Nonparametric inference for Le´vy-driven
Ornstein–Uhlenbeck processes
G . J O N G B L O E D * , F. H . VA N D E R M E U L E N * * and A.W. VAN DER
VAART†
Vrije Universiteit Amsterdam, Department of Mathematics, De Boelelaan 1081a,
1081 HV Amsterdam, The Netherlands. E-mail: *geurt@cs.vu.nl; **meulen@cs.vu.nl;
†aad@cs.vu.nl
We consider nonparametric estimation of the Le´vy measure of a hidden Le´vy process driving a
stationary Ornstein–Uhlenbeck process which is observed at discrete time points. This Le´vy measure
can be expressed in terms of the canonical function of the stationary distribution of the Ornstein–
Uhlenbeck process, which is known to be self-decomposable. We propose an estimator for this
canonical function based on a preliminary estimator of the characteristic function of the stationary
distribution. We provide a suppport-reduction algorithm for the numerical computation of the
estimator, and show that the estimator is asymptotically consistent under various sampling schemes.
We also define a simple consistent estimator of the intensity parameter of the process. Along the way,
a nonparametric procedure for estimating a self-decomposable density function is constructed, and it is
shown that the Ornstein–Uhlenbeck process is �-mixing. Some general results on uniform
convergence of random characteristic functions are included.
Keywords: Le´vy process; self-decomposability; support-reduction algorithm; uniform convergence of
characteristic functions
1. Introduction
For a given positive number º and a given increasing Le´vy process Z without drift
component, consider the stochastic differential equation
dX (t) ¼ �ºX (t)dt þ dZ(ºt), t > 0: (1:1)
A solution X to this equation is called a Le´vy-driven Ornstein–Uhlenbeck (OU) process, and
the process Z is referred to as the background driving Le´vy process (BDLP). The
autocorrelation of X at lag h can be expressed in terms of the ‘intensity parameter’ º as e�ºjhj.
By the Le´vy–Khinchine representation theorem (Sato 1999, Theorem 8.1), the
distribution of Z is characterized by its Le´vy measure r. If
Ð1
2
log xr(dx) ,1, then a
unique stationary solution to (1.1) exists (Sato 1999, Theorem 17.5 and Corollary 17.9).
Moreover, the stationary distribution � of X (1) is self-decomposable with characteristic
function
ł(t) :¼
ð
ei tx�(dx) ¼ exp
ð1
0
(ei tx � 1) k(x)
x
dx
� �
, (1:2)
Bernoulli 11(5), 2005, 759–791
1350–7265 # 2005 ISI/BS
where k(x) ¼ r(x, 1). This shows that � is characterized by the decreasing function k,
which is called the canonical function. Conversely, if we presuppose that � satisfies (1.2),
then there exists an increasing Le´vy process Z, unique in law, such that (1.1) holds for all
º . 0. Due to the special scaling in (1.1), � does not depend on º.
Assume that we have discrete-time observations X 0, X˜, . . . , X (n�1)˜ (˜ . 0) from
(X t, t > 0), as defined by (1.1), where the sampling interval ˜ may depend on n. Based on
these observations, we aim to estimate the parameters of the model. From the previous
remarks this comes down to (i) estimating the intensity parameter º and (ii) estimating the
canonical function k. In this paper we deal with both estimation problems. Our approach to
(ii) is nonparametric, although parametric submodels can be handled with our method as
well (see Jongbloed and van der Meulen 2004).
One motivation for studying this problem comes from stochastic volatility models in
financial mathematics. Barndorff-Nielsen and Shephard (2001a) model stock price as a
geometric Brownian motion. The diffusion coefficient of this motion, referred to as the
volatility, is assumed to be a Le´vy-driven OU process. Based on stock prices, the objective
is to estimate the Le´vy measure of the BDLP and º. Although related, this estimation
problem is intrinsically harder than the one we consider, since volatility is unobservable in
practice. Despite this, the present work may be extended to handle these models by the
addition of a deconvolution step, and hence may provide a first step towards estimating
these models nonparametrically. Another motivation comes from storage theory, where
equation (1.1) is often referred to as the ‘storage equation’ (see, for example, C¸inlar and
Pinsky 1971).
