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):

(2.1) \begin{equation}{\text{d}} X(t) = b(X(t),\varLambda(t))\,{\text{d}} t + \sigma(X(t),\varLambda(t)) \,{\text{d}} W(t) + \int_{U}c(X(t^{-}),\varLambda(t^{-}),u)\tilde{N}({\text{d}} t,{\text{d}} u),\end{equation}

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

\begin{equation*} \mathbb{P}\{\varLambda(t+\Delta)=l\mid \varLambda(t)=k, X(t)=x\} =\begin{cases}q_{kl}(x)\Delta +{\text{o}} (\Delta) & \text{if $ k\neq l$,} \\1 + q_{kl}(x)\Delta +{\text{o}} (\Delta) & \text{if $ k=l$,}\end{cases}\end{equation*}

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

\begin{align*}\Delta_{12}(x) &= [0,q_{12}(x)),\\\Delta_{13}(x) &= [q_{12}(x), q_{12}(x) + q_{13}(x)),\\&\ \ \vdots \\\Delta_{21}(x) &= [q_{1}(x), q_{1}(x) + q_{21}(x)),\\\Delta_{23}(x) &= [q_{1}(x) + q_{21}(x), q_{1}(x) + q_{21}(x) + q_{23}(x)),\\&\ \ \vdots \\\Delta_{31}(x) &= [q_{1}(x) + q_{2}(x), q_{1}(x) + q_{2}(x) + q_{31}(x)),\\&\ \ \vdots\end{align*}

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

(2.2) \begin{equation}h(x,k,r) = \sum_{l\in \mathbb{S}}(l-k)1_{\Delta_{kl}(x)}(r).\end{equation}

In other words, we set

\begin{equation*}h(x,k,r) =\begin{cases}l - k & \text{if $ r \in \Delta_{kl}(x)$,} \\0 & \text{otherwise.}\end{cases}\end{equation*}

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

(2.3) \begin{equation}\varLambda(t) = \varLambda(0) + \int_{0}^{t}\int_{\mathbb{R}_{+}}h(X(s^{-}), \varLambda(s^{-}),r)N_{1}({\text{d}} s,{\text{d}} r),\end{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.

1. (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

(2.4) \begin{equation}P_{t} f(x,k)\;:\!=\; \mathbb{E}_{x,k}[f(X(t), \varLambda(t)]= \mathbb{E}\bigl[X^{(x,k)}(t),\varLambda^{(x,k)}(t)\bigr], \quad t\ge 0, \ f\in \mathfrak{B}_{b}(\mathbb{R}^{d}\times \mathbb{S}).\end{equation}

The corresponding transition probability is defined by

\begin{equation*}P(t,(x,k), B)\;:\!=\; P_{t} 1_{B} (x,k)\end{equation*}

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

\begin{equation*} \mathscr{A}\;f(x,k)\;:\!=\; \mathscr{A}_df(x,k) + \mathscr{A}_j f(x,k) + Q(x)f(x,k),\end{equation*}

where

(2.5) \begin{align} \mathscr{A}_d f(x,k) \;:\!=\; \dfrac{1}{2}\text{tr} (a(x,k)\nabla^{2}f(x,k)) + \langle b(x,k),\nabla f(x,k)\rangle, \end{align}

(2.6) \begin{align} \mathscr{A}_j f(x,k) \;:\!=\; \int_{U}(f(x+c(x,k,u), k) - f(x,k) -\langle\nabla f(x,k), c(x,k,u\rangle)\nu({\text{d}} u), \end{align}

(2.7) \begin{align} Q(x)f(x,k) \;:\!=\; \sum_{l\in\mathbb{S}}q_{kl}(x)[f(x,l) - f(x,k)]. \end{align}

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

\begin{equation*}\nabla f(x,k) \;:\!=\; \biggl(\frac{\partial}{\partial x_1}f(x,k),\ldots,\frac{\partial}{\partial x_d}f(x,k)\biggr)^\top\end{equation*}

denote the gradient, and let

\begin{equation*}\nabla^{2} f(x,k)\;:\!=\; \biggl[\frac{\partial^2}{\partial x_i\partial x_j}f(x,k)\biggr]_{i,j}\end{equation*}

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

\begin{equation*}\pi(A) = \pi P_{t}(A)\;:\!=\; \int_{\mathbb{R}^d\times\mathbb{S}}P(t, (x,k),A)\pi({\text{d}} x,{\text{d}} k) \quad \text{{for all}}\ A\in\mathfrak{B} (\mathbb{R}^d\times\mathbb{S}) \text{ and } t \ge 0.\end{equation*}

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

\begin{equation*}\|\mu\|_{f} \;:\!=\; \sup\{|\mu(g)| \colon \text{all measurable} \, g(x,k) \, \text{with} \, |g| \leq f\},\end{equation*}

where

\begin{equation*}\mu(g) \;:\!=\; \int_{\mathbb{R}^d\times\mathbb{S}}g(x,k)\mu({\text{d}} x,{\text{d}} k).\end{equation*}

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

\begin{equation*} \|P(t,(x,k),\cdot) - \pi(\!\cdot\!)\|_{f} \leq \Theta(x,k)\theta^{t}\end{equation*}

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.

