2. The affine process

Let us introduce more precisely the (1+1)-affine process. Here we adopt the framework of Duffie et al. [Reference Duffie, Filipović and Schachermayer6]; see also Dawson and Li [Reference Dawson and Li4]. Let $D= \mathbb{R}_{+}\times\mathbb{R}$ be endowed with its Borel $\sigma$ -algebra.

Let $U= \mathbb{C}_{-}\times i\mathbb{R}$ , where $\mathbb{C}_{-}= \{a+ib\,{:}\, a\in\mathbb{R}_{-},b\in\mathbb{R}\}$ and $i\mathbb{R}= \{ia\,{:}\, a\in \mathbb{R}\}$ . Given a set of admissible parameters $(a,(\alpha_{ij}), (b_1,b_2), (\beta_{ij}), \mu, \nu)$ , we define the functions F and R on U by the Lévy–Khintchine type representations

(2.1) \begin{eqnarray}F(u)= \langle b,u\rangle + au_2^2 + \int_D \big(\textrm{e}^{\langle u,v\rangle} - 1 - v_2u_2\big) \nu(\textrm{d} v) \end{eqnarray}

and

(2.2) \begin{eqnarray}R(u)= \langle \beta_{\cdot 1},u\rangle + \langle u,\alpha u\rangle + \int_D \big(\textrm{e}^{\langle u,v\rangle} - 1 - \langle u,v\rangle\big) \mu(\textrm{d} v). \end{eqnarray}

The (1+1)-affine process $\{X_t\,{:}\,t\ge 0\}$ is a Markov process with state space D and Feller transition semigroup $(P_t)_{t\ge 0}$ defined by

(2.3) \begin{eqnarray}\int_D\textrm{e}^{\langle u,v\rangle} P_t(x,\textrm{d} v) =\exp\big\{\langle x,\psi(t,u)\rangle+\phi(t,u)\big\}, \quad u\in U, \end{eqnarray}

where $(\phi,\psi)$ is the unique solution of the system of equations

(2.4) \begin{eqnarray}\left\{\begin{array}{ll} \partial_t\phi(t,u)= F(\psi(t,u)),\quad \phi(0,u)= 0, \\ \\[-8pt] \partial_t\psi_1(t,u)= R(\psi(t,u)),\quad \psi_1(0,u)= u_1, \\ \\[-8pt] \psi_2(t,u)= \textrm{e}^{\beta_{22}t}u_2.\end{array}\right. \end{eqnarray}

The uniqueness of the solution implies that

(2.5) \begin{eqnarray}\psi(s+t,u)= \psi(s,\psi(t,u)), \quad s,t\ge 0,\ u\in U. \end{eqnarray}

Clearly, we can rewrite (2.3) as

(2.6) \begin{eqnarray}\int_D\textrm{e}^{\langle u,v\rangle} P_t(x,\textrm{d} v) =\exp\Big\{\langle x,\psi(t,u)\rangle + \int_0^tF(\psi(t,u))\textrm{d} s\Big\}, \quad u\in U. \end{eqnarray}

The Feller property implies that $\{X_t\,{:}\,t\ge 0\}$ has a càdlàg realization.

Suppose that $\{X_t\,{:}\,t\ge 0\}= \{(Y_t,Z_t)\,{:}\,t\ge 0\}$ is a (1+1)-affine process with transition semigroup $(P_t)_{t\ge 0}$ defined by (2.3) and (2.4). Then $\{Y_t\,{:}\,t\ge 0\}$ is a Markov process on $\mathbb{R}_+$ with Feller transition semigroup $(P^{(1)}_t)_{t\ge 0}$ defined by

(2.7) \begin{eqnarray}\int_{\mathbb{R}_+}\textrm{e}^{-\lambda v} P^{(1)}_t(x,\textrm{d} v) =\exp\Big\{x\psi_1(t,-\lambda,0) + \int_0^tF(\psi_1(t,-\lambda,0),0)\textrm{d} s\Big\}, \end{eqnarray}

where $\lambda\ge 0$ . It is known that $\{Y_t\,{:}\,t\ge 0\}$ is a continuous-state branching process with immigration (CBI process) with branching mechanism $\lambda\mapsto -R(\!-\lambda,0)$ and immigration mechanism $\lambda\mapsto -F(\!-\lambda,0)$ . It is known that

(2.8) \begin{eqnarray}\int_{\mathbb{R}_+} v P^{(1)}_t(x,\textrm{d} v)= \textrm{e}^{\beta_{11}t}y + \big[b_1+m_1(\nu)\big] \int_0^t \textrm{e}^{\beta_{11}s}\textrm{d} s, \end{eqnarray}

where

\begin{align*}m_1(\nu)= \int_D v_1\nu(\textrm{d} v); \end{align*}

see, e.g., the formula (79) in Li [Reference Li17].

In particular, when $F(\!-\lambda,0)\equiv 0$ for all $\lambda\ge 0$ , the CBI process $\{Y_t\,{:}\,t\ge 0\}$ reduces to a continuous-state branching process (CB process) with branching mechanism $\lambda\mapsto R(\!-\lambda,0)$ . Then a CB process has Feller transition semigroup $(Q_t)_{t\ge 0}$ defined by

(2.9) \begin{eqnarray}\int_{\mathbb{R}_+}\textrm{e}^{-\lambda v} Q_t(x,\textrm{d} v) =\exp\big\{x\psi_1(t,-\lambda,0)\big\}, \qquad \lambda\ge 0. \end{eqnarray}

As a special case of (2.8) we have

(2.10) \begin{eqnarray}\int_{\mathbb{R}_+} v Q_t(y,\textrm{d} v) =y\textrm{e}^{\beta_{11}t}, \qquad t\ge 0,\ y\ge 0. \end{eqnarray}

Then Jensen’s inequality implies

(2.11) \begin{eqnarray}-\psi_1(t,-\lambda,0) \le\textrm{e}^{\beta_{11}t}\lambda, \qquad t\ge 0,\ \lambda\ge 0. \end{eqnarray}

From (2.9) it is easy to see that zero is a trap for the CB process. For a càdlàg realization of the CB process $\{Y_t\,{:}\,t\ge 0\}$ , we define its extinction time

\begin{align*}\tau_0\,{:\!=}\, \inf\{t\ge 0\,{:}\, Y_t= 0\}. \end{align*}

The reader may refer to Kyprianou [Reference Kyprianou14] and Li [Reference Li15, Reference Li17] for compact treatments of CB and CBI processes.

