5. Spot volatility estimation in the semimartingale model 53
5.2. Estimation of the wavelet coefficients
The content of this section is subdivided into two steps. First we modify pre-averaging, in order to make it suitable for our purposes. Later, we define and discuss the estimators of the wavelet coefficients.
First step: Let us begin with a definition.
Definition 18 (Pre-average function). A function λ : [0,2] → R that is piecewise Lipschitz continuous and satisfies λ(t) = −λ(2−t)is called pre-average function. Given a pre-average function λ, let
λ:=
2 Z 1
0
Z s 0
λ(u)du2
ds1/2
and define the (normalized) pre-average function eλ:=λ/λ.
Note that the graph of λ is point symmetric with respect to (1,0).
Example 3. Let us give a few examples of normalized pre-average functions.
(i) eλ(s) = (k+ 1/2)πcos(s(k+ 1/2)π), k = 0,1, . . . (ii) eλ(s) = p
3/2(I[0,1)(s)−I(1,2](s)). This leads us to the generalized quadratic varia-tion that has already been discussed in Secvaria-tion 5.1.
(iii) eλ(s) = 3−1/2kπsin (kπs), k = 1,2. . . . (iv) eλ(s) = 2−1p
(2k+ 3)(4k+ 5) (1−s)(2k+1), k = 0,1. . . . As in Chapter 4 we set
M =bcn1/2c. (5.2.1)
For the observation vectorY, we introduce the modified pre-averaged observations by Yi,M(λ) : = M fori= 2, . . . , M.There are two changes compared to the original pre-averaging procedure defined in (5.1.1). First, a weighted binning is defined and second we directly consider differences over successive, averaged blocks, without defining first local means. Hence, following the observations of Section 5.1 it is plausible that
Yi,M(λ) = −(X(i−1)/M −X(i−2)/M) +Op(M−1/2+ (M/n)1/2) (5.2.3) and the increments of X are of the same order as the noise.
On the other hand, Yi,M(λ) may also be represented (up to a small error) by weighted increments ofY, due to
Yi,M(λ)≈ −M
Second step: Let φ be an L2-function. The estimator for the scalar product hφ, σ2i is given by
where
In particular, given a multiresolution analysis ((Vj)j, φ) as well as the corresponding wavelet ψ, the estimators of the basis coefficients hφj0,k, σ2i and hψj,k, σ2i are given by hφ\j0,k, σ2i and hψ\j,k, σ2i, respectively.
Explanation of (5.2.4): Going back to (5.2.3) and the discussions in Section 5.1, it is clear that Y2i,M = (X(i−1)/M −X(i−2)/M)2+Op(M−1+ (M/n)) = M1 σ2(i−1M ) +Op(M−1+ (M/n)). Now, the noise is of the same order as the signal. However, by the definition of a pre-average function and imposing smoothness on σ, the influence of the bias due to the approximation can be reduced to smaller order, such that we only need to adjust for the bias induced by the pre-averaged noise. Careful calculations reveal that this can be accomplished by subtracting b(λ, Y). Let us mention, that if τ and φ are sufficiently smooth, we might approximate
b(λ, Y)i,M ≈c21 and hence (5.2.4) can be written as
hφ, σ\2i=
up to some small approximation error. This can be compared directly to (4.2.10).
Furthermore, since Y2i,M −b(λ, Y)i,M has mean σ2(i−1M )/M and variance of order n−1 (as shown later) we may think of M(Y2i,M −b(λ, Y)i,M) as observations coming from a nonparametric regression model (2.1.3), with regression function σ2 and almost centered (but dependent) errors. As mentioned in (2.1.4),
1 is then the natural estimator for the scalar product hφ, σ2i.
Since we will deal with wavelet and approximation coefficients simultaneously, let us introduce h`k(·) = 2`/2h(2` · −k) for a given function h (for which we set h = φ and h=ψ later on).
Assumption 4 (Assumption on h). Suppose that the function h:R→ R is compactly supported, bounded, and has piecewise Lipschitz derivative.
Furthermore, for a function class D, we define ED[·] :=E[· I{σ2∈D}], provided {σ2 ∈ D}
is measurable. In particular, D1 ⊂ D2 implies
ED1[U]≤ED2[U], for non-negative random variablesU. (5.2.6) First, we evaluate the (thresholded) moments of hh\`k, σ2i. This result will allow us to obtain rates of convergence in the sense of Definition 11 for estimation of the spot volatility. Before we can do so, the precise conditions on the noise process are given.
Assumption 5 (Refinement on the noise assumption for model (1.1.3)). Let i,n satisfy Assumption 1. Additionally, assume that E[|ηi,n|p
< ∞ for any p > 0 and that the function (x1, x2)7→τ(x1, x2) is continuous and bounded.
