Central limit theorems for the nonparametric estimation of time-changed L evy...

Central limit theorems for the

nonparametric estimation of

time-changed Levy models

Jose E. Figueroa-Lopez∗

Purdue University

Department of Statistics

West Lafayette, IN 47906, USA

[email protected]

Abstract: Let {Zt}t≥0 be a Levy process with Levy measure ν and let

τ(t) :=∫ t0g(r(u))du be a random clock, where g is a non-negative func-

tion and {r(t)}t≥0 is an ergodic diffusion independent of Z. Time-changed

Levy models of the form Xt := Zτt are able to incorporate several im-

portant stylized features of asset prices, such as leptokurtic distributions

and volatility clustering. In our former paper [19], we proposed consistent

estimators for the integral parameter β(ϕ) :=∫ϕ(x)ν(dx) based on high-

frequency discrete observations of the process X. In this paper, we prove

central limit theorems for our estimators, valid when both the sampling

frequency and time-horizon get larger. Our results combine the long-run

ergodic properties of the diffusion process r with the short-term ergodic

properties of the Levy process Z via central limit theorems for martingale

differences. The performance of the estimators are illustrated numerically

for Normal Inverse Gaussian process Z and a CIR process r.

AMS 2000 subject classifications: 60J75; 60F05; 62M05.

Keywords and phrases: Levy processes, nonparametric estimation, high-

frequency based inference, stochastic volatility.

1. Introduction

Accurate modeling of the stylized features of stock prices has been a funda-mental problem in mathematical finance for a long time, leading to importantadvances in the application of continuous-time stochastic processes as well asin the statistical methods for such processes. Some of the most relevant styl-ized features are leptokurtic distributions of the returns on equally spaced timeintervals, volatility clustering, and leverage (see, e.g., [13]). Many models havebeen proposed to incorporate these features. In this paper we consider a sub-ordinated or time-changed Levy model for the log-return process. Concretely,

∗The author’s research was partially supported by the NSF grant no. DMS 0906919. The

author is indebted to two anonymous referees as well as the editors for their many suggestions

that improved the paper considerably.

1

J.E. Figueroa-Lopez/CLT for time-changed Levy models 2

if {St}t≥0 is the stock price process, the log return process Xt := logSt/S0 isassumed of the form

Xt = Zτ(t), (1.1)

with Z being a Levy process and {τ(t)}t≥0 being a nondecreasing absolutelycontinuous process. The model (1.1), originally proposed by Carr et. al. [11](see also [12]), is able to incorporate many of the statistical features commonlyobserved in asset prices. The flexible marginal distributions of the Levy processZ allow to account for the leptokurtic distributions of the log returns, while therandom clock τ incorporates volatility clustering. Indeed, suppose that

τ(t) :=∫ t

0

r(u)du, (1.2)

with {r(t)}t≥0 being a positive mean-reverting process. Then, when r takes a“high” value (resp. “low” value), the “business time” τ runs faster (resp. slower),resulting in more frequent jump times and greater variability of the Wienercomponent of Z, if present. Hence, the rate process r controls the volatility ofthe price process S. For further motivation and justification of the model (1.1-1.2) we refer to the introductory section in our former paper [19], the originalpaper of Carr et. al. [11], or the last Chapter of the monograph of Cont andTankov [14].

Following [19], we are assuming that the Levy measure of Z admits a densitys and we are interested in estimating it, in nonparametric fashion, based ondiscrete observations of X. As a key intermediate step we consider estimatorsfor the integral parameter

β(ϕ) :=∫ϕ(x)s(x)dx, (1.3)

where ϕ is a given “test function”. The parameters (1.3) serve as building blocksof several approximation methods of functions (e.g. orthogonal expansion seriesor kernels). Hence, the estimators β(ϕ) can be combined with these approxima-tion methods to construct nonparametric estimators s of s. One of such methodsis Grenander’s method of sieves, which was illustrated in [18] in the case of apurely Levy model and which can be readily extended for the time-changedLevy model using the results in this paper. For further motivation and detailsabout this important application of our results, we refer to Section 4.1 below.

Throughout the paper, we assume that the process (1.1) is sampled over afinite time horizon [0, Tn] at the discrete times 0 = tn0 < · · · < tnn := Tn. Thechallenge of drawing inferences for the Levy density s of Z lies in the fact thatthe jumps are unobservable from the discrete-time observations Xtn0

, . . . , Xtnn.

Nevertheless, it is expected that consistent inference of the Levy measure ν

is feasible if the time horizon Tn gets large and simultaneously the sampling


mesh δn := max{tnk − tnk−1} gets smaller. The literature of statistical inferenceof continuous-time processes based on high-frequency data (HFD) has growngreatly in the last few years. For the case of a pure jump Levy process, Wo-erner [35] and Figueroa-Lopez [15] proposed nonparametric methods for theLevy measure based on HFD. In the context of stochastic volatility models,Barndorff-Nielsen and Shephard [4] introduced their bipower and multipowervariations to estimate the stochastic volatility based on HFD. Truncated powervariations were introduced in Mancini [27] (see also [28]). For a pure Levy pro-cesses, Jacod [23] gives a very complete study of the asymptotic properties ofrealized variations with emphasis in the cases where Tn ≡ T and δn → 0 (thegeneral semimartingale case was subsequently considered in [22]). Mutipowervariations have been applied as well to general semimartingale models in prob-lems such as estimation of the integrated volatility process

∫ t0σ2sds (see, e.g.,

[35, 36], [30], [6]), testing for jumps ([6], [30]), and estimation in the presence of“microstructure” noise ([3], [31, 32]). Truncated versions of power variations inthe spirit of Mancini have been recently considered for testing purposes ([1]) andfor the estimation of the jump-activity activity index β of the semimartingale([2]).

In this paper, following our former paper [19], the main building blocks forour estimation method are the statistics

βn(ϕ) :=1Tn

n∑k=1

ϕ(Xtn

k−Xtn

k−1

), (1.4)

for a given deterministic test function ϕ. We can think of (1.4) as the realizedϕ-variation of the process X per unit time, based on the sampling observationsXtn0

, . . . , Xtnn. In the case of a true Levy process X = Z, we have

βn(ϕ) =1Tn

n∑k=1

ϕ(Ztn

k− Ztn

k−1

), (1.5)

and under certain conditions on ϕ necessary for the small-time asymptotic prop-erty

limt→0

1t

Eϕ(Zt) = β(ϕ), (1.6)

to hold (see Condition 2.1 below), (1.5) is known to be a consistent estimatorfor β(ϕ) whenever

Tn →∞, and δn := max{tnk − tnk−1} → 0; (1.7)

see, e.g., [35, Theorem 5.1] and [19, Theorem 2.5].Under the subordinated model (1.1-1.2), with {r(t)}t≥0 being independent

of Z such that

ζ := limT→∞

1T

∫ T

0

r(u)du ∈ (0,∞), a.s., (1.8)


we showed in [19] that βn(ϕ) is a consistent and asymptotically unbiased esti-mator for ζβ(ϕ) when (1.7) holds; that is,

limn→∞

βn(ϕ) P= ζ · β(ϕ), limn→∞

E βn(ϕ) = ζ · β(ϕ). (1.9)

The limit (1.9) is quite natural (seemingly obvious at first) since (1.4) can bewritten as

βn(ϕ) =

∫ Tn

0r(u)duTn

1τnn

n∑k=1

ϕ(Zτn

k− Zτn

k−1

),

where we fixed τni := τ(tni ) =∫ tni

0r(u)du. Conditioning on τn0 = tn0 , . . . , τ

nn = tnn,

we can then apply the consistency of βn(ϕ) in the pure-Levy estimator (1.5),provided that

(i) maxk{τnk −τnk−1} = max

k

∫ tnk

tnk−1

r(u)du→ 0, and (ii) τnn →∞, (1.10)

a.s., as n → ∞. The later two conditions will follow whenever δn → 0 andTn →∞ if r is assumed uniformly bounded away from both∞ and 0. However,without this strong constraint on r, the validity of (1.10) has to do with howfast δn should converge to 0, compared to the speed at which Tn goes to ∞.The precise results and regularity assumptions on ϕ and r for (1.9) to hold willbe reviewed in Section 2.3 (see Theorem 2.1 below).