Rubin and Tucker (1959) considered nonparametric estimation for general Le´vy
processes, based on both continuous- and discrete-time observations, and Basawa and
Brockwell (1982) considered estimation for the subclass of continuously observed increasing
Le´vy processes. In this paper we consider indirect estimation through the observation of the
OU process X at discrete time instants. Thus we deal with an inverse problem, and our
estimation techniques are correspondingly quite different from the ones in these papers.
Another paper on estimation for OU processes is Roberts et al. (2004), in which Bayesian
estimation for parametric models is considered. Other papers on empirical characteristic
function procedures include Knight and Satchell (1997), Feuerverger and McDunnough
(1981) and, in a more general framework, Luong and Thompson (1987).
In section 2 we discuss self-decomposability via the Le´vy–Khinchine representation
theorem. We show that a self-decomposable distribution is characterized by the logarithm of
its characteristic function, which is called the cumulant function. Furthermore, we state the
close relationship between self-decomposability and Le´vy-driven OU processes. Additional
details on this can be found in Sato (1999, Section 17). We show that the process
(X t, t > 0) is a Feller process (Proposition 2.1) and hence satisfies the strong Markov
property. We also give some examples of self-decomposable distributions and related OU
processes. In section 3 we prove that the OU process is �-mixing. In the proof, we use
theory as developed in Meyn and Tweedie (1993a; 1993b) and a result from Shiga (1990).
Section 4 explains our method for estimating the canonical function. The method uses a
given preliminary, consistent estimator ~łłn for the characteristic function ł0 of X (1), a
typical example being the empirical characteristic function of the observations. Any
760 G. Jongbloed, F.H. van der Meulen and A.W. van der Vaart
characteristic function ł without zeros possesses a unique (distinguished) logarithm, its
associated cumulant function, which we denote by Tł. Our estimator of the cumulant
function Tł0 is now defined as the projection of the preliminary estimate T ~łłn onto the
class of cumulant functions of self-decomposable distributions, relative to a weighted L2-
distance. The estimates of ł0 and its associated canonical function are defined by inverting
the respective maps. Under a ‘compactness condition’ on the set of canonical functions, this
cumulant M-estimator exists and is unique (Theorem 4.5). In Section 5 we prove two
uniform convergence results on random characteristic functions, which may be of
independent interest. We then use these results to provide conditions under which the
cumulant M-estimator is consistent (Theorem 5.3). The estimator can numerically be
approximated by a support-reduction algorithm, as discussed in Groeneboom et al. (2003).
In Section 6 we explain how this algorithm fits within our set-up.
Section 7 contains applications and examples of estimators under different observation
schemes and presents some simulation results. We also consider the estimation of a self-
decomposable distribution based on independent and identically distributed (i.i.d.) data. This
problem is difficult to handle by standard estimation techniques, as there exists no general
closed-form expression for the density of a self-decomposable distribution. The approach is
to first estimate the canonical function by our cumulant M-estimator and then apply Fourier
inversion.
For the intensity parameter º, a simple explicit estimator is defined in Section 8. This
estimator is shown to be asymptotically consistent, although biased upward.
The appendix contains proofs of some more technical lemmas.
2. Preliminaries
In this section we discuss self-decomposable distributions on Rþ and Le´vy-driven OU
processes. Furthermore, we introduce notation that will be used throughout the rest of the
paper.
2.1. Self-decomposable distributions on Rþ
A random variable X , with distribution function F, is said to be self-decomposable if for
every c 2 (0, 1) there exists a random variable X c, independent of X , such that
X ¼d cX þ X c. In particular, all degenerate random variables are self-decomposable. Since
the concept of self-decomposability only involves the distribution of a random variable, we
define a probability measure or a characteristic function to be self-decomposable if its
corresponding random variable is self-decompsable.
The class of self-decomposable distributions is a subclass of the class of infinitely
divisible distributions. For the latter type of distributions, there is a powerful
characterization in terms of characteristic functions: the Le´vy–Khinchine representation.
