Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching

In this paper, we consider long time behavior of the Cox-Ingersoll-Ross (CIR) interest rate model driven by stable processes with Markov switching. Under some assumptions, we prove an ergodicity-transience dichotomy, namely, the interest rate process is either ergodic or transient. The sufficient and necessary conditions for ergodicity and transience of such interest model are given under some assumptions. Finally, an application to interval estimation of the interest rate processes is presented to illustrate our results.


1.
Introduction. In 1985, Cox, Ingersoll and Ross (CIR, for short) [7] first established that the instantaneous interest rate satisfies the following stochastic differential equation (SDE) dX t = (δ − βX t )dt + σ X t dW t , where W t is a standard Brownian motion and δ, β, σ are positive constants. Since then, the SDE (1) is called CIR model which has almost perfect properties: nonnegativity, ergodicity, with explicit transition densities. However, as a one factor model, a problem with the model (1) is that it describes interest rate movements driven by only one source of market risk. Some empirical analyses show that processes with Markov switching can better capture the reality data. For example, Zhou and Mamon [24] show that the CIR model with Markov switching provides better fits for the three-month zero-coupon yields of Canada from January 1986 to December 1995 than the CIR model (1). In addition, Wu [22] finds that diffusion processes with Markov switching can well capture the monthly exchange rates of six major Asia-Pacific currencies from January 2000 to December 2011. Moreover, Smith [20] finds that the Markov-switching model is better than the stochastic-volatility model in predicting interest rate volatility according to monthly observations on 30-day treasury bills from June 1964 to December 1996 in U.S. Therefore, many researchers extend CIR model to CIR model with Markov switching. For instance, Zhang, Tong and Hu [23] consider the long-term behavior of the CIR model with Markov switching, that is dX t = (2β(r t )X t + δ(r t ))dt + σ(r t ) X t dW t , where (r t ) t≥0 is a Markov chain taking the state space S = {1, 2, ..., n}. They show that under some conditions, the model (2) has a unique stationary distribution. Furthermore, Tong and Zhang [21] find that the transition semigroup corresponding to the SDE (2) converges to the stationary distribution at an exponential rate in the Wasserstein distance. Although SDE (2) has modeled effects of exogenous noise on interest rate, this model can not characterize some classes of noise distributions with infinite variance, which extensively exist in many economic phenomena [6]. For example, Mandelbrot [14] finds that the distributions of the changes of monthly wool prices from 1890 to 1937 follow α stable distribution with α = 1.7. Stable distributions and power distributions are frequently found in analyses of critical behavior and financial data. According to the generalized central limit theorem, summation of a sequence of independent and identically distributed variables with infinite variance converges weakly to a stable distribution. Hence, it is natural to replace the Brownian motion with an α-stable process in the model (2).
The main aim of this paper is to study the CIR model driven by symmetric α-stable processes with Markov switching. To be precise, we consider the following SDE dX t = (2β(r t )X t + δ(r t ))dt + σ(r t ) α |X t |dZ t , where (Z t ) t≥0 is a symmetric α-stable process with index 1 < α < 2 and its Lévy measure ν α (dz) := C α |z| 1+α dz.
Here C α = α2 α−1 Γ((1+α)/2) π 1/2 Γ(1−α/2) and Γ(·) is the Gamma function defined by The Markov chain (r t ) t≥0 is the same as the model (2). Then, given r t ≡ i, i ∈ S, the SDE (3) reduces to Roughly speaking, this system (3) is operated as follows: Assume r 0 = i 0 and let τ 1 be the jump time when the Markov chain r t jumps from i 0 to i 1 . During the time period [0, τ 1 ), the system evolves according to Eq. (5) with i = i 0 . Then in [τ 1 , τ 2 ), the system behaves like Eq. (5) with i = i 1 . The system will continue to switch as long as the Markov chain jumps. In other words, Eq. (3) can be regarded as Eqs.
(5) switching from one to another according to the state of the Markov chain. If the number of states of the Markov chain is only one, then the model (3) reduces to CIR driven by stable processes without Markov switching, which have been extensively considered by many researchers. For instance, Li and Ma [13] give some asymptotic properties of estimators in a CIR model driven by a spectrally positive stable-Lévy process. Handa [9] considers ergodic properties of the CIR model driven by stable processes without Markov switching. Very recently, Jiao, Ma and Scotti [10] obtain an explicit formula for the bond price and some distributions for large jumps of the model (3) with n = 1. Besides, they also show the ergodicity of this model and present the Laplace transform of the stationary distribution.
Compared model (3) with (2), at first glance, there is only one noise change. One can not help asking whether there are some essential differences? Yes, from the perspective of generator of the Markov process, the main difference is that the second order differential operator corresponding to the SDE (3) is replaced by the fractional Laplace operator. The former is a local operator, whereas, the latter is a non-local operator. To be precise, the infinitesimal generator A of the Markov process (X t , r t ) t≥0 is given by for all where Moreover, one can see from (6) that there is an integral term, which is a nonlocal operator. This leads to the essential difficulty. Besides, note that the coefficients σ(i) α √ X t are Hölder continuous with exponent 1 α ∈ ( 1 2 , 1), which are non-uniformly elliptic. To the best of our knowledge, all the existing literature is out of work to infer the recurrence and transience of the model (3) with n ≥ 2. With help of the Gauss hypergeometric function, the Khasminskii Lemma and the theory of nonsingular M -matrix, we overcome these difficulties and give an almost sufficient and necessary condition for ergodicity of the model (3).
Compared with existing literature, our contributions of this manuscript are the following.
• We prove the nonnegativity of the SDE model (3).
• An almost necessary and sufficient condition of ergodicity of the SDE model (3) is given. • We construct a class of Lyapunov functions to deduce the recurrence and transience of the SDE model (3). • A new method is presented to prove the following limit where 1 < α < 2 and θ ∈ (0, 1). The remaining of this manuscript is organized as follows. In Section 2, we give some notation and prove the existence of a unique solution of the CIR model (3). The sufficient conditions for existence of a unique stationary distribution are presented in Section 3. Conditions for transience of the CIR model (3) are obtained in Section 4. Finally, two examples are given to illustrate our results in Section 5.
2. Preliminaries. Throughout this paper, let (Ω, F, {F t } t≥0 , P) be a complete probability space with filtration {F t } t≥0 satisfying the usual conditions (i.e. it is right continuous and increasing while F 0 contains all P-null sets). We denote by X x,i t the interest rate process X t with initial value X 0 = x, r 0 = i.
where ∆ > 0. The Q-matrix Q = (q ij ) n×n is assumed to be irreducible and conservative. Hence q ij > 0 if i = j and Moreover, the Markov chain (r t ) t≥0 and the stable process (Z t ) t≥0 are assumed to be independent. Since the Markov chain (r t ) t≥0 is irreducible and conservative, it has a unique stationary distribution π = (π(1), π(2), ..., π(n)) which can be obtained by solving the following system of equations To guarantee the existence of a unique solution, we impose the following assumptions (A): Since the CIR model (3) describes the evolution of interest rates, X t should be pathwise unique. Next, we prove the pathwise uniqueness of the solution defined by (3) in the following Lemma. However, if α = 2, the process (X t ) t≥0 can take the negative value, which is different from the CIR model (2). Proof. Given r t ≡ i, according to Theorem 4 of Fournier [8], there is a pathwise unique solution to the equation The idea of this proof, roughly speaking, is that the process (X t ) t≥0 can be regarded as Eqs. (9) switching from one to another according to the state of Markov chain. Based on the pathwise uniqueness of (9), we show the CIR model (3) has a pathwise unique solution.
Let N T be the number of jumps for the Markov chain on the interval [0, T ], τ n be the n-th jump time of the Markov chain r t on the interval [0, T ] and τ 0 = 0. For simplicity, we denote by Y y,i0 t the solution of (Y t ) t≥0 with initial value Y 0 = y, r 0 = i 0 . Now, we construct the sample path of X t as follows.
(i) Given initial value