The main focus of the present paper is to prove the asymptotic normality ofthe estimators (1.4). In the true Levy case X = Z, Figueroa-Lopez [18] showsthe CLT

T 1/2n

(βn(ϕ)− β(ϕ)

)D−→ N

(0, β

(ϕ2)), (1.11)

when Tn → ∞ and δnTn → 0, assuming that ϕ is bounded, ν-continuous, andsuch that ϕ(x) = o(|x|) as x → 0. There are two issues to deal with in thepresence of a random clock. First, as before, the random clock case does notfollow from the CLT (1.11) for the pure Levy case by conditioning on the randomclock, since (1.10) can not be guaranteed whenever δn → 0 and Tn → ∞. Itwill be necessary that δn is small enough compared to Tn. Second, the randomclock introduces dependence between the increments of the process and, hence,Central Limit Theorems (CLT) for row-wise independent arrays are not directlyapplicable. Instead, we will use the more general theory of CLT for martingaledifferences, for which we will assume that r is an ergodic diffusion. In particular,the following well-known CLT for ergodic diffusions will play a key role in ourproof:

√T

(1T

∫ T

0

r(u)du− ζ

)D−→ N (0,Γ), as T →∞, (1.12)


where Γ is a certain positive constant. Our main result shows that

T 1/2n

(βn(ϕ)− ζ · β(ϕ)

)D−→ N

(0, ζβ(ϕ2) + β2(ϕ)Γ

), (1.13)

whenever Tn → ∞ and Tnδn → 0, provided that certain technical conditions

are satisfied. On one hand, the CLT (1.13) shows that the rate of convergence ofβn(ϕ) is Op

(T−1/2n

)and, on the other hand, it opens the door to build asymp-

totic confidence intervals (CI) for β(ϕ) when Γ is estimated simultaneously.Given the linearity of the operator ϕ→ β(ϕ), we can easily deduce a multi-

variate CLT for [β(ϕ1), . . . , β(ϕ1)]. To illustrate their relevance, we will considertwo statistical applications in Section 4. First, we recall the method of sieves andobtain a CLT for the sieve estimator. Second we build asymptotic CI for a ratioparameter of the form β(ϕ1)/β(ϕ0). The later parameter could be useful, forinstance, to estimate the expected proportion of those jumps with size smallerthan a bound b < 0 that will be smaller than another bound a < b. Negativejumps are typically associated with negative losses and the ratio could be usedto devise measures of riskiness. As it was said before, we can also constructasymptotic CI for β(ϕ) itself by first estimating Γ. We will briefly discuss thelater issue in Section 4.2.

Let us finish with a couple of remarks. From a financial point of view, theindependence assumption between Z and r could be undesirable in light of theleverage phenomenon exhibited by real stock prices. We could think of naturalapproaches to incorporate certain degree of dependence such as common drivingfactors for r and Z, but we will not explore this direction in this paper. Also,it is important to note that ζ can be fixed to 1 in (1.9) and (1.13), without anyloss of generality. This is so since the model (1.1-1.2) can be written as

Zτ(t) = Zτ(t),

with Zt := Zζt and τ(t) =∫ t

0r(u)du, fixing r := r/ζ. Thus, the process Z is

again a Levy process with triplet (ζb, ζ2σ2, ζs(x)dx) and (1/T )∫ T

0r(u)du→ 1,

as T →∞. Throughout the paper, we will assume this canonical time-changedLevy model (1.1) where ζ = 1.

The rest of the paper is organized as follows. In Section 2.3, we review someneeded results from [19] and introduce the terminology and assumptions usedthroughout the paper. Section 3 presents the main results about the asymptoticnormality of the estimators (1.4). The multivariate versions of our theorem aswell as some statistical applications are given in Section 4. Some numericalillustration in the case of a Normal Inverse Gaussian process are presented inSection 5. The proofs of the main results are given in an Appendix section.


2. The model and some known preliminary results

As in the introduction, the process X in (1.1) is sampled over a finite timehorizon [0, Tn] at discrete times 0 = tn0 < · · · < tnn = Tn. We assume that{Zt}t≥0 is a Levy process with generating triplet (b, σ2, ν), and {τ(t)}t≥0 is asin (1.2) for a non-negative process r that is independent of Z. We now proceedto describe the different elements of our estimation scheme and to introducesome needed assumptions and results throughout the paper.

2.1. The random clock

The speed process r of the random clock τ is assumed of the form

r(t) := g(r(t)), (2.1)

where r is an ergodic diffusion taking values on an interval I := (a, b) ⊂ R withinvariant probability measure ζ, and g is a measurable non-negative functionsuch that

ζ(g) :=∫g(x)ζ(dx) = 1.

Furthermore, we assume that r solves an SDE of the form

dr(t) = b(r(t))dt+ σ(r(t)) dWt, (2.2)

where W is a Wiener process independent of Z, and b and σ are deterministicfunctions satisfying the standard linear growth conditions

|b(x)|+ |σ(x)| ≤ K (1 + |x|) , (2.3)

for all x and some constant K < ∞. Under the later condition the solution r

exists and satisfies the following short-term uniform bound:

E[

sup0≤s≤h

|r(s0 + s)− r(s0)|2m]≤ kmhm

(1 + E |r(s0)|2m

)ekmh, (2.4)

for any s0 ≥ 0, 0 < h ≤ 1, and m ≥ 1, where km is a constant depending onlyon m and K (see, e.g., Appendix B in [19]). Sometimes, we will also need thefollowing moment conditions:

(i) m2 := supt≥0

E r2(t) <∞, (ii) m2+ε(g) := supt≥0

E |g(r(t))|2+ε <∞, (2.5)

for ε ≥ 0. Note that (2.5) are trivially satisfied if r is stationary and∫Ix2ζ(dx) <

∞ and∫I|g(x)|2+εζ(dx) <∞.


For future reference, we recall the ergodic theorem,

limT→∞

1T

∫ T

0

h(r(u))du =∫I

h(x)ζ(dx) := ζ(h), (2.6)

valid for any h ∈ L1(ζ) (cf. [25, Theorem 20.14]), and the corresponding CLT

√T

(1T

∫ T

0

h(r(u))du− ζ(h)

)D−→ N (0,Γ(h)), (2.7)

valid for h ∈ L1(ζ) such that

Γ(h) := 4m(I)∫I

(∫ x

a

h(y)ζ(dy)− ζ((−∞, x])ζ(h))2

ds(x) <∞, (2.8)

and where m and s are respectively the speed measure and the scale functionof r (see [29]).

We finish with some further needed notation. Throughout the paper, Tt de-notes the transition operator of {r(t)}t≥0,

Ttf(x) := E [f(r(t))| r(0) = x] ,

defined as usual on the space B of bounded measurable functions on (I,B(I))equipped with the sup norm ‖f‖ := sup{|f(x)| : x ∈ I}. Also, A denotes theinfinitesimal generator of r. The extension of A to L2(I, ζ) is denoted by A.Concretely, denoting

B0 := {f ∈ L2(I, ζ) : ‖Ttf − f‖2 → 0, as t→ 0},

we set Af := h, when f and h are as in the following definition:

Dom(A) := {f ∈ B0 : ∃h ∈ B0 s.t.

∥∥∥∥Ttf − ft− h∥∥∥∥

2

→ 0, as t→ 0}.

This extension is valid since

‖Ttf‖22 =∫I

(Ttf(x))2ζ(dx) ≤∫I

(Ttf2)(x)ζ(dx) =∫I

f2(x)ζ(dx) = ‖f‖22,

and thus, {Tt}t≥0 is a contraction semigroup on L2(I, ζ).

2.2. The test functions

Let us introduce the class of test functions ϕ in the integral parameter β(ϕ).To this end, consider the following class of locally bounded (but potentiallyunbounded) functions (first considered in [33]):

S(ν) := {h : R→ R+ :∫|x|>1

h(x)ν(dx) <∞, h(x) = p(x)q(x), (2.9)

where p is subadditive, and q is submultiplicative}.


Recall that a nonnegative locally bounded function g is submultiplicative (resp.subadditive) if there exists a constant K > 0 such that g(x + y) ≤ Kg(x)g(y)(resp. g(x+ y) = K(g(x) + g(y))), for all x, y. We are ready to characterize thetest functions ϕ:

Assumption 2.1.

