1. Introduction
Aregime-switching jump diffusion is a two-component stochastic process
$(X(t),\Lambda(t))$
, where the dynamic of the continuous component X(t) depends on the current state of the discrete component
$\Lambda(t)$
. Intuitively, the continuous component X(t) represents the state of the phenomenon while the discrete component
$\Lambda(t)$
refers to the structural changes in the system. In recent years, regime-switching jump diffusions have received growing attention due to their ability to model and analyze complex systems in which both structural changes and small fluctuations as well as big spikes coexist and are intertwined. We refer to [Reference Cloez and Hairer1], [Reference Mao and Yuan5], [Reference Nguyen and Yin10], [Reference Nguyen and Yin11], [Reference Shao13], [Reference Shao14], [Reference Shao and Xi15], [Reference Xi, Yin and Zhu18], [Reference Yin and Zhu19], and the references therein for more discussions and recent developments in this research area.
In this paper we examine the exponential ergodicity of regime-switching jump diffusions with countably many regimes. A central concern of ergodic theory is the long-term behavior of dynamic systems. This leads to the study of the existence and uniqueness of an invariant distribution as well as the convergence rate towards invariant distributions. The exponential ergodicity of regime-switching (jump) diffusions has been extensively studied in the literature. For example, Shao [Reference Shao13] studied the ergodicity of regime-switching diffusions, in both finite and infinite regime cases, by using Wasserstein distances and M-matrix theory. Cloez and Hairer [Reference Cloez and Hairer1] studied the exponential ergodicity of Markov processes with random switching, a general setting but with finitely many regimes, by using a weak form of Harris’s theorem. Under the regularity assumption on the jump rates and recurrence, the existence of a Foster–Lyapunov function was also investigated. Nguyen and Yin [Reference Nguyen and Yin10, Reference Nguyen and Yin11] studied the recurrence and ergodicity of regime-switching diffusions (no jumps). Interestingly, these two papers treat regime-switching diffusions where the transition rates of the switching component
$\Lambda$
depend on the entire past history of the continuous component X.
The series of papers by Meyn and Tweedie [Reference Meyn and Tweedie6–Reference Meyn and Tweedie8] provides a powerful criterion for exponential ergodicity of Markov processes; see Theorem 6.1 of [Reference Meyn and Tweedie8]. The criterion relies on the existence of a Foster–Lyapunov function and the property that all compact subsets are small in some sense for some skeleton chain. In view of Theorem 3.4 of [Reference Meyn and Tweedie6], the smallness assumption can be verified by establishing the Feller property and
$\varphi$
-irreducibility. Then the existence of a Foster–Lyapunov function becomes the key to establishing exponential ergodicity. Indeed, this type of function plays a vital role in the study of stability and long-term behaviors of stochastic systems; see for example [Reference Hairer, Mattingly and Scheutzow2], [Reference Khasminskii3], [Reference Mao and Yuan5], and [Reference Yin and Zhu19], to name just a few. However, in practice it is usually not easy to find such a function. For regime-switching (jump) diffusions, it is even harder to find an appropriate Foster–Lyapunov function due to the interactions between the continuous and discrete components.
This paper focuses on establishing Foster–Lyapunov-type criteria for exponential ergodicity of regime-switching jump diffusion processes. We aim to provide a set of sufficient conditions for constructing a suitable Foster–Lyapunov function. More precisely, let us first assume that there is a common norm-like function V(x) such that in each discrete state k we have
$\mathcal{L}_{k} V (x) \leq \alpha_{k} V (x)+\beta_{k}$
, where
$\alpha_{k}\in \mathbb{R}$
and
$\beta_{k}> 0$
are bounded constants and
$\mathcal{L}_{k}$
is the infinitesimal generator of the jump-diffusion in regime k. If
$\alpha_{k} < 0$
, in addition to the assumptions of smallness of compact sets, then the subsystem in regime k is exponentially ergodic. For the regime-switching jump diffusion process
$(X,\Lambda)$
, one can intuitively expect that the exponential ergodicity of the system depends not only on
$\{ \alpha_{k} \}$
but also on the generator Q(x) of the switching component
$\Lambda$
. With this idea in mind, we construct a Foster–Lyapunov function U(x, k) for the regime-switching jump diffusion
$(X,\Lambda)$
. The details are spelled out in Theorem 4.1. As a result, this paper establishes an explicit criterion for exponential ergodicity of regime-switching jump diffusions based on the relationship between
$\{ \alpha_{k} \}$
and Q(x). In particular, it indicates that regime-switching jump diffusions can be exponentially ergodic even though some subsystems are not; see Remark 4.1. Moreover, our results can be applied to stochastic differential equations with non-Lipschitz coefficients; see Remark 3.1 for more discussion.
The rest of this paper is organized as follows. In Section 2 we formulate the stochastic differential equations for regime-switching jump diffusion processes. In Section 3 we obtain our main result on exponential ergodicity for regime-switching jump diffusion processes; see Theorem 3.2. In Section 4 we present sufficient conditions and derive the existence of a Foster–Lyapunov function in Theorem 4.1. Moreover, an example is given for illustration. Finally, some applications to feedback control problems will be presented in Section 5.
2. Formulation
To begin, let us give the precise definition of a regime-switching jump diffusion. A regime-switching jump diffusion is a two-component stochastic process
$(X,\varLambda) \;:\!=\; \{(X(t),\varLambda(t))\colon t\in\mathbb{R}_{+}\}$
and it can be defined as follows. Let
$(U,\mathfrak{U})$
be a measurable space,
$\nu$
a
$\sigma$
-finite measure on U, and
$\mathbb{S} =\{1,2,\ldots \}$
the switching state space. Let
$d\geq1$
be an integer. Assume that
$b\colon \mathbb{R}^{d}\times\mathbb{S}\to \mathbb{R}^{d}$
,
$\sigma\colon \mathbb{R}^{d}\times\mathbb{S}\to \mathbb{R}^{d\times d}$
, and
$c\colon \mathbb{R}^{d}\times\mathbb{S}\times U\to \mathbb{R}^{d}$
are Borel-measurable functions.We suppose that
$(X,\varLambda)$
is a right continuous strong Markov process with left-handlimits on
$\mathbb{R}^{d}\times\mathbb{S}$
such that the first component X satisfies the following stochastic differential equation (SDE):

