Given a positive number and an increasing Lévy process without drift component, an Ornstein-Uhlenbeck (OU) process driven by is defined by a solution to the following stochastic differential equation
We refer to Sato (1999) and Bertoin (1996) as standard references on Lévy processes. In this paper, we consider nonparametric inference of the Lévy measure of the back-driving Lévy process in (1.1) from discrete observations of . The Lévy measure is a Borel measure on such that
is self-decomposable with the characteristic function
In this paper, we focus on the case that the Lévy process in (1.1) is a compound Poisson process, that is, is of the form
where is a Poisson process with intensity and
is a sequence of independent and identically distributed positive-valued random variables with common distribution. In this case, has a characteristic function of the form
and the Lévy measure is given by . We also work with macroscopic observation set up, that is, we have discrete observations at frequency with and as .
The goal of this paper is to develop nonparametric inference on the Lévy measure of Lévy-driven Ornstein-Uhlenbeck process (1.1). For this, we propose a spectral (or Fourier-based) estimator of the -function and derive multivariate central limit theorems for the estimator over finite design points. As extension of these results, we also derive high-dimensional central limit theorems for the estimator in the case that design points over an compact interval included in increases as the sample size goes to infinity. Build upon those limit theorems, we develop methods for the implementation of confidence bands. Since confidence bands provide a simple graphical description of the accuracy of a nonparametric curve estimator quantifying uncertainties of the estimator simultaneously over design points, they are practically important in statistical analysis. Moreover, we propose a practical method for bandwidth selection inspired by Bissantz et al. (2007). As a result, this paper contributes the literature on nonparametric inference of Lévy-driven stochastic differential equations, and to the best of our knowledge, this is the first paper to establish limit theorems for nonparametric estimators for the Lévy measure of Lévy-driven OU processes.
Lévy-driven OU processes are widely used in modeling phenomena where random events occur at random discrete times. See, for example, Albrecher et al. (2001), Kella and Stadje (2001) and Noven et al. (2014) for applications to insurance, dam theory, and rainfall models. Several authors investigate parametric inference on Lévy-driven OU processes driven by subordinators. We refer to Hu and Long (2009), Masuda (2010) and Mai (2014) under the high-frequency set up (i.e., as ) and Brockwell et al. (2007) under the low-frequency set up (i.e., is fixed). There are also a large number of studies on parametric and nonparametric estimation and inference on Lévy processes. Recent contributions include Duval and Hoffmann (2011) and Duval (2014) for parametric inference, and Duval and Kappus (2018) for nonparametric estimation on compound Poisson processes under macroscopic observations, Nickl et al. (2016) for inference on Lévy measures, Figueroa-López (2011), Konakov and Panov (2016), and Kato and Kurisu (2017) for inference on Lévy densities, and Kappus and Reiß (2010) for estimation on Lévy densities under the high-frequency set up, van Es et al. (2007), Gugushvili (2009, 2012), Neumann and Reiß (2009), Kappus and Reiß (2010), Belomestny (2011), Duval (2013), Kappus (2014), Belomestny and Reiß (2015), and Belomestny and Schoenmakers (2016) for estimation of Lévy densities, and Nickl and Reiß (2012) and Coca (2018) for inference on Lévy measures, under the low-frequency set up. We also refer to Belomestny et al. (2017) for nonparametric estimation on Lévy measures of moving average Lévy processes under low-frequency observations, and Bücher and Vetter (2013), Vetter (2014), Bücher et al. (2017), and Hoffmann and Vetter (2017) for inference on Lévy measures under high-frequency observations. Jongbloed et al. (2005) and Ilhe et al. (2015) investigate nonparametric estimation of the Lévy-driven OU processes. Jongbloed et al. (2005) derives consistency of their estimator for a class of Lévy-driven OU processes which include compound Poisson-driven OU processes. Ilhe et al. (2015) establish consistency of their estimator of the Lévy density of (1.1) with compound Poisson subordinator in uniform norm at a polynomial rate. However, they does not derive limit distributions of their estimators.
investigate probability density deconvolution problems for-mixing sequences and they derive convergence rate and asymptotic distributions of deconvolution estimators. Since the Lévy-driven OU process (1.1) is -mixing under some condition (see Masuda (2004) for details), our analysis can be interpreted as a deconvolution problem for a -mixing sequence. However, we need non-trivial analysis since we have to take account of additional structures which come from the properties of the Lévy-driven OU process.
