A note on central limit theorems for quadratic variation in case of endogenous observation times

This paper is concerned with a central limit theorem for quadratic variation when observations come as exit times from a regular grid. We discuss the special case of a semimartingale with deterministic characteristics and finite activity jumps in detail and illustrate technical issues in more general situations.


Introduction
High-frequency statistics has attracted a lot of attention in recent years. Given observations of a semimartingale X, one is often interested in estimation of its quadratic variation (or parts thereof, such as integrated volatility) and with associated central limit theorems.
A natural way is to work in a setting where observations come at regular times, that is we have data X j/n , j = 0, . . . , n, over the interval [0, 1], say. The most general paper on asymptotics in this setting is Jacod (2008) where various (stable) central limit theorems for functionals of discretely observed Itō semimartingales are stated, including those for realized variance RV (X, X) n t with RV (X, Y ) n t = nt j=1 (X j/n − X (j−1)/n )(Y j/n − Y (j−1)/n ) for arbitrary processes X and Y ; see also Jacod and Protter (1998) for earlier results on related statistics in the case of Lévy processes. Suppose, X is defined on the filtered probability space (Ω, F, (F t ) t≥0 , P) and can be decomposed as κ(δ(s, x))(µ − ν)(ds, dx) (1.1) where a, σ and δ are adapted processes and a truncation function κ,κ(x) = x−κ(x), separates large from compensated small jumps. The compensating intensity measure ν of the Poisson random measure µ admits the form ν(ds, dx) = ds ⊗ λ(dx) for a σ-finite measure λ. In this case, the quadratic variation process becomes Under essentially no extra conditions, Theorem 2.11 in Jacod (2008) gives the Fstable central limit theorem pointwise in t, where the limiting process Z t = U t + V t consists of two parts: The first one, is due to the continuous martingale part of X only. Here, W denotes an independent Brownian motion defined on an extension (Ω , F , (F t ) t≥0 , P ) of the original space. The process V t takes a more complicated form and comes from both the continuous and the jump part of X. It is given by where the sequence (S p ) p denotes an enumeration of the jump times of X, and where the R p and R p are standard normal and the κ p are uniform random variables on [0, 1], all defined on the same extension as W and all mutually independent. In the case of a continuous σ, the limit obviously reduces to For details on stable convergence we refer to Section VIII.5 of Jacod and Shiryaev (2003). Over the last decade, a lot of work has been connected with extensions of (1.2) whenever the ideal setting of observations of X at equidistant times is not realistic. Typically, the focus has been on a version of X with continuous paths, which means that (1.1) reduces to (1.4) Regarding non-equidistant observations, authors have typically worked in settings where observations times come either in a deterministic way or are (essentially) independent of X. In this case, one still has stable convergence of the standardized realized variance to a limiting process U t , but it differs from U t by an extra factor which accounts for the variability of time. See for example Hayashi et al. (2011), Mykland and Zhang (2012) or Koike (2014). A similar result can be obtained if jumps are present, but in this case with a different process V t as well. See Bibinger and Vetter (2015).
Even though important from a practical point a view, the situation with endogenous observation times has found much less attention. This is certainly the case because the proofs of the corresponding results become much more complicated then. For model (1.4), central limit theorems are provided in Fukasawa (2010) and Li et al. (2014), once certain extra conditions are satisfied which might be difficult to check in practice. As a specific example, hitting times of a grid are discussed in Fukasawa and Rosenbaum (2012).
However, to the best of our knowledge no results exist in the general model (1.1) involving jumps. The aim of this work therefore is to shed some light on this issue. In particular, we will prove a result similar to (1.2), but only in the less general case of a continuous Itō semimartingales plus finite activity jumps with deterministic characteristics, observed at hitting times of a regular grid. This is a bit unsatisfactory from a practical point of view, but we will discuss the reasons for this slight limitation and possible guidelines for future research in order to solve the problem in a general framework.