where W is a standard d-dimensional Brownian motion, N is a Poisson random measure on
$[0,\infty)\times U$
with intensity
${\text{d}} t\nu({\text{d}} u)$
, and
$\tilde{N}$
is the associated compensated Poisson random measure. As in [Reference Nguyen and Yin10], [Reference Nguyen and Yin12], [Reference Xi and Zhu17], and [Reference Xi, Yin and Zhu18], we suppose that
$\varLambda$
is a continuous-time stochastic process taking values in the set
$\mathbb{S}$
and satisfies

for all
$x \in \mathbb{R}^d$
. On the other hand, the process
$\varLambda$
can be defined rigorously as follows. Consider the family of disjoint intervals
$\{\Delta_{kl}(x) \colon k, l \in \mathbb{S}\}$
defined on the positive half of the realline by

where
$q_{k}(x) \;:\!=\; \sum_{l\in \mathbb{S}\backslash\{k\}}q_{kl}(x)$
. If
$q_{kl}(x) = 0$
and
$k\neq l$
, then we set
$\Delta_{kl}(x) = \emptyset$
. We note that
$\{\Delta_{kl}(x) \colon k, l \in \mathbb{S}\}$
are disjoint intervals and the length of the interval
$\Delta_{kl}(x)$
is equal to
$q_{kl}(x)$
. Let
$h\colon \mathbb{R}^d\times\mathbb{S}\times\mathbb{R}_{+}\to \mathbb{R}$
be defined by

In other words, we set

Hence the process
$\varLambda$
can be described as a solution to the stochastic differential equation

where
$N_{1}$
is a Poisson random measure on
$[0,\infty)\times[0,\infty)$
with characteristic measure
$\mathfrak{m}({\text{d}} z)$
, the Lebesgue measure.
We make the following standing assumption throughout the paper.
-
(H) For any
$(x,k) \in \mathbb{R}^d\times\mathbb{S}$ , the system of stochastic differential equations (2.1) and (2.3) has a non-explosive weak solution
$(X^{(x,k)},\varLambda^{(x,k)})$ with initial condition (x, k), and the solution is unique in the sense of probability law.
Consider the semigroup
$(P_{t})$
of operators
$P_t$
given by

The corresponding transition probability is defined by

for
$t \ge0, (x,k) \in \mathbb{R}^{d}\times \mathbb{S}$
, and
$B\in \mathfrak{B}(\mathbb{R}^{d}\times \mathbb{S})$
.
Given a function
$f\colon \mathbb{R}^d\times\mathbb{S} \rightarrow \mathbb{R}$
with
$f(\cdot,k) \in C^{2}_{c}(\mathbb{R}^d)$
for each
$k\in \mathbb{S}$
, let
$a(x,k) \;:\!=\; \sigma(x,k)\sigma(x,k)^{T}$
. Then the infinitesimal generator of the regime-switching jump diffusion
$(X,\varLambda)$
is given by

where



In (2.5), (2.6), and hereafter, we let

denote the gradient, and let

be the Hessian matrix of f with respect to x.
3. Exponential ergodicity
We are interested in whether the semigroup
$(P_{t})$
defined in (2.4) has an invariant probability measure
$\pi(\!\cdot\!)$
and, if it does, what is the convergence rate of the transition probability
$P(t,(x,k),\cdot)$
to
$\pi(\!\cdot\!)$
. Recall that a
$\sigma$
-finite measure
$\pi(\!\cdot\!)$
on the Borel
$\sigma$
-algebra
$\mathfrak{B} (\mathbb{R}^d\times\mathbb{S})$
is called invariant for the semigroup
$(P_{t})$
if

For any function
$f\colon \mathbb{R}^d \times\mathbb{S}\longrightarrow [1,\infty)$
and any signed measure
$\mu$
on
$\mathfrak{B} (\mathbb{R}^d\times\mathbb{S})$
, we set

where

We note that the total variation norm
$\|\mu\|_{\text{TV}}$
is the special case of
$\|\mu\|_{f}$
when
$f =1$
. Using the terminology in [Reference Meyn and Tweedie8], we say that the process
$(X,\varLambda)$
is f-exponentially ergodic if there exist a probability measure
$\pi(\!\cdot\!)$
, a constant
$\theta$
in (0,1), and a finite-valued function
$\Theta(x,k)$
such that

