3.3 Proof of the stable central limit theorem
3.3.1 Discretization error of the synchronous approximation
Proposition 3.3.1. On the Assumptions 1, 2(a) and (3.8a) the discretization error of the closest synchronous approximation converges stably in law to a centred mixed Gaussian distribution:
s N
TDNT st N 0, Z T
0
G0(t)σtXσtY2(ρ2t + 1)dt
!
. (3.10)
Proof. First note that on Assumption 1, by Girsanov’s Theorem 1.2 we may without loss of generality further suppose that µXt =µYt = 0 identically. This is possible with the stability of the weak convergence which is ensured by asymptotic independence between the limiting normal distribution and the martingale parts of the observed processes using Jacod’s Theorem 1.6 (cf. the discussion in Subsection 1.1.2 and Section 1.2).
As before, we often omit the superscripts (N) for the sampling times to increase the readability.
Let Mt and Lt be the continuous martingales Lt = R0tσXs dWsX , Mt = R0tσYs dWsY with standard Brownian motions WX, WY with quadratic covariation hWX, WYit = Rt
0ρsσXs σYs ds, that represent the transformed martingale processes, andLi=R0TiσsXdWsX, Mi =R0TiσsY dWsY.
Proposition 3.3.2. On the same Assumptions as in Proposition 3.3.1, the process DNt defined by
DNt :=
s N
T X
Ti(N)≤t
(Li−Li−1)(Mi−Mi−1)− Z t
0
ρsσsXσsYds
for 0≤t≤T converges as N → ∞ stably in law:
DNt st Z t
0
√vDsdWs⊥ (3.11)
where W⊥ is a Brownian motion independent of Fand
vDs =G0(s)(σXs σYs)2(ρ2s+ 1) . (3.12) Proof. We will prove the proposition by application of Jacod’s Theorem 1.6. It would also be possible to use the discrete-time version of this Theorem 1.3.5 which will be applied in the next subsection. Here, we prefer to consider the continuous-time version to gain a better understanding of the key elements that lead to the stable central limit
3.3. STABLE LIMIT THEOREM CHAPTER 3.
theorem in Proposition 3.3.2.
Using the definition of the quadratic covariation process of martingales (or integration by parts formula) we can find an illustration of the discretization error by a sum of stochastic integrals and an asymptotically negligible term:
X We calculate the corresponding quadratic variation process at timet:
hφ(N)it= N
CHAPTER 3. 3.3. STABLE LIMIT THEOREM
The third equality including a remainder term of Op(1) is proved in the next lemma.
In this calculation we further used the integration by parts formula (1.1.3) in the following step and then the change of variables Theorem 1.1.5 for the integrals with quadratic covariation integrators that are of finite variation. The second last equality is an application of the mean value theorem (the volatility and the correlation processes are continuous and thus also bounded on compact sets) where the constants (σX)i, (σY)i and (ρσXσY)i come from. The Riemann sum converges and using Definition 3.2.3 and Assumption 3.1, the convergence in probability of the quadratic variation to Rt
0G0(s)(ρ2s+ 1)(σsXσsY)2ds=R0tvDs follows.
Lemma 3.3.3. Using the notation as above the following relations hold true:
X
3.3. STABLE LIMIT THEOREM CHAPTER 3.
X
Ti(N)≤t
Z Ti
Ti−1
((Ms−Mi−1)(Ls−Li−1)− hM−Mi−1, L−Li−1is)dhM, Lis=Op(1) (3.13c) N
T X
Ti(n)≤t
(∆Ti)2
(ρσXσY)2i + (σX)2i(σY)2i −ρTi−1σTXi−1σTYi−12−σXTi−1σTYi−12
=Op(1) (3.13d) Proof. The proofs of (3.13a) and (3.13b) are completely analogous. We prove (3.13a).
By Itô’s formula
(Ls−Li−1)2= 2 Z s
Ti−1
(Lr−Li−1)dLr+hL−Li−1is holds. The left-hand side of (3.13a) equals
X
Ti(N)≤t
Z Ti
Ti−1
2 Z s
Ti−1
(Lr−Li−1)dLr
!
dhM−Mi−1ir
!
= X
Ti(N)≤t
2 Z Ti
Ti−1
(Ls−Li−1)(hM−Mi−1iTi)dLs−2 Z Ti
Ti−1
(Ls−Li−1)(hM−Mi−1is)dLs
!
by application of the integration by parts formula (1.1.3) in the way ZTihM−Mi−1iTi =
Z Ti
0
ZtdhM−Mi−1it+ Z Ti
0
hM −Mi−1itdZt
with Zt:=RTt
i−12(Ls−Li−1)dLs for Ti−1 ≤t≤T to the addends. Therefore, we can write the left-hand side of (3.13a) in the way M(N)1 +M(N)2 with two centred continuous martingalesM(N1 ),M(N)2 defined in the fashion ofφ(N) above and calculate the quadratic covariation processes at timet:
hM(N2 )it= 4 X
Ti(N)≤t
Z Ti
Ti−1
(Ls−Li−1)2(hM−Mi−1is)2dhLis
!