NECESSARY AND SUFFICIENT CONDITIONS FOR ERGODICITY OF CIR MODEL 2437
Consequently, we get for t ∈ [0, T ] Recalling that the process Y Xτ k ,rτ k t has a pathwise unique solution on t ∈ [τ k , τ k+1 ] for k = 0, 1, ..., N T − 1, we immediately obtain that the process (X t , r t ) t≥0 has a pathwise unique solution.
Proof. If there exists some t 0 such that X t0 = 0, by the positivity of δ(i), we have X t + 0 = 0. One can obtain the desired result.
3. Stationary distribution. In this section, we mainly find sufficient conditions such that the CIR model (3) has a unique stationary distribution based on theory of M -matrix and Foster-Lyapunov criterion. For convenience for readers, we cite some equivalence conditions about nonsingular M -matrix. For more information on theory of M -matrix, the readers can see Berman and Plemmons [4].
The proof for the existence of a unique stationary distribution is rather technical, so we first present a useful lemma.
Proof. The proof is similar to that of Theorem 4.6 of Mao [15] and Theorem 3.4 in reference [16], we omit it here.
If Re(c) > Re(b) > 0, the Gauss hypergeometric function will be analytically continuous on C\(1, ∞) as where Γ(·) is the Gamma function. Here Z + , Z − , C represent non-negative integers, non-positive integers and complex numbers respectively. For further properties about the hypergeometric functions, please see [1].
To show the existence of a unique stationary distribution, we follow the main idea of Tong and Zhang [21] and Zhang et al. [23]. Let us present a similar result from Khasminskii [11] (pp. 107-109) for stable processes with Markov switching. smooth) boundary such that its closureŪ ⊂ R ≥0 , having the following conditions: Proof. The proof is similar to that of Theorem 4.1 of Khasminskii [11] and Theorem 5.1 of Arapostathis, Biswas and Caffarelli [3]. We omit it here.