(1) ϕ is ν-continuous and locally bounded;(2) There exists a function h in S(ν) such that

lim sup|x|→∞

|ϕ(x)|h(x)

<∞; (2.10)

(3) As x→ 0,

(a) ϕ(x) = o(|x|2);(b) ϕ(x) = O(|x|r), for some r ∈ (1, 2) such that

∫(|x|r ∧ 1) ν(dx) <∞,

and σ = 0;(c) ϕ(x) = o(|x|),

∫(|x| ∧ 1) ν(dx) <∞, and σ = 0;

(d) ϕ(x) = O(|x|r), for some r ∈ (0, 1) such that∫

(|x|r ∧ 1) ν(dx) <∞,σ = 0, and b := b−

∫|x|≤1

xν(dx) = 0;

Under the stated conditions, [17] shows the following useful result:

limt→0

1t

Eϕ(Xt) = β(ϕ), (2.11)

It turns out that the above limit will be β(ϕ) + σ2 if ϕ(x) ∼ x2, while it will beβ(ϕ) + |b| if ϕ(x) ∼ |x|, and σ = 0 (see also [17]).

2.3. Consistent estimation of integrals of the Levy measure

The consistency of (1.4) was studied in detail in [19]. Under the setting outlinedabove, the following result was proved there.

Theorem 2.1. Suppose that (2.5-ii) holds for some ε > 0, and ϕ is ν-continuoussatisfying the Standing Assumptions 2.1 and the bound

Mϕ := supt>0

1t

E |ϕ(Zt)| <∞. (2.12)

Then, the estimators (1.4) are consistent and asymptotically unbiased for β(ϕ)whenever Tn →∞ and δn := max

k{tnk − tnk−1} → 0.

The bound (2.12) holds true if the Standing Conditions 2.1 are satisfied, andeither ϕ is bounded, or has linear growth and additionally Z has bounded firstmoment (see Remark 2.4 in [19]). Let us also remark that when g(x) = x1x≥0

and r is assumed to satisfy only (2.5-i), the conclusion of the above theoremstill holds, but under the stronger sampling scheme Tn → ∞ and Tn(δ2)2 → 0(see Theorem 3.3 in [19]).


3. The main results

In this section we prove the asymptotic normality of the estimators (1.4). Through-out the rest of the paper, we shall use the following notation:

δnk := tnk − tnk−1, τnk := τ(tnk ), Hϕ(t) := Eϕ(Zt), (3.1)

∆nkX := Xtn

k−Xtn

k−1, ∆n

kτ := τ(tnk )− τ(tnk−1). (3.2)

As it was explained in the introduction, a random clock τ as in (2.1-2.2)introduces some dependence in the increments of the process. To deal with thisissue, we will use CLT for martingale differences (e.g., Billingsley [9, Theorem18.1]). Concretely, we consider the martingale difference sequence

ξk,n := T−1/2n

{ϕ (∆n

kX)− E[ϕ(∆n

kX)| Fnk−1

]}, (3.3)

for 1 ≤ k ≤ n, relative to the filtration

Fnk := σ (Xu : u ≤ tnk ) ∨ σ (τ(u) : u ≤ tnk ). (3.4)

Note that if ϕ satisfies (2.12) and E τ(t) <∞, for all t ≥ 0, then E |Hϕ(∆nkτ)| <

∞, and moreover,

E [ϕ(∆nkX)| Fnk−1] = E [Hϕ(∆n

kτ)|Fτtk−1], (3.5)

where Fτt := σ(τu : u ≤ t).The standard method for showing the CLT (1.13) will consist of two steps.

First, show the CLT for Sn :=∑nk=1 ξk,n, which can be written as

Sn = T 1/2n

(βn(ϕ)− µn(ϕ)

),

with

µn(ϕ) :=1Tn

n∑k=1

E [Hϕ(∆nkτ)|Fτtk−1

]. (3.6)

Second, show thatlimn→∞

µn(ϕ) P= β(ϕ), (3.7)

at a rate faster than T−1/2n . However, the last claim is actually not true as it

can be shown that

T 1/2n (µn(ϕ)− β(ϕ)) D−→ N

(0, β(ϕ)2Γ(g)

), (3.8)

with Γ(g) defined as in (2.7) (see Proposition 5.1 in [19]). Instead, we shall seethat T 1/2

n

(βn(ϕ)− β(ϕ)

)can be decomposed as

T 1/2n

(βn(ϕ)− µn(ϕ)

)+ β(ϕ)T−1/2

n

∫ Tn

0

(g(r(u))− 1) du+ op(1), (3.9)


and show that the first two terms in the above expression converge jointly to aGaussian distribution.

A crucial result in the sequel is the limit (3.7), which was shown in [19] forLipschitz functions g, under the additional condition (2.5). However, it will alsobe useful to consider the following conditions (see the end of Section 2.1 for thenotation):

Assumption 3.1. The rate process (2.1) satisfies any of the following con-straints:

(1) g, g2 ∈ Dom(A);(2) g, g2 ∈ Dom(A) and r is stationary;

Regarding the CLT (3.8), we shall require the following standing condition:

Assumption 3.2. There exists a t0 > 0 such that∣∣∣∣1t Eϕ(Xt)− β(ϕ)∣∣∣∣ ≤ k0t, (3.10)

for any 0 < t < t0 and for a constant k0 independent of t.

The previous condition holds for a wide class of functions ϕ. For instance,if ϕ is supported on an interval [c, d] ⊂ R\{0}, where ϕ is continuous withcontinuous derivative (cf. [16]), or if ϕ ∈ C2 vanishes in a neighborhood of theorigin, and for each i = 0, 1, 2, |ϕ(i)| is bounded by an element gi in the classS(ν) of (2.9) (cf. [17]).

The following result summarizes and extends our preliminary findings in [19]to the class of functions g satisfying the Standing Assumption 3.1 instead of aLipschitz type condition. Its proof is outlined in Appendix A.

Theorem 3.1. Assume that the conditions of Theorem 2.1 are satisfied as wellas any of the above Standing Assumptions 3.1. Then, when Tn →∞ and δn → 0,

(1) the limit (3.7) holds true;(2) the CLT

T 1/2n

(βn(ϕ)− µn(ϕ)

)D−→ N

(0, β(ϕ2)

), (3.11)

holds true if, additionally, ϕ2 satisfies Assumption 2.1 and (2.12);(3) the CLT

T 1/2n (µn(ϕ)− β(ϕ)) D−→ N

(0, β(ϕ)2Γ(g)

), (3.12)

holds true when Tnδn n→∞−→ 0 if, additionally to the conditions in (2), the

Standing Assumption 3.2 and (2.8) are satisfied.

As it was explained above, our goal will be to show the joint convergence ofthe first two terms in (3.9). Regarding the second term therein, there are twogeneral methods for showing its convergence in distribution: renewal theory and


martingale method. The former divides∫ t

0g(r(u))du into independent and iden-

tically distributed contributions (see [29, Section IV.5]). The martingale methodimposes more restrictions to the rate function g (see, e.g., [24, Section VIII.3f]or [8]), but it can be combined more easily with the martingale method leadingto (3.11). Due to this reason, we adopt the second approach. The following isthe key assumption for the martingale method:

Assumption 3.3. The rate function g in (2.1) and the diffusion r satisfy eitherone of the following conditions:

(1) g(x)− ζ(g) = Af(x), for a function f ∈ Dom(A) such that f2 ∈ Dom(A).(2) g(x) − ζ(g) = Af(x), for a function f ∈ Dom(A) such that f2, f3 ∈

Dom(A). Also, r is stationary.

Remark 3.2. The previous conditions can be analyzed in particular popularmodels of r such as CIR or OU processes. There are also some general results(see [8]). For instance, if the domain of A is dense in L2(R, ζ) and the Markovprocess r is ergodic, then the range of A is dense in {h ∈ L2(R, ζ) : ζ(h) = 0};and if, in addition, the resolvent (λ− A)−1 is compact, then the range is closedand condition (2) above will hold for any g ∈ L2(R, ζ). Also, if

A =12

∑i,j

ai,j(x)∂2

∂xi∂xj+∑i

bi(x)∂

∂xi,

is an elliptic operator, with [ai,j(x)]i,j positive definite for each x, and bi and

ai,j being continuous functions bounded on compact supports, then Dom(A) =L2(Rk, ζ) and the range of A is dense in {h ∈ L2(R, ζ) : ζ(h) = 0}.