The estimation problem of Lévy measures is generally ill-posed in the sense of inverse problems and the ill-posedness is induced by the decay of the characteristic function of a Lévy process as explained in Nickl and Reiß (2012). In our case, the ill-posedness is induced by the decay of the characteristic function of the stationary distribution of the Lévy-driven OU (1.1). In this sense the problem in this paper is a (nonlinear) inverse problem. Trabs (2014a) investigates conditions that a self-decomposable distribution is nearly ordinary smooth, that is, the characteristic function of the self-decomposable distribution decays polynomially at infinity up to a logarithmic factor. Trabs (2014b) applies those results to nonparametric calibration of self-decomposable Lévy models of option prices. As a refinement of a result in Trabs (2014a), we will show that the characteristic function of a self-decomposable distribution is regularly varying at infinity with some index . This enables us to derive asymptotic distributions of the spectral estimator proposed in this paper.
Kato and Sasaki (2016) is a recent contribution in the literature on the construction of uniform confidence bands in probability density deconvolution problems for independent and identically distributed observations. They develop methods for the construction of uniform confidence bands build on applications of the intermediate Gaussian approximation theorems developed in Chernozhukov et al. (2014a, b, 2015, 2016) and provides multiplier bootstrap methods for the implementation of uniform confidence bands. Kato and Kurisu (2017) also develops confidence bands for Lévy densities based on the intermediate Gaussian and multiplier bootstrap approximation theorems. Our analysis is related to these papers but we adopt different methods for the construction of confidence bands. We derive high-dimensional central limit theorems based on intermediate Gaussian approximation for
-mixing process. We can show that the variance-covariance matrix of the Gaussian random vector appearing in multivariate and high-dimensional central limit theorems is the identity matrix. Therefore, we do not need bootstrap methods to compute critical values of confidence bands.
The rest of the paper is organized as follows. In Section 2, we define a spectral estimator for -function. We give multivariate central limit theorems of the spectral estimator in Section 3 and high-dimensional central limit theorems of the estimator are also given in Section 4. Procedures for the implementation of confidence bands are described in Section 5. In Section 6, we propose a practical method for bandwidth selection and report simulation results to study finite sample performance of the spectral estimator. All proofs are collected in Appendix A.
For any non-empty set and any (complex-valued) function on , let , and for , let for . For any positive sequences , we write if there is a constant independent of such that for all , if and , and if as . For , let . For and , we use the shorthand notation . The transpose of a vector is denoted by . We use the notation as convergence in distribution. For random variables and , we write if they have the same distribution. For any measure on , its support is denoted by .
2. Estimation of function
In this section, we introduce a spectral estimator of the Lévy measure (-function) of the Lévy-driven Ornstein-Uhlenbeck process (1.1). First, we consider a symmetrized version of the -function, that is,
A simple calculation yields that
Therefore, we have that
This yields that
Here, is a sequence of constants such that as (in the rest of this paper, we set ). Let be an integrable function (kernel) such that
and its Fourier transformis supported in (i.e., for all ). Then the spectral estimator of for is defined by
for , where is a sequence of positive constants (bandwidths) such that as , and
In the following sections we develop central limit theorems for .
For a complex value , let be the complex conjugate of . We note that is real-valued. In fact, since and , by a change of variables, we have that
Another natural estimator of for would be
but this estimator have large bias than . We need to symmetrize -function to use global regularity of for suitable bound of the deterministic bias. Note that is continuous at the origin and if has bounded th derivative on for some , the deterministic bias of is (Lemma A.10 in Appendix A). However, the deterministic bias of is because of the discontinuity of at the origin.
