Quadratic Variations along Irregular Subdivisions for Gaussian Processes

In this paper we deal with second order quadratic variations along general subdivisions for processes with Gaussian increments. These have almost surely a deterministic limit under conditions on the mesh of the subdivisions. This limit depends on the singularity function of the process and on the structure of the subdivisions too. Then we illustrate the results with the example of the time-space deformed fractional Brownian motion and we present some simulations.


Introduction
In 1940 Paul Lévy (see [17]) proves that if W is the Brownian motion on [0, 1] then: Then in [2] and in [11] this result is extended to a large class of processes with Gaussian increments. In these cases the subdivisions are regular and the mesh is fixed (equal to 1/2 n ). More general subdivisions are used later in [10] and [15]. The subdivisions can be irregular and the optimal condition is that the mesh must be at most o(1/ log n).
In the case of the fractional Brownian motion, the quadratic variation is used to construct some estimators of the Hurst index H. But these estimators are not asymptotically normal when H > 3/4 (see [12]). To solve this problem in [14], [3], [8] and [9] authors introduce generalized quadratic variations. The most common variation is the second order quadratic variation, which will be defined in the next section. But one more time subdivisions are regular.
In this paper we extend the theorem of [9] to a large class of subdivisions which may be irregular. We obtain that the limit of the second order quadratic variation depends on the structure of the sequence of subdivisions, and one more time that the mesh must be at most o(1/ log n).
In the first section we state our notations. In the second one we prove our theorem. The third one is a discussion about the assumptions made on the structure of the subdivisions. In the fourth one we apply the results to the example of the time-space deformed fractional Brownian motion. In the last one we illustrate the examples with some simulations.
Let (π n ) n∈N be a sequence of subdivisions of the interval [0, 1]. One denotes by N n the number of subintervals of [0, 1] generated by π n . We suppose that π n can be written: One sets the upper mesh of π n : and the lower mesh: Note that one has: ∀n ∈ N, p n ≤ 1 N n ≤ m n .
Definition 1. We say that the sequence of subdivisions (π n ) n∈N is regular if we have: ∀n ∈ N, m n = p n = 1 N n .
Or equivalently: ∀n ∈ N, ∀k ∈ {0; . . . ; N n }, t For fractional processes the following second order quadratic variation has been used: which is taken along the regular subdivision with mesh 1/N n . The real number γ is in ]0, 2[ and depends on the regularity of the process, as we will see later.
If f : [0, 1] −→ R is of class C 2 in a neighborhood of the point t then the Taylor formula yields: This motivates the term X k+1 Nn (1). If we want to use irregular subdivisions we must generalize (2). One more time the Taylor formula yields: So one defines the second order increments of X: where (we drop the super-index in t (n) k whenever it is possible): We will see later that E (∆X k ) 2 is asymptotically of the same order as: That is why we define the second order quadratic variation of X along a general subdivision by: where γ ∈]0, +∞[. Note that when the sequence (π n ) n∈N is regular, one has: Let us recall the Landau notations. Let (u n ) n∈N and (v n ) n∈N be two sequences of real numbers such that ∀n, v n = 0. We will say that: if the sequence (u n /v n ) n∈N goes to zero when n → +∞.
To study the almost sure convergence of V πn (X) we will use the following assumption on the sequence (π n ) n∈N . Definition 2. Let (l k ) 1≤k be a sequence of reals in the interval ]0, +∞[. We say that (π n ) n∈N is a sequence of subdivisions with asymptotic ratios (l k ) 1≤k if it satisfies the following assumptions: 2.
The set L = {l 1 ; l 2 ; . . . ; l k ; . . . } will be called the range of the asymptotic ratios of the sequence (π n ) n∈N .
It is clear that if the sequence (π n ) n∈N is regular, then it is a sequence with asymptotic ratios (l k ) 1≤k where: ∀k ≥ 1, l k = 1.