Condition 2.2. (Grey’s condition.) There exists a constant $\lambda_0> 0$ such that

\begin{align*}-R(\!-\lambda,0)>0 \ \ {for}\ \ \lambda\ge \lambda_0 \ \ {and}\ \ -\int_{\lambda_0}^\infty R(\!-\lambda,0)^{-1}\textrm{d}\lambda\lt \infty. \end{align*}

Proposition 2.3. Suppose that $F(\!-\lambda,0)\equiv 0$ and Condition 2.2 holds. Let $\{Y_t\,{:}\,t\ge 0\}$ be a càdlàg realization of the CB process with $Y_0= y$ . Then we have

(2.12) \begin{eqnarray}\mathbb{P}(\tau_0> t) =\mathbb{P}(Y_t> 0) = 1-\textrm{e}^{-y\bar{v}_t}, \qquad t>0, \end{eqnarray}

where $t\mapsto \bar{v}_t\,{:\!=}\, -\lim_{\lambda\to \infty}\psi_1(t,-\lambda,0)$ is the unique positive solution of

\begin{align*}\partial_t\bar{v}_t= -R(\!-\bar{v}_t,0),\qquad \bar{v}_{0+}= \infty. \end{align*}

The above proposition follows from Theorem 3.4 and Corollary 3.14 in Li [Reference Li17]. By (2.5) and (2.11), for any $t\ge \delta> 0$ we have

(2.13) \begin{align}\bar{v}_t &= -\lim_{\lambda\to \infty}\psi_1(t-\delta,-\psi_1(\delta,-\lambda,0),0) \nonumber\\[3pt] &= -\psi_1(t-\delta,-\bar{v}_\delta,0) \le\textrm{e}^{\beta_{11}(t-\delta)}\bar{v}_\delta. \end{align}

Then for $\beta_{11}< 0$ the probability in (2.12) decays exponentially fast as $t\to \infty$ .

A càdlàg realization of the general (1+1)-affine process can be constructed as the unique strong solution to a system of stochastic integral equations. Let $\sigma_0= \sqrt{a}$ , and let $(\sigma_{ij})$ be a $(2\times2)$ matrix satisfying $(\sigma_{ij})= (\alpha_{ij})(\alpha_{ij})^{\tau}$ . Suppose that $({\Omega},\mathscr{F},\mathscr{F}_t,\mathbb{P})$ is a filtered probability space satisfying the usual hypotheses. Let $W_0(t)$ be a standard $(\mathscr{F}_t)$ -Brownian motion. Let $W_i(\textrm{d} s,\textrm{d} u)$ , $i=1,2$ , be $(\mathscr{F}_t)$ -Gaussian white noises on $(0,\infty)^2$ with intensity $\textrm{d} s\textrm{d} u$ . Let $M(\textrm{d} s,\textrm{d} u,\textrm{d} v)$ be an $(\mathscr{F}_t)$ -Poisson random measure on $(0,\infty)^2\times D$ with intensity $\textrm{d} s\textrm{d} u\mu(\textrm{d} v)$ , and let $N(\textrm{d} s,\textrm{d} v)$ be an $(\mathscr{F}_t)$ -Poisson random measure on $(0,\infty)\times D$ with intensity $\textrm{d} s\nu(\textrm{d} v)$ . The corresponding compensated measures are denoted by $\tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v)$ and $\tilde{N}(\textrm{d} s,\textrm{d} v)$ . We assume these random elements are independent of each other. Given an $\mathscr{F}_0$ -measurable random variable $(Y_0,Z_0)\in D$ , we consider the following system of stochastic integral equations:

(2.14) \begin{align}Y_t &= Y_0 + \int_0^t (b_1+\beta_{11}Y_s) \textrm{d} s + \sqrt{2}\sigma_{11}\int_0^t\int_0^{Y_s} W_1(\textrm{d} s,\textrm{d} u) \nonumber \\[3pt]&\quad + \sqrt{2}\sigma_{12}\int_0^t\int_0^{Y_s} W_2(\textrm{d} s,\textrm{d} u) + \int_0^t\int_0^{Y_{s-}}\int_D v_1\tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v) \nonumber \\[3pt]&\quad + \int_0^t\int_D v_1 N(\textrm{d} s,\textrm{d} v) \end{align}

and

(2.15) \begin{align}Z_t & = Z_0 + \int_0^t (b_2+\beta_{21}Y_s+\beta_{22}Z_s) \textrm{d} s + \sqrt{2}\sigma_0W_0(t) \nonumber\\[3pt]&\quad + \sqrt{2}\sigma_{21} \int_0^t\int_0^{Y_s}W_1(\textrm{d} s,\textrm{d} u) + \sqrt{2}\sigma_{22} \int_0^t\int_0^{Y_s} W_2(\textrm{d} s,\textrm{d} u) \nonumber \\[3pt]&\quad + \int_0^t\int_0^{Y_{s-}}\int_D v_2 \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v) + \int_0^t\int_D v_2\tilde{N}(\textrm{d} s,\textrm{d} v). \end{align}

Here and in the sequel, we understand that, for any $a\leq b\in\mathbb{R}$ ,

\begin{align*}\int_a^b=\int_{(a,b]} \quad\mbox{and}\quad \int_a^{\infty}= \int_{(a,\infty)}. \end{align*}

The existence and pathwise uniqueness of the solution to (2.14) follows from Theorem 8.5 in Li [Reference Li17]. A weakly equivalent stochastic equation was first introduced by Dawson and Li [Reference Dawson and Li4]; see also Dawson and Li [Reference Dawson and Li5]. The existence and pathwise uniqueness of the solution to (2.15) are straightforward. In fact, by (2.14)–(2.15), one can see using integration by parts that

(2.16) \begin{align}\textrm{e}^{-\beta_{22}t}Z_t & = Z_0 + \int_0^t\textrm{e}^{-\beta_{22}s} (b_2+\beta_{21}Y_s) \textrm{d} s + \sqrt{2}\sigma_0\int_0^t \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(s)\nonumber \\[3pt] &\quad + \sqrt{2}\sigma_{21}\int_0^t\int_0^{Y_s} \textrm{e}^{-\beta_{22}s} W_1(\textrm{d} s,\textrm{d} u) + \sqrt{2}\sigma_{22}\int_0^t\int_0^{Y_s} \textrm{e}^{-\beta_{22}s} W_2(\textrm{d} s,\textrm{d} u) \nonumber\\[3pt] &\quad + \int_0^t\int_0^{Y_{s-}}\int_D \textrm{e}^{-\beta_{22}s}v_2 \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v)+ \int_0^t\int_D \textrm{e}^{-\beta_{22}s}v_2 \tilde{N}(\textrm{d} s,\textrm{d} v). \end{align}

By Theorem 6.2 of Dawson and Li [Reference Dawson and Li4], the process $\{(Y_t,Z_t)\,{:}\,t\ge 0\}$ defined by (2.14)–(2.15) is a (1+1)-affine process with transition semigroup $(P_t)_{t\ge 0}$ given by (2.3) and (2.4).

Let us remark that $\{Z_t\,{:}\,t\ge 0\}$ reduces to a one-dimensional OU-type process if $\sigma_{21}= \sigma_{22}= \beta_{21}= 0$ and $\mu$ is supported on $(0,\infty)\times \{0\}$ . For instance, in this case, from (2.16) we have

(2.17) \begin{eqnarray}Z_t= Z_0 + \int_0^t (b_2+\beta_{22}Z_s) \textrm{d} s + \sqrt{2}\sigma_0W_0(t) + \int_0^t\int_D v_2\tilde{N}(\textrm{d} s,\textrm{d} v). \end{eqnarray}

A number of moment estimates for general finite-dimensional affine processes were given in Friesen et al. [Reference Friesen, Jin and Rüdiger8]. Since more accurate estimates are needed in this work, we here present the following result.