3. Multivariate Central Limit Theorems
In this section we present multivariate central limit theorems for .
We assume the following conditions.
for some .
Let , and let be the integer such that . The function is -times differentiable, and is -Hölder continuous, that is,
and , where is the Fourier transform of .
Let be an integrable function such that
where is the Fourier transform of .
, , and
for some as .
Conditions (i) and (ii) imply that the stationary distribution has a bounded continuous density (we also denote the density by ) such that and (see Lemma A.1). In this case, the stationary Lévy-driven Ornstein-Uhlenbeck process defined by (1.1) is exponentially -mixing (Theorem 4.3 in Masuda (2004)), that is, the -mixing coefficients for the stationary continuous-time Markov process
Condition (iii) is concerned with smoothness of . Condition (iv) is satisfied if is two-times continuously differentiable on and . In fact, by Condition (i), we have that for , and by integration-by-parts and the Riemann-Lebesgue theorem, we also have that
Condition (v) is concerned with the kernel function . We assume that is a -th order kernel, but allow for the possibility that . Note that since the Fourier transform of has compact support, the support of the kernel function is necessarily unbounded (see Theorem 4.1 in Stein and Weiss (1971)).
Condition (vi) is concerned with the sampling frequency, bandwidth, and the sample size. The condition implies that we work with macroscopic observation scheme and this is a technical condition for the inference on . We note that for the estimation of uniformly on an interval , we do not need the condition and we can work with low-frequency set up (i.e., is fixed). Masry (1993b) investigates deconvolution problems for mixing sequences and assume that for a mixing sequence , joint densities of and are uniformly bounded for any and to show the asymptotic independence of their estimators at different design points. Although we also observe -mixing sequence , we cannot assume such a condition in the paper directly in our situation since the transition probability of has a point mass at , does not have a transition density function. Moreover, for each fixed , the absolutely continuous part of is uniformly bounded for any , however, it would be difficult to check uniform boundedness of for any and . Zhang et al. (2011) investigates the transition law of compound Poisson-driven OU processes. Define the function
and let be its -th convolution. Note that
is a probability density function (Lemma 1 inZhang et al. (2011)). They derives the concrete form of the transition probability , that is,
This implies that does not have a Lebesgue density and it has a point mass at . In this case, can be written as
Suppose for fixed . Then we can show that for any and we have that
Therefore, in this case, we have that . In particular,
for a wide class of compound Poisson-driven OU processes.
Suppose that the jump distribution has a probability density function such that
Then (3.1) is satisfied.
For example, the density functions of exponential, half-normal, and half- distributions satisfy the assumption of Lemma 3.1. In general, probability density functions which is non-increasing on satisfy the assumptions.
A compound Poisson-driven OU process with a gamma jump distribution with shape parameter and rate parameter satisfies (3.1).
From a theoretical point of view, we could assume and in that case, can be fixed, that is, we can work with low-frequency setup. However, it would be difficult to check the condition directly or to give some sufficient conditions which can be checked for a given compound Poisson-driven OU process. We can avoid the problem if we assume the condition since we do not need the uniform bound of under the condition to show the asymptotic validity of (defined by (3.5)) and the asymptotic independence of and for with (see the proof of Lemma A.8, Proposition A.2, and Proposition A.3). From a practical point of view, our methods can be applied to low-frequency data and would work well if we suitably rescale the time scale of the data and the sample size is sufficiently large. In our simulation study, we consider the case when and our method works well in this case. We also need the condition (vi) to derive the lower bound of for the uniform consistency of for , with . We need the upper bound of for the undersmoothing condition. See also Remark A.1 after the proof of Proposition A.2 in Appendix A for the further discussion on , and Remark 3.5 in the present paper for a comment on the condition on .
To state a multivariate central limit theorem for , we introduce the notion of regularly varying functions.