This work is organised as follows: Section 2 gives a short review on the case of deterministic observation times, and we give a heuristic explanation why the limiting process Z t is of the form as stated in (1.2). Section 3 contains the main theorem of this work, as well as a comparison with the standard result and remarks on further extensions of the model. All proofs are gathered in Section 4.

The standard case: Deterministic observations
Let us shortly sketch the strategy which leads to the main result (1.2) from Jacod (2008). The key idea is to decompose X for each integer q as denotes the large jumps of X, corresponds to the continuous martingale part of X, and the remainder X(q) t involving drift and small jumps is defined implicitly. One uses here the mild assumption that the process |δ(s, x)|, which is responsible for the jump sizes, is bounded by some deterministic function γ(x). Using (2.1) and the binomial theorem one obtains √ nRV (C, X(q)) n t + 2 √ nRV (N (q), X(q)) n t for each integer q. A huge part of the proof of (1.2) deals with negligibility of most terms in the decomposition above. This is the case for all terms involving the remainder X(q) which can be shown to converge to zero in probability if we first let n → ∞ and then q → ∞. The proof relies heavily on martingale techniques when dealing with the compensated small jumps of X. See for example Appendix B in Bibinger and Vetter (2015). Moreover, since N (q) is a finite activity jump process, for any fixed q we have √ n (RV (N (q), N (q)) n t − [N (q), N (q)] t ) = 0 identically, with a probability converging to one, as well. This explains why only two processes appear in the limit in (1.2). In case of continuous paths, the relevant term is √ n (RV (C, C) n t − [C, C] t ), for which the proof of stable convergence to U t is very well understood by now. The main condition, due to conditional Gaussianity, is the convergence of the empirical conditional variance which determines the distribution of the limiting Brownian martingale. Let us therefore focus on the mixed part. Since N (q) t is a finite activity jump process, we can write where i + (s) and i − (s) denote the index of the first observation past or equal to s and the last observation prior to s, respectively. Even though the process looks already like V t , there are several steps necessary in order to get to the final result. First, one uses a similar discretization argument as for the conditional variance in order to write up to an error of size o P (n −1/2 ). Second, for each p we have the equality follows, since the jump times of N (q) are uniformly distributed over [0,1]; see the proof of Lemma 6.2 in Jacod and Protter (1998). All of these steps can be shown to hold jointly for 1 ≤ p ≤ k, for an arbitrary integer k. This is sufficient to prove for any fixed q. Letting q → ∞, stable convergence to V t follows. Finally, some more technicalities are needed in order to prove the joint stable convergence of the two sequences leading to U t and V t .

The endogenous case: Exit times from a regular grid
What makes an extension of (1.2) to random observation times τ n j difficult in general, is in particular the equality in distribution stated in (2.3). As long as the times are exogenous, i.e. independent of X, or satisfy some predictability condition, this equality still holds because at least the local behaviour of W is independent of the observation times. The central limit theorem for then looks similar to (1.2), with some minor modifications because one has to find suitable conditions under which a result like (2.4) holds. See e.g. Bibinger and Vetter (2015) for details.
In the purely endogenous case, however, the observation times depend strongly on the process X (and in particular on the process W ), and therefore one cannot expect a version of (2.3) to remain valid. For this reason, we will work in a specific setting, which makes a computation of the distribution of increments of the Brownian part between successive observations possible. Suppose therefore in the following that we observe where a s and σ s are deterministic continuous processes, W t is a standard Brownian motion and J t is a finite activity jump process with a deterministic compensator. We also assume that the function σ is positive everywhere. Observations are coming as exit times from a regular grid. That is, for a given constant c > 0 and a sequence n → 0, which governs the asymptotics, we observe X at the stopping times τ n 0 = 0 and if no such k exists. A sketch of this observation scheme can be found in Figure 1. Roughly speaking, data is collected whenever the unit of the price is changing, e.g. from one cent to another. No stopping, however, takes place at the previously reached barrier, which is necessary because Brownian motion hits it infinitely often after starting from there. This scheme is a particular case of the setting in Fukasawa and Rosenbaum (2012), who derived a central limit theorem for realized variance of a continuous semimartingale. Note that in their setting observations came at hitting times (not necessarily of a regular grid), which is essentially similar to exit times for processes with continuous paths.