In the sequel we only consider process with Gaussian increments. 1] has Gaussian increments if, for any subdivision In the proof of the next section we use the following notations: And: We remark that: 3 The results We prove the almost sure convergence of V πn (X) to a deterministic limit under some conditions on the covariance function of the process X and on the mesh of the subdivisions π n . For a function g :]0, +∞[×]0, 1[→ R we need the following assumption of continuity: Theorem 4. Let (π n ) n∈N be a sequence of subdivisions with asymptotic ratios (l k ) 1≤k and range of the asymptotic ratios L. Let (X t ) t∈[0,1] be a square integrable process, with Gaussian increments, verifying: 2. the covariance function R(s, t) has the following properties: (a) R is continuous on [0, 1] 2 , (b) the derivative ∂ 4 R ∂s 2 ∂t 2 exists and is continuous on ]0, 1] 2 \{s = t}, and there exists one constant C > 0 and one real γ ∈]0, 2[ such that : (c) We assume that there exists a function g :]0, +∞[×]0, 1[→ R satisfying assumption (10) and such that: where: and such that the following limit exists and is finite: where l n (t) = Nn−1 k=1 l k 1 ]t (n) The lower mesh of the subdivisions π n satisfy: Then one has almost surely: Remarks.
(i) If assumptions (11), (12) is satisfied for γ 0 then they are satisfied for all γ > γ 0 too, but the corresponding function g γ is equal to zero. When γ 0 is chosen as the infimum of real satisfying (11) and (12), g γ 0 can be viewed as a generalization of the singularity function of X (see [2]).
(ii) As we will see later, in most cases one is able to compute explicitly the limit lim n→+∞ 1 0 g(l n (t), t) dt. (iii) We assume that the covariance function has a singularity on the set {s = 0}∪{t = 0} which is the case of the FBM. If the singularity is at another point than 0, we can use a new parameterization in order to have a the singularity at point 0.
Proof of theorem 4. Along the proof K denotes a generic positive constant, whose value does not matter.
First we assume that the theorem is true in the case of X centered.
If X is not centered, one sets X t = X t − M t . Using Baxter's arguments (see [2]) and (16) However the theorem is true in the case of X not centered too. So one can assume that the process X is centered without loss of generality.
First we study the asymptotic properties of EV πn (X). One remarks that: and: EV πn (X) = 2 Moreover assumptions (12) and (5) yield: Therefore: Assumption (10) implies: This with (19) and (6) yield: Moreover: So (13) and (20) imply: Hence the proof is reduced to verify: From Borel-Cantelli lemma, it is enough to find one sequence of positive real ( n ) n∈N satisfying: For that, we will proceed like in [15] and use Hanson and Wright's bound (see [13]).
One remarks that V πn (X) is the square of the Euclidean norm of one (N n −1)-dimensional Gaussian vector which components are: So by the classical Cochran theorem, one can find a n nonnegative real numbers (λ 1,n , . . . , λ an,n ) and one a n -dimensional Gaussian vector Y n , such that its components are independent Gaussian variables N (0, 1) and: Then the Hanson and Wright's inequality yields: where A 1 , A 2 are nonnegative constants, λ * n is defined by λ * n = sup 1≤j≤an λ j,n .
Since an j=1 λ j,n = EV πn (X), and EV πn (X) is a convergent sequence, the sums an j=1 λ j,n are bounded.
Moreover, one has: Therefore, the inequality (24) becomes : We use the following elementary result of linear algebra. Let S = (s ij ) 1≤i,j≤n be a n × n symmetric matrix. We note λ max its higher eigenvalue. Then one has: This with inequality (9) yield: So one must study the asymptotic properties of the d jk . For that we proceed in three steps, according to the value of k − j.
One has (17): If f :]0, 1] −→ R is a two times differentiable function one has for 0 < h 1 < t and 0 < h 2 ≤ 1 − t: Hence if j = 1 and k = 1: Here one uses assumption (11) which yields: And on the integration set one has: Hence one gets: If j = 1 or k = 1 one has by continuity of R: where for 0 < < 1: and: Using the same techniques as above one gets for near 0 + : One makes tends to 0 + in this inequality. It yields that (29) is still true when j = 1 or k = 1. Therefore one has: Thanks to (6) one gets the following estimate: So the preceding three steps and (26) yield: Hence (25) and the preceding estimates yield for 0 < ≤ 1: Now one sets: Then (31) becomes: Now we give briefly two cases where the limit (13) exists and can be easily computed.