For simplicity, throughout we shall write A instead of A to signify that thearguments and results hold for the two above settings. The following processwill play a key role in the sequel:

Mt := f(r(t))− f(r(0))−∫ t

0

Af(r(u))du, (3.13)

which is well-known to be a square-integrable martingale relative to {F rt }t≥0

(see, e.g., [8] and [24, Lemma VIII.3.68]). Note that M will also be a martingalerelative to {FXt ∨ Fτt }t≥0. It is also proved in [24, Lemma VIII.3.68] that thequadratic variation of M is given by

〈M,M〉t =∫ t

0

γ(r(s))ds, (3.14)

where γ = Af2 − 2fAf. The following assumption will also be needed:

Assumption 3.4. The function γ(x) := Af2(x) − 2f(x)Af(x) satisfies eitherone of the following properties:


(a) γ is Lipschitz on R;(b) γ, γ2 ∈ Dom(A).(c) γ, γ2 ∈ Dom(A) and r is stationary.

We are ready to state the CLT for the statistics βn(ϕ).

Theorem 3.3. Suppose that all the conditions of Theorem 3.1 hold as well asStanding Assumptions 3.3 and 3.4. Let Γ := −2

∫Af(x)f(x)ζ(dx). Then,

T 1/2n

(βn(ϕ)− β(ϕ)

)D−→ N

(0, β(ϕ2) + β(ϕ)2Γ

), (3.15)

when Tn →∞ and δn → 0 such that Tnδn → 0.

Remark 3.4.

1. The CLT (3.15) will also hold true when ϕ(x) ∼ x2 as x→ 0 (resp. whenϕ(x) ∼ |x| as x → 0 and σ = 0) by replacing β(ϕ) with σ2 + β(ϕ) (resp.with β(ϕ) + |b|).

2. If ζ(g) is not assumed to be 1, then (3.15) will take the form

T 1/2n

(βn(ϕ)− ζ(g)β(ϕ)

)D−→ N

(0, ζ(g)β(ϕ2) + β(ϕ)2Γ

), (3.16)

3. Note that (3.15) can be written as

T 1/2n

(βn(ϕ)β(ϕ)

− 1

)D−→ N

(0,β(ϕ2)β(ϕ)2

+ Γ).

In particular, if one takes ϕ(x) = 1[a,b](x), with 0 /∈ [a, b], then

T 1/2n

(βn(ϕ)β(ϕ)

− 1

)D−→ N

(0,

1∫ bas(x)dx

+ Γ

).

Hence, the standard error of βn(ϕ) decreases with the size of the interval,but will never be 0 as there will be a constant component coming from theunobservable latent random clock.

4. Multivariate versions and some statistical applications

In light of the linearity of the operator ϕ → βn(ϕ), the multivariate version of(4.1) can be easily obtained. Concretely, given test functions ϕ1, . . . , ϕd satisfy-ing the conditions of Theorem 3.3, it follows that

T 1/2n

([βn(ϕ1), . . . , βn(ϕd)

]− [β(ϕ1), . . . , β(ϕd)]

)D−→ N (0,Σ) , (4.1)

with covariance matrix Σ = [σ2i,j ] given by

σ2i,j := σ2(ϕi, ϕj) := β(ϕiϕj) + Γβ(ϕi)β(ϕj). (4.2)

We now proceed to briefly discuss two application of this multivariate CLT, andalso, outline some methods for estimating the parameter Γ.


4.1. Sieve estimation and other nonparametric methods

As explained in the introduction, the integral parameter β(ϕ) =∫ϕ(x)s(x)dx

can be used to obtain approximations of the function s, leading to naturalestimation methods for s. For instance, in kernel estimation, we aim at es-timating the value of s at each point xi of a regular partition of a domainD := [a, b] ∈ R\{0} by using test functions of the form ϕi,h(x) = 1

hK((x−xi)/h),with a small enough bandwidth h > 0 and a kernel K : R → R+ such that∫K(x)dx = 1 and supp(K) ⊂ [−1, 1]. Then, s(xi) is estimated by

sn,h(xi) := βn(ϕi,h) =1hTn

n∑i=1

K

(∆nkX − xih

).

Another important method is the so-called method of sieves, originally pro-posed by Grenander [21] and implemented by Birge, Massart, and others (see,e.g., [7] & [10]) in several classical nonparametric problems. This approach con-sists of the following general steps. First, choose a family of finite-dimensionallinear models of functions, called sieves, with good approximation properties.Common sieves are splines, trigonometric polynomials, or wavelets. Second,specify a distance between functions relative to which the best approximationto s, in a given linear model, is going to be defined and characterized. Finally,devise an estimator, called the projection estimator, for the best approximationof s in the given linear model.

A linear model has the generic form

Sd := {β1ϕ1 + · · ·+ βdϕd : β1, . . . , βd ∈ R}, (4.3)

where ϕ1, . . . , ϕd are given functions, typically taken to be orthonormal withrespect to the inner product 〈p, q〉 :=

∫Dp(x)q(x)dx. Relative to the distance

induced by the norm ‖p‖2 :=∫Dp2(x)dx, the element of S closest to s, that is,

the orthogonal projection of s on S, is s⊥d (x) :=∑dj=1 β(ϕj)ϕj(x). Hence, the

following is a natural estimator for s:

sn,d(x) =d∑j=1

βn(ϕj)ϕj(x),

where the dimension of the linear model needs to be chosen appropriately. Notethat the larger the linear model Sd is, the smaller the bias of sn,d, but the largerthe variance of the estimator. A common method for choosing an adequatemodel S is a type of cross-validation, which consists of choosing the dimensiond such that −

∑di=1 βn(ϕi)2 + 2βn

(∑di=1 ϕ

2i

)is minimized (see [18, Section 5]

for further motivation).


Using (4.1), the following CLT for the sieve estimator sn,d can be easilydeduced:

T 1/2n

(sn,d(x)− s⊥d (x)

) D−→ N (0,Σd(x)) ,

with covariance matrix Σd(x) = [vdi,j(x)] given by

vdi,j(x) := β

( d∑i=1

ϕi(x)ϕi(·)

)2+ Γ ·

(d∑i=1

β(ϕi)ϕi(x)

)2

.

4.2. Estimation of the parameter Γ

In principle, Γ can be determined from the random clock. The problem of re-covering of the random clock τ(t) =

∫ t0g(r(u))du was recently considered in the

literature (cf. [37], [2], and [20]) under the sampling scheme δn → 0 and Tn = T .In all these papers, some assumptions about the divergence of ν(|x| ≥ a) whena→ 0 are needed. One popular assumption is to take a Levy density s that hasa stable-like behavior near the origin, such as

s(x) =1

xα+1(c0 +O(|x|)) , as x→ 0, (4.4)

for constants c0 > 0 and 0 < α ≤ 2. Once the random clock {τ(t)}t≥0 has beenrecovered, the following limits can be exploited for estimating ζ(g) and Γ(g):

limT→∞

E(

1Tτ(T )

)= ζ(g), lim

T→∞

1T

Var (τ(T )) = Γ(g),

using a type of “batch means method”. Concretely, for n and ∆ > 0 “large”such that n∆ = T , one could use the estimates:

ˆζ(g) :=1T

N∑i=1

Y(∆)i , Γ(g) :=

1T

N∑i=1

(Y

(∆)i − Y (∆)

i

)2

,

where Y (∆)i := τ(i∆)−τ((i−1)∆). However, we can expect that the convergence

will be slow and a large sampling time horizon T would be required.Another method is to consider a semiparametric model where the diffusion r

is modeled parametrically, while still describing Z non-parametrically. One canthen determine the parameters of the model r using the sample moments of thestationary sequences

I∆k := Xk∆ −X(k−1)∆, k = 1, . . . , n,

where we are assuming that r is stationary. For instance, the mean and varianceof I∆

k can be computed as follows:

E I∆k = c1(Z1)∆, Var

(I∆k

)= c2(Z1)∆ + 2c21(Z1)

∫ ∆

0

ρg(t)dt,


where ρg(t) = Cov (g(r(0)), g(r(t))), ρg(t) =∫ t

0ρg(s)ds, and ci := ci(Z1) is

the ith-cumulant of Z1. The first two cumulants are c1 = b +∫|x|≥1

xν(dx)and c2 = σ2 +

∫x2ν(dx). In the light of Remark 3.4, these cumulants can be

estimated via

c∆1,n :=1∆I∆n =

1n∆

Xn∆, and c∆2,n := βn(x2),

where I∆n is the sample mean of the I∆

k ’s. Note that, under mild conditions,

lim∆→0

1∆

∫ ∆

0

ρg(t)dt = 0,

lim∆→∞

1∆

Var (τ(∆)) = lim∆→∞

2∆

∫ ∆

0

ρg(t)dt = Γ(g).

