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Estimation of quadratic variation for two-parameter diffusions 8 0 0 2 Anthony Réveillac ∗ n Laboratoire de Mathématiques a J Université de La Rochelle 9 1 Avenue Michel Crépeau 17042 La Rochelle Cedex ] R France P . h February 2, 2008 t a m [ Abstract 1 v In this paper we give a central limit theorem for the weighted quadratic vari- 7 ations process of a two-parameter Brownian motion. As an application, we 2 0 show that the discretized quadratic variations [in=s1] j[n=t1] |∆i,jY|2 of a two- 3 parameter diffusion Y = (Y ) observed on a regular grid G is an (s,t) (s,t)∈[0,1]2 n 1. asymptotically normal estimator of the quadratPic variPation of Y as n goes to 0 infinity. 8 0 : v Keywords: Weightedquadraticvariationsprocess,functionallimittheorems,two-parameter i X stochastic processes, Malliavin calculus. r Mathematics Subject Classification: 62G05, 60F05, 62M40, 60H07. a 1 Introduction Many statistical properties of stochastic processes can be deduced fromtheir weighted p-power variations. For a one parameter process (Z ) observed at regular times t t∈[0,1] i/n, 0 i n , this quantity is defined as, { ≤ ≤ } n f Zi−1 (∆iZ)p, ∆iZ := Zi Zi−1. n n − n Xi=1 (cid:16) (cid:17) ∗ [email protected] 1 For example, the study of the power variations process has been used by Barndorff- Nielsen and Shephard in [5, 6] to solve some financial econometric problems (application to econometrics are also given in [7]). These theoretical results were also used in several fields of application such as the estimation of the integrated volatility (see for example [1] and references therein), testing for jumps of a process observed at discrete times like for example in [3]. Inthispaper wegive acentrallimit theorem fortheweighted quadraticvariationspro- cess of a two-parameter Brownian motion. More precisely consider a two-parameter Brownian motion W = (W ) and a deterministic and regular enough func- (s,t) (s,t)∈[0,1]2 tion f : R R, we show that → [n·] [n•] 1 n f W ∆ W 2 law(S) √2 f W dB , Xi=1 Xj=1 (cid:16) (i−n1,j−n1)(cid:17)(cid:18)| i,j | − n2(cid:19) n−→→∞ Z[0,·]×[0,•] (cid:0) (u,v)(cid:1) (u,v) (1.1) where B is a two-parameter Brownian motion independent of W and ∆ W denotes i,j the increment of the process W on the subset ∆ := i−1, i j−1, j of [0,1]2 i,j n n × n n defined by (cid:2) (cid:3) (cid:2) (cid:3) ∆ W := W +W W W . (1.2) i,j (i−n1,j−n1) (ni,nj) − (i−n1,nj) − (ni,j−n1) The notation law( ) used above in (1.1) means that the convergence is in the sense S of stable convergence in law in the two-parameter Skorohod space. Furthermore we stress that the limiting process is defined on an extension of the considered probability basis. Note also that usual techniques of proof used in the one-parameter setting are no longer suitable to the two-parameter case. For example the Itô formula for two-parameter diffusion processes cannot be applied as in the one-parameter setting due to the presence of an additional term. Consequently we chose to replace the usual stochastic calculus by the Malliavin calculus which is valid in general Gaussian context. As an application we deduce a central limit theorem for the quadratic variations process of a two-parameter diffusion Y = (Y ) observed on a regular grid (s,t) (s,t)∈[0,1]2 G which allows us to construct an asymptotically normal consistent