Error due to non-synchronicity

+_Op(1) =_Op(1)

converges to zero. The same holds true for every boundedFt-martingale orthogonal to M. Applying Theorem 1.6, we deduce that Proposition 3.3.2 holds true.

Proposition 3.3.1 is a direct consequence of Proposition 3.3.2 since for t = T the marginal distribution is simply a mixed normal distribution independent of F. The stable convergence assures that the convergence also holds under the original probability measure and non-zero drift terms with the same asymptotic law.

3.3.2 Error due to non-synchronicity

Proposition 3.3.4. Let Assumptions 1, 2(a) and (3.8b)-(3.8c) from Assumption 3.1 be satisfied. The error A^N_T due to the lack of synchronicity converges stably in law to a centred mixed Gaussian distribution:

s N

TA^N_T ^st N(0, vAT) , (3.14)

with asymptotic variance vAT =

Z T 0

F⁰(t)σ_t^Xσ_t^Y2dt+ Z T

0

2H⁰(t)ρtσ_t^Xσ_t^Y2dt . (3.15) Proof. First, we write the ith increments occurring as factors in the addends of the estimator (3.2) as the sum of the next-tick interpolation atT_i, the increments ∆X_Ti = XTi−XTi−1 and ∆YTi =YTi−YTi−1, respectively, and the previous-tick interpolation at Ti−1 and multiply out the addends.

hX, Y\i_T=

N

X

i=1

X_gi−X_Ti+X_Ti−X_Ti−1+X_Ti−1−X_li Y_γi−Y_Ti+Y_Ti−Y_Ti−1+Y_Ti−1−Y_λi

=D_T^N+

N

X

i=1

(Xgi−X_Ti)∆YTi+ (Yγi−Y_Ti)∆XTi+ (XTi−1−X_li)∆YTi

+(YTi−1−Yλi)∆XTi+ (Xgi−XTi)(YTi−1−Yλi) + (Yγi−YTi)(XTi−1 −Xli) The indicator functions in (3.6) have been dropped since the corresponding addends are zero if the indicator functions were zero. Since at least one of the next-tick interpolation errors is zero and as well one of the previous-tick interpolation errors, too, two addends, namely the products of next-tick interpolation errors and the product of previous-tick

3.3. STABLE LIMIT THEOREM CHAPTER 3.

interpolation errors, equal zero. Thus, the error due to asynchronicity can be written as the sum of the remaining six terms (where at least another three equal zero in each addend). We conclude, that the errorA^N_T can be expressed in the following way:

A^N_T =

N−1

X

i=1

((X_gi−X_Ti)(Y_Ti−Y_λi) + (Y_γi−Y_Ti)(X_Ti−X_li)

(X_Ti+1−X_Ti)(Y_Ti−Y_λi+1) + (X_Ti −X_li+1)(Y_Ti+1−Y_Ti)+_Op(1). In this equality an index shift has been applied to the partial sum of previous-tick interpolated errors multiplied with ∆X_Ti and ∆Y_Ti, respectively, leading to the structure that in the ith addend the factors contain next- and previous-tick interpolated errors to the timeTi. The asymptotically negligible term emerges from the first and the last addend of the non-shifted original sum.

In the last illustration ofA^N_T consecutive addends of the sum are uncorrelated in contrast to the non-shifted sum. The reason is that, if without loss of generalityγi =Ti holds, (X_gi−X_Ti)∆Y_Ti and (X_Ti−X_li+1)∆Y_Ti+1 have in general a non-zero correlation whereas (Xgi−XTi)∆YTi and (XTi−1 −Xli)∆YTi are uncorrelated. Furthermore, the fact that γi = Ti ⇒ λi+1 = Ti assures that the addends in the last illustration of A^N_T are uncorrelated.

As in the foregoing proof of Proposition 3.3.1, it is sufficient to prove the stable convergence result for the zero-drift case. We denote, as before, the corresponding transformed processesL_t=^R₀^tσ_s^XdW_s^X andM_t=^R₀^tσ^Y_sdW_s^Y.

Consider the sum A^N_t = ^X

T_i+1^(N)≤t

∆A^N_i :=

s N

T X

T_i+1^(N)≤t

((L_gi−L_Ti)(M_Ti−M_λi) + (M_γi−M_Ti)(L_Ti−L_li) +(L_Ti−L_li+1)(M_Ti+1−M_Ti) + (M_Ti−M_λi+1)(L_Ti+1−L_Ti) (3.16) for fixed 0≤t≤T.