for all
$t \geq 0$
and all
$(x,k) \in \mathbb{R}^d \times \mathbb{S}$
.
We obtain the following result as a direct consequence of Theorem 6.1 of [Reference Meyn and Tweedie8]. Similar results for regime-switching jump diffusions can also be found in [Reference Xi16] and [Reference Xi and Zhu17] when the switching state
$\mathbb{S}$
is finite and infinite, respectively.
Theorem 3.1. Suppose that all compact subsets of
$\mathbb{R}^d\times\mathbb{S}$
are petite for some skeleton chain of
$(X(t),\varLambda(t))$
. If there exists a Foster–Lyapunov function
$U\colon \mathbb{R}^d\times\mathbb{S}\rightarrow [0,\infty)$
, that is, U satisfies
-
(i)
$U(x,k)\rightarrow \infty$ as
$ |x|\vee k\rightarrow \infty$ ,
-
(ii)
$\mathscr{A}\; U(x,k) \leq - \alpha U(x,k) + \beta$ for all
$ x \in \mathbb{R}^d$ ,
$k\in \mathbb{S}$ ,
where
$\alpha, \beta > 0$
are constants, then the process
$(X,\varLambda)$
is f-exponentially ergodic with
$f(x,k)\;:\!=\; U(x,k) + 1$
and
$\Theta(x,k) = B(U(x,k)+1)$
, where B is a finite constant.
As in [Reference Meyn and Tweedie6,Reference Meyn and Tweedie7], a set
$B\in \mathfrak{B}(\mathbb{R}^{d}\times \mathbb{S})$
and a sub-probability measure
$\varphi$
on
$\mathfrak{B}(\mathbb{R}^{d}\times \mathbb{S})$
are called petite for the h-skeleton chain
$\{(X(nh),\varLambda(nh))\colon n=0,1,\ldots\}$
(
$h> 0$
) if, for some probability a on
$\mathbb{Z}_{+}$
, we have

Moreover, we say that the semigroup
$(P_{t})$
, or equivalently the process
$(X,\varLambda)$
, is
$\varphi$
-irreducible if
$\varphi$
is a
$\sigma$
-finite measure on
$\mathcal{B}(\mathbb{R}^d\times\mathbb{S})$
and

Similarly, the h-skeleton chain
$\{(X(nh),\varLambda(nh))\colon n=0,1,\ldots\}$
is said to be
$\varphi$
-irreducible if
$\varphi$
is a
$\sigma$
-finite measure on
$\mathfrak{B}(\mathbb{R}^d\times\mathbb{S})$
and

The semigroup
$(P_{t})$
or the process
$(X,\varLambda)$
is said to be irreducible or open set irreducible if, for any
$t > 0$
and
$(x,k) \in \mathbb{R}^d \times \mathbb{S}$
, we have

for all
$l \in \mathbb{S}$
and every non-empty open set
$B \in \mathfrak{B}(\mathbb{R}^d)$
.
According to Theorem 3.4 (ii) of [Reference Meyn and Tweedie6], if the semigroup
$(P_{t})$
is
$\varphi$
-irreducible with the Feller property and if the support of
$\varphi$
has non-empty interior, then all compact subsets of
$\mathbb{R}^d\times\mathbb{S}$
are petite. To treat the petiteness of compact sets we employ the results from [Reference Kunwai and Zhu4]. To this end, let us state the following assumptions.
Assumption 3.1. For each
$k \in \mathbb{S}$
and
$x \in \mathbb{R}^d$
, the stochastic differential equation