Definition 3.1 (Regularly varying function).
A measurable function is regularly varying at with index (written as ) if for ,
Assume Condition (ii) in Assumption 3.1. There exist a function slowly varying at and a constant such that
Condition (ii) in Assumption 3.1 is concerned with smoothness of the stationary distribution of the Lévy-driven OU process. Condition (ii) implies that the stationary distribution is nearly ordinary smooth, that is, the characteristic function (1.2) decays polynomially fast as (Lemma 3.3) up to a logarithmic factor. Since , the finiteness of is equivalent to the finiteness of the total mass of the Lévy measure of the Lévy process and this means that the Lévy process is of finite activity, i.e., it has only finitely many jumps in any bounded time intervals. It is known that a Lévy process with a finite Lévy measure is a compound Poisson process. If , the Lévy process is of infinite activity, i.e., it has infinitely many jumps in any bounded time intervals. In this case, the characteristic function (1.2) decays faster than polynomials. In particular, it decays exponentially fast as if the Blumenthal-Getoor index of is positive, that is, if
Inverse Gaussian, tempered stable, normal inverse Gaussian processes are included in this case, for example. Condition (ii) rules out those cases since we could not construct confidence bands based on Gaussian approximation under our observation scheme (see also the comments after Assumption 10 in Kato and Sasaki (2016)). Therefore, we need the nearly ordinary smoothness of in our situation to drive practical asymptotic theorems for the inference on .
Lemma 3.3 implies that is a regularly varying function at with index . A slowly varying function may go to as but it does not glow faster than any power functions, that is,
for any . In fact, if , from Proposition 1 in Trabs (2014a), we have that
for any . Such a tail behavior of is related to Condition (vi) in Assumption 3.1. If the stationary distribution is ordinary smooth, that is, satisfies the relation
for some , we can set in Condition (vi). However, we need to introduce to take into account the effect of the slowly varying function .
As shown in (A.8) and the comments below, without the condition
we have that
where the second term of the right hand side comes from the deterministic bias. For central limit theorems to hold, we have to choose bandwidth so that the bias term is asymptotically negligible relative to the first term or “variance” term. The right hand side is optimized if . Since for any by Lemma 3.3 (see Remark 3.4), we have that if we take to satisfy Condition (vi) in Assumption 3.1.
Let . Under Assumption 3.1, we have that
where is the by diagonal matrix with
In Theorem 3.1, we also need for . However, this condition is automatically satisfied under Assumption 3.1. Let be the Lévy measure of . As explained in the introduction of the present paper, the stationary distribution is a self-decomposable distribution with Lévy density , which is a Lebesgue density of . By the definition of -function, we have that and . In this case, by Theorem 24.10 in Sato (1999). Moreover, the density of the stationary distribution satisfies that for any by Theorem 2 in Yamazato (1978) since and the density is continuous on (see Lemma A.1).
Under Assumption 3.1, we can show that can be approximated by
where . By a change of variables, we may rewrite the right hand side of (3.3) as
where is a function defined by
is well-defined and real-valued. To construct confidence interval of, we estimate the variance of , which is , by
Now we can show the next feasible multivariate central limit theorem.
Assume Assumption 3.1. Then for any , we have that
where is the by identity matrix and .
4. High-dimensional Central Limit Theorems
In Section 3, we give multivariate (or finite-dimensional) central limit theorems of . In this Section, we give high-dimensional central limit theorems as refinements of Theorems 3.1 and 3.2. As an application of those results, we propose some methods for the construction of confidence bands for -function in Section 4.2.
4.1. High-dimensional central limit theorems for
For and , let
and let an interval with finite Lebesgue measure , , , . We assume that
and this implies that . Therefore, can go to infinity as .
for any , Lemma 4.1 implies that
Theorem 4.1 can be shown in two steps. As a first step, we approximate the distribution of by that of . Here, is a centered normal random vector with covariance matrix where is a sequence of integers with and as , and