Proposition 3.3.5. Assume the same conditions as in Proposition 3.3.4. For fixed 0 ≤ t ≤ T the transformed error due to non-synchronicity A^N_t is the endpoint of a discrete, centred, square-integrable martingale with respect to the filtration Fi,N :=F_T^(N)

i+1

. The process A^N_t converges as N → ∞ stably in law:

A^Nt

st At= Z t

0

√v_AsdW_s^⊥ (3.17)

where W^⊥ is a Brownian motion independent of F and

v_As =F⁰(s)σ_s^Xσ_s^Y2+ 2H⁰(s)ρsσ_s^Xσ^Y_s² . (3.18) Proof. The expectation of the absolute value of the sum is bounded for allt∈[0, T] and

CHAPTER 3. 3.3. STABLE LIMIT THEOREM

∆A^N_i , i= 0, . . . , N are Fi,N =F_T^(N)

i+1

-measurable. Since

E

h∆A^N_i |Fi−1,N

i=E

∆A^N_i |F_T^(N)

i

=E[(L_gi−L_Ti)(M_Ti−M_λi) + (M_γi−M_Ti)(L_Ti−L_li) +(L_Ti−L_li+1)∆M_Ti+1+ (M_Ti−M_λi+1)∆L_Ti+1|F_T^(N)

i

=E[L_gi−L_Ti] (M_Ti−M_λi) +E[M_γi −M_Ti] (L_Ti−L_li)

+ (L_Ti−L_li+1)E∆M_Ti+1+ (M_Ti−M_λi+1)E∆L_Ti+1= 0 for the conditional expectation of the increments holds, A^Nt is the endpoint of a Fi,N -martingale.

The stable weak convergence to a limiting Brownian motion is proven with Corollary 1.3.5 to Jacod’s Theorem 1.6.

First, we verify the conditional Lindeberg condition (C–LB) that is implied by the stronger conditional Lyapunov condition (C–LY). Therefore, we proof the following lemma:

Lemma 3.3.6. The sum of the conditional fourth moments of the martingale increments A^N_i converges to zero in probability:

E





 X

T_i+1^(N)≤t

∆A^N_i ⁴Fi−1,N





=_Op(1) .

Proof. Throughout the proofC denotes a generic constant that does not depend on N. We consider the addends of the fourth conditional moments consecutively. The sum of conditional fourth moments incorporates partial sums of the following types:

• fourth-order moments:

N² T²

X

T_i+1^(N)≤t

E

h(L_gi−L_Ti)⁴ⁱ(M_Ti−M_λi)⁴ ,

• second-order moments:

N² T²

X

T_i+1^(N)≤t

E h

(Lgi−LTi)²(∆MTi+1)²ⁱ(LTi−Lli+1)²(MTi−Mλi)² ,

• third- and first-order moments:

N² T²

X

T_i+1^(N)≤t

4(MTi−M_λi)³(LTi −L_li+1)³E∆MTi+1(Lgi −LTi) .

3.3. STABLE LIMIT THEOREM CHAPTER 3.

For the partial sum of the first type including fourth-order moments an application of the Burkholder-Davis-Gundy inequalities 1.1.4 yields

N²

The last inequality can be deduced by the result that the convergence (N/(3T))^P_i (∆M_Ti)⁴ →^R₀^t(σ^Y_s)⁴dsholds almost surely asN → ∞for the so-called realized quarticity (Barndorff-Nielsen and Shephard [2002]) and that (gi−Ti) ≤δN. Without the result about the convergence of the realized quarticity, the asymptotic order in probability can be derived by the convergence to zero of the expectation of the above sum and calculating the second moment that is bounded from above by a constant timesN⁴δ_N⁷.

For the partial sums incorporating second-order moments we obtain an upper bound by application of the Cauchy-Schwarz inequality and the Burkholder-Davis-Gundy inequali-ties:

The stochastic order follows, since the term has the expectation

C N²

CHAPTER 3. 3.3. STABLE LIMIT THEOREM

where again the Cauchy-Schwarz and BDG-inequalities have been applied. The variance is bounded from above by a constant timesN⁴δ_N⁷ what can be shown by a similar calculation where thanks to the fact thatTi=γi⇒λi+1 =Ti the addends are uncorrelated and the variance of the sum equals the sum of variances.

We treat the third type of addends occurring in the sum of conditional fourth moments in the same way. Itô isometry yields

N²

This term has expectation N²

and an analogous calculation as before yields that the variance is of asymptotic order N⁴δ⁷_N.

Since all addends in the sum of conditional fourth moments are of one of the three above considered types, the sum converges to zero in probability and hence the conditional Lyapunov condition of Lemma 3.3.6 holds true.

Next, we consider the sum of conditional variances of the increments of the discrete martingale.