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

\begin{equation*}K_{a}((x,k), \cdot) \;:\!=\; \sum_{n=1}^{\infty} a(n)P(nh,(x,k), \cdot)\ge \varphi(\!\cdot\!) \quad \text{for all }(x,k) \in B.\end{equation*}

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

\begin{equation*}\varphi(A) > 0 \Rightarrow \int_{0}^{\infty} P(t,(x,k), A)\,{\text{d}} t > 0 \quad \text{for all } (x,k) \in \mathbb{R}^d\times\mathbb{S}.\end{equation*}

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

\begin{equation*}\varphi(A) > 0 \Rightarrow \sum_{n=1}^{\infty}P(nh,(x,k),A) > 0 \quad \text{for all } (x,k) \in \mathbb{R}^d\times\mathbb{S}.\end{equation*}

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

\begin{equation*}P(t,(x,k),B\times\{l\}) > 0\end{equation*}

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

(3.1) \begin{align} X^{(k)}(t)& = x+ \int_{0}^{t}b\bigl(X^{(k)}(s), k \bigr) \,{\text{d}} s + \int_{0}^{t} \sigma\bigl(X^{(k)}(s), k\bigr) \,{\text{d}} W(s) \notag \\ & \quad + \int_{0}^{t}\int_{U} c\bigl(X^{(k)}(s-), k, u\bigr)\tilde{N}({\text{d}} s, {\text{d}} u) \end{align}

has a non-explosive weak solution $X^{(k)}$ with initial condition x and the solution is unique in the sense of probability law.

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.

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

\begin{equation*}P((n+1)h,(y,i),A) \geq \int_{U} P(nh,(y,i),{\text{d}} z\times {\text{d}} j) P(h,(z,j),A)> 0.\end{equation*}

Summingthis up gives $\sum_{n=1}^{\infty}P(nh,(y,i),A) > 0$ . This completes the proof.

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.

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

\begin{equation*}\mathcal{L}_{k} V(x) \le \alpha_{k} V(x) + \beta_{k}\quad \text{for all }(x,k) \in \mathbb{R}^{d}\times \mathbb{S},\end{equation*}

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 4.2. There exists a twice continuously differentiable and norm-like function $V\colon \mathbb{R}^d\rightarrow [1,\infty)$ such that, for each $k\in\mathbb{S}$ ,

(4.3) \begin{equation}\mathcal{L}_kV(x) \leq \alpha_kV(x) + \beta_k \quad \textit{for all } x\in \mathbb{R}^d,\end{equation}

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

(4.4) \begin{equation}\sum_{k\in \mathbb{S}}\alpha_k\nu_k < 0.\end{equation}

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

\begin{equation*} \sum_{k=1}^{\infty}(\alpha_k + \gamma)\nu_k =\sum_{k=1}^{\infty}\alpha_k \nu_k + \sum_{k=1}^{\infty}\gamma \nu_k =0. \end{equation*}

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

(4.5) \begin{equation}\sum_{j\in \mathbb{S}}\hat{q}_{kj}\gamma_j = \alpha_k + \gamma \quad \text{for all } k \in \mathbb{S}.\end{equation}

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

(4.6) \begin{equation}p|\gamma_k| \leq 0.5\end{equation}

and

(4.7) \begin{equation}p|\gamma_k\alpha_k| \leq 0.5\gamma.\end{equation}

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

\begin{equation*} U(x,k) \;:\!=\; (1-p\gamma_k)V^p(x) + \phi(k).\end{equation*}

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

\begin{equation*}\mathscr{A}\;U(x,k) = \mathscr{A}_dU(x,k) + \mathscr{A}_j U(x,k) + Q(x)U(x,k).\end{equation*}

First, observe that

\begin{equation*}\nabla U(x,k) = p(1-p\gamma_k)V^{p-1}(x)\nabla V(x)\end{equation*}

and

\begin{equation*}\nabla^{2} U(x,k) = p(1-p\gamma_k)V^{p-1}(x)\nabla^{2} V(x) - p(1-p)(1-p\gamma_k)V^{p-2}(x)\nabla V(x)\nabla V(x)^\top.\end{equation*}