estimator of the n 2 quadratic variation of Y. More precisely, consider a two-parameter stochastic process (Y ) defined by, (s,t) (s,t)∈[0,1]2 Y := σ W dW + M dudv, (s,t) [0,1]2, (1.3) (s,t) (u,v) (u,v) (u,v) ∈ Z[0,s]×[0,t] Z[0,s]×[0,t] (cid:0) (cid:1) where (W ) is a two-parameter Brownian motion, σ : R R is a bounded (s,t) (s,t)∈[0,1]2 → sufficiently smoothdeterministic functionand(M ) isa continuous adapted (s,t) (s,t)∈[0,1]2 process. Assume that (Y ) is observed on the regular grid (s,t) (s,t)∈[0,1]2 G := (i/n,j/n), 0 i,j n , n { ≤ ≤ } let [ns] [nt] Vn := ∆ Y 2, (s,t) [0,1]2, n 1, (1.4) (s,t) | i,j | ∈ ≥ i=1 j=1 XX and C := σ2 W dudv, (s,t) [0,1]2. (1.5) (s,t) (u,v) ∈ Z[0,s]×[0,t] (cid:0) (cid:1) Using (1.1) we show in Lemma 4.4 that, [n·] [n•] n ∆ Y 2 σ2 W dudv (1.6) i,j (u,v)  | | −  i=1 j=1 Z[0,·]×[0,•] XX (cid:0) (cid:1)   law(S) √2 σ2 W dB , (u,v) (u,v) n−→→∞ Z[0,·]×[0,•] (cid:0) (cid:1) which is then used to prove that the consistent estimator Vn of C is asymptotically normal (cf. Proposition 4.2). Similar results have been recently established in the one-parameter setting [1, 12, 14, 15], let us mention some of them. Consider a one-parameter semimartingale (Z ) t t∈[0,1] observed at regular times i/n, 0 i n with { ≤ ≤ } t t Z = z + σ(B ) dB + b(B ) ds, t [0,1] (1.7) t 0 s s s ∈ Z0 Z0 where (B ) is a standard Brownian motion and σ : R R and b : R R are t t∈[0,1] → → sufficiently regular deterministic functions. 3 Gradinaru and Nourdin have shown in [12] that, [n·] · · √n ∆ Z 2 σ2(B ) ds law √2 σ2(s,β(1)) dβ(2), (1.8)  | i | − s  n−→→∞ s s i=1 Z0 Z0 X   where β(1) and β(2) are two independent Brownian motions. Note that the convergence obtained in (1.8) hold in the Skorohod space. Property (1.8) has been used by Gradinaru and Nourdin in [12] to construct of a goodness-of-fit test for the integrated volatility (their results are even more general since they can be applied to diffusion processes of the form (1.7) where the diffusion and the drift terms depend on the observed process Z and not only on B). In [14, 15], Jacod proved functional limit theorems similar to (1.8) in a larger setting than in (1.7) since he considered quite general functions of the increments and the process (Z ) was supposed to belong to the class of Itô semimartingales which t t∈[0,1] contains some non-continuous processes and Lévy processes. Werefer to[1]for similar results established by A¨ıt Sahalia and Jacod. We also mention that Nourdin in [20] and Nourdin, Nualart and Tudor in [21] have studied weighted power variations of a one-parameter fractional Brownian motion. Furthermore Nourdin and Peccati in [22] have investigated the asymptotic behavior of weighted p-power variations for the iterated Brownian motion. We proceed as follows. First we recall in Section 2 some elements of stochastic analysis of two-parameter processes. Actually we present some definitions concerning stochastic calculus of two-parameter processes taken from [13] and the definition of the two-parameter Skorohod space initially introduced in [23] and in [31]. Secondly, in Section 3 we establish the central limit theorem (Theorem 3.1) for the weighted quadratic variations process of the two-parameter Brownian motion briefly presented in(1.1). Asanapplicationweprove inSection4thattheconsistent estimatorVn (1.4) of the quadratic variation C (1.5) is asymptotically normal (Proposition 4.2). Finally we present in an appendix (Section 5) some background on set-indexed processes, extension of probability bases and on the Malliavin calculus for the two-parameter 4 Brownian motion which are used in Sections 3 and 4. 