder–Davis–Gundy inequalities and again the ℱT-measurability of Δ, E(|δ(T,Δ)|r|ℱT)≤KrΔ−r/2 E((∫TT+Δ|bs|ds)r+(∫TT+Δ|as|2ds)r/2|ℱT)≤KT 1Δ ∫TT+ΔE(|bs|rΔr/2+|as|r |ℱT) ds. The first inequality in Equation (7.2) holds if we take χr(u,v)=Kr limv′↓v supt≤u E(sups≤u+v′(|bs|rvr/2+|as|r)|ℱT), which is finite-valued by (H-r) and Doob’s inequality for martingales. b. Observing that δ(t,s)−δ′(t,s)=s−1/2(∫tt+s(au−at)dWu+∫tt+sbudu), the same argument as above shows that E(|δ(T,Δ)−δ′(T,Δ)|r|ℱT)≤KT 1Δ ∫TT+ΔE(|bs|rΔr/2+|as−aT|r |ℱT) ds. (7.4) Then if β(u,v)=sup(|at+s−at|: 0≤t≤u,0≤s≤v), Equation (7.3) holds with χ′r(u,v)=Kr limv′↓v supt≤u E(sups≤u+v′(|bs|rvr/2+β(u,v′)r)|ℱT), (7.5) Further β(u,v)→0 as v→0 by (H), and this convergence also takes place in Lr if (H-r) holds. Then Doob’s inequality again gives χ′r(u,0)=0. □ Lemma 17 Assume (H), (H-r) for all r<∞, and (K1). Then with γt as in (K), for all r<∞ there is an increasing process χ′′r on IR+3, with χ′′r(u,0,w)=0 a.s. and such that for all (T,Δ)∈S: E(|g(T,δ(T,Δ))−g(T,δ′(T,Δ))|r|ℱT)≤χ′′r(T,Δ,γT). (7.6) Proof Let (T,Δ)∈S and q<∞. Set δ=δ(T,Δ) and δ′=δ′(T,Δ) and γ=γT. By (K1), for all p<∞, ɛ>0 there is a strictly positive random variable ν(ɛ,p) such that |x|≤p, |y|≤p and |x−y|≤ν(ω,ɛ,p) imply |g(ω,t,x)−g(ω,t,y)|≤ɛ. Then by (K): β:=|g(T,δ)−g(T,δ′)|r≤{Krγr(1+|δ|rγ+|δ′|rγ)ɛr if |δ|,|δ′|≤p, |δ−δ′|≤ν(ɛ,p). Then for some constant Kr, for all ɛ,θ,u,v,w>0 we have on {T≤u,Δ≤v,γ≤w}: E(β|ℱT)≤ɛr+KrwrE((1+|δ|rw+|δ′|rw) 1{|δ|>p}∪{|δ′|>p}∪{|δ−δ′|>ν(ɛ,p)} |ℱT)≤ɛr+Krwr(1+χ2rw(u,v))1/2 (2p2 χ2(u,v)+χ′2(u,v)/θ +Z(ɛ,p,θ))1/2, (7.7) where Z(ɛ,p,θ)=supt P(ν(ɛ,p)≤θ|ℱT) (use Equations (7.2), (7.3), and the inequalities of Cauchy–Schwarz and Bienaymé–Tchebicheff). If Y(ɛ,p,θ,u,v,w) is the right-hand side of Equation (7.7), then Equation (7.6) holds with χ′′r(u,v,w)=limv′↓v infɛ,p,θ>0Y(ɛ,p,θ,u,v′,w). Further, there exist finite variables Z′(u,w) such that for all ɛ,p,θ>0 and v∈[0,1], we have χ′′r(u,v,w)≤ɛr+Z′(u,w)(p−2χ′2(u,2v)/θ +Z(ɛ,p,θ))1/2. Since P(ν(ɛ,p)≤θ)→0 as θ→0 we clearly have Z(ɛ,p,θ) →P 0 as θ→0 for all ɛ,p>0, while χ′2(u,2v)→0 as v→0. Then by choosing first p, then θ, then v, it is clear that χ′′r(u,v,w)→0 as v→0. □ Next, we will assume (H′) and the following (implying (H-r) for all r<∞: Assumption H′- ∞ The processes b and a′,b′ of Equations (6.15) are bounded by a constant C, and |a0| belongs to Lr for all r. □ By definition a′ takes its values in IRd⊗IRm⊗IRm, and we define the IRd-valued variables Y(t,s)=(Y(t,s)i)1≤i≤d by Y(t,s)i=bti+1s∑1≤j,k≤dat′ijk∫tt+s(Wuj−Wtj)dWuk. (7.8) Lemma 18 Assume (H′) and (H′- ∞). For all r<∞ there is an increasing process χ¯r′ on IR+2, with χ¯r′(u,0)=0 a.s. and such that for all (T,Δ)∈S, E(|Δ−1/2(δ(T,Δ)−δ′(T,Δ))−Y(T,Δ)|r |ℱT)≤χ¯r′(T,Δ). (7.9) Proof It is enough to prove the result for r≥2. Observe first that Δ−1/2(δ(T,Δ)−δ′(T,Δ))−Y(T,Δ)=A(T,Δ)+B(T,Δ), where (see Equations (7.1), (6.15), and (7.9)): A(T,Δ)=1Δ∫TT+ΔDs(T)dWs, where