Proposition 2.4. Let $\{(Y_t,Z_t)\,{:}\,t\ge 0\}$ be a (1+1)-affine process with $Y_0= y\in \mathbb{R}_+$ and $Z_0= z\in \mathbb{R}$ . Let $D_1= \mathbb{R}_+\times [\!-1,1]$ and $D_1^c= D\setminus D_1$ . Then we have

\begin{align*}\mathbb{E}(|Z_t|) & \le \textrm{e}^{\beta_{22}t}|z| + \Big(|b_2| + 2\int_{D_1^c}|v_2| \nu(\textrm{d} v)\Big) \int_0^t\textrm{e}^{\beta_{22}(t-s)}\textrm{d} s \nonumber\\[3pt]&\quad + \Big(|\beta_{21}| + 2\int_{D_1^c}|v_2| \mu(\textrm{d} v)\Big) \int_0^t \textrm{e}^{\beta_{22}(t-s)}\mathbb{E}(Y_s)\textrm{d} s \end{align*}

(2.18) \begin{align} \nonumber \\[-30pt] &\quad + \Big[\sqrt{2}\sigma_0 + \Big(\int_{D_1}v_2^2 \nu(\textrm{d} v)\Big)^{1/2}\Big]\Big(\int_0^t \textrm{e}^{2\beta_{22}(t-s)} \textrm{d} s\Big)^{1/2} \nonumber\\ &\quad +\, \sqrt{2}(\sigma_{21} + \sigma_{22}) \Big(\int_0^t\textrm{e}^{2\beta_{22}(t-s)}\mathbb{E}(Y_s)\textrm{d} s\Big)^{1/2} \nonumber\\ &\quad +\, \Big(\int_{D_1}v_2^2 \mu(\textrm{d} v)\Big)^{1/2} \Big(\int_0^t\textrm{e}^{2\beta_{22}(t-s)}\mathbb{E}(Y_s)\textrm{d} s\Big)^{1/2}. \end{align}

Proof. We may assume that $\{(Y_t,Z_t)\,{:}\,t\ge 0\}$ is defined by (2.14)–(2.15). In view of (2.16) we have

\begin{align*}\mathbb{E}\big(\textrm{e}^{-\beta_{22}t}|Z_t|\big) & \le |z| + |\beta_{21}|\int_0^t \textrm{e}^{-\beta_{22}s}\mathbb{E}(Y_s) \textrm{d} s + \sqrt{2}\sigma_0 \mathbb{E}\Big(\Big|\int_0^t \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(s)\Big|\Big) \\[3pt]& \quad +\, |b_2|\int_0^t \textrm{e}^{-\beta_{22}s}\textrm{d} s + \sqrt{2}\sigma_{21}\Big[\mathbb{E}\Big(\Big|\int_0^t\int_0^{Y_s} \textrm{e}^{-\beta_{22}s} W_1(\textrm{d} s,\textrm{d} u)\Big|^2\Big)\Big]^{1/2} \\[3pt]& \quad +\, \sqrt{2}\sigma_{22}\Big[\mathbb{E}\Big(\Big|\int_0^t\int_0^{Y_s} \textrm{e}^{-\beta_{22}s} W_2(\textrm{d} s,\textrm{d} u)\Big|^2\Big)\Big]^{1/2} \\[3pt]& \quad +\,\Big[\mathbb{E}\Big(\Big|\int_0^t\int_0^{Y_{s-}}\int_{D_1} \textrm{e}^{-\beta_{22}s}v_2 \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v)\Big|^2\Big)\Big]^{1/2} \\[3pt]& \quad +\, \mathbb{E}\Big(\Big|\int_0^t\int_0^{Y_{s-}}\int_{D_1^c} \textrm{e}^{-\beta_{22}s}|v_2| \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v)\Big|\Big) \\[3pt] & \quad +\,\Big[\mathbb{E}\Big(\Big|\int_0^t\int_{D_1} \textrm{e}^{-\beta_{22}s}v_2 \tilde{N}(\textrm{d} s,\textrm{d} v)\Big|^2\Big)\Big]^{1/2} \\[3pt] & \quad +\, \mathbb{E}\Big(\Big|\int_0^t\int_{D_1^c} \textrm{e}^{-\beta_{22}s}|v_2| \tilde{N}(\textrm{d} s,\textrm{d} v)\Big|\Big) \\[3pt]& \le |z| + \Big(|b_2| + 2\int_{D_1^c}|v_2| \nu(\textrm{d} v)\Big)\int_0^t\textrm{e}^{-\beta_{22}s}\textrm{d} s \\[3pt] & \quad +\, \Big(|\beta_{21}| + 2\int_{D_1^c}|v_2| \mu(\textrm{d} v)\Big)\int_0^t\textrm{e}^{-\beta_{22}s}\mathbb{E}(Y_s)\textrm{d} s \\[3pt] & \quad +\, \Big[\sqrt{2}\sigma_0 + \Big(\int_{D_1}v_2^2 \nu(\textrm{d} v)\Big)^{1/2} \Big]\Big(\int_0^t \textrm{e}^{-2\beta_{22}s} \textrm{d} s\Big)^{1/2} \\[3pt] & \quad +\, \sqrt{2}(\sigma_{21} + \sigma_{22}) \Big(\int_0^t\textrm{e}^{-2\beta_{22}s}\mathbb{E}(Y_s) \textrm{d} s\Big)^{1/2} \\[3pt] & \quad +\, \Big(\int_{D_1}v_2^2 \mu(\textrm{d} v)\Big)^{1/2} \Big(\int_0^t\textrm{e}^{-2\beta_{22}s}\mathbb{E}(Y_s) \textrm{d} s\Big)^{1/2}. \end{align*}

Then (2.18) follows.

Proposition 2.5. Suppose that $\beta_{11}< 0$ and $\beta_{22}< 0$ . Then the transition semigroup $(P_t)_{t\ge 0}$ defined by (2.3) and (2.4) has a unique stationary distribution $\pi$ , which is given by

(2.19) \begin{eqnarray}\int_D\textrm{e}^{\langle u,v\rangle} \pi(\textrm{d} v) =\exp\Big\{\int_0^\infty F(\psi(t,u))\textrm{d} s\Big\}, \quad u\in U. \end{eqnarray}

Moreover, the distribution $\pi$ has finite first moment; that is,

\begin{align*}\int_D (v_1+|v_2|)\pi(\textrm{d} v)< \infty. \end{align*}

Proof. By Theorem 2.7 of Jin et al. [Reference Jin, Kremer and Rüdiger11], the affine process has a unique stationary distribution $\pi$ given by (2.19). In particular, we have

\begin{align*}\int_{D}\textrm{e}^{-\lambda v_1} \pi(\textrm{d} v) =\exp\Big\{\int_0^\infty F(\psi_1(t,-\lambda,0),0)\textrm{d} s\Big\}, \quad \lambda\ge 0. \end{align*}