+Op(1)
≤4 max
i sup
s∈(Ti−1,Ti]
(Ls−Li−1)2max
i sup
s∈(Ti−1,Ti]
hM−Mi−1i2s X
Ti(N)≤t
Z Ti
Ti−1
dhLis+Op(1).
The first addend is up to a logarithmic factor Op(δN3) and hence M(N)2 = Op(1) on Assumption 2. That M(N)1 =Op(1) is proved analogously. This proves (3.13a).
The strategy to prove (3.13c) follows the same approach. For the sake of completeness
CHAPTER 3. 3.3. STABLE LIMIT THEOREM
we give the first part of the proof in the following. We begin with the equation (Ls−Li−1)(Ms−Mi−1) = and apply the integration by parts formula as above with Zt=RTt
i−1(Ls−Li−1)d(M −
Now one can proceed as for (3.13a) to prove (3.13c).
We complete the proof of the convergence of the quadratic variation with the proof of approximation (3.13d). Denote (ρσXσY)2i =(σ^X)2i·(σ^Y)2i·(ρ)g2i to distinguish between the values from the application of the mean value theorems to the two different addends.
An upper bound of the left-hand side of (3.13d) can be found by elementary algebra and the triangle inequality for the absolute value:
N
For the martingales φ(N) there are representations as time-changed Brownian motions
3.3. STABLE LIMIT THEOREM CHAPTER 3.
Bhφ(DDS,N)(N)it =φ(N)t by the Dambis-Dubins-Schwarz Theorem 1.3. The sequence of mar-tingales φ(N) or associated time-changed Dambis-Dubins-Schwarz Brownian motions converges weakly to a limiting Brownian motion by Theorem 1.4. This convergence is stable and the limiting Brownian motion is defined on an orthogonal extension of the original probability space. To obtain the stable convergence result, we are left to verify conditions (1.3a) and (1.3b) of Jacod’s Theorem 1.6.
Consider the quadratic covariation process ofφ(N) and the reference martingaleL hL, φ(N)it=
s N T
X
Ti(N)≤t
Z Ti
Ti−1
(Ls−Li−1)dhM, Lis+ Z Ti
Ti−1
(Ms−Mi−1)dhLis
!
+Op(1).
The term of smaller order than 1 in probability comes from the increment of the covariation process on [T(t), t]. As before, this equality holds true for all t, since for t < T1 the covariation isOp(1). Integration by parts yields:
hL, φ(N)it= s
N T
X
Ti(N)≤t
"
(hM, LiTi− hM, LiTi−1)(Li−Li−1)− Z Ti
Ti−1
hM, Lisd(Ls−Li−1)
+(hLiTi − hLiTi−1)(Mi−Mi−1)− Z Ti
Ti−1
hLisd(Ms−Mi−1)
# .
It remains to show that this term converges to zero in probability. The term is centred and using Itô isometry we find the following upper bound for the second moment:
E
hL, φ(N)it2
≤2N TE
X
Ti(N)≤t
(hM, LiTi− hM, LiTi−1)2(Li−Li−1)2
+(hLiTi− hLiTi−1)2(Mi−Mi−1)2
+ max
i∈{1,...,N} sup
s∈(Ti−1,Ti]
(hM, Lis− hM, LiTi−1)2X
i
Z Ti
Ti−1
dhL−Li−1it
+ max
i∈{1,...,N} sup
s∈(Ti−1,Ti]
(hLis− hLiTi−1)2X
i
Z Ti
Ti−1
dhM −Mi−1it
#
=ON δN2 .
The term is bounded by a constant times N δN2 since squared increments, cross products of increments and increments of the quadratic (co-)variations of L and M over time instants ∆Ti(N) are bounded by ∆Ti(N) times a constant. To sums with products of time instants we can apply Hölder’s inequality with the supremum norm to obtain upper bounds. There are at most order δN−1 time instants ∆Ti(N) of order supi∆Ti(N) =δN sincePi∆Ti(N)≤T and the time spanT is fixed.
CHAPTER 3. 3.3. STABLE LIMIT THEOREM
Hence,hL, φ(N)it=Op(1) ∀t ∈[0, T]. With the same strategyhM, φ(N)it=Op(1) ∀t ∈ [0, T] can be proven.
For every bounded Ft-martingaleL⊥ satisfying hL, L⊥i ≡0 the covariation hL⊥, φ(N)it= N
T X
Ti(N)≤t
Z Ti
Ti−1
(L⊥s −L⊥i−1)dhM, L⊥is
!
+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.