Independent Poisson observation schemes

In this section, we consider the model in which the sequences of observation times are supposed to be realizations of two homogeneous Poisson processes that are mutually independent and independent of the processes ˜X and ˜Y. Although this model can be criticized for its flaw that sampling schemes of two correlated processes follow two independent processes and time homogeneity, what might seem to be unrealistic in financial applications, independent and homogeneous Poisson sampling times designs constitute the most commonly used model in this research area (cf. Zhang [2006], Hayashi and Yoshida [2005] among others).

Let ˜n⁽ⁿ⁾(t) and ˜m⁽ⁿ⁾(t) be sequences of two independent homogeneous Poisson processes with parametersT n/θ1 and T n/θ2 (n∈N), such that the waiting times between jumps of ˜n⁽ⁿ⁾ and ˜m⁽ⁿ⁾ are exponentially distributed with expectations E

h∆t⁽ⁿ⁾_i ⁱ = θ1/n and E

h∆τ_j⁽ⁿ⁾ⁱ = θ₂/n , i ∈ N, j ∈ N. Thus, ˜n⁽ⁿ⁾(T) and ˜m⁽ⁿ⁾(T) correspond to the sequences giving the numbers of observation times of ˜X and ˜Y in the time span [0, T].

The increments of the sampling times of the closest synchronous approximation (3.4) correspond to the maxima of the exponentially distributed waiting times:

∆T_k⁽ⁿ⁾ ∼F(t) = 1−exp sampling schemes affecting the asymptotics of both, the synchronized realized covariance estimator (3.2) from Chapter 3 and the generalized multiscale estimator (4.2) from

5.2. POISSON SAMPLING CHAPTER 5.

Chapter 4. Namely those quantities of interest are the quadratic (co-)variations of times defined in Definition 3.2.3 and the degree of regularity of non-synchronicity from Definition 4.2.1.

Proposition 5.2.1. In the independent homogeneous Poisson model, it holds true that G^N(t)−→^p 2 1− 2θ₁²θ₂² Proof. We make use of the basic properties of mutually independent homogeneous Poisson processes in this proof. Those are Markovian and the exponential distributions of the increments between arrival times are memoryless. Wald’s identity ensures that E processes and further information we refer to Cox and Isham [1980].

First of all we ascertain that t⁽ⁿ⁾_i 6= τ_j⁽ⁿ⁾∀(i, j) ∈ {1, . . . ,n˜⁽ⁿ⁾(T)} × {1, . . . ,m˜⁽ⁿ⁾(T)}

almost surely. For arbitrarily fixedi, the expected values of next-tick, previous-tick and refresh time instants yield

E

CHAPTER 5. 5.2. POISSON SAMPLING

E h

T_i⁽ⁿ⁾−λ⁽ⁿ⁾_i+1ⁱ= 1 n

θ1θ₂² (θ1+θ2)² ,

E

hT_i+1⁽ⁿ⁾−T_i⁽ⁿ⁾ⁱ= θ1

n + θ2

n − 1 n

θ1θ2

θ₁+θ₂ .

The conditional expectations given that theith refresh time T_i⁽ⁿ⁾ = γ_i⁽ⁿ⁾ is an arrival time of ˜m⁽ⁿ⁾ yield E

hT_i+1⁽ⁿ⁾−T_i⁽ⁿ⁾|T_i⁽ⁿ⁾=γ_i⁽ⁿ⁾ⁱ=E

hT_i+1⁽ⁿ⁾−T_i⁽ⁿ⁾ⁱ and E

h

T_i⁽ⁿ⁾−l_i+1⁽ⁿ⁾|T_i⁽ⁿ⁾=γ_i⁽ⁿ⁾ⁱ=E h

T_i⁽ⁿ⁾−l⁽ⁿ⁾_i+1ⁱ,

since the latter previous-tick interpolation is zero with probability 1 ifT_i⁽ⁿ⁾6=γ⁽ⁿ⁾_i . Only for (T_i⁽ⁿ⁾−λ⁽ⁿ⁾_i ) the conditional expectation differs from the unconditional and can be calculated by further conditioning