(C2) If the sequence of functions (l n (t)) n∈N converges uniformly to l(t) on the interval [0, 1], then: thanks to the Riemann theorem. So assumption (13) is fulfilled.
In the next section we give a method to construct irregular subdivisions of which range of asymptotic ratios can be chosen infinite.

Construction of irregular sequences of subdivisions
We give a necessary and sufficient condition to find a sequence of subdivisions (π n ) n∈N which has a given sequence of asymptotic ratios (l k ) k≥1 ⊂]0, +∞[ . Proposition 5. Let (l k ) k≥1 be a sequence of real numbers in ]0, +∞[ and (N n ) n∈N be an increasing sequence of positive integer numbers such that: Then (l k ) k≥1 is the sequence of asymptotic ratios of a sequence of subdivisions π n = t (n) . This sequence is unique under the condition: The converse is true: if (π n ) n∈N is a sequence of subdivisions, (N n ) n∈N is the associated sequence of number of subintervals and (l k ) k≥1 is the sequence of the asymptotic ratios of (π n ) n∈N , then the sequences (l k ) k≥1 and (N n ) n∈N satisfy the conditions (32) and (33).
Proof of proposition 5. We begin with the proof of the converse. We take (π n ) n∈N , (l k ) k≥1 and (N n ) n∈N as in the second part of the proposition.
One has: m n n→+∞ = O(p n ). Therefore there exists a constant K > 0 such that: Moreover, for all L ∈ N * : And: which shows that condition (32) is satisfied with D = K.
Now we prove the first part of the proposition. We take two sequences (l k ) k≥1 and (N n ) n∈N satisfying conditions (32) and (33).
For n ∈ N, we define the subdivision π n = t (n) 0 One gets a sequence of subdivisions with asymptotic ratios (l k ) k≥1 . Indeed: so the sequence satisfies (5).
We note k n and j n the integer such that: We envisage three cases. First case: k n ≥ j n . Then: One sets ∀L ∈ N * , P L = L i=1 l i and P 0 = 1. One has: m n p n = P jn P kn , and because of (32): Therefore: m n p n ≤ D 2 .
Second case: k n < j n . We use the same arguments. One has: Third case: k n = j n . Then: Hence in all cases: m n p n ≤ D 2 , so the sequence (π n ) n∈N satisfies (4) too.
Now we must show the unicity of (π n ) n∈N under the assumptions of the proposition. Necessarily one has Nn−1 k=0 ∆t (n) k = 1 which yields: With (34) one gets: And with (35) and (34), one gets the recursive procedure used to define (π n ) n∈N . Since this procedure defines a unique sequence, (π n ) n∈N is the unique solution of our problem.
2 Example. One sets N n = n and l k = 1 + 1 k 2 . Note that the sequence (P L ) is increasing and lower bounded by 1. So condition (32) is satisfied with D = +∞ j=1 1 + 1 j 2 . Moreover: so condition (33) is satisfied too.
In next section we illustrate the conclusion of theorem 4 with one example: the time-space deformed fractional Brownian motion. 5 Example: time-space deformed fractional Brownian motion

The fractional Brownian motion
The fractional Brownian motion (FBM) with Hurst's index H ∈]0, 1[ is the centered Gaussian process B H t t∈R , vanishing at the origin, with covariance function given by: For H = 1/2, B H t t∈R is the Brownian motion.
The process B H is self-similar with index H: and its increments are stationary.
Moreover the Hölder critical exponent of its sample paths is equal to H (see [1] th.8.3.2 and th.2.2.2 ), in the following sense: Definition 6. Let β ∈]0, 1[. A process (X t ) t∈R is said to have Hölder critical exponent β whenever it satisfies the two following properties: • for any β * ∈]0, β[, the sample paths of X satisfy a.s. a uniform Hölder condition of order β * on any compact set, i.e; for any compact set K of R, there exists a positive r.v. A such that a.s.: ∀s, t ∈ K, |X s − X t | ≤ A|s − t| β * ; • for any β * ∈]β, 1[, a.s. the sample paths of X fails to satisfy any uniform Hölder condition of order β * .