has a non-explosive weak solution
$X^{(k)}$
with initial condition x and the solution is unique in the sense of probability law.
Assumption 3.2.
-
(i) There exist positive constants
$\lambda$ and
$\kappa$ such that, for each
$x\in \mathbb{R}^d$ and
$k \in \mathbb{S}$ , we have
(3.2)and\begin{equation}2\langle x, b(x,k)\rangle \leq \kappa\bigl(|x|^2+1\bigr), \|\sigma(x,k)\|^2 + \int_{U}|c(x,k,u)|^2\nu({\text{d}} u) \leq \kappa\bigl(|x|^2+1\bigr)\end{equation}
(3.3)\begin{equation}\langle \xi, a(x,k)\xi\rangle \geq \lambda|\xi|^2, \quad \xi \in \mathbb{R}^d.\end{equation}
-
(ii) Thereexists a non-negative function
$g\in C (0,\infty) $ satisfying
\begin{equation*} \int_{0}^{1}g(r)\,{\text{d}} r < \infty,\end{equation*}
(3.4)for all\begin{align}&\|\sigma_{\lambda}(x,k) - \sigma_{\lambda}(z,k)\|^2 + 2\langle x-z,b(x,k)-b(z,k)\rangle \notag \\& \quad + \int_{U}|c(x,k,u)-c(z,k,u)|^2v({\text{d}} u) \leq 2\kappa |x-z|g(|x-z|)\end{align}
$x, z \in \mathbb{R}^d$ with
$|x-z| \leq \delta_{0}$ , where
$\delta_{0}$ is a positive constant and
$\sigma_{\lambda}$ theunique symmetric non-negative definite matrix-valued function such that
$\sigma_{\lambda}^2(x,k) = a(x,k) - \lambda I$ .
Remark 3.1. Assumption 3.1 requires each subsystem to be well-posed in the weak sense. The linear growth condition (3.2) and the uniform ellipticity (3.3) are required to derive the irreducibility in Theorem 1.12 of [Reference Kunwai and Zhu4]. Assumption 3.2 (ii) allows us to deal with regime-switching jump diffusions with non-Lipschitz coefficients. For example, we can treat the case of Holder-continuous coefficients by taking
$g(r) = r^{-p}$
for some
$0\leq p < 1$
in (3.4). Therefore Theorem 3.2 improves many results in the literature such as Theorem 6.3 of [Reference Xi16] and Theorem 6.3 of [Reference Xi and Zhu17], which require the coefficients to be Lipschitz-continuous.
Assumption 3.3.
-
(i) For each
$k \in \mathbb{S}$ there exists a concave function
$\gamma_{k}\colon \mathbb{R}_{+} \mapsto \mathbb{R}_{+}$ with
$\gamma_{k}(0) = 0$ such that, for each
$x, y \in \mathbb{R}^d$ with
$|x| \vee |y| \le R$ , we have
(3.5)for some positive constant\begin{equation}\sum_{l\in \mathbb{S}\backslash\{k\}}|q_{kl}(x)-q_{kl}(y)| \leq \kappa_{R} \gamma_{k}(|x-y|)\end{equation}
$\kappa_{R}$ .
-
(ii) There exists a positive constant
$\kappa_{0} > 0$ such that
(3.6)for all\begin{equation}0 \leq q_{kl}(x) \leq \kappa_{0} l 3^{-l}\end{equation}
$x \in \mathbb{R}^d$ and
$k \neq l \in \mathbb{S}$ .
-
(iii) For any
$k, l \in \mathbb{S}$ , there exist
$k_0, k_1,\ldots,k_n \in \mathbb{S}$ with
$k_i \neq k_{i+1}$ ,
$k_0=k$ , and
$k_n = l$ such that the set
$\{x\in \mathbb{R}^d \colon q_{k_ik_{i+1}}(x) > 0\}$ has positive Lebesgue measure for all
$i=0, 1,\ldots,n-1$ .
Remark 3.2. Condition (3.5) plays a vital role in deriving the Feller and strong Feller properties of regime-switching jump diffusions with countable regimes; see [Reference Kunwai and Zhu4] and [Reference Xi, Yin and Zhu18]. Since we are working with regime-switching jump diffusions, one can expect that the switching component
$\Lambda$
must satisfy some sort of irreducibility conditions so that the process
$(X,\Lambda)$
is irreducible. Assumption 3.3 (iii) is imposed to guarantee that
$\Lambda$
is irreducible. Condition (3.6) is a technical assumption and it is imposed so that we can express the transition probability
$P(t,(x,k),\cdot)$
in a suitable form; see the proof of Theorem 4.8 of [Reference Xi, Yin and Zhu18].
We obtain the following result as a direct consequence of Propositions 6.1.5 and 6.1.6 of [Reference Meyn and Tweedie9]. For the sake of completeness we give the proof here.
Proposition 3.1. Suppose that Assumptions 3.1, 3.2, and 3.3 hold. Then the h-skeleton chain
$\{(X(nh),\varLambda(nh))\colon n=0,1,\ldots\}$
is
$\varphi$
-irreducible, where
$\varphi = P(h,(x,k),\cdot)$
.
Proof. From Theorem 1.7 of [Reference Kunwai and Zhu4], the process
$(X,\Lambda)$
has the strong Feller property and so does the chain
$\{(X(nh),\varLambda(nh))\colon n=0,1,\ldots\}$
. Then
$ P(h,\cdot, A)$
is lower semicontinuous for every
$A \in \mathcal{B}(\mathbb{R}^d\times\mathbb{S})$
; see for example Proposition 6.1.1 of [Reference Meyn and Tweedie9]. Given a measurable set
$A \in \mathcal{B}(\mathbb{R}^d\times\mathbb{S})$
with
$P(h,(x,k),A)>0$
, since
$P(h,\cdot, A)$
is lower semicontinuous then there exists a neighborhood U of (x, k) such that
$P(h,(z,j),A)>0$
for all
$(z,j) \in U$
. From Theorem 1.12 of [Reference Kunwai and Zhu4], the semigroup
$(P_{t})$
is open set irreducible, and hence every point in
$\mathbb{R}^d\times\mathbb{S}$
is reachable. In particular, the (x, k) is reachable. Hence, for any
$(y,i) \in \mathbb{R}^d\times\mathbb{S}$
, there exists
$n\geq 1$
such that
$P(nh,(y,i),U) > 0$
. Then we have