Dt(T)=∫TT+t(a′s−a′T)dWs+∫TT+tb′sds,B(T,Δ)=1Δ∫TT+Δ(bs−bT)ds. Then, it is enough to prove the result separately for E(|B(T,Δ)|r|ℱT) and for E(|A(T,Δ)|r|ℱT). In the first case, it holds with χ¯r′(u,v)=limv′↓v supt≤u E(sups≤u,s′≤v′|bs+s′−bs|r|ℱt), which has χ¯r′(u,0)=0 because here t↦bt is continuous and uniformly bounded (same argument as in (b) of Lemma 16). Next, as in Lemma 16: E(|A(T,Δ)|r|ℱT)≤KrΔ−r/2−1∫TT+δE(|Dt(T)|r|ℱT) dt. Since a′ and b′ are uniformly bounded and a′ is continuous, exactly as in the proof of Lemma 16 again we obtain an increasing process ζr on IR+2 with ζr(u,0)=0, such that E(|Dt(T)/t|r|ℱT)≤ζr(T,t). Then if (T,Δ)∈S, E(|A(T,Δ)|r|ℱT)≤Kr 1Δ∫TT+δE(|Dt(T)/t|r|ℱT) dt≤Krζr(T,Δ) and the result follows. □ Lemma 19 Assume (H′), (H′- ∞), and (K′1). Then with γt satisfying Equation (1.5) for both g and ∇g, for all r<∞ there is an increasing process χ¯′′r on IR+3 with χ¯′′r(u,0,w)=0 a.s. and such that for all (T,Δ)∈S: E(|Δ−1/2(g(T,δ(T,Δ))−g(T,δ′(T,Δ)))−∇g(T,δ′(T,Δ))Y(T,Δ)|r |ℱT)≤χ¯r″(T,Δ,γT). (7.10) Proof Here again it is enough to prove the result for r≥2. Due to our assumptions, we can apply Lemma 7.1 to the process a instead of X, hence with the same notation χT we get any finite stopping time T: E(|t−1/2(aT+t−aT)|r|ℱT)≤χr(T,t). (7.11) Plugging this into Equation (7.4) gives, instead of Equation (7.5): χ′r(u,v)=vr/2ζr(u,v), where ζr is the following increasing process on IR+2: ζr(u,v)=Kr limv′↓v supt≤u E(sups≤u+v′|bs|r|ℱt)+χr(u,v)). b. Let (T,Δ)∈S. Set δ=δ(T,Δ), δ′=δ′(T,Δ), Y=Y(T,Δ), Z=δ−δ′−Δ Y. Taylor’s formula yields Δ−1/2(g(T,δ)−g(T,δ′))−∇g(T,δ′)Y=A(T,Δ)+B(T,Δ), with A(T,Δ)=Δ−1/2∇g(T,δ′)Z, and B(T,Δ)=Δ−1/2(∇g(T,δ″)−∇g(T,δ′))(δ−δ′) and δ″=δ′+θ(δ−δ′) for a random variable θ taking values in [0,1]. Our assumptions imply (H-r) for all r, hence we can reproduce the proof of Lemma 17 with ∇g instead of g and δ″ instead of δ, after observing that |δ″−δ′|≤|δ−δ′|. We obtain E(|∇g(T,δ″)−∇g(T,δ′)|r|ℱT)≤χ′′r(T,Δ,γT). Combining this and Equation (7.3) and (a) above, Cauchy–Schwarz inequality gives E(|B(T,Δ)|r|ℱT)≤(χ′′2r(T,Δ,γT) ζ2r(T,Δ))1/2. (7.12) c. Finally, Equation (7.6) for ∇g and Equation (7.2) yield E(|∇g(T,δ′)|r|ℱT)≤ζ′r(T,Δ,γT) for some other increasing process ζ′r. This and Equation (7.9) give us E(|A(T,Δ)|r|ℱT)≤(χ¯′2r(T,Δ) ζ′2r(T,Δ,γT))1/2. (7.13) Then adding Equations (7.12) and (7.13) gives Equation (7.14) with the required properties for χ¯′′r. □ We end this section with an estimate for functions g: IRd→IRq that are continuously differentiable and have for some r: |∇g(x)|≤r(1+|x|r). (7.14) Set also U(t,s)=ρt+s(g)−ρt(g). Then, Lemma 20 Assume (H′), (H′- ∞), and (7.14). There are increasing processes ζ and ζ′ on IR+2 with ζ(u,0)=0 a.s. and such that for all (T,Δ)∈S: |E(U(T,Δ)|ℱT)|≤Δ ζ(T,Δ), (7.15) E|(U(T,Δ)|2|ℱT)|≤Δ ζ′(T,Δ). (7.16) Proof Below the constant K changes from line to line. We fix u<∞ and set θ=1+supt|at| and θ¯p=suptE(θp|ℱt), which is integrable for all p<∞. We