In [9] authors proved that almost surely: We are able now to sharpen this result. Let (π n ) n∈N be a sequence of subdivisions with asymptotic ratios (l k ) k≥1 .
On the one hand we compute the derivative ∂ 4 R ∂s 2 ∂t 2 . One has: ∀s, t ∈]0, 1] 2 \{s = t}, On the other hand we must compute the singularity function of B H . We use the notations of the theorem 4. One has: where λ = h 2 /h 1 .
To check (12) one sets g(l, t) = 1+l 2H−1 −(1+l) 2H−1 l H− 1 2 and one remarks: The function φ is continuous on ]0, +∞[, so it is uniformly continuous on compact subsets of ]0, +∞[. This property proves the limit (12). Moreover it is obvious that g satisfies assumption (10). Therefore theorem 4 can be applied to B H . By example, for a regular sequence of subdivisions one has a.s.: with N n such that 1 Nn = o 1 log n .
We give as well an example of non regular subdivision. We set: N n = 2n, and: This is a sequence of subdivisions with asymptotic ratios (l k ) 1≤k where: ∀k ≥ 1, l 2k = 2, Its range of asymptotic ratios is L = {1/2; 2} and g(l, t) is invariant in l on this set (which is condition (C1)), so one is able to compute the limit (13). Therefore theorem 4 yields: Note that when H = 1/2, the value of this limit is different from (38). Indeed its depends not only on the singularity function of the process but on the asymptotic ratios of the subdivisions too.
For H = 1/2 the singularity function g does not more depend on l. Indeed: It is classical that the standard quadratic variation of the Brownian motion does not depend on the sequence of subdivisions; we obtain the same result for the second order quadratic variation.

Time-space deformed fractional Brownian motion
Let σ and ω be two functions from R to R. We define the (σ, ω)-time-space deformed fractional Brownian motion (Z H t ) t∈R by the formula: This is a centered Gaussian process with covariance function: We want to study this process in the following way: does it have the same properties as the FBM? It is clear that it has no more stationary increments. In the sequel, we study its properties of self-similarity and Hölder regularity, and we apply theorem 4 to this process.
The following lemma will be useful: Lemma 7. Let I, J, K be three subintervals of R and α, β two real numbers in [0,1]. We will say that a function f : I −→ J is α-Hölderian on I if: 1. Let f : I −→ J and g : I −→ K be two functions. Assume that f is α-Hölderian and bounded on I, and g is β-Hölderian and bounded on I. Then the function f g : I −→ R is min(α, β)-Hölderian.
2. Let f : I −→ J and g : J −→ K be two functions. Assume that f is α-Hölderian an g is β-Hölderian. Then the function f • g : I −→ K is αβ-Hölderian.
¿From this lemma we can deduce immediately the regularity of the sample paths of Z H .
Proposition 8. We assume that σ is Σ-Hölderian and ω is Ω-Hölderian on a compact interval I of R with Σ, Ω ∈ [0, 1]. Then for all H < H the sample paths of Z H are a.s. min(Σ, ΩH )-Hölderian on I.
Proof of proposition 8. It is a straightforward consequence of lemma 7.

2
Now we show a property of self-similarity of the process Z H . We begin with the definition of this property (see [4]) in the case of a Gaussian field. Definition 9. Let d ∈ N * and β > 0. A process (X(t)) t∈R d is locally asymptotically selfsimilar (l.a.s.s.) of order β at point t 0 ∈ R d if the finite dimensional distributions of the field converge to the finite dimensional distributions of a non trivial Gaussian field when → 0 + . The limit field is called the tangent field at point t 0 .
We give assumptions on the functions σ, ω so that the process Z H is l.a.s.s..