Summingthis up gives
$\sum_{n=1}^{\infty}P(nh,(y,i),A) > 0$
. This completes the proof.
Proposition 3.2. Suppose that Assumptions 3.1, 3.2, and 3.3 hold. Then all compact sets of
$\mathbb{R}^{d}\times \mathbb{S}$
are petite for any h-skeleton chain of
$(X,\varLambda)$
.
Proof. The chain
$\{(X(nh),\varLambda(nh))\colon n=0,1,\ldots\}$
is
$\varphi$
-irreducible, where
$\varphi = P(h,(x,k),\cdot)$
by Proposition 3.1. This chain has the strong Feller property. Moreover, it is open set irreducible and hence every point in
$\mathbb{R}^d\times\mathbb{S}$
is reachable. Then we have
$\text{supp}(\varphi) = \mathbb{R}^d\times\mathbb{S}$
; see Lemma 6.1.4 of [Reference Meyn and Tweedie9]. Therefore every compact subset of
$\mathbb{R}^d\times\mathbb{S}$
is petite by Theorem 3.4 (ii)of [Reference Meyn and Tweedie6].
In order to obtain the exponential ergodicity, we still need to determine the existence of a Foster–Lyapunov function; this will be investigated in the next section under Assumptions 4.1 and 4.2. However, to keep the flow of the paper, let us now state and prove our main result in this section, as follows.
Theorem 3.2. Suppose that Assumptions 3.1, 3.2, 3.3, 4.1, and 4.2 hold; then the process
$(X,\varLambda)$
is f-exponentially ergodic.
Proof. Thanks to Proposition 3.2, under Assumptions 3.1, 3.2, and 3.3, all compact sets of
$\mathbb{R}^{d}\times \mathbb{S}$
are petite for some h-skeleton chain. Under Assumptions 4.1 and 4.2, a Foster–Lyapunov function U exists by Theorem 4.1. Then the desired f-exponential ergodicity follows from Theorem 3.1, where
$f(x,k)\;:\!=\; U(x,k) + 1$
.
4. Existence of Foster–Lyapunov functions
Having established sufficient conditions for petite compact subsets of
$\mathbb{R}^{d}\times \mathbb{S}$
, it remains to find an appropriate Foster–Lyapunov function. In practice it is not easy to find the right Foster–Lyapunov function for an underlying regime-switching jump diffusion, especially when dealing with countably many regimes. Motivated by the recent paper [Reference Nguyen and Yin12] in which the stability of regime-switching diffusion was investigated, we develop a novel approach to constructing a Foster–Lyapunov function for regime-switching jump diffusions. Let us briefly sketch the idea here. Suppose that there exists a common ‘nice’ function
$V\colon \mathbb{R}^{d}\mapsto \mathbb{R}_{+}$
so that

where
$\alpha_{k}$
and
$\beta_{k}$
are real numbers. Suppose also that the generator Q(x) of the discrete component is ‘close’ to a strongly exponentially ergodic (see [Reference Nguyen and Yin12]) constant q-matrix in the neighborhood of
$\infty$
. Then, under some additional assumptions, we construct a Foster–Lyapunov function for the process
$(X,\Lambda)$
.
To proceed, we make the following assumptions.
Assumption 4.1.
-
(a) There exists an increasing function
$\phi\colon \mathbb{S}\rightarrow [0,\infty)$ such that
\begin{equation*} \lim_{k\to\infty}\phi(k) = \infty \end{equation*}
(4.1)where\begin{equation}\sum_{j\in \mathbb{S}}q_{kj}(x)[\phi(j)-\phi(k)] \leq C_{1} - C_{2}\phi(k)\quad \textit{for all } k \in \mathbb{S},\ x \in \mathbb{R}^d,\end{equation}
$C_{1} \ge 0$ and
$C_{2} > 0$ are constants.
-
(b) There exists a bounded and x-independent q-matrix
$\hat{Q} = (\hat{q}_{ij})_{i,j\in\mathbb{S}}$ that is strongly exponentially ergodic with invariant measure
$\nu=(\nu_1,\nu_2,\ldots )$ such that
(4.2)\begin{equation}\sup\limits_{k\in\mathbb{S}} \sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| \rightarrow 0 \quad \textit{as $ x\rightarrow \infty$.}\end{equation}
Assumption 4.2. There exists a twice continuously differentiable and norm-like function
$V\colon \mathbb{R}^d\rightarrow [1,\infty)$
such that, for each
$k\in\mathbb{S}$
,

where
$\{\alpha_k\}_{k\in\mathbb{S}}$
and
$\{\beta_k\}_{k\in\mathbb{S}}$
are bounded sequences of real numbers such that
$\beta_k \geq 0$
and

Remark 4.1. It is worth noticing that if
$\alpha_{k} < 0$
in (4.3) then V(x) is a Foster–Lyapunov function for the corresponding subsystem
$X^{(k)}$
defined in (3.1). However, to obtain the existence of a Foster–Lyapunov function for the process
$(X,\Lambda)$
, we only require (4.4) to be satisfied. In other words, we can still obtain exponential ergodicity of
$(X,\Lambda)$
as long as ‘most’ of the subsystems
$X^{(k)}$
are nice in some sense; for example, in this case the ‘average’ in (4.4) issatisfied.
Theorem 4.1. Suppose that Assumptions 4.1 and 4.2 hold. Then there exists a Foster–Lyapunov function
$U\colon \mathbb{R}^{d}\times \mathbb{S} \mapsto \mathbb{R}_{+} $
satisfying the following properties:
-
(i)
$ U(x,k)\rightarrow \infty$ as
$ |x|\vee k\rightarrow \infty$ ,
-
(ii)
$\mathscr{A}\;U(x,k) \leq - \alpha U(x,k) + \beta $ for all
$ x \in \mathbb{R}^d$ ,
$k\in \mathbb{S}$ ,
where
$\alpha, \beta > 0$
are constants.
Proof. Let
$\gamma \;:\!=\; -\sum_{k\in \mathbb{S}}\alpha_k\nu_k > 0$
. Since
$\{\alpha_{k}\}_{k\in \mathbb{S}}$
is bounded and
$\sum_{k\in \mathbb{S}}\nu_k = 1$
, the series
$\sum_{k=1}^{\infty}(\alpha_k + \gamma)\nu_k $
is absolutely convergent and hence

Since
$\hat{Q} = (\hat{q}_{ij})_{i,j\in\mathbb{S}}$
is strongly exponentially ergodic, it follows from Lemma A.1 of [Reference Nguyen and Yin12] that there exists a bounded sequence of real numbers
$\{\gamma_k\colon k \in \mathbb{S}\}$
such that