Hence, in principle, c2(Z1) can be estimated by taking

c∆2,n :=1n∆

n∑i=1

(I∆k − I∆

n

)2= c∆2,n −∆

(c∆1,n

)2,

with ∆ “small” and, then, estimate Γ(g) via

Γ∆′

n′ (g) :=c∆′

2,n′ − c∆2,n(c∆1,n)2

by making ∆′ “large” such that T = ∆′n′ = ∆n. As before, we will expect thata large time horizon T will be needed (about 6 years according to preliminarynumerical experiments). A more efficient method could be to impose a paramet-ric model in r and estimate its parameters using a method of moments or thequasi likelihood estimation in the paper [5], which concentrates in the statisticalproperties of the realized variation for estimation purposes in a fully parametrictime-changed Levy model.

4.3. Estimation of the ratio of integral parameters

In this section, we apply the CLT (4.1) to “estimate” the parameters ϕ→ β(ϕ)up to a common normalizing constant β(ϕ0). Concretely, we set

β(ϕ) :=β(ϕ)β(ϕ0)

.

If, for instance, ϕ0(x) = 1(−∞,−a)(x) and ϕ(x) = 1(−∞,−b)(x) (0 < a < b), thenβ0 will represent the expected percentage of the jumps smaller than −a thatare smaller than −b per unit time. Note also that estimating ϕ → β(ϕ) willallow us to estimate non-parametrically the Levy density s up to a constant.This information could be useful for estimating the rate of convergence of s at∞ or at the origin if we consider a semi-parametric model such as (4.4).

The following result gives a multivariate CLT for such ratio parameters:


Proposition 4.1. Suppose that ϕ0, . . . , ϕn satisfy the conditions of Theorem3.3 and that β(ϕ0) ∈ (0,∞). Then, when Tn →∞ and Tnδn → 0,

T 1/2n

([βn(ϕ1)

βn(ϕ0), . . . ,

βn(ϕd)

βn(ϕ0)

]−[β(ϕ1)β(ϕ0)

, . . . ,β(ϕn)β(ϕ0)

])D−→ N

(0, Σ

), (4.5)

where Σ = [vi,j ]di,j=1 is given by

vi,j =1

β(ϕ0)(β(ϕ2

0)β(ϕi)β(ϕj)− β(ϕi)β(ϕ0ϕj)− β(ϕj)β(ϕ0ϕi) + β(ϕiϕj)).

Proof. We apply the δ-method (see, e.g., [34]) with the function f(θ0, . . . , θd) =[θ1/θ0, . . . , θd/θ0]. Then,

T 1/2n

(f(βn(ϕ0), . . . , βn(ϕd)

)− f (β(ϕ0), . . . , β(ϕd))

)D−→ N (0,∇fΣ∇f ′) ,

where Σ = [β(ϕiϕj) + Γβ(ϕi)β(ϕj)]di,j=0. Clearly,

∇f =

−β(ϕ1)β(ϕ0)

1β(ϕ0) 0 0 . . . 0

−β(ϕ2)β(ϕ0) 0 1

β(ϕ0) 0 . . . 0...

......

......

...−β(ϕd)β(ϕ0) 0 0 . . . . . . 1

β(ϕ0)

Thus, the (i, j) entry in Σ∇f ′ is given by

− β(ϕj)β(ϕ0)2

(β(ϕiϕ0) + Γβ(ϕ0)β(ϕi)) +1

β(ϕ0)(β(ϕiϕj) + Γβ(ϕi)β(ϕj))

= −β(ϕj)β(ϕiϕ0)β(ϕ0)2

+β(ϕiϕj)β(ϕ0)

,

for i = 0, . . . , d, j = 1, . . . , d, independently of Γ. Hence, the (i, j) entry of thecovariance matrix Σ = ∇fΣ∇f ′ has the form:

β(ϕ20)β(ϕi)β(ϕj)β(ϕ0)4

− β(ϕi)β(ϕ0ϕj)β(ϕ0)3

− β(ϕj)β(ϕ0ϕi)β(ϕ0)3

+β(ϕiϕj)β(ϕ0)2

.

The result is now clear.

Let us given a few consequences. Note that if ϕ0(x) = 1[a,b](x) for an interval[a, b] ⊂ R\{0} and supp(ϕi) ⊂ [a, b], i = 1, . . . , d, then

β(ϕ20) = 1, β(ϕ0ϕi) = β(ϕi), vi,j =

1β(ϕ0)

(β(ϕiϕj)− β(ϕi)β(ϕj)

).

Equivalently, if J is a random variable with density proportional to s(x)1[a,b](x),then vi,j = Cov (ϕi(J), ϕj(J)) /β(ϕ0). In that case, we also have that β(ϕi) =Eϕi(J).


One common application of a CLT is to devise asymptotic confidence inter-vals. For instance, in the above context, we have that

P

Tn( βn(ϕ1)

βn(ϕ0)− β(ϕ1)β(ϕ0)

)2

≤z2α/2

βn(ϕ0)

{βn(ϕ2

1)

βn(ϕ0)−(β(ϕ1)β(ϕ0)

)2} ≈ 1− α.

After some algebraic manipulations, an approximated CI for β(ϕ1)/β(ϕ0) canbe easily obtained. When Tn →∞ so that Tnβn(ϕ0) is large, such a CI behavesas follows:

βn(ϕ1)

βn(ϕ0)±zα/2T

−1/2n

βn(ϕ0)1/2

√√√√ βn(ϕ21)

βn(ϕ0)−

(βn(ϕ1)

βn(ϕ0)

)2

. (4.6)

5. Numerical illustrations

In this part we illustrate the finite-sample performance of our estimators usingsimulation experiments. All the figures in this part are deferred to the AppendixB. We assume that Z is a Normal Inverse Gaussian (NIG) process:

Z(t) = σZB(U(t)) + θ

ZU(t) + bt,

where σZ

and θZ

are constants, B is a Wiener process, and U is an InverseGaussian subordinator such that EU(t) = vt, for some v > 0. The NIG modelis an infinite-jump activity process with Levy density of the form

s(x) =C

|x|eAxK1(B|x|),

where K is the modified Bessel function of second kind, and

A =θ

Z

σ2Z

, B =1σ2

Z

√θ2

Z+σ2

Z

v, C =

1πσ

Z

√θ2

Z

v+σ2

Z

v2;

see, e.g., [14]. To simulate the increment of Z, we use the rejection method givenin [14], pp. 183.

The process r is assumed to be a Cox-Ingersoll-Ross (CIR) process

dr(t) = α(m− r(t))dt+ σr√r(t) dBr(t), (5.1)

where α, σr andm are positive constants such that αm/σ2r > 1/2, and {Br(t)}t≥0

is a standard Brownian motion independent of the Levy process Z. Under thestated conditions, (5.1) admits a unique weak positive solution with stationary


Γ( 2mασ2

r,σ2

r

2α ) distribution. Also, the conditional mean and variance of r(t) aregiven by

E [r(t)|r(0)] = r(0)e−αt +m(1− e−αt

),

Var (r(t)|r(0)) = r(0)σ2r

α

(e−αt − e−2αt

)+m

σ2r

2α(1− e−αt

)2.

We take g(x) := x. Clearly, (5.1) satisfies the linear growth condition (2.3) andall the conditions of Theorem 2.1 with ζ(g) = m. The infinitesimal generator Aof (5.1) is

Af(x) =12σ2rx∂2f

∂x2+ α(m− x)

∂f

∂x,

which domain includes all C2 functions with compact support and, hence, thedomain of A is dense in L2((0,∞), ζ). Note also that h(x) := g(x) − ζ(g) =x −m = Af , with f(x) = −x/α. In particular, the constant Γ in Theorem 3.3is determined by

Γ(g) =2α

∫(x−m)xζ(dx) =

2α

(2mασ2r

)(σ2r

2α

)2

=mσ2

r

α2.

