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

Then we have that

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]).

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

where, by assumption (A2). Since increases to as increases to, for any, there exist and with, such that

Clearly,

Combining Equations (4)-(6), the drift condition for holds.

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

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

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.

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

-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.

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.