A random variable Y with values in Rþ (¼ [0, 1)) is infinitely divisible if and only if its
characteristic function has the form
Nonparametric inference for Le´vy-driven Ornstein–Uhlenbeck processes 761
ł(t) ¼ Ei tY ¼ exp iª0 t þ
ð1
0
(ei tx � 1)�(dx)
� �
, 8t 2 R, (2:1)
where ª0 > 0. The measure � is called the Le´vy measure of Y and satisfies the integrability
condition
Ð1
0
(x ^ 1) �(dx) ,1. The parameter ª0 is called the drift.
If Y is self-decomposable, the measure � takes a special form. It has a density with
respect to Lebesgue measure (Sato 1999, Corollary 15.11) and
�(dx) ¼ k(x)
x
dx,
where k is a decreasing function on (0, 1), known as the canonical function. We take
this function to be right-continuous. The integrability condition on � is given byÐ 1
0
k(x)dx þ Ð1
1
x�1 k(x)dx ,1. By Proposition V.2.3 in van Harn and Steutel (2004), the
class of self-decomposable distributions on Rþ is closed under weak convergence. By
Theorem 27.13 in Sato (1999), the distribution of Y is either absolutely continuous with
respect to Lebesgue measure or degenerate.
Thus each non-degenerate positive, self-decomposable random variable is characterized
by a couple (ª0, k) consisting of a non-negative number ª0 and decreasing function k. In
the next section we shall see that the variable X (1) of the process X solving (1.1) is self-
decomposable. Due to our assumption that the BDLP Z in (1.1) possesses no drift, the
parameter ª0 corresponding to X (1) is zero.
Next, we introduce some notation. Define a measure � on the Lebesgue measurable sets
in (0, 1) by
�(dx) ¼ 1 ^ x
x
dx, x 2 (0, 1):
Let L1(�) be the space of �-integrable functions on (0, 1). Define a semi-norm k:k� on
L1(�) by kkk� ¼
Ð jkjd�. Note that the definition of the measure � precisely suits the
integrability condition on k, which can now be formulated as kkk� ,1.
Define a set of functions by
K :¼ fk 2 L1(�) : k(x) > 0, k is decreasing and right-continuousg:
The set K � L1(�) is a convex cone which contains precisely the canonical functions of all
non-degenerate self-decomposable distributions on Rþ and the degenerate distribution at 0.
Let � be the corresponding set of characteristic functions
� :¼ ł : R! Cjł(t; k) ¼ exp
ð1
0
(ei tx � 1) k(x)
x
dx
� �
for some k 2 K
� �
: (2:2)
By the definition of � the mapping Q : K 7! �, assigning to each function k 2 K its
corresponding characteristic function in �, is onto. As a consequence of the Le´vy–Khinchine
theorem, Q is also one-to-one.
The following result from complex analysis can be found, for example, in Chung (2001,
Section 7.6). Suppose j : R! C is continuous, j(0) ¼ 1 and j(x) 6¼ 0 for all x 2 [�T , T ].
Then there exists a unique continuous function f : [�T , T ] ! C such that f (0) ¼ 0 and
exp( f (x)) ¼ j(x). The corresponding statement when [�T , T ] is replaced by (�1, 1) is
762 G. Jongbloed, F.H. van der Meulen and A.W. van der Vaart
also true. The function f is referred to as the distinguished logarithm. If j is a
characteristic function, then f is called a cumulant function.
Since an infinitely divisible characterstic function has no real zeros (see Sato 1999,
Lemma 7.5), we can attach to each ł 2 � a unique continuous function g such that
e g( t) ¼ ł(t) and g(0) ¼ 0. Since we will switch between sets of characteristic functions and
cumulant functions throughout, we define a mapping T from � onto its range by
[T (ł)](t) ¼ g(t), ł 2 �, t 2 R,
By the uniqueness of the distinguished logarithm and the Le´vy–Khinchine representation it
follows that
G :¼ T (�) ¼ g : R! C j g(t) ¼
ð1
0
(ei tx � 1) k(x)
x
dx, for some k 2 K
� �
:
We have thus defined three sets, each parametrizing the class of self-decomposable
distributions: (i) K, the set of canonical functions; (ii) �, the set of characteristic functions;
(iii) G, the set of cumulant functions. Typical members of each will be denoted by k, ł and
g respectively.