3.3. STABLE LIMIT THEOREM CHAPTER 3.

CHAPTER 3. 3.3. STABLE LIMIT THEOREM

In the first equality Itô isometry has been used. The proofs of the following three equalities are postponed to the next two lemmas. In the last step we have involved Definition 3.2.3. The Riemann sum converges on the Assumption 3.1 (in particular (3.8b) and (3.8c)) in probability as N → ∞to the expression ^R₀^tv_Asdswith v_As given in

Proposition 3.3.5.

Lemma 3.3.8. On the same assumptions as before, the following equations hold true:

N

3.3. STABLE LIMIT THEOREM CHAPTER 3.

Proof. We restrict ourselves to the proof of the first two equalities, since all other terms can shown to converge to zero in probability in an analogous way. The left-hand side of the first equality has an expectation equal to zero what can be concluded directly by Itô isometry: In order to derive the stochastic order of the term, consider the second moment:

E

CHAPTER 3. 3.3. STABLE LIMIT THEOREM

where the asymptotic order is deduced by Itô isometry and the fourth moment of Brownian increments (or application of the BDG inequalities). Since the error induced by this term in the approximation of the conditional variance before is centred and has a variance converging to zero as N → ∞, the error is asymptotically negligible in the sense that it converges to zero in probability.

In the second equality we consider the error when the expected increment of the quadratic variation ofX over the next-tick interpolated time interval is substituted by the integral itself. We proceed as before for the first approximation. Since

E and the second moment

E

is bounded from above by a constant times N²δ_N³ again, the approximation error is asymptotically negligible. The fact that γi = Ti ⇒ λi+1 = Ti has been used that guarantees that the addends of the sum are uncorrelated.

Lemma 3.3.8 has been applied in the second and third equality in the sum of the conditional variances and the following Lemma 3.3.9 will complete the proof of Lemma 3.3.7.

Lemma 3.3.9. On the same assumptions as before, the following equation holds true N

and analogously the errors in the five other addends converge to zero in probability when replacing the product of increments of quadratic (co-)variations by the values of ρTi, σ_T^X_i, σ^Y_Ti multiplied with the corresponding times increments.

Proof. We prove the equality explicitly given in the lemma. The five remaining terms can be handled with the same strategy. By an application of the mean value theorem,

3.3. STABLE LIMIT THEOREM CHAPTER 3.

elementary algebra and the triangle inequality for the absolute values, we deduce N

what is assured by the conditions of Assumption 1 on the volatility processes.

For the stable convergence in Proposition 3.3.5 it remains to show that the discrete covariations ofA^Nt with the F-generating underlying martingalesL_t and M_tconverge to zero in probability.

Proof. Both relations are proven similarly and we leave out the second one. The left-hand side of the first equation equals

s

CHAPTER 3. 3.3. STABLE LIMIT THEOREM Γ is centred and calculating the variance using Itô isometry yields

N

We are left to verify the last condition of the discrete-time version to Jacod’s Theorem 1.6 in Corollary 1.3.5. It suffices to prove that the discrete covariation of the martingale with every boundedFt-adapted martingale, that is orthogonal toLtandMtconverges to zero in probability. From this result, we are able to conclude the stability of the conver-gence and the asymptotic independence of the limiting Brownian motion is established that is defined on an orthogonal extension of the original underlying probability space.

In the next lemma, we can even prove the stronger result, that the discrete covariation of our considered martingale with every bounded Ft-martingale that is orthogonal to L_t or Mt, converges to zero in probability. Hence, this lemma will complete the proof of Proposition 3.3.5.

Proof. As in the preceding lemma, we only prove the first part of the result. The left-hand

3.3. STABLE LIMIT THEOREM CHAPTER 3.

side of the first equation equals s

This term is centred and the has the variance N Thus, the covariations converge to zero in probability.

The Lemma completes the proof of Proposition 3.3.5.

The mixed normal limit in Proposition 3.3.4 can be obtained as the marginal distribution ofA^N_T settingt=T. Proposition 3.3.4 is implied by the stronger result of Proposition 3.3.5 and hence the stable convergence of the error due to the lack of synchronicity has been proved.

Proposition 3.3.4 for the error of the approximation by the discretization error of the closest synchronous approximation (3.6) and the stable limit theorem for this synchronous discretization error (3.5) given in Proposition 3.3.1 suffice to imply Theorem 3.1. The

CHAPTER 3. 3.3. STABLE LIMIT THEOREM

multivariate stable convergence Theorem 1.2.4 applies to the vector of the two uncorrelated terms. Since the covariations converge to zero the stable convergence to the mixed Gaussian limit with the sum of the two asymptotic variances is concluded.

3.4 The synchronized realized covariance under the influence

Im Dokument Estimating the quadratic covariation from asynchronous noisy high-frequency observations (Seite 73-85)