always take below (T,Δ) in T(u). Equation (7.14) implies |g(x)−g(y)|≤K(1+|x|r+|y|r)|x−y|, so |g(aT+Δx)−g(aTx)|≤K(1+|x|r)θr|aT+Δ−aT| and integrating w.r.t. the normal measure G gives |U(T,Δ)|≤Kθr|aT+Δ−aT|. Then, Equation (7.11) readily gives Equation (7.16) with ζ′(u,v)=K(θ¯4r χ4(u,v))1/2 for a suitable constant K. Taylor’s formula gives g(y)−g(x)=(∇g(x)+α(x,y))(y−x) with |α(x,y)|≤K(1+|x|r+|y|r) and α(x,y)→0 as y→x, uniformly in x on each compact subset of IRd. Therefore, there are reals ν(ɛ,p)>0 such that |x|≤p and |y−x|≤ν(ɛ,p) imply |α(x,y)|≤ɛ. By definition of U(T,Δ) we have U(T,Δ)=U1+U2, where Ui=∫ui(x)ρ(dx) and u1(x)=∇g(aTx)(aT+Δ−aT)x, u2(x)=α(aTx,aT+Δx)(aT+Δ−aT)x. It is enough to prove Equation (7.15) separately for U1 and U2. We have |u2(x)|≤Kθr(1+|x|r+1)|aT+Δ−aT| and, as soon as θ|x|≤p and |aT+Δ−aT| |x|≤ν(ɛ,p), then |u2(x)|≤c|aT+Δ−aT| |x|. Integrating w.r.t. ρ, we obtain for all ɛ,p>0, as for Equation (7.7) (recall that K changes from line to line): |U2|≤K(ɛ+θr+1(1p+|aT+Δ−aT|ν(ɛ,p)))|aT+Δ−aT|. We deduce from Equation (7.11) that |E(U2|ℱT)|≤Δ Y(ɛ,p,T,Δ), where Y(ɛ,p,u,v)=K((ɛ+θ¯2r+21/2/p) χ2(u,v)+θ¯2r+21/2v χ4(u,v)/ν(ɛ,p)). This is true for all ɛ,p>0. Then Equation (7.15) is satisfied by U2 with ζ(u,v)=limv′↓v infɛ,p>0 Y(ɛ,p,u,v′), and that ζ(u,0)=0 is easily checked by choosing first p, then ɛ, the v. Finally, Equation (5.15) allows us to write (recall that a′ and b′ are bounded): |E(U1|ℱT)|=|∫∇g(aTx)(∫TT+ΔE(b′s|ℱT) ds)xρ(dx)|≤KΔ∫|∇g(aTx)||x||ρ(dx)≤ KθrΔ use Equation (7.14)). Then Equation (7.15) holds for U1, with ζ(u,v)=kθrv. □ 7 PROOF OF THE RESULTS OF SECTION 5 Proof of Theorem 4 In view of Theorem 3 it is enough to prove the claim (a) of Theorem 4. We do that in several steps. Step 1 First we prove that under the assumptions of Theorem 4, U1,n(g) is well-defined. First assume (i), and let γt be as in (K). Set T=T(n,i) and Δ=Δ(n,i), so that on the ℱT-measurable set {γT≤p} we have |g(T,ΔinX)|≤p(1+|ΔinX|p). Then, E(|g(T,ΔinX)| |ℱT)≤p(1+χp(T,Δ))<∞ by Equation (7.2), and since {γT≤p}↑Ω as p→∞, the conditional expectations in Equation (6.2) are well defined. In cases (ii) and (iii), the same argument works, with γt as in (K2-r) (with r = 0 in case (iii)), so that |g(T,ΔinX)|≤p(1+|ΔinX|r). Step 2 Now we prove Equation (6.13) under (i). Set χin=g(T(n,i),ΔinX)−g(T(n,i),aT(n,i)ξin) (8.1) Gtn=δn ∑i∈Σ(n,t)E(|χin|2|ℱT(n,i)). (8.2) Then since Δ(n,i) and S(n, i) are ℱT(n,i)-measurable, Ytn:=Ut1,n(g)−Ut3,n(g)=δn∑i∈Σ(n,t) (χin−E(χin|ℱT(n,i))). As in part (a) of the proof of Theorem 2, we get Equation (6.13) if ∑i∈Σ(n,t)E(ξinξin,T|ℱT(n,i))→P0 with ξin=δn (χin−E(χin|ℱT(n,i))). In view of Equation (8.2) it is then enough to prove that Gtn→P 0. (8.3) Then with γt as in (K), we deduce from Equation (7.6) that (recall that χ′′2 is increasing in each of its arguments, and that δncard (Σ(n,t))=μn([0,t])): Gtn≤δn∑i∈Σ(n,t) χ′′2(T(n,i),Δ(n,i)),γT(n,i))≤μn([0,t]= χ′′2(t,δn,γt)+χ′′2(t,t,γt)∑i∈Σ(n,t) 