In order to switch easily between canonical functions and cumulants, we define the
mapping L : K ! G by L ¼ T � Q. That is, for k 2 K,
[L(k)](t) ¼
ð1
0
(ei tx � 1) k(x)
x
dx, t 2 R:
The following diagram may help to clarify the relations between the operators defined so far:
� �
�
�
� �
Next, we give a few examples of positive self-decomposable distributions.
Example 2.1. (i) Let X be Gamma(c, Æ) distributed with density f given by f (x) ¼
(Æc=ˆ(c))xc�1e�Æx1fx.0g, c, Æ . 0. The characteristic and canonical functions are given by
ł(t) ¼ (1 � Æ�1it)�c and k(x) ¼ ce�Æx, respectively.
(ii) Let X be an Æ-stable distribution with Æ 2 (0, 1). Then X has support [0, 1) if and
only if its characteristic function is
ł(t) ¼ exp �jtjÆ 1� i tan �Æ
2
� �
sgn(t)
h i� �
:
Its corresponding canonical function is given by k(x) ¼ cÆx�Æ, where cÆ ¼ Æ=
(ˆ(1 � Æ) cos (�Æ=2)). Note that c1=2 ¼ 1=
ffiffiffiffiffiffi
2�
p
. The density function of X permits a known
closed-form expression in terms of elementary functions only if Æ ¼ 1
2
. In this case
f (x) ¼ (2�)�1=2x�3=2e�1=(2x)1fx.0g. The probability distribution with this density is called the
Le´vy distribution. If Z has a standard normal distribution, then W , defined by W ¼ 1=Z2 if
Z 6¼ 0 and W ¼ 0 otherwise, has a Le´vy distribution.
Nonparametric inference for Le´vy-driven Ornstein–Uhlenbeck processes 763
(iii) The inverse Gaussian distribution with parameters � and ª, IG(�, ª), has probability
density function
f (x) ¼ 1ffiffiffiffiffiffi
2�
p �e�ªx�3=2 exp(�(�2x�1 þ ª2x)=2)1fx.0g, � . 0, ª > 0:
See, for example, Barndorff-Nielsen and Shephard (2001b). Its canonical function is given by
k(x) ¼ (2�)�1=2�x�1=2 exp(�ª2x=2)1fx.0g. The case (�, ª) ¼ (1, 0) corresponds to the Le´vy
distribution.
2.2. Le´vy-driven Ornstein–Uhlenbeck processes
In this section we discuss some properties of Le´vy-driven OU processes. We can assume
that the driving Le´vy process Z ¼ (Z t, t > 0) has right-continuous sample paths, with
existing left-hand limits. It is easily verified that a (strong) solution X ¼ (X t, t > 0) to
(1.1) is given by
X t ¼ e�º t X 0 þ
ð
(0, t]
e�º( t�s)dZ(ºs), t > 0: (2:3)
Up to indistinguishability, this solution is unique (Sato 1999, Section 17). Furthermore, since
X is given as a stochastic integral with respect to a cadlag semi-martingale, the OU process
(X t, t > 0) can be assumed cadlag itself. The stochastic integral in (2.3) can be interpreted
as a pathwise Lebesgue–Stieltjes integral, since the paths of Z are almost surely of finite
variation on each interval (0, t], t 2 (0, 1) (Sato 1999, Theorem 21.9). Figure 1 shows a
simulation of an OU process with Gamma(2, 2) marginal distribution.
Denote by (F 0t ) t>0 the natural filtration of (X t). That is, (F 0t ) ¼ � (X u, u 2 [0, t]). As
noted in Shiga (1990, Section 2), (X t, F 0t ) is a temporally homogeneous Markov process.
Denote by (E, E) the state space of X , where E is the Borel � -field on E. We take
E ¼ [0, 1). The transition kernel of (X t), denoted by Pt(x, B) (x 2 E, B 2 E), has
characteristic function (Sato 1999, Lemma 17.1).ð
eizy Pt(x, dy) ¼ exp ize�º t x þ º
ð t
0
g(eº(u� t)z)du
� �
, z 2 R, (2:4)
where g is the cumulant of Z(1).