In the following numerical experiments we take the following parameter values(units are years and a count convention of 252 days per year):

θZ

= −.080, σZ

= .091, v = .210,

b = .143, α = 1.763, m = 1, σr = 0.563.

These parameters are motivated by the recent empirical study of Li [26], wherethe previous model was fit to daily log returns of the S&P 500 stock index fromJanuary 1986 to December 2000 using a MCMC Bayesian estimation method.However, the speed of mean-reversion in [26] was found to be 71.93, a valuethat is relatively high (this corresponds to a half-time mean reversion of about4 days). Such a high value of α would typically result in a random clock thatfollows quite closely the identity function. To better assess the statistical meth-ods proposed here, we prefer to choose a much smaller value of 1.763, whichcorresponds to a half-time mean reversion of 4.7 months. Figure 1 shows a re-alization of the rate of time change r and the price process St = S0 exp(Xt)during a time interval of 12 years.

To illustrate the performance of the estimators, we consider test functions ofthe form ϕ(x) = 1x≥a for different values of a > 0. These test functions willallow us to estimate the right tail of the spectral Levy function a → ν([a,∞))of the process Z. We consider three different estimators for the spectral func-tion. The first estimator, called hereafter Est 1, is given by (1.4). The secondestimator, called hereafter Est 2, substitutes the time-horizon Tn in (1.4) by


the true (unobservable) random clock τ(Tn) =∫ Tn

0r(u)du. The last estimator,

called from now on Est 3, replaces the time horizon Tn by the estimator ofthe random clock τ(Tn) proposed in Woerner [37] based on bipower variations,which requires the knowledge of the Blumenthal-Getoor index β of Z and thelimit value c0 := limx→0 s(x)|x|1+β ∈ (0,∞), which existence is assumed. Notethat out of these three estimators, the only one that is feasible is the first one.The other two are possible here because we know the simulated true clock andalso, we know the true parameters of the model.

We first consider a short time horizon of 1 year and sampling frequenciesof 1 and 2 observations per day. Figure 2 shows the true spectral functiona→ ν([a,∞)) and the sample mean values of the three estimators described inthe previous paragraph based on 500 simulations. Concretely, we generate 500paths of the log return process t ∈ [0, 1] → Xt. For each path, we apply eachof the three estimators to estimate ν([aj ,∞)) for every aj in a partition of thetime interval [.003, .03] (mesh is .00001). We then computed the sample means

βn(1[aj ,∞)) =1

500

500∑i=1

β(i)n (1[aj ,∞)),

corresponding to each of the three estimators an each aj ∈ [.003, .03]. Figure2 shows the graphs of the means of the three estimators against true spectralfunction ν([aj ,∞))). We also compute the sampling standard error based onthe 500 simulations:

SEn,j =

√√√√ 1500

500∑i=1

(β

(i)n (1[aj ,∞))− βn(1[aj ,∞))

)2

,

The graphs of aj → βn(1[aj ,∞)) + SEn,j and a → βn(1[a,∞)) − SEn,j for Est1 are also shown in Figure 2. On average, the estimators are close to the truespectral measure (e.g. the mean of βn(1[aj ,∞))− β(1[aj ,∞)) over all ajs for theEst 1 is about .4160). However, the standard error is relatively large, especiallyfor large values of aj (e.g. the mean value of SEn,j over all aj for Est 1 is about1.8539). This is due to the short time horizon of 1 year. Note that the biasof the estimators reduces significantly when taking higher frequencies (from anaverage of 0.4160 for daily observations to 0.2152 for two observations per day);however, the standard error of the estimators does not seem be affected by achange in frequency.

We repeat the previous experiment for a time horizon of T = 3 years andsampling frequencies of 1 and 2 observations per day. Figure 3 shows the truespectral function aj → ν([aj ,∞)) and the sample mean values βn(1[aj ,∞)) of thethree estimators described above based on 500 simulations. Again, on average,the estimators are close to the true spectral measure. The bias of the estimators


does not reduced significantly compare to those of 1 year (e.g. the mean ofβn(1[aj ,∞))−β(1[aj ,∞)) over all ajs for the Est 1 changes from 0.4160 to 0.3946).However, the standard error of the estimators reduces significantly (e.g. themean value of SEn,j over all aj for Est 1 changes from 1.8539 for T = 1Yto 1.1179 for T = 3Y in the case of daily frequency). Again, the bias of theestimators reduces when taking higher frequencies (from 0.4160 for one obs. perday to 0.2152 for 2 obs. per day in the case of Est 1). The three estimatorsseems to be performing similarly.

For a long time horizon of 12 years (such as the one used in Li [26]), thestandard error of the estimator is reduced significantly (see Figure 4). For in-stance, in the case of daily sampling frequency, the mean of the standard errorof Est 1 reduces to 0.5694 (from a value of 1.1179 in the case of T = 3Y ). Al-though, the biases do not seem to improve significantly when T increases. Thestandard error of the estimators does not seem to vary greatly with the valueof aj . Of course, the relative error of the estimators will perform worst for largevalues of aj where ν([aj ,∞) is small. Again, the bias of the estimators reducessignificantly when taking higher frequencies (from 0.4590 for one obs. per dayto 0.1639 for 2 obs. per day in the case of Est 1). One important conclusion ofour results is that the bias of the estimators is affected mainly by the samplingfrequency, while the standard error is affected mainly by the time-horizon T .

To illustrate the normal-like distribution of the estimators, we show in Figure6 the sampling distribution of (1.4) taking ϕ(x) = 1x>.0212. It turns out thatβ(ϕ) = ν([.0212,∞)) is approximately equal to 1. Finally, we assess the cover-age probability of the asymptotic confidence interval (4.6). In order to do this,we construct confidence intervals for ν([aj ,∞)/ν([.003,∞)) for each aj in a par-tition of [0.003, .03] with a mesh of .0001. The coverage ratios of these CI’s fortime horizons T = 6 and T = 12 using a sampling frequency of 2 observationsper day are shown in Figure 5. The nominal value is .95. We observer in generalthat the intervals are below their nominal value especially for larger values aj .

Appendix A: Proofs of the main results

Proof of Theorem 3.1.

(1) For simplicity, we will drop the superscript n in the sampling times tnk .The first part of the proof follows closely the proof of Lemma 5.1 in [19] and isomitted. In there, it is shown that it suffices to check that

limn→∞

1Tn

n∑k=1

∫ tk−tk−1

0

E (r(tk−1 + s)− r(tk−1))2ds = 0 (A.1)

We now show that any of the Conditions 3.1 implies (A.1). First, let us considerassumption 3.1-(1). By the Markov’s property, E (r(tk−1 + s)− r(tk−1))2 can


be written as

E[Tsg

2(r(tk−1)) + g2(r(tk−1))− 2g(r(tk−1))Tsg(r(tk−1))], (A.2)

for s ∈ [0, tk− tk−1]. Using that Tsf(x) = f(x) + sAf(x) + o(1)s, with o(1)→ 0as s→ 0 uniformly in x, (A.2) can be written as

E[Ag2(r(tk−1))− 2g(r(tk−1))Ag(r(tk−1))

]s+ o(1)s,

where o(1) → 0 as n → 0 uniformly in k = 1, . . . , n and in s ∈ [0, tk − tk−1],provided that δn → 0. Since g, g2 ∈ Dom(A), the functions g, Ag and Ag2 arebounded, and (A.1) can be bounded by an expression like K

∑nk=1 (δnk )2

/Tn,

for large n and some constant K <∞. Hence, it vanishes as n→ 0.Let us now assume that Condition 3.1 (2) holds. Using Markov’s property,

the stationarity of r, and the definition of A, for any s > 0,

E (r(s+ tk−1)− r(tk−1))2 =∫ (

Tsg2(x) + g2(x)− 2g(x)Tsg(x)

)ζ(dx) (A.3)

= s

∫(Ag2 − 2gAg)ζ(dx) +O(1)s, (A.4)

where as before o(1) → 0 as n → 0, uniformly in k = 1, . . . , n and in s ∈[0, tk − tk−1]. This implies (A.1) as before.

(2) See [19, Proposition 5.2] for its proof.