Then

(4.8) \begin{align}\mathscr{A}_d U(x,k)&= p(1-p\gamma_k)V^{p-1}(x)\dfrac{1}{2}\text{tr}(a(x,k)\nabla^{2}V(x) ) \notag\\& \quad - p(1-p)(1-p\gamma_k)V^{p-2}(x)\dfrac{1}{2}\text{tr}(a(x,k)\nabla V(x)\nabla V(x)^\top ) \notag \\&\quad + p(1-p\gamma_k)V^{p-1}(x)\langle b(x,k),\nabla V(x) \rangle \notag\\&= p(1-p\gamma_k)V^{p-1}(x)\biggl[\dfrac{1}{2}\text{tr}(a(x,k)\nabla^{2}V(x) ) + \langle b(x,k),\nabla V(x) \rangle \biggr] \notag\\& \quad - p(1-p)(1-p\gamma_k)V^{p-2}(x)\dfrac{1}{2}|\nabla V(x)^\top\sigma(x,k)|^2\notag \\&\leq p(1-p\gamma_k)V^{p-1}(x)\biggl[\dfrac{1}{2}\text{tr}(a(x,k)\nabla^{2}V(x) ) + \langle b(x,k),\nabla V(x) \rangle \biggr].\end{align}

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

\begin{align*}U& (x+c(x,k,u),k)) - U(x,k) -\langle\nabla U(x,k), c(x,k,u)\rangle \\& = (1-p\gamma_k)V^p(x+c(x,k,u)) - (1-p\gamma_k)V^p(x) - p(1-p\gamma_k)V^{p-1}(x)\langle\nabla V(x), c(x,k,u)\rangle \\&\leq p(1-p\gamma_k)V^{p-1}(x)[V(x+c(x,k,u)) - V(x) - \langle\nabla V(x), c(x,k,u)\rangle ].\end{align*}

Hence

(4.9) \begin{equation}\mathscr{A}_j U(x,k)\leq p(1-p\gamma_k)V^{p-1}(x)\int_{U}[V(x+c(x,k,u)) - V(x) - \langle\nabla V(x), c(x,k,u)\rangle ]\nu({\text{d}} u).\end{equation}

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

\begin{align*}\sum_{j\in \mathbb{S}}q_{kj}(x)|\phi(j)-\phi(k)| &= \sum_{j<k}q_{kj}(x)|\phi(j)-\phi(k)| + \sum_{j>k}q_{kj}(x)[\phi(j)-\phi(k)]\\&\leq -\sum_{j<k}q_{kj}(x)[\phi(j)-\phi(k)] + C_{1}-C_{2}\phi(k) - \sum_{j<k}q_{kj}(x)[\phi(j)-\phi(k)]\\&= C_{1}- C_{2} \phi(k) - 2\sum_{j<k}q_{kj}(x)[\phi(j)-\phi(k)]\\&< \infty.\end{align*}

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

\begin{align*}\sum_{j\in\mathbb{S}}|q_{kj}(x)[pV^p(x)(\gamma_k-\gamma_j)]|&= pV^p(x)\sum_{j\neq k}q_{kj}(x)|\gamma_k-\gamma_j|\\&\leq 2 pV^p(x)\sup\limits_{j \in \mathbb{S}}\{| \gamma_j|\} q_{k}(x)\\& < \infty.\end{align*}

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

\begin{align*}\sum_{j\in\mathbb{S}}q_{kj}(x)[pV^p(x)(\gamma_k-\gamma_j)]&= pV^p(x)\gamma_k\sum_{j\in\mathbb{S}}q_{kj}(x) - pV^p(x)\sum_{j\in\mathbb{S}}q_{kj}(x)\gamma_j\\&= 0 - pV^p(x)\sum_{j\in\mathbb{S}}q_{kj}(x)\gamma_j\\&= - pV^p(x)\sum_{j\in\mathbb{S}}q_{kj}(x)\gamma_j.\end{align*}

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

\begin{align*}Q(x)U(x,k) &= \sum_{j\in\mathbb{S}}q_{kj}(x)[U(x,j) - U(x,k)]\\&= \sum_{j\in\mathbb{S}}q_{kj}(x)[(1-p\gamma_j)V^p(x) +\phi(j)-(1-p\gamma_k)V^p(x) -\phi(k)]\\&= \sum_{j\in\mathbb{S}}q_{kj}(x)[pV^p(x)(\gamma_k-\gamma_j) + \phi(j) -\phi(k)]\\&= \sum_{j\in\mathbb{S}}q_{kj}(x)[pV^p(x)(\gamma_k-\gamma_j)] + \sum_{j\in\mathbb{S}}q_{kj}(x)[\phi(j) -\phi(k)]\\&= - pV^p(x)\sum_{j\in\mathbb{S}}q_{kj}(x)\gamma_j + \sum_{j\in\mathbb{S}}q_{kj}(x)[\phi(j) -\phi(k)]\\&\leq - pV^p(x)\sum_{j\in\mathbb{S}}q_{kj}(x)\gamma_j + C_{1} -C_{2} \phi(k),\end{align*}