E

hT_i⁽ⁿ⁾−λ⁽ⁿ⁾_i |T_i⁽ⁿ⁾=γ⁽ⁿ⁾_i ⁱ= E

hT_i⁽ⁿ⁾−λ⁽ⁿ⁾_i |T_i⁽ⁿ⁾ =γ_i⁽ⁿ⁾, T_i−1⁽ⁿ⁾ =λ⁽ⁿ⁾_i ⁱP

T_i−1⁽ⁿ⁾ =λ⁽ⁿ⁾_i |T_i⁽ⁿ⁾=γ_i⁽ⁿ⁾ +E

hT_i⁽ⁿ⁾−λ⁽ⁿ⁾_i |T_i⁽ⁿ⁾=γ_i⁽ⁿ⁾, T_i−1⁽ⁿ⁾ =l_i⁽ⁿ⁾ⁱP

T_i−1⁽ⁿ⁾ =l⁽ⁿ⁾_i |T_i⁽ⁿ⁾=γ_i⁽ⁿ⁾

=

θ1+θ2− θ1θ2

θ₁+θ₂ θ1

θ₁+θ₂ + 2θ1

θ2

θ₁+θ₂ ,

where the factor 2θ₁ in the second addend is simply the expectation of the waiting time for two jumps of ˜n. Here, we have used some simplifying symmetry aspects, a rigorous proof using the density functions is obtained by calculation of

E

T_i⁽ⁿ⁾−λ⁽ⁿ⁾_i 1_{T⁽ⁿ⁾

i =γ⁽ⁿ⁾_i ,T_i−1⁽ⁿ⁾=λ⁽ⁿ⁾_i }

= Z ∞

0

Z ∞ x

xn θ₁e^−x

n θ1e^−y

n θ2yn

θ₂e^−x

n θ1e^−y

n θ2 dx dy

= 2θ₁θ₂ θ₁+θ₂ .

The conditional expectations onT_i⁽ⁿ⁾=g_i⁽ⁿ⁾ are deduced analogously. Since E

hT_i⁽ⁿ⁾−l_i⁽ⁿ⁾ⁱ=E

hT_i⁽ⁿ⁾−T_i−1⁽ⁿ⁾ⁱ+E

hT_i−1⁽ⁿ⁾−l_i⁽ⁿ⁾ⁱ

and the (conditional) expectations of the products occurring in G^N, F^N, H^N equal the products of (conditional) expectations thanks to the memorylessness of exponential distributions, the latter results suffice to apply the law of large numbers to the empir-ical (co-)variations of times. For the asymptotics of G^N(T), F^N(T) and H^N(T), we conclude for the number of addends ˜N(T)⁽ⁿ⁾, that EN˜(T)⁽ⁿ⁾ = (T /θ)n+_O(n) with θ=θ₁+θ₂−(θ₁θ₂)/(θ₁+θ₂) what follows fromEN˜(T)⁽ⁿ⁾E

h∆T₁^(N)i=T+Op(n⁻¹) and

5.2. POISSON SAMPLING CHAPTER 5.

VarN˜(T)⁽ⁿ⁾=O(n⁻¹) since

Var





N(T˜ )⁽ⁿ⁾

X

k=0

∆T_k⁽ⁿ⁾



=VarN˜(T)⁽ⁿ⁾E

∆T₁⁽ⁿ⁾²

+E

hN˜(T)⁽ⁿ⁾ⁱVar∆T₁⁽ⁿ⁾ .

The exact probability mass functions of the counting processes ˜N(t)⁽ⁿ⁾ associated with the maxima of the waiting times ∆t⁽ⁿ⁾_i ,∆τ_j⁽ⁿ⁾ have a quite complicated form, so that we only give the last two results on the expectation and the variance that are necessary for the proof of the proposition.

From the preceding conclusions, it follows that G^N(t) =

N˜(T)⁽ⁿ⁾ T

X

T_i⁽ⁿ⁾≤t

∆T_i⁽ⁿ⁾²−→^p n² θ²

2θ²₁ n² +2θ²₂

n² −2

θ₁θ₂ (θ1+θ2)

2 1 n²

! t T ,

F^N(t) =

N˜(T)⁽ⁿ⁾ T

X

T_i+1⁽ⁿ⁾≤t

(T_i⁽ⁿ⁾−λ⁽ⁿ⁾_i )(g⁽ⁿ⁾_i −T_i⁽ⁿ⁾) +T_i⁽ⁿ⁾−l_i⁽ⁿ⁾ γ_i⁽ⁿ⁾−T_i⁽ⁿ⁾ + ∆T_i+1⁽ⁿ⁾T_i⁽ⁿ⁾−l_i+1⁽ⁿ⁾+ ∆T_i+1⁽ⁿ⁾T_i⁽ⁿ⁾−λ⁽ⁿ⁾_i+1

−→p t T θ²

θ₁θ₂ (θ1+θ2)