Next we choose
$p \in (0,1)$
so that

and

Define a function
$U\colon \mathbb{R}^d\times\mathbb{S}\rightarrow [0,\infty)$
by

From (4.6) we see that U(x, k) is non-negative and satisfies
$\lim_{ |x|\vee k\rightarrow \infty} U(x,k) =\infty$
.
The rest of the proof is to verify that condition (ii) holds. To proceed, we compute and estimate each term of the generator

First, observe that

and

Then

To estimate the second term we note that the function
$f(r) = r^p$
for
$r > 0$
is concave since
$0 < p < 1$
. Hence
$b^p - a^p \leq pa^{p-1}[b-a]$
for all a,
$b > 0$
. By taking
$b=V(x+c(x,k,u))$
and
$a=V(x)$
, we have

Hence

Finally we estimate the last term Q(x)U(x, k). Note that
$q_{kj}(x) \geq 0$
for all
$k\neq j$
. Since
$\phi$
is increasing and satisfies
$\phi(k)\rightarrow \infty$
as
$k \rightarrow \infty$
, then (4.1) asserts that

Then
$\sum_{j\in \mathbb{S}}q_{kj}(x)[\phi(j)-\phi(k)]$
is absolutely convergent for each
$k \in \mathbb{S}$
. Since
$\{\gamma_{k}\}_{k\in\mathbb{S}}$
is a bounded sequence, we have

Hence
$\sum_{j\in\mathbb{S}}q_{kj}(x)[pV^p(x)(\gamma_k-\gamma_j)]$
is also absolutely convergent. We compute

The absolute convergences allow us to rearrange the summands of Q(x)U(x, k) as follows:

where we use (4.1) to obtain the inequality. Furthermore, since
$\sum_{j\in\mathbb{S}}q_{kj}(x)\gamma_j$
and
$\sum_{j\in\mathbb{S}}\hat{q}_{kj}\gamma_j$
are also absolutely convergent, we have

wherewe use (4.5) to obtain the last equality. It follows from (4.8)–(4.10) that

where the last inequality follows from (4.3). Thanks to (4.6), we have
$1-p\gamma_k > 0$
and it is bounded. Let

Note that

From (4.7), we see that
$-(p\gamma_k\alpha_k + \gamma) \leq - 0.5\gamma$
. Hence

On the other hand, since V is norm-like and
$\{\beta_k\}_{k\in\mathbb{S}}$
is bounded, there exists an
$M_{1} > 0$
such that

Similarly, we can use (4.2) and (4.6) to find an
$M_{2}> 0$
such that

for all
$|x| \ge M_{2}$
and
$ k\in \mathbb{S}$
. Now plugging (4.12)–(4.14) into (4.11) yields

for all
$|x| \ge M_{1}\vee M_{2}$
and
$k\in \unicode{x00DF}$
. Note that we used the fact that
$0.5 \delta p \le C_{2}$
to derive the last inequality. To complete the proof, we choose
$\alpha, \beta > 0$
so that

holds for all
$x \in \mathbb{R}^d$
and
$k\in \mathbb{S}$
. This completes the proof.
Corollary 4.1. Suppose that the q-matrix
$Q(x) =(q_{kl})$
is constant, irreducible, and strongly exponentially ergodic with invariant measure
$\pi=(\pi_1,\pi_2,\ldots)$
. Then, under Assumptions 4.1 (a) and 4.2, a Foster–Lyapunov function exists.
The strongly exponentially ergodic q-matrix
$\hat{Q}$
in Assumption 4.1 (b) plays a very crucial role in the proof of Theorem 4.1 as it allows us, by Lemma A.1 of [Reference Nguyen and Yin12], to find a bounded sequence
$\{\gamma_k\colon k \in \mathbb{S}\}$
satisfying equality (4.5). To demonstrate our results, let us first give some examples of such matrices.
Given a positive constant
$\theta > 0$
, consider the q-matrix
$\hat{Q} = (\hat{q}_{ij})$
given by

that is,

It is clear that
$\nu = (\frac{1}{2},\frac{1}{3},\frac{1}{3^2},\ldots )$
solves the equation
$\nu \hat{Q} =0$
and
$\nu 1 = 1$
. Then
$\nu$
is an invariant probability measure. Solving the Kolmogorov backward equation
$\hat{P}'(t) = \hat{Q}\hat{P}(t)$
gives

that is,

For each
$i \in \mathbb{S}$
and
$t \geq 0$
, we see that

Then, for arbitrary but fixed
$\theta > 0$
, any Markov chain generated by
$\hat{Q}$
is strongly exponentially ergodic.
Example 4.1. Consider the SDE

where W is a standard two-dimensional Brownian motion and
$\tilde{N}$
is the compensated Poisson random measure on
$[0,\infty)\times U$
with intensity
${\text{d}} t\nu({\text{d}} u)$
in which
$U = \{u\in \mathbb{R}^2 \colon 0 < |u| < 1\}$
and
$\nu({\text{d}} u)\;:\!=\; {{\text{d}} u }/{|u|^{2+\delta}}$
for some
$\delta \in (0,2)$
. The coefficients of (4.15) are given by

where
$\gamma$
is a positive constant so that
$\gamma^2\int_{U}|u|^2\nu({\text{d}} u) = 1$
. The
$\varLambda$
component takes values in
$\mathbb{S}\;:\!=\; \{1,2,\ldots \}$
and is generated by
$Q(x) = (q_{kj}(x))$
given by