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

(4.10) \begin{align}& Q(x)U(x,k) \notag\\&\quad \leq - pV^p(x)\sum_{j\in\mathbb{S}}q_{kj}(x)\gamma_j + C_{1} -C_{2}\phi(k) \notag\\&\quad = -pV^p(x)\sum_{j\in\mathbb{S}}(q_{kj}(x) -\hat{q}_{kj})\gamma_j -pV^p(x)\sum_{j\in\mathbb{S}}\hat{q}_{kj}\gamma_j + C_{1} -C_{2} \phi(k) \notag\\&\quad \leq pV^p(x)\sup\limits_{j\in\mathbb{S}}|\gamma_j|\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| -pV^p(x)\sum_{j\in\mathbb{S}}\hat{q}_{kj}\gamma_j + C_{1} -C_{2} \phi(k) \notag\\&\quad = pV^p(x)\sup\limits_{j\in\mathbb{S}}|\gamma_j|\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| - pV^p(x)(\alpha_k + \gamma) + C_{1} -C_{2} \phi(k),\end{align}

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

(4.11) \begin{align}\notag &\mathscr{A}\;U(x,k) \\\notag &\quad \leq p(1-p\gamma_k)V^{p-1}(x)\biggl[\dfrac{1}{2}\text{tr}(a(x,k)\nabla^{2}V(x) ) + \langle b(x,k),\nabla V(x) \rangle \biggr] \\\notag&\quad\quad + p(1-p\gamma_k)V^{p-1}(x)\int_{U}[V(x+c(x,k,u)) - V(x) - \langle\nabla V(x), c(x,k,u)\rangle ]\nu({\text{d}} u)\\\notag&\quad\quad + pV^p(x)\sup_{j\in\mathbb{S}}|\gamma_j|\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| - pV^p(x)(\alpha_k + \gamma) + C_{1} -C_{2} \phi(k)\\\notag &\quad = p(1-p\gamma_k)V^{p-1}(x)[\mathcal{L}_kV(x)] + pV^p(x)\sup\limits_{j\in\mathbb{S}}|\gamma_j|\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| \\\notag &\quad\quad - pV^p(x)(\alpha_k + \gamma) + C_{1} -C_{2} \phi(k) \\\notag &\quad \le p(1-p\gamma_k)V^{p-1}(x)[\alpha_k V(x) +\beta_k] + pV^p(x)\sup\limits_{j\in\mathbb{S}}|\gamma_j|\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| \\\notag &\quad\quad - pV^p(x)(\alpha_k + \gamma) + C_{1} -C_{2} \phi(k) \\&\quad = p(1-p\gamma_k)V^{p}(x)\biggl[ \dfrac{\beta_k}{V(x)} - \dfrac{p\alpha_k \gamma_{k}+ \gamma}{1-p\gamma_k} + \dfrac{\sup_{j\in\mathbb{S}}|\gamma_j|\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}|}{1-p\gamma_k}\biggr] \notag\\&\quad\quad + C_{1} -C_{2}\phi(k),\end{align}

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

\begin{equation*}\delta\;:\!=\; 1\wedge \frac{0.5\gamma}{1-p\gamma_{k}}\wedge \frac{2C_{2}}{p}.\end{equation*}

Note that

\begin{equation*}0 < \delta \le \frac{0.5\gamma}{1-p\gamma_{k}}.\end{equation*}

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

(4.12) \begin{equation} -\dfrac{p\gamma_k\alpha_k + \gamma}{1-p\gamma_{k}} \le \dfrac{-0.5 \gamma}{1-p\gamma_{k}} \le\dfrac{-\delta (1-p\gamma_{k})}{1-p\gamma_{k}}=-\delta.\end{equation}

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

(4.13) \begin{equation} \dfrac{\beta_k}{V(x)} \leq 0.25\delta \quad \text{for all } |x| \ge M_{1} \text{ and } k\in \mathbb{S}.\end{equation}

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

(4.14) \begin{equation}\dfrac{\sup_{j\in\mathbb{S}}|\gamma_j|\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}|}{1-p\gamma_k} \le 2\sup\limits_{j\in\mathbb{S}}|\gamma_j|\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| \leq 0.25\delta\end{equation}

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

\begin{align*}\mathscr{A}\; U(x,k) &\le p(1-p\gamma_k)V^{p}(x)[0.25 \delta -\delta +0.25\delta] + C_{1} - C_{2} \phi(k)\\&= -0.5\delta p(1-p\gamma_k)V^{p}(x) + C_{1} - C_{2}\phi(k)\\&= -0.5\delta p[(1-p\gamma_k)V^{p}(x) + \phi(k)] + 0.5\delta p \phi(k)+ C_{1} - C_{2}\phi(k)\\&\leq -0.5\delta p U(x,k) + C_{1}\end{align*}

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

\begin{equation*}\mathscr{A}\; U(x,k) \leq -\alpha U(x,k) + \beta\end{equation*}

holds for all $x \in \mathbb{R}^d$ and $k\in \mathbb{S}$ . This completes the proof.

