Exponential Ergodicity and β-Mixing Property for Generalized Ornstein-Uhlenbeck Processes ()
1. Introduction
Many continuous time processes are suggested and studied as a natural continuous time generalization of a random recurrence equation, for example, diffusion model of Nelson [2], continuous time GARCH (COGARCH) (1,1) process of Klüppelberg et al. [3] and Lévy-driven Ornstein-Uhlenbeck (OU) process of Barndorff-Nielsen and Shephard [4] etc. Continuous time processes are particularly appropriate models for irregularly spaced and high frequency data [5]. We consider the generalized Ornstein-Uhlenbeck (GOU) process 
which is defined by
(1)
where 
is a two-dimensional Lévy process and the starting random variable 
is independent of
. Lévy processes are a class of continuous time processes with independent and stationary increments and continuous in probability. Since Lévy processes 
and 
are semimartingales, stochastic integral in Equation (1) is well defined.
The GOU process is a continuous time version of a stochastic recurrence equation derived from a bivariate Lévy process (de Haan and Karandikar [1]). The GOU process has recently attracted attention, especially in the financial modelling area such as option pricing, insurance and perpetuities, or risk theory. Stationarity, moment condition and autocovariance function of the GOU process are studied in Lindner and Maller [6]. Fasen [7] obtain the results for asymptotic behavior of extremes and sample autocovariance function of the GOU process. For related results, we may consult, e.g. Masuda [8], Klüppelberg et al. [3,9], Maller et al. [5] and Lindner [10] etc.
Mixing property of a stochastic process describes the temporal dependence in data and is used to prove consistency and asymptotic normality of estimators. For a stationary process 
and
, let
where the supremum takes over 
if 
and
. If 
as
, then 
is called β-mixing. 
is called exponentially β-mixing if 
for some 
and all
.
In this paper we prove the exponential ergodicity and exponentially β-mixing property of the GOU process
of Equation (1) and obtain the β-mixing property of the Lévy-driven OU process as a special case.
For more information on Markov chain theory, we refer to Meyn and Tweedie [11]. We refer to Bertoin [12] and Sato [13] for basic results and representations concerning Lévy processes.
2. Exponential Ergodicity of 
2.1. The Model
A bivariate Lévy process 
defined on a complete probability space 
is a stochastic process in
, with càdlàg paths, 
and stationary independent increments, which is continuous in probability.
Consider the GOU process 
given by
Assume that 
is independent of
. Let
(2)
Then we have that
(3)
Let n denote an integer and 
a real number. We can easily show that 
in Equation (2) is a sequence of independent and identically distributed random vectors and 
in Equation (1) is a time homogeneous Markov process with t-step transition probability function
where 
is a Borel σ-field of subsets of real numbers R.
We temporally assume that 
is fixed. 
in Equation (3) can be considered as a discrete time Markov process with n-step transition probability function
. 
is called the h-skeleton chain of
. A Markov process 
is 
-irreducible if, for some 
-finite measure
, 
for all 
whenever
. 
is said to be simultaneously 
-irreducible if any h-skeleton chain is 
-irreducible. It is known that if 
is simultaneously 
-irreducible, then any h-skeleton chain is aperiodic (Proposition 1.2 of Tuominen and Tweedie [14]).
For fixed
, we make the following assumptions:
(A1) 
and
.
(A2) 
for some 
Theorem 2.1 Under the assumption (A1), 
defined by Equation (3) converges in distribution to a probability measure 
which does not depend on
. Further, 
is the unique invariant initial distribution for
.
Proof. The conclusion follows from Theorem 3.1 and Theorem 3.4 in de Haan and Karandikar [1]. Note that if the assumption (A1) holds, then it is obtained that
and
.
Remark 1 Assume that
. Then 
is also necessary for the existence of a strictly stationary solution. (See Theorem 2.1 in Lindner and Maller [6].)
Remark 2 Suppose that there exist 
and 
with 
such that
where 
denotes the Lévy exponent of the Lévy process
: 
If in addition, 
then assumptions (A1) and (A2) hold (Proposition 4.1 in Lindner and Maller [6]).
2.2. Drift Condition for 
A discrete time Markov process 
is said to hold the drift condition if there exist a positive function g on R, a compact set K, and constants 
and 
such that
and 
Theorem 2.2 Under the assumptions (A1) and (A2), 
given in Equation (3) satisfies the drift condition.
Proof. For notational simplicity, let
. From assumptions, we have that 
and 
for some
. Then
as 
( Hardy et al. [15]). Here 
implies the existence of
, 
such that
. Now define a nonnegative test function g on R by
. Then we have that
(4)
where
, by assumption (A2). Since 
increases to 
as 
increases to
, for any
, there exist 
and 
with
, such that
(5)
Clearly,
(6)
Combining Equations (4)-(6), the drift condition for 
holds.
2.3. Simultaneous 
-Irreducibility of 
For reader’s convenience, we state the following theorems which play important roles to prove our main results.