By differentiating both sides of the above equality at $\lambda= 0+$ and using (2.1), one may see that

\begin{align*}m_1(\pi)\,{:\!=}\, \int_D v_1 \pi(\textrm{d} v) =[b_1 + m_1(\nu)]\int_0^\infty \textrm{e}^{-|\beta_{22}|s}\textrm{d} s< \infty. \end{align*}

By (2.8) and (2.18) there is a constant $C\ge 0$ such that

\begin{align*}\int_{D} |v_2| P_t(x,\textrm{d} v) \le\textrm{e}^{-|\beta_{22}|t}|z| + C(1+y), \quad t\ge 0,\ x=(y,z)\in D. \end{align*}

Since $\pi$ is a stationary distribution for $(P_t)_{t\ge 0}$ , it follows that

\begin{align*}\int_D (|v_2|\land k) \pi(\textrm{d} v) &= \int_D \pi(\textrm{d} u)\int_D (|v_2|\land k) P_t(u,\textrm{d} v) \\[3pt] &\le \int_D \big[(\textrm{e}^{-|\beta_{22}|t}|u_2|\land k) + C(1+u_1)\big]\pi(\textrm{d} u) \\[3pt] &\le \int_D (\textrm{e}^{-|\beta_{22}|t}|u_2|\land k) \pi(\textrm{d} u) + C[1+m_1(\pi)]. \end{align*}

Then, letting $t\to \infty$ and $k\to \infty$ , we obtain

\begin{align*}\int_D |v_2| \pi(\textrm{d} v)\le C[1+m_1(\pi)]< \infty. \end{align*}

This proves the result.

Proposition 2.6. For $i=1,2$ , let $x_i= (y_i,z_i)\in D$ , and let $\{X_i(t)\,{:}\,t\ge 0\}= \{(Y_i(t),Z_i(t))\,{:}\,t\ge 0\}$ be defined by (2.14) and (2.15) with $(Y_i(0),Z_i(0))= x_i$ . Then, for any $t\ge 0$ ,

\begin{align*} &\mathbb{E}\Big(\sup_{0\le s\le t}|Z_1(s)-Z_2(s)|\Big) \\[3pt] &\qquad\le \textrm{e}^{\beta_{22}t}|z_1-z_2| + |\beta_{21}||y_1-y_2|\int_0^t \textrm{e}^{\beta_{11}s}\textrm{e}^{\beta_{22}(t-s)} \textrm{d} s \\[3pt] &\quad\qquad+\, 2|y_1-y_2| \int_{D_1^c} |v_2| \mu(\textrm{d} v) \int_0^t \textrm{e}^{\beta_{11}s}\textrm{e}^{\beta_{22}(t-s)}\textrm{d} s \\[3pt] &\quad\qquad+\, 2\sqrt{2}(\sigma_{21}+\sigma_{22})|y_1-y_2|^{1/2}\Big(\int_0^t \textrm{e}^{\beta_{11}s}\textrm{e}^{2\beta_{22}(t-s)}\textrm{d} s\Big)^{1/2} \\[3pt] &\quad\qquad+\, 2|y_1-y_2|^{1/2}\Big(\int_{D_1} v_2^2 \mu(\textrm{d} v)\Big)^{1/2} \Big(\int_0^t \textrm{e}^{\beta_{11}s}\textrm{e}^{2\beta_{22}(t-s)} \textrm{d} s\Big)^{1/2}. \end{align*}

Proof. Without loss of generality, we may assume $y_1\ge y_2$ . By Theorem 10.1 in Li [Reference Li17] one can see that $\{Y_1(t)\ge Y_2(t)$ for every $t\ge 0\}= 1$ and $\{Y_1(t)-Y_2(t)\,{:}\,t\ge 0\}$ is a CB process with transition semigroup $(Q_t)_{t\ge 0}$ . Then we apply (2.16) to $Z_1(t)$ and $Z_2(t)$ and take the difference to see

\begin{align*}Z_1(t)-Z_2(t) &= \textrm{e}^{\beta_{22}t}\Big\{(z_1-z_2) + \beta_{21}\int_0^t \textrm{e}^{-\beta_{22}s}[Y_1(s)-Y_2(s)] \textrm{d} s \\[3pt] &\quad + \sqrt{2}\sigma_{21}\int_0^t\int_{Y_1(s)}^{Y_2(s)}\textrm{e}^{-\beta_{22}s} W_1(\textrm{d} s,\textrm{d} u) \\[3pt] &\quad + \sqrt{2}\sigma_{22}\int_0^t\int_{Y_1(s)}^{Y_2(s)} \textrm{e}^{-\beta_{22}s} W_2(\textrm{d} s,\textrm{d} u) \\[3pt] &\quad + \int_0^t\int_{Y_1(s-\!)}^{Y_2(s-\!)}\int_D \textrm{e}^{-\beta_{22}s}v_2 \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v)\Big\}. \end{align*}

It follows that

\begin{align*} &\mathbb{E}\Big(\sup_{0\le s\le t}|Z_1(s)-Z_2(s)|\Big) \\[3pt] &\qquad\le \textrm{e}^{\beta_{22}t}\Big\{|z_1-z_2| + |\beta_{21}|\int_0^t \textrm{e}^{-\beta_{22}s} \mathbb{E}[Y_1(s)-Y_2(s)] \textrm{d} s \\[3pt] &\qquad\qquad+\, 2\sqrt{2}\sigma_{21}\Big[\mathbb{E}\Big(\Big|\int_0^t\int_{Y_1(s)}^{Y_2(s)}\textrm{e}^{-\beta_{22}s} W_1(\textrm{d} s,\textrm{d} u)\Big|^2\Big)\Big]^{1/2} \\[3pt] &\qquad\qquad+\, 2\sqrt{2}\sigma_{22}\Big[\mathbb{E}\Big(\Big|\int_0^t\int_{Y_1(s)}^{Y_2(s)} \textrm{e}^{-\beta_{22}s} W_2(\textrm{d} s,\textrm{d} u)\Big|^2\Big)\Big]^{1/2} \\[3pt] &\qquad\qquad+\, 2\Big[\mathbb{E}\Big(\Big|\int_0^t\int_{Y_1(s-\!)}^{Y_2(s-\!)}\int_{D_1} \textrm{e}^{-\beta_{22}s}v_2 \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v)\Big|^2\Big)\Big]^{1/2} \\[3pt] &\qquad\qquad+\, \mathbb{E}\Big(\Big|\int_0^t\int_{Y_1(s-\!)}^{Y_2(s-\!)}\int_{D_1^c} \textrm{e}^{-\beta_{22}s}|v_2| \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v)\Big|\Big)\Big\} \\[3pt] &\qquad\le \textrm{e}^{\beta_{22}t}\Big\{|z_1-z_2| + |\beta_{21}|\int_0^t \textrm{e}^{-\beta_{22}s} \mathbb{E}[Y_1(s)-Y_2(s)] \textrm{d} s \\[3pt] &\qquad\qquad+\, 2\sqrt{2}(\sigma_{21}+\sigma_{22})\Big(\int_0^t \textrm{e}^{-2\beta_{22}s}\mathbb{E}[Y_1(s)-Y_2(s)]\textrm{d} s\Big)^{1/2} \\[3pt] &\qquad\qquad+\, 2\Big(\int_{D_1} v_2^2 \mu(\textrm{d} v)\Big)^{1/2} \Big(\int_0^t \textrm{e}^{-2\beta_{22}s} \mathbb{E}[Y_1(s)-Y_2(s)]\textrm{d} s\Big)^{1/2} \\[3pt] &\qquad\qquad+\, 2 \int_{D_1^c} |v_2| \mu(\textrm{d} v) \int_0^t \textrm{e}^{-\beta_{22}s}\mathbb{E}[Y_1(s)-Y_2(s)]\textrm{d} s\Big\}, \end{align*}