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

\begin{equation*} \hat{q}_{ij} \;:\!=\; \begin{cases} -\frac{1}{2}\theta & \text{if $ j=1$, $i=1$,}\\[3pt]\frac{1}{2}\theta & \text{if $ j=1$, $ i\neq j$,}\\[3pt]\frac{1}{3^{j-1}}\theta & \text{if $ j>1$, $i \neq j$,}\\[3pt]-\frac{3^{j-1} - 1}{3^{j-1}}\theta & \text{if $ j>1$, $i = j$,} \end{cases} \end{equation*}

that is,

\begin{equation*}\hat{Q} = \theta \begin{bmatrix}-\frac{1}{2}\;\;\;\;\; & \frac{1}{3}\;\;\;\;\; & \frac{1}{3^2}\;\;\;\;\; & \cdots \\[3pt]\frac{1}{2}\;\;\;\;\; &- \frac{2}{3}\;\;\;\;\; & \frac{1}{3^2}\;\;\;\;\; & \cdots \\[3pt]\frac{1}{2}\;\;\;\;\; & \frac{1}{3}\;\;\;\;\; & -\frac{8}{3^2}\;\;\;\;\; & \cdots \\[3pt]\vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \vdots\end{bmatrix}\!.\end{equation*}

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

\begin{equation*} \hat{P}_{ij}(t) =\begin{cases}\frac{1}{2} + \frac{1}{2}{\text{e}}^{-\theta t} & \text{if $ j=1$, $i=1$,}\\[3pt]\frac{1}{2} - \frac{1}{2}{\text{e}}^{-\theta t} & \text{if $ j=1$, $i \neq j$,}\\[3pt]\frac{1}{3^{j-1}} -\frac{1}{3^{j-1}}{\text{e}}^{-\theta t} & \text{if $ j>1$, $i \neq j$,}\\[3pt]\frac{1}{3^{j-1}} + \frac{3^{j-1} - 1}{3^{j-1}}{\text{e}}^{-\theta t} & \text{if $j>1$, $i = j$,}\end{cases}\end{equation*}

that is,

\begin{equation*}\hat{P}(t) = \begin{bmatrix}\frac{1}{2} + \frac{1}{2}{\text{e}}^{-\theta t}\;\;\;\;\; & \frac{1}{3} - \frac{1}{3}{\text{e}}^{-\theta t}\;\;\;\;\; & \frac{1}{3^2} - \frac{1}{3^2}{\text{e}}^{-\theta t}\;\;\;\;\; & \cdots \\[3pt]\frac{1}{2} - \frac{1}{2}{\text{e}}^{-\theta t}\;\;\;\;\; & \frac{1}{3} + \frac{2}{3}{\text{e}}^{-\theta t}\;\;\;\;\; & \frac{1}{3^2} - \frac{1}{3^2}{\text{e}}^{-\theta t}\;\;\;\;\; & \cdots \\[3pt]\frac{1}{2} - \frac{1}{2}{\text{e}}^{-\theta t}\;\;\;\;\; & \frac{1}{3} - \frac{1}{3}{\text{e}}^{-\theta t}\;\;\;\;\; & \frac{1}{3^2} + \frac{8}{3^2}{\text{e}}^{-\theta t}\;\;\;\;\; & \cdots \\[3pt]\vdots & \vdots & \vdots\end{bmatrix}\!.\end{equation*}

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

\begin{align*}\sum_{j=1}^{\infty}|\hat{P}_{ij}(t) - \nu_{j}| &= \dfrac{1}{2}{\text{e}}^{-\theta t} + \sum_{j>1,j\neq i}\dfrac{1}{3^{j-1}}{\text{e}}^{-\theta t} + \bigg|\dfrac{3^{i-1} - 1}{3^{i-1}}{\text{e}}^{-\theta t}\bigg| \\&\leq \dfrac{1}{2}{\text{e}}^{-\theta t} + \sum_{j=1}^{\infty}\dfrac{1}{3^j}{\text{e}}^{-\theta t} + {\text{e}}^{-\theta t} \\&= 2{\text{e}}^{-\theta t}.\end{align*}