Theorem 2.3 (Meyn and Tweedie [11]) Suppose that a Markov chain 
has the Feller property. If 
satisfies the drift condition for a compact set
, then there exists an invariant probability measure. In addition, if the process is 
-irreducible and aperiodic, then the given process is geometrically ergodic.
Theorem 2.3 shows that the crucial step to prove the geometric ergodicity of a Markov process is to show that the given process is 
-irreducible and holds the drift condition. In many cases, however, proving irreducibility of a Markov process is an awkward task. Consulting the following Theorem 2.4, irreducibility of the process can be derived from connection between 
-irreducibility and the uniform countable additivity condition. A Markov chain 
is said to hold the uniform countable additivity condition (Liu and Susko [16]) if its one-step transition probability function satisfies that for any decreasing sequence 
inside compact sets,
Theorem 2.4 (Tweedie [17]) Suppose that the drift condition holds with a test set K and the uniform countable additivity condition holds for the same set K. Then there is a unique invariant measure for 
if and only if 
is 
-irreducible.
Let 
be the compact set defined in the proof of Theorem 2.2.
Theorem 2.5 Under the assumptions (A1) and (A2), 
is simultaneously π-irreducible if for any
,
has a probability density function 
(with respect to the Lebesgue measure
), which is uniformly bounded on compacts for
.
Proof. Let 
be any decreasing sequence inside compact sets with
. Then
(7)
where 
.
The inequality in Equation (7) and the condition that 
is any sequence inside compact sets in 
with 
imply that
.
Therefore the uniform countable additivity condition holds for the compact set K. Theorem 2.4 and the existence of a unique invariant initial distribution for
yield the 
-irreducibility of any h-skeleton chain
.
To complete the proof, we need to show that the assumption (A1) and (A2) hold for all
. Since Lévy processes have stationary and independent increments, it is easy to show that the assumption (A1) and 
hold for all
. It remains to prove that
for all 
with some
. We first define a finite Lévy process 
as follows:
Then it is shown that
,
(See Proposition 2.3 in Lindner and Maller [6]). Without loss of generality, we may assume that
. Choose any
. Then
, where n is a nonnegative integer, 
and 
is in the assumptions (A1) and (A2), we have that
(8)
The first inequality in Equation (8) follows from stationary and independent increments property of Lévy processes 
and
.
Therefore for any
, h-skeleton chain 
is 
-irreducible and hence 
is simultaneously 
-irreducible and 
is aperiodic.
2.4. Exponential Ergodicity of 
The next theorem is our main result.
Theorem 2.6 Suppose that the assumptions of Theorem 2.5 hold. Then the GOU process 
in Equation (1) is exponentially ergodic and holds the exponentially 
-mixing property.
Proof. Theorem 2.5 shows that any h-skeleton chain
is 
-irreducible and aperiodic. Note that 
is a Feller chain, that is, 
is a continuous function of x whenever f is continuous and bounded. Therefore any nontrivial compact set is a small set. Theorem 2.2 ensures that 
holds the drift condition and hence Theorem 2.5 and Theorem 2.3 imply that 
is geometrically ergodic, that is, there exists a constant 
such that
(9)
-a.a. x as
, where 
denotes the total variation norm. Under simultaneous 
-irreducibility condition of
, Equation (9) and Theorem 5 in Tuominen and Tweedie [14] guarantee the exponential ergodicity of 
in the following sense:
as
, for some 
and 
-a.a. x. 
-mixing property for the continuous time GOU process 
is also obtained.
2.5. Examples
In this example, we assume that
. If 
is any Lévy process, then 
in Equation (1) is the Lévydriven OU process which is studied by Barndorff-Nielsen and Shephard [4]. In particular, if 
is a subordinator, that is, 
has nondecreasing sample path, finite variation with nonnegative drift and Lévy measure concentrated on
, then 
is called the Lévy-driven stochastic volatility model. For the case that 
is a Brownian motion, 
is the classical OU process. Let 
be the Lévy measure for the process 
and assume that 
for some 
and
.
Then
. Here we can easily show that the assumptions (A1)and (A2) hold. Theorem 2.2 implies that 
holds the drift condition. Moreover, it is known that 
admits a 
density 
for each 
(Sato and Yamazato [18]) and by Theorem 2.5, 
is 
-irreducible. Above statements hold for any 
and hence 
is simultaneously 
-irreducible. Therefore exponential ergodicity and exponential 
-mixing property of 
follow from Theorem 2.6.
3. Conclusion
Recently, time series models in finance and econometrics are suggested as continuous time models which are particularly appropriate for irregularly spaced and high frequency data. The GOU process is a continuous time stochastic process driven by a bivariate Lévy process. The stationarity, moment conditions, autocovariance function and asymptotic behavior of extremes of the process are studied in [6,7], but exponential ergodicity does not seem to have been investigated as yet. In this paper, we give sufficient conditions under which the process is exponentially ergodic and 
-mixing. The drift condition and the simultaneous 
-irreducibility of the process that is induced from uniform countable additivity condition play a crucial role to prove the results. Our results are used to show, in particular, consistency and asymptotic normality of estimators.
4. Acknowledgements
This research was supported by KRF grant 2010- 0015707.