where we have used Doob’s $L^2$ inequality for the square-integrable martingales; see, e.g., Revuz and Yor [Reference Revuz and Yor25, Theorem 1.7, p. 54]. Then the desired estimate follows by (2.10).

A Markov process $\{(X_1(t),X_2(t))\,{:}\,t\ge 0\}$ with state space $D^2$ is called a Markov coupling of the (1+1)-affine process with transition semigroup $(P_t)_{t\ge 0}$ defined by (2.3) and (2.4) with coupling time $\tau\,{:\!=}\, \inf\{t\ge 0\,{:}\, X_1(t)= X_2(t)\}$ if both $\{X_1(t)\,{:}\,t\ge 0\}$ and $\{X_2(t)\,{:}\,t\ge 0\}$ are Markov processes with transition semigroup $(P_t)_{t\ge 0}$ and $X_1(\tau+t)= X_2(\tau+t)$ for every $t\ge 0$ .

The method of couplings provides an efficient way to estimate the variations of the transition probabilities of the affine process. Let $\|\cdot\|_{\textrm{var}}$ denote the total variation norm of signed measures. Let $\mathscr{B}_1$ be the set of Borel functions f on D satisfying $|f|\le 1$ . Then we have

(2.20) \begin{eqnarray}\big\|P_t(x_1,\cdot)-P_t(x_2,\cdot)\big\|_{\textrm{var}} =\sup_{f\in \mathscr{B}_1}\big[P_tf(x_1) - P_tf(x_2)\big]. \end{eqnarray}

Let $\{(X_1(t),X_2(t))\,{:}\,t\ge 0\}$ be a Markov coupling of the affine process with initial state $(X_1(0),X_2(0))= (x_1,x_2)$ and coupling time $\tau$ . From (2.20) it follows that

(2.21 \begin{eqnarray}\big\|P_t(x_1,\cdot)-P_t(x_2,\cdot)\big\|_{\textrm{var}} =\sup_{f\in \mathscr{B}_1}\mathbb{E}\big[f(X_1(t))-f(X_2(t))\big] \le2\mathbb{P}(\tau> t). \end{eqnarray}

By the pathwise uniqueness for (2.14)–(2.15), the process $\{(X_1(t),X_2(t))\,{:}\,t\ge 0\}$ defined in Proposition 2.6 is a Markov coupling of the affine process with transition semigroup $(P_t)_{t\ge 0}$ . Based on this coupling, several different couplings of the affine process will be given in the next two sections. We shall see that the stochastic equations (2.14)–(2.15) and (2.16) provide an efficient method for the pathwise construction and analysis of those couplings. The approach of stochastic equations has also played an important role in other recent developments concerning branching processes in continuous state spaces; see, e.g., Bansaye and Méléard [Reference Bansaye and Méléard1], Li [Reference Li17], Pardoux [Reference Pardoux22], and the references therein.

3. Estimates for variations of probabilities

In this section, we study the strong Feller property and the exponential ergodicity of the total variation distance of the affine process. Let $\|\cdot\|_{\textrm{var}}$ denote the total variation norm of signed measures. Our strategy is to establish some estimates for the differences of the transition probabilities in the form (2.20). The proofs of the estimates are based on couplings of the affine process constructed in terms of strong solutions of stochastic equations. From those estimates we deduce the strong Feller property and the exponential ergodicity under natural conditions.

We first consider the case $\sigma_0> 0$ . Write $x_1= (y_1,z_1)$ and $x_2= (y_2,z_2)$ , where $y_1,y_2\in \mathbb{R}_+$ and $z_1,z_2\in \mathbb{R}$ . For $i=1,2$ let $(Y_i(t),Z_i(t))$ be defined by (2.14)–(2.16) with $(Y_i(0),Z_i(0))= (y_i,z_i)$ , and write $X_i(t)= (Y_i(t),Z_i(t))$ . Then $\{(X_1(t),X_2(t))\,{:}\,t\ge 0\}$ is a Markov coupling of the affine process. The pathwise uniqueness of the solution for (2.14) implies $Y_1(\tau_0+t)= Y_2(\tau_0+t)$ for $t\ge 0$ . In fact, by Theorem 10.1 in Li [Reference Li17] it is easy to see that $\{|Y_1(t)-Y_2(t)|\,{:}\,t\ge 0\}$ is a CB process with transition semigroup $(Q_t)_{t\ge 0}$ . Let $\tau_0= \inf\{t\ge 0\,{:}\, Y_1(t)= Y_2(t)\}$ be the extinction time of the process. By Proposition 2.3 we have

(3.1) \begin{eqnarray}\mathbb{P}(\tau_0> t) =1-\textrm{e}^{-|y_1-y_2|\bar{v}_t} \le|y_1-y_2|\bar{v}_t, \quad t\ge 0. \end{eqnarray}

Let $a(\tau_0)= [Z_1(\tau_0) - Z_2(\tau_0)]/2\sqrt{2}\sigma_0$ and

\begin{align*}\tau= \inf\Big\{t\ge 0\,{:}\, \int_0^t \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(\tau_0+s)= -a(\tau_0)\Big\}. \end{align*}

Then we define the process $\{Z_2'(t)\,{:}\,t\ge 0\}$ by

\begin{align*}\textrm{e}^{-\beta_{22}t}Z_2'(t) &= z_1 + \int_0^t \textrm{e}^{-\beta_{22}s}[b_2+\beta_{21}Y_2(s)] \textrm{d} s + \sqrt{2}\sigma_{21}\int_0^t\int_0^{Y_2(s)} \textrm{e}^{-\beta_{22}s} W_1(\textrm{d} s,\textrm{d} u) \\[3pt]&\quad + \sqrt{2}\sigma_{22}\int_0^t\int_0^{Y_2(s)} \textrm{e}^{-\beta_{22}s} W_2(\textrm{d} s,\textrm{d} u) + \sqrt{2}\sigma_0\Big[\int_0^{t\land \tau_0} \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(s) \\[3pt] &\quad - \int_{t\land \tau_0}^{t\land (\tau_0+\tau)} \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(s) + \int_{t\land (\tau_0+\tau)}^t \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(s)\Big] \\[3pt] &\quad + \int_0^t\int_0^{Y_2(s-\!)}\int_D \textrm{e}^{-\beta_{22}s}v_2 \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v) + \int_0^t\int_D \textrm{e}^{-\beta_{22}s}v_2 \tilde{N}(\textrm{d} s,\textrm{d} v). \end{align*}