1{Δ(n,i)>δn}. We have ∑i∈Σ(n,t)Δ(n,i)≤t: hence, the last sum above is smaller than t/δn. That is, Gtn≤μn([0,t]= χ′′2(t,δn,γt)+tδn χ′′2(t,t,γt). Since δn→0 and χ′′2(t,v,γt)→0 a.s. as v→0 and since the sequence μn([0,t]) is bounded in probability by (A2), we deduce Equations (8.3) and (6.13). Step 3 Here, we assume (ii) or (iii) of Theorem 4. In order to apply Step 2, although (H-r) does not hold for all r, we “localize” the coefficients: since a and b are locally bounded, there exists an increasing sequence (τl) of stopping times satisfying τl=0 if |a0|+|b0|>l and |at|+|bt|≤l if t≤τl and τl>0, and τl↑+∞ a.s. as l→∞. (8.4) Set a(l)=at∧τl and b(l)=bt∧τl if τl>0, and a(l)t=b(l)t=0 if τl=0, and X(l)t=x0+∫0ta(l)sdWs+∫0tb(l)sds. (8.5) We denote by Ui,n(l,g) the processes defined by Equations (6.2), (6.3), and (6.4), with (a(l),X(l)) instead of (a, X). Now, a(l) and b(l) satisfy (H) and (H-r) for all r<∞, hence Step 2 implies sups≤t|Us1,n(l,g)−Us3,n(l,g)| →P 0 as n→∞, for all l<∞. (8.6) Further, on {τl≥t}, Usi,n(g)=Usi,n(l,g) for all s∈[0,t], i = 1, 2, 3 (this is obvious for i = 2 and i = 3; for i = 1 it comes from the fact that S(n, j) is ℱT(n,i)-measurable). Then Equation (6.13) readily follows from Equations (8.4) and (8.6). Proof of Corollary 3 Assume (H) and (K1). In view of Equation (6.7) it is enough to prove that, for each t<∞ and with χin defined by Equation (8.1), Gtn=δn∑i∈Σ(n,t)|ξin| →P 0. Because (χin:i∈Σ(n,t)) are the same for X and for X(l) on {Rn≥t} and because of Equation (8.4), we can in fact work with each process X(l), or equivalently assume (H-r) for all r<∞. Further, with θ(n,t) as in part (a) of the proof of Theorem 2 and Xin=∑j≤iδn|χjn|, we have Gtn=Xθ(n,t)n and the predictable compensator of Xn for the filtration (ℱT(n,i+1))i≥0 is X˜in=∑j≤iδn E(|χjn| |ℱT(n,j)). Then by Lenglart’s inequality, X˜θ(n,t)n →P 0 implies Xθ(n,t)n →P 0 (because θ(n,t) is a stopping time). Now, we can reproduce the proof of Step 2 in the previous proof to obtain X˜θ(n,t)n=δn∑i∈Σ(n,t)E(|χjn| |ℱT(n,j)) →P 0 (substituting |χin|2 with |χin|, and thus χ′′2 with χ′′1). □ Proof of Theorem 5 Note that if Un, Yn, U, Y are IRk-valued random variables, with Yn going to Y in probability and Un going to U stably in law, then Un+Yn converge stably in law to U + Y. The same holds for the Skorokhod topology if Un, Yn, U, Y are càdlàg processes and further Y is continuous in time. Therefore, if we set Ytn=Ut2,n(g)−Ut3,n(g), (8.7) Yt=∫0t(ρsX(∇g)bs+ρ¯sX(∇g)) μ⋆(ds), (8.8) in order to deduce Theorem 5 from Theorem 3, it is enough to prove that sups≤t|Ysn−Ys| →P 0 (8.9) under (A1), (A2), (H′), and (K′). The proof goes through several steps. Step 1 We wish to show that for every (small enough) function f on IRd and every pair (T,Δ) in S (see Section 6, recall also that δjk is the Kronecker symbol), we have E((f(WT+Δ)−f(WT)) ∫TT+Δ(Wsj−WTj)dWsk |ℱT)=12 E((f(WT+Δ)−f(WT))((WT+Δj−WTj)(WT+Δk−WTk)−Δδjk)|ℱT). (8.10) When