Evidently (4.15) possesses a unique strong solution
$(X,\varLambda) = \{(X(t),\varLambda(t)), 0\le t < \infty \}$
(see Theorem 2.5 of [Reference Xi, Yin and Zhu18]). Assumptions 3.1, 3.2, and 3.3 are trivially satisfied. Next we verify Assumptions 4.1 and 4.2. To show (4.1) we consider the function
$\phi(k) = k$
and observe that

For (4.2) we consider the following. Let
$\hat{\Lambda}$
be a continuous-time Markov chain with state space
$\mathbb{S}$
and generated by
$\hat{Q} =\{\hat{q}_{kj}\}$
, where

As shown above, with
$\theta = 1$
,
$\hat{\Lambda}$
is strongly exponentially ergodic with invariant measure
$\nu = (\frac{1}{2},\frac{1}{3},\frac{1}{3^2},\ldots)$
. We see that

Hence

thus establishing (4.2). As a result, Assumption 4.1 is verified.
To verify Assumption 4.2, we consider function
$V(x) \;:\!=\; |x|^2$
and observe that
$\nabla V(x) = 2x$
and
$\nabla^{2}V(x) = 2I$
. We compute

Then we set
$\beta_k = 2$
for all
$k \in \mathbb{S}$
and

We also see that

Then Theorem 4.1 ensures the existence of a Foster–Lyapunov function U(x, k). Moreover, the process
$(X,\varLambda)$
is exponentially ergodic by Theorem 3.2.
5. Application to feedback controls
Inthis section we illustrate an application of Theorem 3.2. To proceed, we start with the system of SDEs

where b,
$\sigma$
, and c are appropriate measurable functions. Motivated by the study of feedback controls for weak stabilization studied in [Reference Zhu and Yin20], we raise and try to answer the following question: If a regime-switching jump diffusion is not exponentially ergodic or even not ergodic, can we find a suitable control so that the controlled regime-switching jump diffusion becomes exponentially ergodic? To this end, we consider the SDE

where
$\xi\colon \mathbb{R}^{d}\times\mathbb{S}\to \mathbb{R}^{d}$
denotes the feedback control which will be determined later on. We let
$(\tilde{X},\tilde{\varLambda})$
denote the solution to the system of SDEs

where
$\tilde{h}$
can be defined in a similar way to (2.2). In other words, if
$\Lambda$
is determined by the probability rate matrix Q(x), then
$\tilde{\Lambda}$
will be determined by the matrix
$Q(\tilde{x})$
.
In practice we usually decompose the switching state space
$\mathbb{S}$
into the union of two disjoint subsets, namely
$\mathbb{S} = \mathbb{S}_{\text{int}}\cup\mathbb{S}_{ab}$
. To be more precise,
$\mathbb{S}_{ab}$
consists of states in which no intervention is allowed while
$\mathbb{S}_{\text{int}}$
consists of those states when any intervention can be implemented. It is reasonable and easy to consider the feedback controls of the form

where
$L(k) \in \mathbb{R}^{d\times d}$
is a constant matrix for
$k \in\mathbb{S}_{\text{int}}$
. Of course, we take
$L(k)=0$
for each
$k \in \mathbb{S}_{ab}$
. For simplicity, we set
$\tilde{b}(y,k) \;:\!=\; b(y,k) - L(k)y$
. Then (5.2) becomes

We let
$\tilde{X}^{(k)}$
denote a non-explosive solution of the subsystem

Before we proceed, one may ask whether or not the control process
$\xi$
in (5.3) is admissible, that is, whether the system (5.2) or equivalently (5.4) has a unique non-explosive solution. To tackle this issue, we make the following assumption.
Assumption 5.1. Assume the following conditions hold.
-
If
$d=1$ , then there exist a positive number
$\delta_{0}$ and a non-decreasing concave function
$\rho\colon [0,\infty)\to [0,\infty)$ satisfying
\begin{equation*} \int_{0^{+}}\dfrac{{\text{d}} r}{\rho(r)} = \infty,\end{equation*}
$k\in \mathbb{S}$ ,
$R>0$ , and
$x, z \in \mathbb{R}$ with
$|x|\vee |z| \leq R$ and
$|x-z|\leq \delta_{0}$ ,
\begin{gather*} \text{sgn}(x-z)(b(x,k) - b(z,k)) \leq \kappa_{R}\rho(|x-z|),\\|\sigma(x,k)-\sigma(z,k)|^2 +\int_{U}|c(x,k,u)-c(z,k,u)|^2v({\text{d}} u) \leq \kappa_{R}|x-z|,\end{gather*}
$\kappa_{R}$ is a positive constant and
$\text{sgn}(a)=1$ if
$a>0$ and
$-1$ if
$a\leq 0$ . In addition, for each
$k \in \mathbb{S}$ , the function c satisfies
\begin{equation*}\textit{the function } x\mapsto x+c(x,k,u) \textit{ is non-decreasing for all } u \in U,\end{equation*}
$\beta > 0$ such that
\begin{equation*}|x-z +\theta(c(x,k,u)-c(z,k,u))| \geq \beta|x-z|, \forall(x,z,u,\theta) \in \mathbb{R}\times\mathbb{R}\times U\times [0,1].\end{equation*}
-
If
$d\geq 2$ , then there exist a positive number
$\delta_{0}$ and a non-decreasing concave function
$\rho\colon [0,\infty)\to [0,\infty)$ satisfying
\begin{equation*} 0 < \rho(r) \leq (1+r)^2\rho(r/(1+r))\quad \text{for all $ r>0$,}\quad \textit{and}\quad \int_{0^{+}}\dfrac{{\text{d}} r}{\rho(r)} = \infty,\end{equation*}
$k\in \mathbb{S}, R>0$ and
$x, z \in \mathbb{R}^d$ with
$|x|\vee |z| \leq R$ and
$|x-z|\leq \delta_{0}$ ,
\begin{align*}&2\langle x-z, b(x,k)-b(z,k)\rangle + |\sigma(x,k)-\sigma(z,k)|^2 \notag \\& \quad + \int_{U}|c(x,k,u)-c(z,k,u)|^2v({\text{d}} u)\leq \kappa_{R}\rho(|x-z|^2),\end{align*}
$\kappa_{R}$ is a positive constant.
Theorem 5.1. Suppose that Assumptions 3.2, 3.3, 4.1, and 5.1 hold, and we are given a constant matrix L(k) for each
$k \in \mathbb{S}_{\text{int}}$
. If there exists a bounded sequence of real numbers
$\{\alpha_{k}\}$
such that