In order to state the stable central limit theorem, we again assume the process X to live on the filtered probability space (Ω, F, (F t ) t≥0 , P), and we consider a second Figure 1: The observation scheme used: The first observations τ n j to τ n j+2 are due to movements of the underlying Brownian motion only; the jump at S p is big enough to cause the observation τ n j+3 . Remark that between τ n j+2 and τ n j+3 the path hits (k + 2) n c several times without generating an observation. Last, the jump at S p+1 is too small for the path to exit the interval [(k + 5) n c, (k + 3) n c].
probability space (Ω , F , (F t ) t≥0 , P ) which supports a Brownian motion W and a sequence (η p ) p≥1 of random variables with density All of these are mutually independent and independent of W as well. Limiting variables are then defined on the product ( Ω, F, ( F t ) t≥0 , P) of the two afore-mentioned probability spaces. Finally, we denote with S 1 , S 2 , . . . a specific sequence of stopping times exhausting the jumps of X, namely we order the sequence in chronological order. This is possible, as there are only finitely many jumps almost surely. In principle, other enumerations were possible as well, but this choice facilitates some parts of the proof.
Theorem 3.1. Suppose that X is given by (3.1) and is observed at the random instants coming from the sampling scheme described above. Then, as n → ∞, we have the F-stable convergence with the Brownian motion W and the sequence (η p ) p≥1 as above.
(i) In contrast to (1.3), the distribution of V t does not involve a factor σ Sp . Intuitively, when finding analogues of (2.3) and (2.4) we are concerned with the distribution of the random variables −1 , where it is sufficient to take S p as the right end point because a jump of X causes an exit from the current interval with probability converging to one. (Here and below, τ − n (t) is the last observation prior to t.) After rescaling, each distribution essentially equals the one of a Brownian motion σW (starting in zero and with σ = σ τ − n (Sp) ) at time t, when it is known to not having left the interval [−c, c] by that time. This additional knowledge makes the distribution independent of σ. See Lemma 4.2 for details.
(ii) A similar effect is known from Fukasawa and Rosenbaum (2012), from which we borrow the stable convergence of the continuous martingale part. Whereas in the case of exogeneous observation times σ enters to second order in the conditional standard deviation, as seen e.g. from (2.2), it only contributes to first order to the (conditional) standard deviation of U . The missing order is due to a factor c in our situation, which enters likewise in the mixed term.
(iii) We believe that a generalization to the case of random characteristics and irregular grids holds as well, but in this situation the proof becomes even more involved. If the grid takes the form we conjecture that apart from U , which changes according to the results in Fukasawa and Rosenbaum (2012) under their assumptions, the second summand becomes where the distribution of η p (x) is a suitably weighted mixture of the limiting distribution of two independent Brownian motions starting at the grid points right above and right below x, respectively, and which are known to not having caused another observation.
(iv) Using our strategy of proof it is extremely difficult to prove (a version of) Lemma 4.2 and, thus, Theorem 3.1 when one works with infinitely many jumps within X. In our setting, as two jumps are usually far apart, the key steps in the proof regard the exact distribution of exit times of Brownian motion and a result by Freedman (1983) who derives an explicit formula for In the general case, one has to account for exit times of jump processes as well, and very little is known even for Lévy processes. (Asymptotics are provided in Rosenbaum and Tankov (2011), however.) Also, to the best of the authors' knowledge, a generalization of (3.3) does not even exist in the literature for Brownian motions with a constant drift, let alone for more general processes involving jumps.

Appendix: Proof of Theorem 3.1
We start with the proof of Theorem 3.1 when X t = σW t + J t with a constant σ > 0. As above, we set C t = σW t , and we denote with S 1 , S 2 , . . . the sequence of jump times in chronological order. For convenience, S 0 = 0. We write for each p. The main part of the proof is Theorem 3.1 is contained in the following lemma.