2θ1+ 2θ2−2 θ₁θ₂

(θ1+θ2) + 2θ₁θ₂ (θ1+θ2)

+

θ₁+θ₂− θ₁θ₂ (θ1+θ2)

θ²₁θ₂+θ₁θ²₂ (θ1+θ2)²

! ,

H^N(t) =

N˜(T)⁽ⁿ⁾ T

X

T_i+1⁽ⁿ⁾≤t

T_i⁽ⁿ⁾−l⁽ⁿ⁾_i+1 g⁽ⁿ⁾_i −T_i⁽ⁿ⁾+T_i⁽ⁿ⁾−λ⁽ⁿ⁾_i+1 γ_i⁽ⁿ⁾−T_i⁽ⁿ⁾

−→p t T θ²

θ₁²θ²₂(θ₁+θ₂) (θ₁+θ₂)³ .

Insertingθ we obtain formulae (5.8a)-(5.8c). In the evaluation ofG^N we have also used the second moment of ∆T₁⁽ⁿ⁾ which can be calculated using the above given distribution function.

Considering the degree of regularity of non-synchronicity defined in Definition 4.2.1 we have learned in Sections 4.2 and 4.3 that it is due to observation time aggregations distributed according to case2 (cf. Section 4.2) where two jumps of the same process occur in a time interval in that the other process has no jumps. The probability that this is the case for theith step when applying Algorithm 3.1 equals

P

g_i+1⁽ⁿ⁾ =g_i⁽ⁿ⁾=P

g_i⁽ⁿ⁾ > γ_i⁽ⁿ⁾, g_i⁽ⁿ⁾≥γ_i,+⁽ⁿ⁾

CHAPTER 5. 5.2. POISSON SAMPLING

=P

g⁽ⁿ⁾_i ≥γ_i,+⁽ⁿ⁾g_i⁽ⁿ⁾> γ_i⁽ⁿ⁾P

g_i⁽ⁿ⁾> γ_i⁽ⁿ⁾

= θ₂ θ1+θ2

θ₁ θ1+θ2

= θ₁θ₂ (θ1+θ2)² for every i∈ {1, . . . ,N˜(T)⁽ⁿ⁾}and analogously

P

γ_i+1⁽ⁿ⁾ =γ_i⁽ⁿ⁾= θ1θ2

(θ₁+θ₂)² .

Recall that for almost surely totally disjoint sets of arrival times of ˜n and ˜m these probabilities have to be equal. With the law of large numbers

I_X^N(t) = T N˜(T)⁽ⁿ⁾

X

g_j⁽ⁿ⁾≤t

1_{g⁽ⁿ⁾

j =g_j−1⁽ⁿ⁾}

−→p θ₁θ₂t (θ1+θ2)² ,

I_Y^N(t) = T N˜(T)⁽ⁿ⁾

X

γ⁽ⁿ⁾_j ≤t

1_{γ⁽ⁿ⁾

j =γ_j−1⁽ⁿ⁾}

−→p θ1θ2t (θ₁+θ₂)² .

Figure 5.1: Quadratic (Co-)variations of times for homogeneous Poisson sampling.

Figure 5.1 shows the quadratic (co-)variations of times and degrees of regularity of non-synchronicity for simulated mutually independent homogeneous Poisson processes.

On the left-hand side both parameters have been set θ = 1 forT = 1 and n= 30000.

The limits are linear increasing functions on [0,1] with slope 14/9,10/9,2/9 and 1/4, respectively. On the right-hand side we see the (co-)variations of times and degrees of regularity of asynchronicity for T = 1, n= 30000, θ₁ = 1, θ₂ = 0.5. Those tend to linear limiting functions with slope 82/49,44/49,8/49 and 2/9, respectively.

In the model of non-synchronously observed Itô processes X and Y as considered in Chapter 3 and observation times following an independent Poisson sampling scheme of the above given form, we derive the following stable central limit theorem as special case

5.2. POISSON SAMPLING CHAPTER 5.

of Theorem 3.1:

Corollary 5.2.2. The estimation error of the synchronized realized covariance estimator (3.2)converges on the Assumption 1 conditionally on the independent Poisson sampling scheme with 0 < θ1 < ∞ and 0 < θ2 <∞ stably in law to a centred mixed Gaussian distribution:

qN˜(T)⁽ⁿ⁾





N(T˜ )⁽ⁿ⁾

X

i=0

Xg⁽ⁿ⁾_i −X

l⁽ⁿ⁾_i Y

γ⁽ⁿ⁾_i −Y

λ⁽ⁿ⁾_i

− hX , Yi_T





st N(0, v_T) , (5.9)

with the asymptotic variance v_T = 2

Z T 0

ρ_tσ_t^Xσ_t^Y2 dt+

2 θ₁θ₂

θ(θ₁+θ₂) + 1 Z T

0

σ^X_t σ^Y_t ²

where the two addends come from the asymptotic variances of the discretization error D_T^N of the closest synchronous approximation (3.5)and the additional error A^N_T due to interpolations (3.6), respectively, and θ=θ₁+θ₂−_θ^θ1θ2

1+θ2.