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

Example 4.1. Consider the SDE

(4.15) \begin{equation}{\text{d}} X(t) = b(X(t),\varLambda(t))\,{\text{d}} t + \sigma(X(t),\varLambda(t)) \,{\text{d}} W(t) + \int_{U}c(X(t^{-}),\varLambda(t^{-}),u)\tilde{N}({\text{d}} t,{\text{d}} u), \end{equation}

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

\begin{equation*} \sigma(x,k) =\begin{pmatrix}1\;\;\;\;\; & 0 \\0\;\;\;\;\; & 1 \\\end{pmatrix}\!, \quad b(x,k) =\begin{cases}-x & \text{if $ k = 1$,}\\\frac{1}{4k}x & \text{if $ k \geq 2$,}\end{cases}\quad c(x,k,u)= \gamma\dfrac{1}{\sqrt{2k}}|u|x,\end{equation*}

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

\begin{equation*}q_{kj}(x) \;:\!=\; \begin{cases}\frac{1}{2}\frac{k}{k + {\text{e}}^{-|x|^2}} & \text{if $ j=1$, $k \neq j$,}\\[3pt]\frac{1}{3^{j-1}}\frac{k}{k + {\text{e}}^{-|x|^2}} & \text{if $ j>1$, $k \neq j$,}\\[3pt]-\sum_{j\neq k}q_{kj}(x) & \text{if $ k = j$.}\end{cases}\end{equation*}

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

\begin{align*}\sum_{j=1}^{\infty}q_{kj}(x)[\phi(j) - \phi(k)]&= \dfrac{1}{2}\dfrac{k}{k + {\text{e}}^{-|x|^2}}[1-k] + \sum_{j>1}\dfrac{1}{3^{j-1}}\dfrac{k}{k + {\text{e}}^{-|x|^2}} [j - k]\\&\leq \dfrac{1}{2} -\dfrac{1}{2}\cdot\dfrac{1}{2}k + \dfrac{1}{3}\cdot\dfrac{k}{k + {\text{e}}^{-|x|^2}} \biggl[ \sum_{j>1}\dfrac{j}{3^{j}} - k\sum_{j>1}\dfrac{1}{3^{j}}\biggr]\\&= \dfrac{1}{2} -\dfrac{1}{4}k + \dfrac{1}{3}\cdot\dfrac{k}{k + {\text{e}}^{-|x|^2}} \biggl[ \biggl( \dfrac{3}{4}-\dfrac{1}{3}\biggr) -k\biggl( \dfrac{1}{2}-\dfrac{1}{3}\biggr)\biggr]\\&= \dfrac{1}{2} -\dfrac{1}{4}k + \dfrac{1}{3}\cdot\dfrac{k}{k + {\text{e}}^{-|x|^2}}\biggl[\dfrac{5}{12}-\dfrac{1}{6}k\biggr] \\&\leq \dfrac{1}{2} -\dfrac{1}{4}k + \dfrac{1}{3}\cdot\dfrac{5}{12} - \dfrac{1}{3}\cdot\dfrac{1}{2}\cdot\dfrac{1}{6}k\\&\leq 2 - \dfrac{5}{18}\phi(k).\end{align*}

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

\begin{equation*} \hat{q}_{ij} \;:\!=\; \begin{cases} -\frac{1}{2} & \text{if $ j=1$, $i=1$,}\\[3pt]\frac{1}{2} & \text{if $ j=1$, $i\neq j$,}\\[3pt]\frac{1}{3^{j-1}} & \text{if $j>1$, $i \neq j$,}\\[3pt]-\frac{3^{j-1} - 1}{3^{j-1}} & \text{if $ j>1$, $i = j$.} \end{cases} \end{equation*}

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

\begin{align*}\sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| &= \sum_{j\neq k}|q_{kj}(x) -\hat{q}_{kj}| + |q_{kk}(x) -\hat{q}_{kk}|\\&= \sum_{j\neq k}|q_{kj}(x) - \hat{q}_{kj}| + \bigg|-\sum_{j\neq k}q_{kj}(x) + \sum_{j\neq k}\hat{q}_{kj}(x) \bigg|\\&\leq \sum_{j\neq k}|q_{kj}(x) -\hat{q}_{kj}| + \sum_{j\neq k}|q_{kj}(x) - \hat{q}_{kj}|\\&= 2\sum_{j\neq k}|q_{kj}(x) -\hat{q}_{kj}|\\&= 2\bigg| \dfrac{1}{2}\dfrac{k}{k + {\text{e}}^{-|x|^2}}-\dfrac{1}{2}\bigg| + 2\sum_{j>1, j\neq k}\bigg|\dfrac{1}{3^{j-1}}\dfrac{k}{k + {\text{e}}^{-|x|^2}} -\dfrac{1}{3^{j-1}}\bigg|\\&\leq \bigg|\dfrac{k}{k + {\text{e}}^{-|x|^2}} - 1\bigg| + 2\sum_{j\geq 1}\dfrac{1}{3^j}\bigg|\dfrac{k}{k + {\text{e}}^{-|x|^2}} - 1\bigg|\\&= 2\biggl[1 - \dfrac{k}{k + {\text{e}}^{-|x|^2}}\biggr] \\&\leq 2{\text{e}}^{-|x|^2}.\end{align*}