2 Stochastic analysis of two-parameter processes In this section we recall some definitions of two-parameter stochastic analysis which will be used in Sections 3 and 4 and we present the two-parameter Skorohod space introduced in [23] and [31]. Some elements of two-parameter stochastic calculus Let (Ω, ,( ) ,P) be a filtered probability space. z z∈[0,1]2 F F We denote the partial order relation on [0,1]2 defined by, (cid:22) z′ z (s′ s and t′ t), z′ = (s′,t′), z = (s,t). (cid:22) ⇔ ≤ ≤ We also define the strong past information filtration on (Ω, ,P). F Definition 2.1. Let z = (s,t) in [0,1]2. F(∗s,t) := F(s′,t′). s′≤s or t′≤t _ Until the end of this paper we assume that the following commutation condition hold. This property is a conditional independence property (CI in short) and corresponds to the condition (F4) of [11]. Assumption (CI): The filtration ( ) is supposed to satisfy the (CI) condition i.e. for all z = (s,t) z z∈[0,1]2 F and z′ = (s′,t′) in [0,1]2 IE IE[ z] [0,z]∩[0,z′] = IE (s∧s′,t∧t′) . ·|F |F ·|F (cid:2) (cid:3) (cid:2) (cid:3) Definition 2.2. An ( ) -adapted process (Y ) is said to be z z∈[0,1]2 z z∈[0,1]2 F i) a martingale if for every z and z′ in [0,1]2 such that z z′ (cid:22) IE[Yz′ z] = Yz, |F 5 ii) a strong martingale if for all z and z′ in [0,1]2 such that z z′ (cid:22) IE[Y[z,z′]|Fz∗] = 0, where Y[z,z′] denotes the increments of Y on the interval [z,z′]. Asanexample, wementionthetwo-parameterBrownianmotion(W ) isastrong z z∈[0,1]2 martingale with respect to its natural filtration and a centered Gaussian process with covariance function, IE[W(s,t)W(s′,t′)] = (s s′)(t t′), (s,t),(s′,t′) [0,1]2. ∧ ∧ ∈ Skorohod space ([0,1]2) D In the one-parameter setting, Skorohod introduced in [30] four topologies known as J , J , M and M . The topology M is the weakest of theses topologies in the 1 2 1 2 2 sense that convergence of a sequence (x ) of functions on [0,1] to x for J , J or n n 1 2 M consists in the convergence of (x ) to x in M plus some additional conditions. 1 n n 2 The M topology has been extended to the general setting of set-indexed functions by 2 Bass and Pyke in [9] whereas the J topology has been extended to multiparameter 1 functions by Neuhaus and Straf respectively in [23] and [31]. The two-parameter Skorohod space (relative to J ) introduced in [23] and [31] is denoted by ([0,1]2) 1 D and give an equivalent to two-parameter functions of the notion of càdlàg functions on [0,1]. The set ([0,1]2) can be equipped with a metric d which makes it a Polish D space and we denote by the Borel σ-algebra on ( ([0,1]2),d). Note that as in the 2 L D one-parameter setting the J topology is stronger than the M topology. Furthermore 1 2 compactsets(relativetoJ )on( ([0,1]2),d, )canbedescribedthankstoamodulus 1 2 D L of continuity w which enables us to use techniques described in [10] for one-parameter functions. We conclude this section by giving the definition of w. Let f : [0,1]2 R → be an element of ([0,1]2) and δ > 0 we define w(f,δ) as, D w(f,δ) := sup f(s,t) f(s′,t′) , (2.1) k(s,t)−(s′,t′)k<δ| − | where (s,t) (s′,t′) := max s s′ ; t t′ for (s,t),(s′,t′) [0,1]2. k − k {| − | | − |} ∈ 6 3 Central limit theorem In this section we state and prove the functional limit theorem (Theorem 3.1) which will allow us to show in Section 4 that the consistent estimator Vn (1.4) of the quadratic variation C (1.5) is asymptotically normal (Proposition 4.2). Let f : R R be a bounded and measurable deterministic function. Let a two- → parameter Brownian motion W = (W ) defined on a probability basis (s,t) (s,t)∈[0,1]2 := Ω, ,( ) ,P . Let also (s,t) (s,t)∈[0,1]2 B F F (cid:0) (cid:1) 1 ξ := n f W ∆ W 2 , 1 i,j n, n 1. i,j (i−n1,j−n1) | i,j | − n2 ≤ ≤ ≥ (cid:18) (cid:19) (cid:16) (cid:17) The re-normalized weighted quadratic variations process Xn = (Xn ) is (s,t) (s,t)∈[0,1]2 defined as, [ns] [nt] Xn := ξ , (s,t) [0,1]2. (3.1) (s,t) i,j ∈ i=1 j=1 XX Stable convergence in law has been introduced by Rényi in [28] and in [29]. It requires someparticularcare,sinceherethelimitingprocessX isnotdefinedontheprobability basis = (Ω, ,( ∗) ,P) on which the Xn, n 1 are defined but an extension B F Fz z∈[0,1]2 ≥ ˜:= (Ω˜, ˜,( ˜ ) ,P˜) of . z z∈[0,1]2 B F F B Theorem 3.1. Assume that the deterministic function f : R R considered above → is bounded. Then (Xn) defined by (3.1) converges -stably in law in the Skorohod n≥1 F space ( ([0,1]2),d, ) to a non-Gaussian continuous process X presented below in 2 D L the proof by (3.2) defined on an extension of the probability basis . B Proof. Let us first describe the extension of on which the limiting process X is B defined. We denote by ′ := (Ω′, ′,( ′) ,P′) the two-parameter Wiener space, that is B F Fz z∈[0,1]2 Ω′ := 0([0,1]2) is the space of real-valued continuous functions on [0,1]2 vanishing C on the set (s,t) [0,1]2, s = 0 or t = 0 . Then P′ is the unique measure on (Ω′, ′) { ∈ } F under which the canonical process (B ) on Ω′ defined by, z z∈[0,1]2 B (ω′) := ω′(z), ω′ Ω′, z [0,1]2, z ∈ ∈ 7 is a standard two-parameter Brownian motion. Let the extension ˜:= (Ω˜, ˜,( ˜∗) ,P˜) defined as, B F Fz z∈[0,1]2 Ω˜ := Ω Ω′, ˜ := × ′,  F( ˜ ) F ⊗F:= ( ∗ ′) ,  P˜F(dzωz,∈d[0y,1)]2:= P(∩dρω>)zPF′(ρd⊗y).Fρ z∈[0,1]2  We will denote by IE (respectively IE˜) the expectation under P (respectively under P˜). On ˜ we define (X ) as, z z∈[0,1]2 B X (ω,ω′) := √2 f(W (ω))dB (ω′), z [0,1]2. (3.2) z ρ ρ ∈ (cid:18)Z[0,z] (cid:19) The process X is a -progressive conditional Gaussian martingale with independent F increments on ˜, which means that X is an ( ˜ ) - adapted process such that for z z∈[0,1]2 B F P almost ω in Ω, X(ω, ) is a Gaussian process on ′ with covariance function · B IEP′ X(s1,t1)(ω,·)X(s2,t2)(ω,·) = 2 (cid:2) f2(W(cid:3) )(ω)dρ, (s ,t ),(s ,t ) [0,1]2. ρ 1 1 2 2 ∈ Z[s1∧s2,s1∨s2]×[t1∧t2,t1∨t2] Note that ˜ is clearly a very good extension of in the sense of Definition 5.4. B B Since ( ([0,1]2),d, ) is a Polish space, by [17, Proposition VIII.5.33], -stable con- 2 D L F vergence in law holds if for every random variable Z on (Ω, ,P) the couple (Z,Xn) n F converges in law. Adapting an argument presented in the proof of [17, Theorem VIII.5.7 b)], the convergence in law of a such couple (Z,Xn) will be obtained as n follows. First we give a tightness property for the sequence (Xn) (relative to the Sko- n rohod space ( ([0,1]2),d, )) and then we make an “identification of the limit”via 2 D L -stable finite-dimensional convergence in law to