Then the process Z H is l.a.s.s. at point t 0 of order H and the tangent process is: Remark. Under these assumptions proposition 8 yields that for all H < H the sample paths of Z H are a.s. H -Hölderian. Proposition 10 yields that they are not H -Hölderian in the neighborhood of t 0 for all H > H. Therefore the Hölder critical exponent of Z H is equal to H.
Proof of proposition 10. Along the proof K will denote a generic positive constant, whose value does not matter.
Let t 0 ∈ R. We denote by R (resp. ρ) the covariance function of the process Z H (resp. B H ). Since we work with centered Gaussian process it is enough to show that: One sets for h 1 , h 2 > 0: Equality (41) is equivalent to: One takes s, t ∈ R. One has: Therefore: One sets: where: and: First we show that: By assumption σ is Σ-Hölderian on a neighborhood of t 0 and ω is differentiable at t 0 , so it is clear that ω is 1-Hölderian on a neighborhood of t 0 . Moreover the function x → |x| 2H is min(1, 2H)-Hölderian on a neighborhood of t 0 . Therefore lemma 7 allows us to conclude that the function x → |ω(x)| 2H σ(x) is min(Σ, 2H)-Hölderian on a neighborhood of t 0 .
Hence there exists K > 0 such that for enough small: Moreover one has assumed that Σ > H, so Σ + min(Σ, 2H) > 2H. This shows (43). Now we show that: If s = 0 or t = 0 equality (44) is obvious. So one can assume that s = 0 and t = 0. Since ω is differentiable at point t 0 one has: This shows equality (44). Since (43) and (44) are sufficient conditions for (42) the proposition is proved.

2
Now we apply theorem 4 in order to generalize previous results about the FBM. We make additional assumptions on σ and ω.
Proposition 11. Assume that: 1. σ is of class C 3 and ω of class C 2 on [0, 1], Let (π n ) n∈N be a sequence of subdivisions with asymptotic ratios (l k ) k≥1 . Assume too that: Then theorem 4 can be applied to Z H with γ = 2 − 2H and: Proof of proposition 11. Along the proof K denotes a generic positive constant, whose value does not matter.
We denote by R the covariance function of Z H . One has: We must show that the assumptions of theorem 4 are satisfied. For 1 and 2.(a) it is obvious.
Secondly we prove: Likewise Taylor formula and compacity of the closure of L imply that the terms and ω(t+h 2 )−ω(t−λh 2 ) (1+λ)h 2 have for limit ω (t) on E δ . Then use the fact that σ and x → |x| 2H are uniformly continuous on compact sets and ω is of class C 2 . One gets (47).
Therefore assumption 2.(c) is fulfilled. Example: the fractional Ornstein-Uhlenbeck process. We consider the Lamperti transform (see [16]) of the FBM: we take σ(t) = e −λt and ω(t) = αe λt H where λ, α > 0. This new process is stationary. If H = 1/2 it is the Ornstein-Uhlenbeck process with parameters √ 2αλ and λ. So for H ∈]0, 1[ it is called the fractional Ornstein-Uhlenbeck process (see [6]) denoted by O H . All the results given above can be applied.
If one uses proposition 11 one gets that the singularity function of the process O H is equal to g(l, t) = αλ . If one takes a regular sequence of subdivisions such that 1/N n = o(1/ log n) one gets a.s.: Like in the case of the FBM it yields a strongly consistent estimator of the parameter H given by: In the last section we illustrate the result obtained about the FBM with some simulations.

Simulations
We illustrate the results with some simulations in the case of the FBM. To simulate the FBM we use the method of the circulant matrix (see [5] and [7]) and Matlab c . We use two sequences of subdivisions: the first one is regular and the second one is not (we use the irregular sequence defined in section 5.1). If one uses the method of the circulant matrix to simulate the FBM one obtains a discretized path along a regular subdivision. Therefore for simulations we use the fact that the irregular sequence of section 5.1 can be refined in a regular sequence of subdivisions.  On figure 4 we do the same but we replace the regular sequence of subdivisions with the irregular sequence constructed in section 5.1. In each case the mean is respectively equal to 2.5535, 1.9966 and 0.9455, and the standard deviation is equal to 0.0880, 0.0650 and 0.0300.