The following assumption will allow us to remove the drift in the proofs by a change of measure. It is of interest to note that this assumption is not essential for our proof. In fact, it is imposed in order to reduce the number of terms we need to estimate when we prove moment bounds later. Recall that by Definition 2, the processes σ and b are c`adl`ag and Ft-adapted.
Assumption 6. Suppose that a weak solution of (1.1.1) is unique and well defined.
Moreover, a weak solution to Xet=Rt
0σsdWs is also unique and well defined, the laws of X and Xe are equivalent on F1 and we have, for some ρ >1
E h
exp ρ Z 1
0
bs
σsdWsi
<∞.
In order to state the following result, we must first introduce the empiricalLp[0,1]-norms with respect to the uniform measure on{i/M :i= 1, . . . , M}, defined by
kfkp,M := 1 M
M
X
i=1
|f(Mi )|p1/p
. (5.2.7)
Proposition 3 (Moment bounds). Suppose that Assumptions 5 and 6 hold and let hh\`k, σ2i as in (5.2.4). Assume further that h satisfies Assumption 4 and 2` ≤ M = bcn1/2c. Let s >1/π, then, for any p≥1, C >0,
EBsπ,∞(C)
hh\`k, σ2i − hh`k, σ2i
p
. M−p/2+M−min{s−1/π,1/2}pkh`kkp1,M, uniformly over `, k.
Proof. Let us first introduce some notation. In the following, eλ always denotes the normalized version of a pre-average function (in the sense of Definition 18). We define the functions Λ,Λ :R→R,
and by using Lemma B.4 also
kΛ M · −(i−2)
k2 =M−1/2. Moreover, for C > 0, we define the L∞-ball
L∞(C) := {f : [0,1]→R, kfk∞ ≤C}. (5.2.11) Some properties deduced from Assumption 4 that will be used extensively can be found in Lemma B.1. and in the spirit of (5.2.2)
Xi,M :=Xi,M(λ) := M
are the natural extensions of applying pre-averaging toX and . Bounding I : In a first step we will show that
EBsπ,∞(C) Note that by the continuous embedding (2.4.8) and the identity (2.4.9) it follows
Bsπ,∞(C)⊂ Cmin(s−1/π,1/2)
where the last inequality follows from Lemma B.4. Let |supp(h`k)| denote the support length ofh`k. Therefore, by H¨older inequality and Lemma B.1
EBsπ,∞(C)
and further by triangle inequality Therefore, on the event σ2 ∈ Bπ,∞s (C), Equation (5.2.20) implies by Lemma B.1 (iii)
Recall that by continuous Sobolev embedding (2.4.8), Bπ,∞s ⊂ Bs−1/π∞,∞ . Since B∞,∞s−1/π ⊂ The moment bound on I, i.e. (5.2.14) follows now by applying successively (5.2.19), (5.2.22), (5.2.23) and (5.2.24).
Bounding II : Combining Lemmas B.6, B.8, B.9 and B.10, we obtain EBsπ,∞
|II|p
.kh`kkp1,MMpn−p+kh`kkp2,MM−3p/2n−p+kh`kkpp,MMp+1n−p .M−p/2, where Lemma B.1 is applied for the last inequality.
Bounding III : Lemma B.7 gives EBπ,∞s (C) By combining the estimates on partsI−III, the proof of Proposition 3 is complete.
In order to apply Theorem 2, we need further a result of the type (2.2.3). This is given in the next Proposition.
Proposition 4 (Deviation bounds). Suppose that Assumptions 5 and 6 hold. Let us further suppose that h satisfies Assumption 4, s > 1/π, and M = bcn1/2c. Further assume that
(i) M2−` ≥Mq, for some q >0 and (ii) M−(s−1/π)kh`kk1,M .M−1/2.
Then for C > 0 and p≥1, we have P
h
hh\`k, σ2i − hσ2, h`kiL2
≥κ plogMM1/2
and σ2 ∈ Bπ,∞s (C)i
.M−max(2,p) for a sufficiently large constant κ and
C:= sup
σ2∈Bsπ,∞(C)
kσ2kL∞.
If X is a driftless continuous Itˆo semimartingale, i.e. b= 0, then κ can be chosen as κ >4C+ 4p
2 C kτk∞ckλk2λ −1+ 4kτk2∞c2kλk22 λ −2. (5.2.25) Remark 2. IndeedC <∞,as it follows from the continuous embedding (5.2.16). More-over, in the case of high smoothness, i.e. s−1/π >1/2, Assumption (ii) in Proposition 4 becomes trivial.