j = k this is just Itô’s formula applied to s↦(WT+sj−WTj)2 and the equality holds even before taking conditional expectations. If j≠k, and since W has stationary independent increments and independent components, it is enough to prove Equation (8.10) when T = 0 and Δ is deterministic and f(x)=exp(iuxj+ivxk) for some u,v∈IR. In other words, we need to prove that if B, B′ are two independent one-dimensional Brownian motion, and Zt=∫0tBsdB′s, E(eiuBs+ivB′s Zs)=12 E(eiuBs+ivB′s BsB′s). (8.11) Set V=eiuB+ivB′. Itô’s formula yields that the process YZ equals a martingale plus the following process: 12 ∫0s(−(u2+v2)VtZt+2ivVtBt)) dt. Hence, if h(s) denotes the left-hand side of Equation (8.11), we have, h(s)=12 ∫0s(−(u2+v2)h(t)+2iv E(VtBt)) dt and, since E(VtBt)=iut e−(u2+v2)t/2, we easily deduce that h(s)=−uvs22 e−(u2+v2)s/2, which is equal to the right-hand side of Equation (8.11). Step 2 Here, we assume in addition (H′- ∞). Recalling Equation (7.8), we set ηin=∇g(T(n,i),aT(n,i)ξin) Y(T(n,i),Δ(n,i)). Then Equations (8.10) and (6.16) yield, E(ηin|ℱT(n,i))=ρT(n,i)X(∇g)bT(n,i)+ρ¯T(n,i)X(∇g). Since t↦ρtX(∇g)bt+ρ¯tX(∇g) is continuous, one proves exactly as in Lemma 8 the following convergence in probability, locally uniform in time: ∑i∈Σ(n,t)δnΔ(n,i) E(ηin|ℱT(n,i))→Yt. Recalling Equation (8.1), we have Ytn=δn∑i∈Σ(n,t)E(χin|ℱT(n,i)). Therefore, the same argument as in the proof of Corollary 3 shows that Equation (8.9) holds, provided we have for all t<∞ Gtn := ∑i∈Σ(n,t)δnΔ(n,i) E(|Δ(n,i)−1/2χin−ηin|ℱT(n,i)) →P 0. We reproduce Step 2 of the proof of Theorem 4, for |Δ(n,i)−1/2χin−ηin| instead of |χin|2: use Equation (7.10) with r = 1 and χ¯′′1 instead of Equation (7.6) and χ′′2, and truncate at Δ(n,i)>δn1/4, so Gtn≤μn⋆([0,t]) χ¯′′1(t,δn1/4,γt)+t3/2δn1/4χ¯′′1(t,t,γt). Step 3 We no longer assume (H′- ∞), but we localize as in Step 3 of the proof of Theorem 4: we have an increasing sequence (τl) of stopping times satisfying Equation (8.4), and τl=0 if |a0|+|b0|+|a′0|+|b′0|>l, and |at|+|a′t|+|b′t|≤l if t≤τl and τl>0. Set a(l)′t=a′t∧τl, b(l)t=bt∧τl, b(l)′t=b′t∧τl and a(l)t=a0+∫0ta(l)′sdWs+∫0tb(l)′sds if τl>0, and a(l)t=0, b(l)t=0, a(l)′t=0, b(l)′t=0 if τl=0. Finally, let X(l) be defined by Equation (8.5), and denote by Y(l)n, Y(l) the quantities associated with these processes indexed by l via Equations (8.7), (8.8). For each l the term (a(l),b(l),a(l)′,b(l)′) satisfies (H′) and (H′- ∞). Hence, Step 1 implies Equation (8.9) for (Y(l)n,Y(l)) for each l, while on {Rl≥t} we have Ys=Ys(l) and Ysn=Ysn(l) for all s≤t. Then, Equation (8.9) for (Yn,Y) follows from Equation (8.4). □ Proof of Corollary 4 We only need to prove the claim (a). Recall that now T(n,i)=i/n and Δ(n,i)=1/n. Observe first that, Ytn:=Ut2,n(g)−n Vtn(g)=n ∑0≤i≤[nt]−1ηin, where ηin=∫i/n(i+1)/n(ρsX(g)−ρi/nX(g))ds. Next, let us localize as in Step 3 of the proof of Theorem 5, and call Ytn(l) the above quantity associated with the localized