Lemma 4.1. As n → ∞, we have the F-stable convergence with U t = 2/3cσW t and the sequence η p as before.
Proof. Using the standard metric on an infinite Cartesian product, it is enough to prove the result for 1 ≤ p ≤ k and some k. Without loss of generality, we assume k ≤ Γ where Γ is the random, almost surely finite number of jumps over [0, t]. Thus, we show which formally means that we have to prove for any bounded Ψ ∈ F and for all bounded Lipschitz functions g, h 1 , . . . , h k . First, the idea is to separate U n t from the other variables. To this end, we need some technical definitions. Fix some integer and set S − p = (S p − 1/ ) + and S + p = S p + 1/ . Then B = ∪ Γ p=1 [S − p , S + p ] denotes the union of all those intervals around the jump times, and we use Λ n ( , t) to define the set of indices j such that τ n j ≤ t and [τ n j−1 , τ n j ] ∩ B = ∅. The increments over all such j then cover Finally, we define which is the same quantity as U n t , but only involves intervals which are far from the jump times. We will see that U ( ) n t and U n t obey the same asymptotic law. It is helpful to introduce the sets Ω n = {ω ∈ Ω : inf{|∆J Sp |; 1 ≤ p ≤ Γ} > 2 n c} and Ω( ) = ω ∈ Ω : inf{|S p − S p−1 |; 1 ≤ p ≤ Γ} > 2 .
On the set Ω n every jump of X leads to an exit from the current interval, thus causes an observation. Obviously, Ω n → Ω as n → ∞, because there are only finitely many jumps almost surely. Therefore, it is equally well possible to prove Let us further remark that on Ω( ) any two jumps are further than 2/ apart, and thus B becomes a disjoint union.
We are now interested in showing as → ∞, and because of Ω( ) → Ω we may assume to live on Ω( ) as well. Let (F t ) t denote the filtration which is the smallest one containing (F t ) t and such that the jump measure µ of J is F 0 -measurable. Note that W remains a standard Brownian motion with respect to this filtration. We discuss On the set Ω n ∩Ω( ) all jump times are observation times, and conditionally on F 0 these are known. Therefore, by successive conditioning and using the Burkholder-Davis-Gundy inequality, Here and below, K denotes an unspecified constant. Note that on Ω( ) all observations after S − p and until S + p are due to exits of the Brownian motion from the respective interval. The final observation τ + n (S + p ), for example, might be due to the next jump S p+1 , however, but apart from these two additional increments -which, as can be seen from (4.5) and (4.8) later, do not alter the asymptotic behaviourwe are in the same situation as in Fukasawa and Rosenbaum (2012). The proof of their Proposition 4.3 then gives which basically equals the variance of realized variance when computed over intervals outside of Λ n ( , t) only. Then, it it easy to deduce because the random variables inside the expectation are uniformly bounded by a constant, thus in L 2 , and because it suffices to bound |U n t − U ( ) n t |, due to the Lipschitz property of g. Since E[Γ] < ∞, we obtain (4.1) as requested. Also, since B decreases to a discrete set we have as → ∞. Combining (4.1) and (4.2) it is certainly enough to show for any fixed integer . Let us introduce Obviously, and we work conditionally on F 0 again. We bound P {τ + n (S p ) ≥ S p + 1 } ∩ Ω n F 0 by distinguishing the two cases that the next observation is due to a jump (which has to be the next one after S p since we are on Ω n ) or to an exit of the Brownian motion. In both situations, it is obvious that no exit of σW t from the interval [− n c, n c] has taken place over (S p , S p + 1/ ). Using Theorem 2.49 in Mörters and Peres (2010), this exit time T (starting at x) satisfies (4.5) Therefore, (4.6) A similar argument, which will be encountered in the proof of Lemma 4.2 in detail (see (4.8)), can be used to discuss the second indicator in (4.4). Thus, for any