Proof. It is a basic result in the theory of extreme values that for the supremum of n i. i. d. exponentially distributed waiting times ∆T_i with E∆T_i=n⁻¹, it holds true that sup_i(∆Ti) =Op(log (n)/n). We refer to de Haan and Ferreira [2006] for a proof. In the setting of mutually independent homogeneous Poisson processes with parametersT n/θ₁ and T n/θ₂, we conclude that sup_{i∈{1,...,N(T˜ )}(n)} = Op

log ˜N(T)⁽ⁿ⁾/N˜(T)⁽ⁿ⁾. Hence, Assumption 2(a) holds for the sampling design where the order in condition (a) holds in probability. Then all findings in the proofs of Propositions 3.3.2 and 3.3.5 stay valid when we insert the (co-)variations of time deduced above in the limits of the variances.

The stable convergence holds conditionally given the observation times (cf. the discus-sion following Assumption 2), what means that endogenous observation times are not covered but Poisson sampling independent to the processes ˜X and ˜Y.

The asymptotic variance of the mixed Gaussian limit is in line with the result by Hayashi and Yoshida [2008] and Hayashi and Yoshida [2011]. We remark that one has to pay attention to the proportionality toθ in the rate ˜N(T)⁽ⁿ⁾ when comparing the asymptotic variances to the one in Hayashi and Yoshida [2011].

The following versions of the stable central limit theorems from Theorem 4.1 and Corollary 4.2.2 complete our analysis of the homogeneous Poisson sampling setting.

Corollary 5.2.3. On the Assumptions 1 and 3, the generalized multiscale estimator (4.2)with noise-optimal weights (4.14), and MN =c_multi·√

N, converges conditionally on the independent Poisson sampling scheme with 0< θ₁ <∞ and 0< θ₂ <∞ stably in law with rate N^1/4 to a mixed normal limit:

N^1/4

hX, Y\i^multi_T − hX, Yi_T st

N0,AVAR^poiss_multi

CHAPTER 5. 5.2. POISSON SAMPLING

with the asymptotic variance AVAR^poiss_multi =c⁻³_multi

24 + 12 2θ₁θ₂ (θ₁+θ₂)²

η_X²η_Y² +c⁻¹_multi 12η_X² η²_Y 5 +c_multi 26

35 Z T

0

2 1− 2θ²₁θ₂²

θ₁²θ²₂+ (θ₁²+θ²₂)(θ₁+θ₂)²

!

(σ^X_t σ_t^Y)²(1 +ρ²_t)dt (5.10) +c⁻¹_multi12

5 η²_Y Z T

0

(1 + θ1θ2

θ₁+θ₂)(σ_t^X)²dt +η_X² Z T

0

(1 + θ1θ2

θ₁+θ₂)(σ^Y_t )²dt

! .

On the same Assumptions, the one-scale subsampling estimator with subsampling frequency iN =csub·N^2/3 converges conditionally on the sampling scheme stably in law with rate N^1/6 to a mixed Gaussian limiting distribution:

N^1/6

hX, Y\i^sub_T − hX, Yi_T st

N0,AVAR^poiss_sub , with the asymptotic variance

AVAR^poiss_sub =c⁻²_sub4η_X²η_Y² (5.11)

+csub

2 3

Z T 0

2 1− 2θ₁²θ₂²

θ²₁θ₂²+ (θ²₁+θ²₂)(θ1+θ2)²

!

(σ_t^Xσ_t^Y)²(1 +ρ²_t)dt .

Proof. Since Assumption 2(b) also holds for the sampling design when the order in condition (b) is in probability, the proofs of Theorem 4.1 and Corollary 4.2.2 stay valid with the according asymptotic degrees of regularities and asymptotic quadratic variation of time for the closest synchronous approximation derived in this section above.

5.3 Sample size dependent noise variances and relaxing some

Im Dokument Estimating the quadratic covariation from asynchronous noisy high-frequency observations (Seite 141-147)