Hence

\begin{equation*}\sup\limits_{k\in\mathbb{S}} \sum_{j\in\mathbb{S}}|q_{kj}(x) -\hat{q}_{kj}| \rightarrow 0 \quad \text{as $ x\rightarrow \infty$,}\end{equation*}

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

\begin{align*}\mathcal{L}_kV(x) &= \dfrac{1}{2}\text{tr}(a(x,k)\nabla^{2}V(x)) + \langle b(x,k),\nabla V(x,k)\rangle \\& \quad + \int_{U}(V(x+c(x,k,u)) - V(x) -\langle\nabla V(x), c(x,k,u\rangle)\nu({\text{d}} u) \\&= 2 + 2\langle b(x,k),x\rangle + \int_{U}(|x+c(x,k,u)|^2 - |x|^2 -2\langle x, c(x,k,u\rangle)\nu({\text{d}} u) \\&= 2 + 2\langle b(x,k),x\rangle + \dfrac{|x|^2}{2k}\\&= \begin{cases}2 - \frac{3}{2}|x|^2 & \text{if $ k=1$,} \\[3pt]2 + \frac{|x|^2}{k} & \text{if $ k \geq 2$.}\end{cases}\end{align*}

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

\begin{equation*} \alpha_{k} =\begin{cases}-\frac{3}{2} & \text{if $ k = 1$,}\\[3pt]\frac{1}{k} & \text{if $ k \geq 2$.}\end{cases}\end{equation*}

We also see that

\begin{equation*}\sum_{k=1}^{\infty}\alpha_{k}\nu_k = -\dfrac{3}{2}\cdot\dfrac{1}{2} + \sum_{k=2}^{\infty}\dfrac{1}{k}\dfrac{1}{3^{k-1}}= -\dfrac{3}{4} + 3\biggl[\log\biggl(\dfrac{3}{2}\biggr) - \dfrac{1}{3}\biggr]< 0.\end{equation*}

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

(5.1) \begin{equation}\begin{aligned}{\text{d}} X(t) &= b(X(t),\varLambda(t))\,{\text{d}} t + \sigma(X(t),\varLambda(t)) \,{\text{d}} W(t) + \int_{U}c(X(t^{-}),\varLambda(t^{-}),u)\tilde{N}({\text{d}} t,{\text{d}} u),\\\varLambda(t) &= \varLambda(0) + \int_{0}^{t}\int_{\mathbb{R}_{+}}h(X(s^{-}), \varLambda(s^{-}),r)N_{1}({\text{d}} s,{\text{d}} r),\end{aligned}\end{equation}

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

\begin{align*} {\text{d}} X(t) &= b(X(t),\varLambda(t))\,{\text{d}} t + \xi(X(t),\varLambda(t))\,{\text{d}} t + \sigma(X(t),\varLambda(t)) \,{\text{d}} W(t) \notag\\ &\quad + \int_{U}c(X(t^{-}),\varLambda(t^{-}),u)\tilde{N}({\text{d}} t,{\text{d}} u),\end{align*}

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

(5.2) \begin{equation} \begin{aligned} {\text{d}} \tilde{X}(t) & = b(\tilde{X}(t),\tilde{\varLambda}(t))\,{\text{d}} t + \xi(\tilde{X}(t),\tilde{\varLambda}(t))\,{\text{d}} t + \sigma(\tilde{X}(t),\tilde{\varLambda}(t))\,{\text{d}} W(t) \\&\quad + \int_{U}c(\tilde{X}(t^{-}),\tilde{\varLambda}(t^{-}),u)\tilde{N}({\text{d}} t,{\text{d}} u),\\\tilde{\varLambda}(t) & = \tilde{\varLambda}(0) + \int_{0}^{t}\int_{\mathbb{R}_{+}}\tilde{h}(\tilde{X}(s^{-}), \tilde{\varLambda}(s^{-}),r)N_{1}({\text{d}} s,{\text{d}} r),\end{aligned}\end{equation}

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

(5.3) \begin{equation}\xi(\tilde{X}(t),\tilde{\varLambda}(t)) = -L(\tilde{\varLambda}(t))\tilde{X}(t),\end{equation}