fixed , P (Ω n ( ) c ) ≤ E[Γ]K 2 n + o(1) = o(1). Note that on Ω( )∩Ω n ( ) all jump times and the previous and the following observations are located in the interior of B . Therefore, all intervals with [τ n j−1 , τ n j ]∩B = ∅ are equal in distribution to lengths of intervals of exit times of σW t from the regular grid. We define and let (F t ) t denote the smallest filtration containing (F t ) t such that W ( ) t is F 0 -measurable. We introduce with Q = Q ω a regular version of the conditional probability with respect to F 0 . We are now in the position to use the stable convergence stated in Fukasawa and Rosenbaum (2012), in particular Theorem 2.4 and Proposition 3.1 therein. This is possible, as for any t the convergence sup i∈N |τ n i+1 ∧t−τ n i ∧t| → 0 holds in probability, with a similar proof as in their paper. From the fact that D n ( , t) shrinks to [0, t]\B (using e.g. (4.6) again) we obtain where Q denotes the corresponding probability measure on the product space. Thus, is another bounded F-measurable random variable, it is certainly enough to prove for all bounded F-measurable Ξ in order to establish (4.3).
On the set Ω n , the random variables α(n, p) only depend on the jump times S p , 1 ≤ p ≤ k, and the Brownian motion over the intervals between those times, as all jumps automatically lead to an observation. By conditioning, it is then sufficient to show (4.7) for where {W u : u ∈ (S p−1 , S p ]} denotes the Brownian motion between the successive jump times and any ϕ p and p are bounded. Also, a similar argument as the one leading to (4.1) shows that it is sufficient to work with Ξ = k p=1 ϕ p ({W u : u ∈ (S p−1 , S p − 1/m]}) p (S p ), for any fixed integer m larger than . (Note that S p − 1/m > S p−1 always, since we are on Ω( ).) Let us first discuss the asymptotics of As in (4.6), we have P(τ − n (S k ) ≤ S k − 1/m) → 0. On the complement the conditional expectation simplifies, as α(n, k) is independent of W until S k − 1/m then. We obtain The idea is to finally use Lemma 4.2 below, which proves One then obtains (4.7) via successive conditioning.
Lemma 4.2. Let t > 0 be arbitrary and let τ − n (t) be the last hitting time of the grid { n kc | k ∈ Z} by σW prior to t. Then, as n → ∞ we have the weak convergence where η has the density h from (3.2).
Proof. Before we begin with the proof, note from self-similarity of Brownian motion that −2 n (t − τ − n (t)) is equal in distribution to −2 n t − υ − n ( −2 n t), where υ − n (s) denotes the last hitting time of the grid {kc | k ∈ Z} by σW t . This distribution is called G n in the following. Also, after a shift to zero, A1.3.0.2 in Borodin and Salminen (2002) shows that the distribution of each hitting time of the grid {kc | k ∈ Z} by σW t equals S with Theorem 1.18 in Mitov and Omey (2014) then proves pointwise in z, where we have used that c 2 /σ 2 is the expected waiting time until the next hit; recall (4.5). By conditioning on t − τ − n (t) we then obtain Of course, we are not in the setting of stopping times here, as t − τ − n (t) = 2 n z states both that t − 2 n z was a hitting time and that no further hitting time has taken place between t − 2 n z and t. However, since we work with a regular grid we can at least assume that the place of the last hit was at zero and, using stationarity of the increments of Brownian motion, we can set t − 2 n z = 0. Overall, we obtain where we have used self-similarity of the Brownian motion again. While the denominator can be written as P(S > z) with S as before, we use Theorem 33 in Freedman (1983) to obtain c], and it vanishes otherwise. Overall, using dominated convergence, Fubini's theorem and a change of variables, This proves the result. Note finally that h is easily seen to be a density, as Let us now finish the proof of Theorem 3.1 when X t = σW t + J t . From (4.5) we obtain