(3) The idea is to consider the following decomposition:

T 1/2n (µn(ϕ)− β(ϕ)) = β(ϕ)T−1/2

n

∫ Tn

0

(g(r(u))− ζ(g)

)du+Rn,

where

Rn := T−1/2n

n∑k=1

E[(

1∆nkτHϕ(∆n

kτ)− β(ϕ))

∆nkτ

∣∣∣∣Fτtk−1

](A.5)

+ β(ϕ)T 1/2n

{1Tn

n∑k=1

E[

∆nkτ | Fτtk−1

]− 1Tn

∫ Tn

0

r(u)du

}. (A.6)

In view of (2.7), it suffices that Rn converge to 0 in probability. The term onthe right-hand side of (A.5) is denoted An. Note that without loss of generality,(3.10) can be assumed for all t > 0. Then,

P [|An| ≥ ε] ≤k0

εT−1/2n

n∑k=1

E (∆nkτ)2

≤ k0

εT−1/2n

n∑k=1

(tk − tk−1) E∫ tk

tk−1

r2(u)du ≤ m2(g)k0

εT 1/2n δn,


which converges to 0. Next, let Bn be the expression in (A.6). Using Jensen’sinequalities together with the inequality

E(r(u)− E

[r(u)| Fτtk−1

])2

≤ E (r(u)− r(tk−1))2, (A.7)

we have that

E |Bn|2 ≤ β(ϕ)2n∑k=1

∫ tk

tk−1

E (r(u)− r(tk−1))2du,

To show that the later expression vanishes when Tnδn → 0, we proceed as beforewhen proving (A.1).

Proof of Theorem 3.3. Consider the martingale difference sequences

ξ(1)k,n = T−1/2

n

(ϕ (∆Xn

k )− E [Hϕ(∆nkτ)|Fτtk−1

]), ξ

(2)k,n = T−1/2

n (Mtnk−Mtn

k−1),

relative to the filtration (3.4). Let

S(1)n =

∞∑k=1

ξ(1)k,n, S(2)

n =∞∑k=1

ξ(2)k,n,

where we had fixed ξ(1)k,n = ξ

(2)k,n = 0, for k > n. Using [9, Theorem 18.1] and the

Cramer-Wold device, we can show that

(S(1)n , S(2)

n ) D−→ N (0, C), (A.8)

with covariance matrix C = [Ci,j ]i,i=1,2 if, for i, j ∈ {1, 2} and any ε > 0,

(i)∞∑k=1

E[|~ξk,n|21|~ξk,n|≥ε

]→ 0, (ii)

∞∑k=1

E[ξ

(i)k,nξ

(j)k,n

∣∣∣Fnk−1

]P→Ci,j , (A.9)

as n→∞, where ~ξk,n = (ξ(1)k,n, ξ

(2)k,n). We shall show (A.9) with

C1,1 := β(ϕ2), C2,2 := Γ = −2∫Af(x)f(x)ζ(dx), C1,2 = C2,1 := 0. (A.10)

Note that the result of the theorem will follow easily from (A.8) with C givenas in (A.10). Indeed,

S(1)n − β(ϕ)S(2)

nD−→ N

(0, β(ϕ2) + β(ϕ)2Γ

). (A.11)

Also, T 1/2n

(βn(ϕ)− β(ϕ)

)can be decomposed as follows:

S(1)n − β(ϕ)S(2)

n + β(ϕ)T−1/2n (f(r(Tn))− f(r(0))) +Rn, (A.12)


where Rn is defined as in (A.5), where it was proved that it converges to 0 inprobability. The second term in (A.12) also vanishes in probability under anyof the Conditions 3.3. For instance, if 3.3-(2) holds, Markov’s inequality andstationarity implies:

P(T−1/2n (f(r(Tn))− f(r(0))) > ε

)≤ 4ζ(f2)ε−2

Tn→ 0.

It is now clear that (A.8) will imply (A.11) and, hence, (3.15). We now proceedto show the sufficient conditions (A.9) for the validity of (A.8).

• Verification of (A.9)-(i):One can easily check that

E[∣∣∣~ξk,n∣∣∣2 1|~ξk,n|≥ε

]≤ 4 E

[(ξ

(1)k,n

)2

1|ξ(1)k,n|≥ ε√

2

]+ 4 E

[(ξ

(2)k,n

)2

1|ξ(2)k,n|≥ ε√

2

]Thus, (A.9)-(i) will follow if

∞∑k=1

E[(ξ

(1)k,n

)2

1|ξ(1)k,n|≥ε

]n→∞−→ 0,

∞∑k=1

E[(ξ

(2)k,n

)2

1|ξ(2)k,n|≥ε

]n→∞−→ 0, (A.13)

for any ε > 0. The first limit above was shown in the proof of Proposition3.1 given in [19] (see Proposition 5.2 therein). If f satisfies Conditions 3.3-(1),then the second limit in (A.13) follows directly since f is bounded and thus,1|ξ(2)

k,n|≥ε ≡ 0 for large n. Suppose that f satisfies Conditions 3.3-(2). First, note

that E [(ξ(2)k,n)21|ξ(2)

k,n|≥ε] can be bounded by

4Tn

E[(f(r(tk))− f(r(tk−1)))2 1|f(r(tk))−f(r(tk−1))|≥ ε

2T1/2n

](A.14)

+4Tn

E

(∫ tk

tk−1

Af(r(u))du

)2

1∣∣∣∫ tk

tk−1Af(r(u))du

∣∣∣≥ ε2T

1/2n

. (A.15)

Let us analyze each of the two above expectations. The term (A.14) can bebounded by the following term

4ε−2

TnE[(f(r(tk))− f(r(tk−1)))4

]=

4ε−2

Tn

(α(f)δnk + o(δn)

),

where α(f) := −4ζ(Af3 ·f)+6ζ(Af2 ·f2)−4ζ(Af ·f3). For the above equality wehave used the stationarity of r and the definition of A (similar to the procedurein (A.3-A.4)). On the other hand,

E

(∫ tk

tk−1

Af(r(u))du

)2

1∣∣∣∫ tk

tk−1Af(r(u))du

∣∣∣≥ ε2T

1/2n

≤ (δnk )2ζ((Af)2).


It is now evident that, for some constants K1,K2 <∞,

∞∑k=1

E[(ξ

(2)k,n

)2

1|ξ(2)k,n|≥ε

]≤ K1

T 2n

n∑k=1

δnk +K2

Tn

n∑k=1

(δnk )2 → 0,

as n→∞.

• Verification of (A.9)-(ii):

(a) Case i = j = 1. This was shown in the proof of Proposition 3.1-(3).

(b) Case i = j = 2. Using the quadratic variation formula (3.14),

∞∑k=1

E[(ξ

(2)k,n

)2∣∣∣∣Fnk−1

]=

1Tn

n∑k=1

E

[∫ tk

tk−1

γ (r(s)) ds

∣∣∣∣∣Fnk−1

].

Let

Gn :=1Tn

n∑k=1

E

[∫ tk

tk−1

γ (r(s)) ds

∣∣∣∣∣Fnk−1

]− 1Tn

∫ Tn

0

γ (r(s)) ds.

Using Jensen’s inequality,

EG2n ≤

1Tn

n∑k=1

∫ tk

tk−1

E{(γ(r(s))− E

[γ (r(s))| Fnk−1

])2}ds

≤ 1Tn

n∑k=1

∫ tk

tk−1

E{

(γ(r(s))− γ (r(tk−1)))2}ds

In order to estimate the above quantity, we shall use the Conditions 3.4. In thecase where the Condition 3.4 (a) holds, and r solves (2.2) under (2.3) and (2.5),we can use the Lipschitz condition of γ and then, apply (2.4) with m = 1 toobtain that

EG2n ≤ K

1Tn

n∑k=1

(δnk )2 n→0−→ 0,

for some K <∞. To show that Condition 3.4 (b) implies also EG2n → 0, note

that, for s ∈ [tk−1, tk] and fixing d = s−tk−1, E{

(γ(r(s))− γ (r(tk−1)))2∣∣∣Fnk−1

}equals

Tdγ2(r(tk−1)) + γ2(r(tk−1))− 2γ(r(tk−1))Tdγ(r(tk−1))

=(Aγ2(r(tk−1))− 2γ(r(tk−1))Aγ(r(tk−1)

)d+ o(δn),

where o(δn) is independent of r(tk−1) and k. Since Aγ2(x) − 2γ(x)Aγ(x) arebounded in x by say K (since by assumption γ and γ2 belong to Dom(A)),

EG2n ≤ K

1Tn

n∑k=1

∫ tk

tk−1

(s− tk−1)ds+ o(δn) n→0−→ 0.