It is clear that $Z_2'(t\land \tau_0)= Z_2(t\land \tau_0)$ for $t\ge 0$ . Write $X_2'(t)= (Y_2(t),Z_2'(t))$ . Then $\{(X_1(t),X_2'(t))\,{:}\,t\ge 0\}$ is also a Markov coupling of the affine process. For $t\ge 0$ let

\begin{align*}\zeta(t)= Z_1(\tau_0+t)-Z_2'(\tau_0+t). \end{align*}

Since $Y_1(\tau_0+t)= Y_2(\tau_0+t)$ for $t\ge 0$ , by the construction of $Z_1(t)$ and $Z_2'(t)$ we have

(3.2) \begin{eqnarray}\zeta(t)= 2\sqrt{2}\sigma_0\textrm{e}^{\beta_{22}t}\Big[a(\tau_0) + \int_0^{t\land \tau} \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(\tau_0+s)\Big]. \end{eqnarray}

It follows that \P $\tau= \inf\{t\ge 0\,{:}\, \zeta(t)= 0\}$ , and so

\begin{align*}\tau_0+\tau= \inf\{t\ge \tau_0\,{:}\, Z_1(t)= Z_2'(t)\} =\inf\{t\ge \tau_0\,{:}\, X_1(t)= X_2'(t)\}. \end{align*}

Then $\tau_0+\tau$ is the coupling time of $\{(X_1(t),X_2'(t))\,{:}\,t\ge 0\}$ .

Theorem 3.1. Suppose that $\sigma_0> 0$ . Then there is a constant $C\ge 0$ such that

\begin{align*} &\|P_t(x_1,\cdot)-P_t(x_2,\cdot)\|_{\textrm{var}} \\[3pt] & \qquad\le 2|y_1-y_2|\bar{v}_{t/2} + C\Big\{\textrm{e}^{\beta_{22}t}|z_1-z_2| + |\beta_{21}||y_1-y_2|\int_0^{t/2} \textrm{e}^{\beta_{11}s}\textrm{e}^{\beta_{22}(t/2-s)} \textrm{d} s \\[3pt] & \qquad\quad+\, 2\sqrt{2}(\sigma_{21}+\sigma_{22})|y_1-y_2|^{1/2}\Big(\int_0^{t/2} \textrm{e}^{\beta_{11}s} \textrm{e}^{2\beta_{22}(t/2-s)} \textrm{d} s\Big)^{1/2} \\[3pt] & \qquad\quad+\, 2\Big[\int_{D_1} v_2^2 \mu(\textrm{d} v)\Big]^{1/2} |y_1-y_2|^{1/2} \Big(\int_0^{t/2} \textrm{e}^{\beta_{11}s}\textrm{e}^{2\beta_{22}(t/2-s)} \textrm{d} s\Big)^{1/2} \\[3pt] & \qquad\quad+\, 2\int_{D_1^c} |v_2| \mu(\textrm{d} v) |y_1-y_2| \int_0^{t/2} \textrm{e}^{\beta_{11}s}\textrm{e}^{\beta_{22}(t/2-s)}\textrm{d} s\Big\}\Big(\int_0^{t/2} \textrm{e}^{-2\beta_{22}s}\textrm{d} s\Big)^{-1/2}, \nonumber\\[3pt] & \qquad\qquad\qquad\qquad\qquad t>0,\ x_i= (y_i,z_i)\in D,\ i=1,2. \end{align*}

Proof. Let $\{(X_1(t),X_2'(t))\,{:}\,t\ge 0\}$ be the coupling of the affine process constructed as above. Then we have

\begin{align*}\mathbb{P}(\tau_0+\tau> t) \le\mathbb{P}(\tau_0> t/2) + \mathbb{P}(\tau_0\le t/2,\tau_0+\tau> t), \end{align*}

where $\mathbb{P}(\tau_0> t/2)\le |y_1-y_2|\bar{v}_{t/2}$ by (3.1). In view of (3.2), there is a standard Brownian motion $\{B(t)\,{:}\,t\ge 0\}$ independent of $\mathscr{F}_{\tau_0}$ such that

\begin{align*}\zeta(t)= 2\sqrt{2}\sigma_0 \textrm{e}^{\beta_{22}t}\big[a(\tau_0) + B(\rho(t\land \tau))\big], \end{align*}

where

\begin{align*}\rho(t)= \int_0^{t} \textrm{e}^{-2\beta_{22}s} \textrm{d} s, \quad t\ge 0. \end{align*}

Since $t-\tau_0$ is measurable relative to $\mathscr{F}_{\tau_0}$ , by the reflection principle for the Brownian motion we get

\begin{align*}\mathbb{P}(\tau_0 &\le t/2,\tau_0+\tau> t) = \mathbb{E}\big[1_{\{\tau_0\le t/2\}} \mathbb{P}(\tau_0+\tau> t|\mathscr{F}_{\tau_0})\big] \nonumber\\[3pt] & \le \mathbb{E}\big[1_{\{\tau_0\le t/2\}} \mathbb{P}(|B(\rho(t-\tau_0))| < |a(\tau_0)|)\big] \nonumber\\[3pt] & \le \mathbb{E}\Big[1_{\{\tau_0\le t/2\}}\frac{2|a(\tau_0)|}{\sqrt{2\pi\rho(t-\tau_0)}}\Big] \\[3pt] & \le \frac{1}{2\sigma_0\sqrt{\pi\rho(t/2)}}\mathbb{E}\big[1_{\{\tau_0\le t/2\}}|Z_1(\tau_0) - Z_2(\tau_0)|\big] \\[3pt] & \le \frac{1}{2\sigma_0\sqrt{\pi\rho(t/2)}}\mathbb{E}\Big(\sup_{0\le s\le t/2}|Z_1(s)-Z_2(s)|\Big). \end{align*}

Then the result follows by Proposition 2.6 and (2.21).

Corollary 3.3. Suppose that $\beta_{11}< 0$ , $\beta_{22}< 0$ , and $\sigma_0> 0$ . Let $\pi$ be the unique stationary distribution for $(P_t)_{t\ge0}$ . Then for every $\delta>0$ there is a constant $C_\delta\ge 0$ such that

\begin{align*}\|P_t(x,\cdot)-\pi\|_{\textrm{var}}\le C_\delta(1+|x|)\textrm{e}^{-\kappa t/2}, \quad t\ge \delta,\ x\in D, \end{align*}

where $\kappa= |\beta_{11}|\land |\beta_{22}|$ .