Let bE denote the space of bounded E-measurable functions. The transition kernel
induces an operator Pt : bE ! bE by
Pt f (x) :¼
ð
f (y)Pt(x, dy) ¼
ð
f (e�º t x þ y)Pt(0, dy): (2:5)
The second equality follows directly from the explicit solution (2.3). We call Pt the transition
operator. Let C0(E) denote the space of continuous functions on E vanishing at infinity (i.e.
for all � . 0 there exists a compact subset K of E such that j f j < � on EnK).
764 G. Jongbloed, F.H. van der Meulen and A.W. van der Vaart
Proposition 2.1. The transition operator of the OU-process is of Feller type. That is,
(i) PtC0(E) � C0(E) for all t > 0,
(ii) 8 f 2 C0(E), 8x 2 E, lim t#0 Pt f (x) ¼ f (x).
For general notions concerning Markov processes of Feller type we refer to Revuz and Yor
(1999, Chapter 3).
Proof. (i) Let f 2 C0(E), whence f is bounded. If xn ! x in E, then
f (e�º t xn þ y) ! f (e�º t x þ y) in R, by the continuity of f , for any y 2 R. By dominated
convergence, Pt f (xn) ! Pt f (x), as n !1. Hence, Pt f is continuous. Again by dominated
convergence, Pt f (x) ! 0, as x !1.
(ii) By dominated convergence
Ð t
0
g(eº(u� t)z)du ¼ Ð t
0
g(e�ºuz)du ! 0, as t # 0. Here we
use the continuity of the cumulant g and g(0) ¼ 0. Then it follows from (2.4) that
� � � � � �� �� �� �� �� ��
�
��
��
��
� � � � � �� �� �� �� �� ��
�
�
�
�
�
� ��� ��� ��� ��� ��� ��� �� ���
�� ����
�
�
�
�
Figure 1. Top: simulation of the BDLP (compound Poisson process of intensity 2 with exponential
jumps of expectation 1
2
). Middle: corresponding OU process with Gamma(2,2) marginal distribution.
Bottom: OU process on longer time horizon.
Nonparametric inference for Le´vy-driven Ornstein–Uhlenbeck processes 765
lim
t#0
ð
eizy Pt(x, dy) ¼ eizx:
Thus Pt(x, �) converges weakly to �x(�) (Dirac measure at x):
lim
t#0
ð
f (y)Pt(x, dy) ¼
ð
f (y)�x(dy) ¼ f (x), 8 f 2 Cb(E):
Here Cb(E) denotes the class of bounded, continuous functions on E. The result follows since
C0(E) � Cb(E). h
The Feller property of (X t) implies (X t) is a Borel right Markov process; see the
definitions in Getoor (1975, Chapter 9). We will need this result in Section 3.
Since Pt is Feller, (X t) satisfies the strong Markov property (Revuz and Yor 1999,
Theorem III.3.1). In order to state a useful form of the latter property, we define a
canonical OU process on the space � ¼ D[0, 1), by setting X t(ø) ¼ ø(t), for ø 2 �
(here D[0, 1) denotes the space of cadlag functions on [0, 1), equipped with its � -algebra
generated by the cylinder sets). By the Feller property, this process exists (Revuz and Yor
1999, theorem III.2.7). Let � be a probability measure on (E, E) and denote by P� the
distribution of the canonical OU process on D[0, 1) with initial distribution �. For
t 2 [0, 1), we define the shift maps Łt : �! � by Łt(ø(�)) ¼ ø(� þ t).
Next, we enlarge the filtration by including certain null sets. Denote by F �1 the
completion of F 01 ¼ � (F 0t , t > 0) with respect to P�. Let (F �t ) be the filtration obtained
by adding to each F 0t all the P�-negligible sets of F �1. Finally, set F t ¼
T
� F �t and
F1 ¼
T
� F �1, where the intersection is over all initial probability measures � on (E, E). In
the special case of Feller processes, it can be shown that the filtration (F t) obtained in this
way is automatically right-continuous (thus, it satisfies the ‘usual hypotheses’). See
Proposition III.2.10 in Revuz and Yor (1999). Moreover, (X t) is still Markov with respect to
this completed filtration (Revuz and Yor 1999, Proposition III.2.14). The strong Markov
property can now be formulated as follows. Let Z be an F1-measurable and positive (or
bounded) random variabl
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