NECESSARY AND SUFFICIENT CONDITIONS FOR ERGODICITY OF CIR MODEL 2439
Proof. To prove the theorem, according to Lemma 3.3, it suffices to verify the conditions (B.1) and (B.2). Let N be a positive real number. Set So, the condition (B.1) in Lemma 3.3 holds.
Next, we will verify the condition (B.2) in Lemma 3.3. To be precise, we shall show that there exists a sufficiently large N > 0 such that for all ( where (14) is divided into two cases.
First, we explain our strategy of the proof in this case. By Itô's formula, for all where Letting t → +∞ yields Next, we define the Lyapunov function V . By Lemma 3.1, there exists a constant θ 1 > 0 such that for all θ ∈ (0, θ 1 ), the matrix A(θ) is a nonsingular M -matrix. So Proposition 1 implies that there exists a vector ξ 0 (a vector ξ 0 means all elements of ξ are positive) such that To proceed, we first consider the case the process (X t ) never hits the origin, that is X t = 0, for all t > 0. Then the Lyapunov function V : R × S → R ≥0 is defined by where θ ∈ (0, min{1, θ 1 }). Furthermore, we will show in the Appendix that V ∈ D(A), where D(A) denotes the domain of the generator A.
Applying (6) gives Then, we will prove that there exists a sufficiently large N > 1 and for all To this end, it suffices to prove We divided the proof of (21) into three steps.
Observing that it is difficult to show the limit (21) directly, we first rewrite the expression for Step 2. We will deal with C(x, i) and show that Let z = y/x. Then As for C 1 (x, i), let x = 1 , > 0. Then Clearly, lim x→+∞ C 1 (x, i) = lim →0 + h( , i). In order to prove lim →0 + h( , i) exists, it suffices to prove For |z| < 1, we have Then As for the integral C 4 (x, i), we have Moreover, we compute the integral C 5 (x, i). It follows that

NECESSARY AND SUFFICIENT CONDITIONS FOR ERGODICITY OF CIR MODEL 2443
Thus Step 3. We now compute lim x→+∞ B(x, i) and the term A(x, i). By the mean value theorem, there exists η ∈ [x − 1, x + 1] such that Noticing that V xx (x, i) is an increasing function and θ < 1, we get For the term A(x, i), by direct computation we have Now, since Then, there exists a sufficiently large N 1 > 0 such that for all x > N 1 , i ∈ S AV (x, i) < −1. (26) Next, we consider the process (X t ) can hit the origin. In this case, define the Lyapunov function where θ ∈ (0, min{1, θ 1 }). Here ϕ(x) ∈ C 2 (R) is a nondecreasing, radial function satisfying ϕ(x) = |x| for |x| > 1 and 0 ≤ ϕ(x) ≤ |x| for |x| ≤ 1. By the similar discussion of the case that the process (X t ) t≥0 never hits the origin, one can obtain the result (27).
Case 2. We show that there exists a sufficiently large N 2 > 0 such that for any Suppose that the statement is false. Then, for any N > 0, there exists ( In other words, for any ∈ [0, The case when x < 0 is treated in the similar way. Hence, we verifies the condition (B.2) in Lemma 3.3. The proof is therefore complete.