Proof. By Proposition 2.5, the stationary distribution $\pi$ possesses a finite first moment. It is well known that

(3.3) \begin{eqnarray}\|P_t(x,\cdot)-\pi\|_{\textrm{var}} \le\int_D\|P_t(x,\cdot)-P_t(x_2,\cdot)\|_{\textrm{var}} \pi(\textrm{d} x_2). \end{eqnarray}

By Theorem 3.1, there is a constant $C\ge 0$ such that, for $x_i= (y_i,z_i)\in D$ , $i= 1,2$ ,

\begin{align*} &\|P_t(x_1,\cdot)-P_t(x_2,\cdot)\|_{\textrm{var}} \nonumber\\[3pt]& \qquad\le C\big(|x_1-x_2| + |y_1-y_2|^{1/2}\big) \big[\bar{v}_{t/2}\vee \textrm{e}^{-|\beta_{22}|t/2} (1-\textrm{e}^{-|\beta_{22}|t})^{-1/2}\big]. \end{align*}

Then the desired estimate follows by (2.13).

Now let us consider the case where $\nu(D)> 0$ . For $\varepsilon> 0$ let $D_\varepsilon= \mathbb{R}_+\times [\!-\varepsilon,\varepsilon]$ and $D_\varepsilon^c= D\setminus D_\varepsilon$ . By choosing sufficiently small $\varepsilon\in (0,1]$ we have $0< \nu(D_\varepsilon^c)< \infty$ . Let $\nu_\varepsilon$ be the finite measure on D defined by

(3.4) \begin{align}\nu_\varepsilon(A) =\bigg\{\begin{array}{l@{\quad}l}{\nu(A)} & \mbox{if} \, \nu(D)< \infty, \\ \\[-7pt] {\nu(A\cap D_\varepsilon^c)} & \mbox{if} \, \nu(D)= \infty,\end{array} \end{align}

where $A\in \mathscr{B}(D)$ . Let $\hat{\nu}_\varepsilon= \nu_\varepsilon(D)^{-1} \nu_\varepsilon$ .

Condition 3.4. There exists $\varepsilon\in (0,1]$ such that

\begin{align*}\limsup_{|z|\to 0} |z|^{-1} \|\hat{\nu}_\varepsilon - \delta_{(0,z)}* \hat{\nu}_\varepsilon\|_{\textrm{var}}< \infty. \end{align*}

The above condition is a slight modification of (10) in Wang [Reference Wang29] for OU-type processes. As in Wang [Reference Wang29, p. 996], one may see that the above condition implies

(3.5) \begin{eqnarray}K_\varepsilon\,{:\!=}\, \sup_{z\in \mathbb{R}} |z|^{-1} \|\hat{\nu}_\varepsilon - \delta_{(0,z)}* \hat{\nu}_\varepsilon\|_{\textrm{var}}< \infty. \end{eqnarray}

Theorem 3.5. Suppose that Condition 3.4 is satisfied for some $\varepsilon\in (0,1]$ . Then we have

\begin{align*} &\|P_t(x_1,\cdot)-P_t(x_2,\cdot)\|_{\textrm{var}} \\ &\qquad\le 2\big(|y_1-y_2|\bar{v}_{t/3} + \textrm{e}^{-\nu(D_\varepsilon^c)t/3}\big) + K_\varepsilon \textrm{e}^{\beta_{22}t/3}\Big\{\textrm{e}^{\beta_{22}t/3}|z_1-z_2| \nonumber\\ &\quad\qquad+\, \Big(\beta_{21}| + 2\int_{D_1^c} |v_2| \mu(\textrm{d} v)\Big)|y_1-y_2|\int_0^{t/3} \textrm{e}^{\beta_{11}s}\textrm{e}^{\beta_{22}(t/3-s)} \textrm{d} s \\ &\quad\qquad+\, 2\sqrt{2}(\sigma_{21}+\sigma_{22})|y_1-y_2|^{1/2}\Big(\int_0^{t/3} \textrm{e}^{\beta_{11}s} \textrm{e}^{2\beta_{22}(t/3-s)}\textrm{d} s\Big)^{1/2} \\ &\quad\qquad+\, 2\Big(\int_{D_1} v_2^2 \mu(\textrm{d} v)\Big)^{1/2} |y_1-y_2|^{1/2} \Big(\int_0^{t/3} \textrm{e}^{\beta_{11}s}\textrm{e}^{2\beta_{22}(t/3-s)} \textrm{d} s\Big)^{1/2}\Big\}, \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad t\ge 0,\ x_i= (y_i,z_i)\in D,\ i=1,2. \end{align*}

Proof. Step 1. Consider the case where $y_1= y_2= y\in \mathbb{R}_+$ . Let $\{Y_t\,{:}\,t\ge 0\}$ be the solution of (2.14) with $Y_0= y$ . Let $z_0= 0$ . For $i=0,1,2$ let $\{Z_i(t)\,{:}\,t\ge 0\}$ be defined by (2.16), with $Z_i(0)= z_i$ . It is easy to see that

(3.6) \begin{eqnarray}Z_i(t)= \textrm{e}^{\beta_{22}t}z_i + Z_0(t), \quad t\ge 0,\ i=1,2. \end{eqnarray}

Let $\{\eta_\varepsilon(t)\,{:}\,t\ge 0\}$ be the compensated compound Poisson process defined by

\begin{align*}\eta_\varepsilon(t)= \int_0^t\int_{D_\varepsilon^c} v_2 \tilde{N}(\textrm{d} s,\textrm{d} v). \end{align*}

Let $\tau_1= \inf\{t\ge 0\,{:}\, \eta_\varepsilon(t)\neq \eta_\varepsilon(t-\!)\}$ be the first jump time of this process. For any $f\in \mathscr{B}_1$ we have

\begin{align*}\big|P_tf(y,z_1)-P_tf(y,z_2)\big| &= \big|\mathbb{E}\big[f(Y_t,Z_1(t))-f(Y_t,Z_2(t))\big]\big| \\[3pt] & \le 2\mathbb{P}(\tau_1> t) + p_\varepsilon(t), \end{align*}

where $\mathbb{P}(\tau_1> t)= \textrm{e}^{-\nu(D_\varepsilon^c)t}$ and

\begin{align*}p_\varepsilon(t)= \Big|\mathbb{E}\big\{\big[f(Y_t,Z_1(t)) - f(Y_t,Z_2(t))\big]1_{\{\tau_1\le t\}}\big\}\Big|. \end{align*}