and

where
$\kappa$
is the positive constant given in (3.2), then the controlled process
$(\tilde{X},\tilde{\varLambda})$
is exponentially ergodic.
Proof. In view of Theorem 3.2, we proceed as follows. Since the control process
$\xi$
defined in (5.3) is linear in the x variable, it is clear that if (5.1) satisfies Assumptions 3.2, 3.3, 4.1, and 5.1, then so does (5.4). Thanks to Lemma 2.4 of [Reference Xi, Yin and Zhu18], under Assumption 5.1 and (3.2), there exists a unique non-explosive strong solution
$\tilde{X}^{(k)}$
to the SDE (5.5).
We observe that (3.2), (3.5), (3.6), and (4.1) constitute Assumption 2.1 of [Reference Xi, Yin and Zhu18] with relaxed condition (3.5). Moreover, Theorem 2.5 of [Reference Xi, Yin and Zhu18] is still valid under this relaxation. Therefore Theorem 2.5 of [Reference Xi, Yin and Zhu18] ensures the existence and uniqueness of a non-explosive strong solution
$(\tilde{X},\tilde{\Lambda})$
to (5.4).
Next, we only need to verify (4.3) for
$\tilde{X}$
. Consider the function
$V(x)\;:\!=\; |x|^2$
. Thanks to (3.2) and (5.7), we verify that

This, together with (5.6), implies that
$(\tilde{X},\tilde{\varLambda})$
satisfies Assumption 4.2. It follows directly from Theorem 3.2 that the controlled process
$(\tilde{X},\tilde{\varLambda})$
is exponentially ergodic.
As in [Reference Zhu and Yin20], one of the simplest example of feedback controls is of the form

where
$\theta(k)$
is a non-negative constant and I is the
$d\times d$
identity matrix. That is, L(k) takes the form

Since
$\kappa$
is a fixed constant and
$\{\alpha_{k}\}$
is bounded, then (5.7) is immediate if we choose
$\theta(k)$
large enough; that is,
$ 2\theta(k) \geq \kappa - \alpha_{k}$
. To summarize this discussion, we state the following corollary.
Corollary 5.1. Suppose that Assumptions 3.2, 3.3, 4.1, and 5.1 hold. Assume further that
$\sum_{k\in \mathbb{S}_{\text{int}}}\nu_k >0$
. Then there exists a feedback control
$\xi$
so that the controlled process
$(\tilde{X},\tilde{\varLambda})$
is exponentially ergodic.
Proof. We take
$\alpha_{k} = \kappa$
for all
$k \in \mathbb{S}_{ab}$
and
$\alpha_{k} = -2\kappa/\sum_{k\in \mathbb{S}_{\text{int}}}\nu_k$
for all
$k \in \mathbb{S}_{\text{int}}$
. Then
$\sum_{k\in \mathbb{S}}\alpha_k\nu_k < 0.$
Choose
$\theta(k)$
large enough so that
$ \theta(k) \geq \kappa - \alpha_{k}$
for
$k \in \mathbb{S}_{\text{int}}$
and
$ \theta(k) = 0$
for
$k \in \mathbb{S}_{ab}$
. So
$\xi(x,k) \;:\!=\; -\theta(k)Ix$
is the desired feedback control.
Example 5.1. Consider the SDE

where W is the standard two-dimensional Brownian motion. Suppose that
$\varLambda$
takes values in the set
$\mathbb{S}\;:\!=\; \{1,2\}$
and is generated by the constant rate matrix

Define the coefficients in (5.8) as follows:

For any matrix A, we let
$\lambda_{\min}(A)$
denote the minimum eigenvalue of A. We verify that

and

In view of Theorem 4.13 of [Reference Zhu and Yin20], we conclude that the process
$(X,\Lambda)$
is transient and hence it is not ergodic. One can show that Assumptions 3.2, 3.3, 4.1, and 5.1 are satisfied. It follows from Corollary 5.1 that there exists a feedback control
$\xi$
for which the controlled process
$(\tilde{X},\tilde{\varLambda})$
is exponentially ergodic.
Acknowledgements
The author would like to express the deepest thanks to Professor Chao Zhu for valuable discussions and encouragements throughout this paper. He would also like to thank the Editor and reviewers for their helpful comments and suggestions.
Funding information
There are no funding bodies to thank relating to this creation of this article.
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.