4.
Transience. In the previous section, we have shown that under the condition n i=1 π(i)β(i) < 0, the CIR model (3) has a unique stationary distribution. The natural question is: what happens if n i=1 π(i)β(i) > 0 ? In this section, we will answer this question. Proof. The proof is similar to that of Theorem 4.6 of Mao [15] or Lemma 3.4 of Li et al. [12]. We omit it here.
Let θ ∈ (0, min{1, θ 2 }) and define the function V : where ϕ(x) is defined in the proof of Theorem 3.4. Clearly, V ∈ C 2 (R × S). Moreover, since V is bounded, we have V ∈ D(A). Our first goal is to show that there exists a sufficiently large N > 0 such that for all x > N, i ∈ S AV (x, i) < 0 (31) and the proof of (31) is divided into three steps.
Step 1. First, we rewrite the expression of AV (x, i). With help of (6), we have Furthermore, recalling the definition of ϕ(x), we have for x > 1 large enough Hence, in order to prove (31), it suffices to show To prove the limit (32), we define Then, we will deal with α(1+x) α+θ Cαθσ α (i)x AV (x, i) by considering A(x, i), B(x, i) and C(x, i) respectively.
Step 2. We first consider C(x, i). In this step, we aim to show that lim x→+∞ C(x, i) = −η i E(α, θ).

NECESSARY AND SUFFICIENT CONDITIONS FOR ERGODICITY OF CIR MODEL 2449
we get For the term D 3 (x, i), by direct computation, we have

NECESSARY AND SUFFICIENT CONDITIONS FOR ERGODICITY OF CIR MODEL 2451
For the term A(x, i), straightforward calculations show that Recalling that It then follows from (33) to (43) that Then, there exists a sufficiently large N > 0 such that for all x > N, i ∈ S AV (x, i) < 0.
Consequently, to prove the transience of the X t in the whole space in this case, we only need to check the transience of X t in the domain D c = (−ε, ε). If the process X t is recurrent in interval (−ε, ε), by the arbitrary of ε, we must have P (lim t→+∞ X t = 0) = 1, which is in contraction to the result of lemma 2.2. This completes the proof.
5. An application to interval estimation. In this section, as an application of our theoretic results, we present an interval estimation of the interest rate process X t defined by (3). Then, by (8), we get π(1) = 1 4 and π(2) = 3 4 .
To verify the theoretical result in Theorem 3.4, we perform a computer simulation of 30000 iterations of the single path of X t with initial value X 0 = 0.3, r 0 = 1. The sample path is shown in Fig. 1. In Theorem 3.4, we give the condition n i=1 π(i)β(i) < 0 for the existence of a unique stationary distribution. It is easy to verify that the coefficients satisfy the condition. Then we compute the average values of the last 20000, 10000 iterations of X t when α = 1.75, which are 0.1628 and 0.1581 respectively. Besides, we compute the variance of the last 20000, 10000 iterations of X t , which are 0.0072 and 0.0076 respectively. We can see that the differences of the two average values and the two variance are very small, which means that the process X t is stationary and this verifies our theoretical result in Theorem 3.4. Furthermore, it is well known that if α becomes bigger, the jump sizes become smaller. So we choose α = 1.25 and α = 1.75 to see the effects that the increase of α causes. The first picture in Fig. 1 is the path with α = 1.25 and the second is α = 1.75. As we can see from Fig. 1, the fluctuations in the first picture of Fig. 1 on some time intervals are much larger than that of the second one. Moreover, if we sort the iterations of X t (α = 1.75) into sorted data from the smallest to the largest, then the 150th and 29850th value in the sorted data are -0.0914 and 0.4676, respectively. Thus, we get a confidence interval of X t , namely, for any sufficiently large t P (−0.0914 < X t < 0.4676) ≈ 0.99.
Similarly, when α = 1.25, for any sufficiently large t, one has P (−0.0682 < X t < 0.5689) ≈ 0.99. We perform a computer simulation of 30000 iterations of the single path of X t with initial value x 0 = 0.3, r 0 = 1. The sample path is shown in Fig. 2. The coefficients satisfy the condition n i=1 π(i)β(i) > 0 in Theorem 4.2. Then, it is obvious in Fig. 2   where A is the generator of (X t , r t ) t≥0 . It is obvious