A similar conclusion will follow if Condition 3.4 (c) holds since, for s ∈ [tk−1, tk]and fixing d = s− tk−1,

E{

(γ(r(s))− γ(r(tk−1)))2}

=∫ (

Tdγ2(x) + γ2(x)− 2γ(x)T

dγ(x)

)ζ(dx)

= d

∫(Aγ2 − 2γAγ)d ζ + o(δn).

In the three cases of Conditions 3.4, GnP→ 0 as n → ∞ and using the Ergodic

theorem,

∞∑k=1

E[(ξ

(2)k,n

)2∣∣∣∣Fnk−1

]=

1Tn

∫ Tn

0

γ (r(s)) ds+GnP−→∫γ(x)ζ(dx).

One can check that∫Af2(x)ζ(dx) = 0 and thus,

∫γ(x)ζ(dx) = −2

∫f(x)g(x)dx.

(c) Case i = 1 and j = 2. We first note that

E[ϕ(Xtk −Xtk−1

)∣∣Fτtk] = Hϕ(∆nkτ).

Conditioning on Fnk−1∨σ(τ(u)− τ(tk−1) : u ∈ [tk−1, tk]) and using the previous

fact and the independence between X and r, E[ξ

(1)k,nξ

(2)k,n

∣∣∣Fnk−1

]can be written

as

1Tn

E[(Mtk −Mtk−1

) (Hϕ(∆n

kτ)− E[Hϕ(∆n

kτ)| Fτtk−1

])∣∣∣Fτtk−1

].

Let

An := E

∣∣∣∣∣n∑k=1

E[ξ

(1)k,nξ

(2)k,n

∣∣∣Fnk−1

]∣∣∣∣∣ , Bk,n := E[(Mtk −Mtk−1

)2],

Ck,n := E[(Hϕ(∆n

kτ)− E[Hϕ(∆n

kτ)| Fτtk−1

])2].

Clearly, An ≤ (1/Tn)∑nk=1B

1/2k,nC

1/2k,n , and also,

Bk,n ≤ 2 E[(f(r(tk))− f(r(tk−1)))2

]+ 2 E

(∫ tk

tk−1

Af(r(u))du

)2 . (A.16)

Under any of the Conditions 3.3, the first term can be bounded by a term ofthe form Kδnk when n is large enough (using arguments similar to those after(A.2)). Using Jensen’s inequality and the boundedness of Af (or stationarity ofr and Af ∈ L2(I, ζ)), the second term in (A.16) can be bounded by a term ofthe form K ′(δnk )2. Thus, for large n, Bk,n ≤ Kδnk , for some constant K <∞. On


the other hand, using the decomposition Hϕ(t) = (Hϕ(t)/t− β(ϕ)) t + β(ϕ)t,(3.10), and (2.12),

Ck,n ≤ K1 E[(∆n

kτ)2]

+K2 E[(

∆nkτ − E

[∆nkτ | Fτtk−1

])2],

for some constants K1,K2 <∞. Using Jensen’s inequality and (A.7)

Ck,n ≤ K1δnk

∫ tk

tk−1

E g2(r(u))du+K2δnk

∫ tk

tk−1

E[(g(r(s)− r(tk−1))2

]ds.

Using (2.5-ii), for larger n, Ck,n ≤ K(δnk )2, for some K <∞. We deduce that

An ≤K

Tn

n∑k=1

(δnk )3/2 → 0,

as n→∞, and thus,∑∞k=1 E

[ξ

(1)k,nξ

(2)k,n

∣∣∣Fnk−1

]P→ 0.

Appendix B: Figures


0 2 4 6 8 10 120.5

1

1.5

2Time−Changed Normal Inverse Gaussian Process

Time (in years)

Sto

ck P

rice

and

Log

Ret

urn

Incr

emen

ts

Stock Price

Log Returns

thetaZ=−.080; σ

Z=.091;

νZ=.210; b=.143;

α=1.763; m=1; v=0.563;N=12*(252); δ=1/252

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5CIR speed process of the Random Clock

Time (in Years)

Spe

ed o

f ran

dom

clo

ck (r

)

Fig 1. (Left Panel) Price process of a NIG timed-changed Levy model. (Right Panel). Rateprocess r of the random clock following a CIR model.


0 0.005 0.01 0.015 0.02 0.025 0.03

0

5

10

15

20

25

Time−Changed Normal Inverse Gaussian ProcessRecovery of Spectral function a −> ν((a,∞))

TrueEst 1Est 2Est 3StdError 1StdError 1

θZ=−.080; σ

Z=.091;

νZ=.210; b=.143;

α=1.763; m=1;v=0.563;T=1 y; Freq.= 1/d;500 smltns.

Mean of True − Mean of Est (Mean of Standard Error).Est 1: 0.4160 (1.8539); Est 2: 0.3872 (1.6507);Est 3: 0.7622 (1.7375)

0 0.005 0.01 0.015 0.02 0.025 0.03

0

5

10

15

20

25


TrueEst 1Est 2Est 3StdError Est 1StdError Est 1

θZ=−.080; σ

Z=.091;

νZ=.210; b=.143;

α=1.763; m=1;v=0.563;T=1 y; Freq.=2/d;500 smltns.

Mean of True − Mean of Est (Mean of Standard Error).Est 1: 0.2152 (1.7623); Est 2: 0.1525 (1.5820);Est 3: 0.3113 (1.5921)

Fig 2. Recovery of the spectral function a → ν([a,∞)) of the Levy process using 1 year ofobservations, and 1 and 2 observations per day.

0 0.005 0.01 0.015 0.02 0.025 0.03

0

5

10

15

20

25

Time−Changed Normal Inverse Gaussian ProcessRecovery of Spectral function a −> ν((a,∞)


Mean of True − Mean of Est(Mean of Standard Error).Est 1: 0.3946 (1.1179); Est 2: 0.4106 (0.9328);Est 3: 0.7932 (0.9983).

θZ=−.080; σ

Z=.091;

νZ=.210; b=.143;


0 0.005 0.01 0.015 0.02 0.025 0.03

0

5

10

15

20

25



θZ=−.080; σ

Z=.091;

νZ=.210; b=.143;



Fig 3. Recovery of the spectral function a → ν([a,∞)) of the Levy process using 3 years ofobservations, and 1 and 2 observations per day.


0 0.005 0.01 0.015 0.02 0.025 0.03

0

5

10

15

20

25



θZ=−.080; σ

Z=.091;

νZ=.210; b=.143;


Mean of True − Mean of Est(Mean of Standard Error).Est 1: 0.4590 (0.5694); Est 2: 0.4061 (0.4485;Est 3: 0.8422 (0.5015).

0 0.005 0.01 0.015 0.02 0.025 0.03

0

5

10

15

20

25



θZ=−.080;

σZ=.091;

νZ=.210; b=.143;



Fig 4. Recovery of the spectral function a→ ν([a,∞)) of the Levy process using 12 years ofobservations, and 1 and 2 observations per day.

Fig 5. Sample coverage rate of the confidence interval 4.6 with a nominal coverage rate of95%. Rates computed based on 100 CIs constructed from two observations of the process perday during time intervals [0, 6] and [0, 12 years, respectively.


0 0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

120

Samplig Distribution for the estimator of ν(x>.0212)=1T=3 years; Frq = 1 obs/day; 500 Observations

Mean = 1.0693 Std Dev = 0.59658

0 0.5 1 1.5 2 2.5 3 3.50

20

40

60

80

100

120


Mean = 1.0993 Std Dev = 0.65054

0 0.5 1 1.5 2 2.50

50

100

150


Sample Mean = 1.0713 Sample Std Error = 0.3138

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

20

40

60

80

100

120


Sample Mean = 1.0508 Sample Std Dev = 0.29995

Fig 6. Sampling distribution of Est 1 (1.4) based on 500 samples of {Xt} with time horizonsT = 3 and T = 12 years and sampling frequencies of 1 and 2 observations per day.

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