processes. Since Ysn=Ysn(l) for all s≤t on {τl≥t}, we see by Equation (8.4) that it is enough to prove sups≤t|Ysn(l)| →P 0 for each l, or in other words we can and will assume (H′- ∞). Now we can apply Lemma 20 with T=i/n and Δ=1/n. Integrating Equations (7.15) and (7.16) against Lebesgue measure on [i/n,(i+1)/n], we get for i≤[nt]−1: |E(ηin|ℱi/n)|≤n−3/2ζ(t,1/n), E(|ηin|2|ℱi/n)|≤n−3ζ′(t,1/n). Therefore, if Atn=n∑0≤i≤[nt]−1E(ηin|ℱi/n) and Btn=Vtn−Atn, we deduce sups≤t|Asn| →P 0 (because ζ(t,v)→0 a.s. as v→0), and the bracket of the (ℱ[nt])-local martingale Bn is |⟨Bn,Bn,T⟩t|≤ζ′(t,1/n)/n. Then Lenglart’s inequality implies that sups≤t|Bsn| →P 0, hence sups≤t|Ysn| →P 0 as well. □ 8 APPLICATIONS AND EXAMPLES We will consider below a Brownian semimartingale X satisfying (H). Our first remark is that the measure ρtX is symmetric about 0. Hence (see Equation (6.9)): If x↦g(ω,t,x) is an even function ,ρtX(g)=0 and ρtX(∇g)=0, and also ρ¯tX(∇g)=0 and U¯(g)=g⋆BX in Equation (6.17) if further (K′) holds.} (9.1) Let us for example consider the even function g(ω,t,x)=xxT (taking values in IRd⊗IRd, hence q=d2). Equation (6.14) yields the following well-known approximation of the quadratic variation: supt|∑1≤i≤[nt](Xi/n−X(i−1)/n)(Xi/n−X(i−1)/n)T −∫0tcsds| →P 0. (9.2) Further, Corollary 4 gives a rate of convergence in Equation (9.2), which is easily proved directly but is not so well-known (apply the easily proved fact that ρs(gjkgil)=csjkcsil+csjicskl+csjlcski. Proposition 5 Assume (H′). The d2-dimensional processes Ytn=n(∑1≤i≤[nt](Xi/n−X(i−1)/n)(Xi/n−X(i−1)/n)T −∫0tcsds) (9.3)converge stably to a process Y defined on a very good extension of the space (Ω,ℱ,(ℱt),P), and which is ℱ-conditionally a continuous Gaussian martingale with “deterministic” bracket given by ⟨Yjk,Yil⟩t=∫0t(csjkcsil+csjicskl+csjlcski)ds. (9.4) Now we assume for simplicity that d=m=1. Consider g(ω,t,x)=xp for some p∈IN. Then if αp denotes the pth moment of the distribution N(0,1), Corollary 4 gives: Proposition 6 Assume (H′). The processes n(np/2−1 ∑1≤i≤[nt](Xi/n−X(i−1)/n)p−αp∫0t(cs)p/2 ds) (9.5)converge stably in law to a process Y defined on a very good extension of the space (Ω,ℱ,(ℱt),P) which is as follows: If p is even, Y is ℱ-conditionally a continuous Gaussian martingale with “deterministic” bracket given by ⟨Y,Y⟩t=(α2p−(αp)2)∫0t(cs)p ds. (9.6) b. If p is odd and p≥3, Y=Y′+Y″ where Y′t=αp+1∫0t(cs)(p−1)/2dXsc+p∫0t(αp−1(bs−a′s/2)+αp+1a′s/2)(cs)(p−1)/2 ds, (9.7)and Y″ is ℱ-conditionally a continuous Gaussian martingale with deterministic bracket given by Equation (9.6). The first summand in Equation (9.7) is a local martingale, but the second one is not: this is a good example of the “drift” introduced in the error term of the approximation (6.14) when the function g is not even. We also deduce results on the approximations of the β-variation of X ( β>0), defined by Var(X,β)tn=∑1≤i≤[nt]|Xi/n−X(i−1)/n|β. This is done by