X. Recall that the latter prop- F erty means that for every integer m 0, for every continuous and bounded function ≥ ψ : Rm+1 R and every elements z ,...,z in a dense subset of [0,1]2, 0 m → IE[Zψ(Xn(z ),Xn(z ),...,Xn(z ))] IE˜ [Zψ(X(z ),X(z ),...,X(z ))]. (3.3) 0 1 m 0 1 m n−→→∞ The proof is decomposed in two steps. In Step 1) we show that (Xn) is tight in n ( ([0,1]2),d, ) and in Step 2) we prove the -stable finite-dimensional convergence 2 D L F 8 in law to X. Step 1) We show the sequence (Xn) is tight in the Skorohod space ( ([0,1]2),d, ). n 2 D L A complete description of ( ([0,1]2),d, ) can be found in [23]. In particular it is 2 D L shown in [23] that the set of conditions (3.4) and (3.5) is necessary and sufficient for the sequence (Xn) to be tight in ( ([0,1]2),d, ), n 2 D L (Xn) converges in distribution, (3.4) 0 n limlimsupP[w(Xn,δ) ε] = 0, ε > 0, (3.5) δ→0 n→∞ ≥ where w is defined in (2.1). Property (3.4) is clear since for every n 1 Xn = X = ≥ 0 0 0, P-a.s.. We will show (3.5) using a method from [10, p. 89]. Let ε > 0, δ > 0 and n 1. Let m := n and v := n . We consider on [0,1]2 the ≥ δ m rectangles Ri,j := min−1, mni × mjn−1, mnj(cid:2) ,(cid:3)(i,j) ∈ {1(cid:2),··(cid:3)· ,v}2 where mi := im, 1 < i < v and mv = n.(cid:2)With thi(cid:3)s no(cid:2)tation th(cid:3)e length of the shortest side of the rectangles R is greater than δ and v 2/δ. We can adapt the proof of [10, Theorem 7.4] to i,j ≤ our case and we have, v v P[w(Xn,δ) 3ε] P sup Xn Xn ε . (3.6) ≥ ≤ z − (mi−1,mj−1) ≥ i=1 j=1 "z∈Ri,j(cid:12) n n (cid:12) # XX (cid:12) (cid:12) (cid:12) (cid:12) Let us give some notations. For (k,j) 1,...(cid:12),n 2 let (cid:12) ∈ { } k l S := f W ∆ W 2 1/n2 , k,l (i−n1,j−n1) | i,j | − Xi=1 Xj=1 (cid:16) (cid:17)(cid:0) (cid:1) that is X(nk,l) = nSk,l. For z in Ri,j we write Sˆki,,jl := Sk,l − Smin−1,mjn−1. Using these notations we can write (3.6) as, v v ε P[w(Xn,δ) 3ε] P sup Sî,j ≥ ≤ k,l ≥ n Xi=1 Xj=1 "mi−1≤k≤mi,mj−1≤l≤mj(cid:12) (cid:12) # (cid:12) (cid:12) (cid:12) (cid:12) We will now use [10, Section 10] which provides maximal inequalities for partial sums of non-independent and non-stationary random variables. For i,j fixed as above, we 9 re-index the random variables appearing in Sî,j to obtain, k,l η(i,j,k,l) Sî,j = τ , k,l p p=1 X with τ equal to some ξ divided by n and η(i,j,k,l) is an integer. p ·,· Let two integers α β. Since f is supposed to be bounded by a non-random function, ≤ let R := sup f(x) non-random. Let K,K˜,K˜˜ denote non-random constants. x∈R| | 4 β β 1 P τ λ IE τ (3.7) "(cid:12) p(cid:12) ≥ # ≤ λ4 (cid:12) p(cid:12)  (cid:12)p=Xα+1 (cid:12) (cid:12)p=Xα+1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) β(cid:12) (cid:12)  1 (cid:12) (cid:12) = (cid:12) IE[ τ 4(cid:12)] p λ4 | | p=α+1 X KR4 1 (β α)2ρ, < ρ < 1. (3.8) ≤ λ4n8 − 2 Using [10, Theorem 10.2] and (3.7) we obtain KR4m4ρ P max Sî,j λ . (3.9) i≤k≤m,j≤l≤m k,l ≥ ≤ n8λ4 (cid:20) (cid:21) Now injecting inequality (3.9) in (3.6) we have, v2K˜R4m4ρ P[w(Xn,δ) 3ε] ≥ ≤ n4ε4 K˜˜m4ρ , since v 2/δ, ≤ ε4n4δ2 ≤ K˜˜m4ρ n4(ρ−1)δ4ρ−2, since m = [nδ], ≤ ε4 which leads to (3.5). Step 2) Here we choose to consider processes Xn and X as set-indexed processes and we use all the notations and definitions of Subsection 5.1. Consequently the -stable finite- F dimensional convergence in law property (3.3) can be rewritten as follows: for every 10