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

(5.4) \begin{equation} \begin{aligned}{\text{d}}\tilde{X}(t) &= \tilde{b}(\tilde{X}(t),\tilde{\varLambda}(t))\,{\text{d}} t + \sigma(\tilde{X}(t),\tilde{\varLambda}(t)){\text{d}} W(t) + \int_{U}c(\tilde{X}(t^{-}),\tilde{\varLambda}(t^{-}),u)\tilde{N}({\text{d}} t,{\text{d}} u),\\\tilde{\varLambda}(t)& = \tilde{\varLambda}(0) + \int_{0}^{t}\int_{\mathbb{R}_{+}}\tilde{h}(\tilde{X}(s^{-}), \tilde{\varLambda}(s^{-}),r)N_{1}({\text{d}} s,{\text{d}} r).\end{aligned}\end{equation}

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

(5.5) \begin{align} \tilde{X}^{(k)}(t) & = x + \int_{0}^{t}\tilde{b}(\tilde{X}^{(k)}(s), k ) \,{\text{d}} s + \int_{0}^{t} \sigma(\tilde{X}^{(k)}(s), k)\,{\text{d}} W(s) \notag \\ &\quad + \int_{0}^{t}\int_{U} c(\tilde{X}^{(k)}(s-), k, u)\tilde{N}({\text{d}} s, {\text{d}} u).\end{align}

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.

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

(5.6) \begin{equation}\sum_{k\in \mathbb{S}}\alpha_k\nu_k < 0\end{equation}

and

(5.7) \begin{equation}(\kappa - \alpha_{k})|x|^2 \leq \langle x, L(k)x\rangle \quad \textit{for all } x \in \mathbb{R}^d,\ k \in \mathbb{S}_{\text{int}}\cup\mathbb{S}_{ab},\end{equation}

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

\begin{align*}\mathcal{L}_{k} V(x) &= \dfrac{1}{2}\text{tr}(a(x,k)\nabla^{2}V(x)) + \langle \tilde{b}(x,k),\nabla V(x)\rangle \\& \quad + \int_{U}(V(x+c(x,k,u)) - V(x) -\langle\nabla V(x), c(x,k,u\rangle)\nu({\text{d}} u)\\&= \|\sigma(x,k)\|^2 + 2\langle x, b(x,k)\rangle + \int_{U}|c(x,k,u)|^2\nu({\text{d}} u) - 2\langle x, L(k)x\rangle\\&\leq 2\kappa(|x|^2 + 1) - 2\langle x, L(k)x\rangle\\&\leq 2\alpha_{k}|x|^2 + 2\kappa\\&= 2\alpha_{k}V(x) + 2\kappa.\end{align*}

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

\begin{equation*}\xi(x,k) = -\theta(k)Ix,\end{equation*}

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

\begin{equation*}L(k) = \theta(k)I.\end{equation*}

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.

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

(5.8) \begin{equation}{\text{d}} X(t) = b(X(t),\varLambda(t))\,{\text{d}} t + \sigma(X(t),\varLambda(t))\,{\text{d}} W(t), \end{equation}

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

\begin{align*} Q(x) =\begin{pmatrix}-\frac{3}{2} & \frac{3}{2} \\[3pt]\frac{3}{2} & -\frac{3}{2}\end{pmatrix}\!.\end{align*}

Define the coefficients in (5.8) as follows:

\begin{equation*} \sigma(x,k) =\begin{pmatrix}1\;\;\;\;\; & 0 \\0\;\;\;\;\; & 1 \\\end{pmatrix}\!,\quad b(x,1) =\begin{pmatrix}2\;\;\;\;\; & -1 \\1\;\;\;\;\; & 1 \\\end{pmatrix}x, \quad b(x,2) =\begin{pmatrix}2\;\;\;\;\; & 2 \\-2\;\;\;\;\; & 3 \\\end{pmatrix}x.\end{equation*}

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

\begin{equation*}\lambda_{\min}\biggl( \begin{pmatrix}2\;\;\;\;\; & -1 \\1\;\;\;\;\; & 1 \\\end{pmatrix} + \begin{pmatrix}2\;\;\;\;\; & -1 \\1\;\;\;\;\; & 1 \\\end{pmatrix}^\top\biggr) = 2 > 0\end{equation*}

and

\begin{equation*}\lambda_{\min}\biggl(\begin{pmatrix}2\;\;\;\;\; & 2 \\-2\;\;\;\;\; & 3 \\\end{pmatrix} + \begin{pmatrix}2\;\;\;\;\; & 2 \\-2\;\;\;\;\; & 3 \\\end{pmatrix}^\top \biggr) = 4 > 0.\end{equation*}

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.