applying the previous results to g(ω,t,x)=|x|β. If α′r=∫G(dx)|x|r (hence α′r=αr if α is an even integer), we have under (H): nβ/2−1 Var(X,β)tn → α′β∫0t(cs)β/2ds uniformly in time, in probability. Further if β>1, (K′) holds and the processes n(nβ/2−1 Var(X,β)tn−α′β∫0t(cs)β/2ds) converge stably to a process which, conditionally on ℱ, is a continuous Gaussian martingale with bracket equal to (α′2β−(α′β)2)∫0t(cs)β ds. Another interesting type of results, closely related to the previous ones, goes as follows. We consider only the situation of the β-variations (which include the quadratic variation of Proposition 5 for β=2). Assume that a does not vanish and take g(ω,t,x)=|x/at(ω)|β. Set Var′(X,β)tn=∑1≤i≤[nt]|(Xi/n−X(i−1)/n)/a(i−1)/n|β. Then nβ/2−1 Var′(X,β)tn → α′βt uniformly in time, in probability. Further if β>1, the processes n (nβ/2−1 Var′(X,β)tn−α′βt) converge stably to a process which, conditionally on ℱ, is a continuous Gaussian martingale with bracket given by |α′2β−(α′β)2|t. (2) The previous examples were concerned with regular schemes. Now consider, again in the case m=d=1, an example of random schemes. Set T(n,0)=0, T(n,i+1)=inf(t>T(n,i): nt∈IN,|Xt|≤hn), Δ(n,i)=1/n, (9.8) where hn is a sequence of positive numbers tending to 0 and such that δn=1/2nhn tends to 0. Clearly (A1) holds, and we have Ltn:=μn([0,t])=12nhn ∑1≤i≤[nt]1{|X−(i−1)/n|≤hn} (9.9) and μn⋆=2hn μn. Then, as is well known, (A2) is met with μ(dt)=dLt and μ⋆=0, where L is the local time of X at 0. We cannot use Corollary 4 here. However, Theorem 3 gives the following result, when g(ω,t,x)=xp for some p∈IN: Proposition 7 Assume (H). The processes, 12nhn ∑1≤i≤[nt] (np/2(Xi/n−X(i−1)/n)p−αp(c(i−1)/n)p/2) 1{|X(i−1)/n|≤hn}converge stably in law to a process Y defined on a very good extension of the space (Ω,ℱ,(ℱt),P), which is ℱ-conditionally a continuous Gaussian martingale with “deterministic” bracket given by ⟨Y,Y⟩t=(α2p−(αp)2)∫0t(cs)p dLs. Although we cannot deduce a rate of convergence of Ln in Equation (9.9) to L, it is interesting to re-state Corollary 6.6 here: take g satisfying (K1), and assume (H). Then the following convergence holds in probability, locally uniformly in time: 12nhn ∑1≤i≤[nt] g(i−1n,n (Xi/n−X(i−1)/n)) 1{|X(i−1)/n|≤hn} → ∫0tρs(g)dLs. Let us mention that results similar to Proposition 7 have already been used in statistics: see Florens-Zmirou (1989). Analogous results when d≥2 have also been proved by Brugière (1992) via a method of moments, but are not consequences of this paper since (A2) is violated in this case by sequence (9.8) (there is no local time when d≥2, and the processes Ln of Equation (9.9) converge in law, but not in probability; note that the normalization in Equation (9.9) should be changed, and it depends on the dimension d. REFERENCES Aldous D. J. , Eagleson G. K. . 1978 . On Mixing and Stability of Limit Theorems . The Annals of Probability 61 : 325 – 331 . Google Scholar CrossRef Search ADS Brugière P. 1992 . Théorème de limite centrale pour un estimateur non paramérique de la variance d’un processus de diffusion multidimensionnel . Annales de l’I.H.P. Probabilités et Statistiques 29 : 357 – 389 . Dohnal G. 1987 . On Estimating the Diffusion Coefficient . Journal of Applied Probability 24 : 105 – 114 . Google Scholar CrossRef Search ADS Fujiwara T. , Kunita H. . 1990 . Limit Theorems for Stochastic Difference–Differential Equations . Nagoya Mathematical Journal 127 : 83 – 116 . Google Scholar CrossRef Search ADS Florens-Zmirou D. 1989 . Estimation de la variance d’un processus de diffusion à partir d’une observation discrètisée . Comptes Rendus de l’Académie des Sciences Paris 309 : 195 – 200 . Genon-Catalot V. , Jacod J. . 1993 . On the Estimation of the Diffusion Coefficient for Multi-dimensional Diffusion Processes . Annales de l’Institut Henri Poincare (B) Probability 29 : 119 – 151 . Jacod J. 1979 . Calcul Stochastique et Problèmes de Martingales . Lecture Notes in Mathematics, vol. 714 . Berlin : Springer Verlag . Google Scholar CrossRef Search ADS Jacod J. , Mémin J. . 1981 . Weak and Strong Solutions for Stochastic Differential Equations: Existence and Stability . In Williams D. (ed.), Stochastic Integrals . Proceedings of the LMS Durham Symposium, Lecture Notes in Mathematics, vol. 851 , pp. 169 – 212 . Berlin : Springer Verlag . Google Scholar CrossRef Search ADS Jacod J. , Shiryaev A. N. . 1987 . Limit Theorems for Stochatisc Processes . Berlin : Springer Verlag . Google Scholar CrossRef Search ADS Kunita H. 1991a . Limits of Random Measures Induces by an Array of Independent Random Variables . In: Rassias T. M. (ed.), Constantin Caratheodory, An International Tribute , pp. 676 – 712 , World Scientific . Kunita H. 1991b . Limits on Random Measures and Stochastic Differential Equations Related to Mixing Arrays of Random Variables . In: Barlow M. T. , Bingham N. H. (eds.), Stocastic Analysis , pp. 221 – 254 , Cambridge University Press . Kurtz T. G. , Protter P. . 1991 . Wong–Zakai Corrections, Random Evolutions, and Simulation Schemes for SDEs . In Mayer-Wolf M. , Schwarz (eds.), Stochastic Analysis , pp. 331 – 346 . New York : Academic Press . Renyi A. 1963 . On Stable Sequence of Events . Sankya A 251 : 293 – 302 . Rootzen H. 1980 . Limit Distributions for the Error in Approximation of Stochastic Integrals . The Annals of Probability 81 : 241 – 251 . Google Scholar CrossRef Search ADS Walsh J. B. 1986 . An Introduction to Stochastic Partial Differential Equations . In Ecole d’été de St-Flour XIV (1984) . Lecture Notes in Mathematics, vol. 1180 , pp. 266 – 437 . Berlin : Springer Verlag . Google Scholar CrossRef Search ADS © The Author, 2017. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com

Journal of Financial Econometrics – Oxford University Press

**Published: ** Aug 2, 2017

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