By the strong Markov property and (2.16),

\begin{align*}p_\varepsilon(t) &= \bigg|\mathbb{E}\Big\{\int_0^t \nu(D_\varepsilon^c)\textrm{e}^{-\nu(D_\varepsilon^c)s} \Big[\int_{D}P_{t-s} f(Y_s,Z_1(s-\!)+r) \hat{\nu}_\varepsilon(\textrm{d} r) \\[3pt] & \qquad- \int_{D}P_{t-s} f(Y_s,Z_2(s-\!)+r) \hat{\nu}_\varepsilon(\textrm{d} r)\Big] \textrm{d} s\Big\}\bigg| \\[3pt] & = \bigg|\mathbb{E}\Big\{\int_0^t \nu(D_\varepsilon^c)\textrm{e}^{-\nu(D_\varepsilon^c)s} \Big[\int_{D}P_{t-s} f(Y_s,Z_1(s)+r) \hat{\nu}_\varepsilon(\textrm{d} r) \\[3pt] & \qquad- \int_{D}P_{t-s} f(Y_s,Z_1(s)+r) \delta_{(0,\textrm{e}^{\beta_{22}s}(z_2-z_1))}* \hat{\nu}_\varepsilon(\textrm{d} r)\Big] \textrm{d} s\Big\}\bigg| \\[3pt] & \le \int_0^t \nu(D_\varepsilon^c)\textrm{e}^{-\nu(D_\varepsilon^c)s} \|\hat{\nu}_\varepsilon - \delta_{(0,\textrm{e}^{\beta_{22}s}(z_2-z_1))} * \hat{\nu}_\varepsilon\|\textrm{d} s \\[3pt] & \le K_\varepsilon\nu(D_\varepsilon^c)|z_2-z_1|\int_0^t\textrm{e}^{-\nu(D_\varepsilon^c)s}\textrm{d} s \le K_\varepsilon|z_2-z_1|. \end{align*}

It follows that

(3.7) \begin{eqnarray}\big|P_tf(y,z_1)-P_tf(y,z_2)\big| \le2\textrm{e}^{-\nu(D_\varepsilon^c)t} + K_\varepsilon|z_1-z_2|. \end{eqnarray}

Then we can use the Markov property and the representation (3.6) to get

\begin{align*} &\big|P_tf(y,x_1)-P_tf(y,x_2)\big| \nonumber\\[3pt]& \qquad= \Big|\mathbb{E}\big[f(Y_t,Z_1(t)) - f(Y_t,Z_2(t))\big]\Big| \\[3pt] & \qquad=\Big|\mathbb{E}\big[P_{t/2}f(Y_{t/2},Z_1(t/2)) - P_{t/2}f(Y_{t/2},Z_2(t/2))\big]\Big| \nonumber\\[3pt] & \qquad\le\, 2\textrm{e}^{-\nu(D_\varepsilon^c)t/2} + K_\varepsilon\mathbb{E}\big[|Z_1(t/2)-Z_2(t/2)|\big] \nonumber\\[3pt] & \qquad\le\, 2\textrm{e}^{-\nu(D_\varepsilon^c)t/2} + K_\varepsilon|z_1-z_2|\textrm{e}^{\beta_{22}t/2}. \end{align*}

Step 2. In the general case, we have $x_1= (y_1,z_1)$ and $x_2= (y_2,z_2)$ , where $y_1,y_2\in \mathbb{R}_+$ and $z_1, z_2\in \mathbb{R}_+$ . It suffices to consider the case of $y_1\ge y_2$ . Let $\{(Y_i(t),Z_i(t))\,{:}\,t\ge 0\}$ be defined by (2.14)–(2.15) with $(Y_i(0),Z_i(0))= x_i$ , $i=1,2$ . Then

\begin{align*}\big|P_tf(x_1)-P_tf(x_2)\big| &= \Big|\mathbb{E}\big[f(Y_1(t),Z_1(t)) - f(Y_2(t),Z_2(t))\big]\Big| \nonumber\\[3pt] &\le 2\mathbb{P}(\tau_0> t/3) + q_\varepsilon(t), \end{align*}

with $\mathbb{P}(\tau_0> t/3)\le |y_1-y_2|\bar{v}_{t/3}$ and

\begin{align*}q_\varepsilon(t)& = \Big|\mathbb{E}\big\{\big[f(Y_1(t),Z_1(t))-f(Y_1(t),Z_2(t))\big]1_{\{\tau_0\le t/3\}}\big\}\Big| \nonumber\\[3pt] & = \Big|\mathbb{E}\big\{1_{\{\tau_0\le t/3\}}\mathbb{E}\big[f(Y_1(t),Z_1(t)) - f(Y_1(t),Z_2(t))\big| \mathscr{F}_{\tau_0}\big]\big\}\Big| \nonumber\\[3pt] & = \Big|\mathbb{E}\big\{1_{\{\tau_0\le t/3\}} \big[P_{t-\tau_0} f(Y_1(\tau_0),Z_1(\tau_0)) - P_{t-\tau_0} f(Y_1(\tau_0),Z_2(\tau_0))\big]\big\}\Big| \nonumber\\[3pt] &\le 2\textrm{e}^{-\nu(D_\varepsilon^c)t/3} + K_\varepsilon \mathbb{E}\big(1_{\{\tau_0\le t/3\}} |Z_1(\tau_0)-Z_2(\tau_0)|\big) \textrm{e}^{\beta_{22}t/3} \nonumber\\[3pt] & \le 2\textrm{e}^{-\nu(D_\varepsilon^c)t/3} + K_\varepsilon \mathbb{E}\Big(\sup_{s\le t/3}|Z_1(s)-Z_2(s)|\Big) \textrm{e}^{\beta_{22}t/3}, \end{align*}

where we have used (3.7) for the first inequality. Then the desired estimate follows by (2.20) and Proposition 2.6.

Proof. Suppose that $\{x_k\}\in D$ is a sequence such that $\lim_{k\to \infty} x_k= x_0\in D$ . By Theorem 3.5, for $t>0$ and $n\ge 1$ we have

\begin{align*}\limsup_{k\to \infty}\|P_t(x_k,\cdot)-P_t(x_0,\cdot)\|_{\textrm{var}} \le2\textrm{e}^{-\nu(D_{\varepsilon_n}^c)t/3}. \end{align*}

The left-hand side vanishes since $\lim_{n\to \infty}\nu(D_{\varepsilon_n}^c)= \infty$ .

Corollary 3.7. Suppose that $\beta_{11}< 0$ , $\beta_{22}< 0$ , and Condition 3.4 is satisfied. Then there is a constant $C_\varepsilon\ge 0$ such that

(3.8) \begin{eqnarray}\|P_t(x,\cdot)-\pi\|_{\textrm{var}}\le C_\varepsilon(1+|x|)\textrm{e}^{-\kappa_\varepsilon t/3}, \quad t\ge 0,\ x\in D, \end{eqnarray}

where $\kappa_\varepsilon= |\beta_{11}|\land |\beta_{22}|\land \nu(D_\varepsilon^c)$ .

Proof. By Theorem 3.5 there is a constant $C_\varepsilon\ge 0$ such that, for $t> 0$ and $x_i= (y_i,z_i)\in D$ , $i= 1,2$ ,

\begin{align*} &\|P_t(x_1,\cdot)-P_t(x_2,\cdot)\|_{\textrm{var}} \nonumber\\[3pt] &\qquad\le 2\textrm{e}^{-\nu(D_\varepsilon^c)t/3} + C_\varepsilon\big(|x_1-x_2| + |y_1-y_2|^{1/2}\big) (\bar{v}_{t/3}\vee \textrm{e}^{-|\beta_{22}|t/3}). \end{align*}

Then the result follows as in the proof of Corollary 3.3.

Theorems 3.1 and 3.5 and their corollaries are natural extensions of the existing results on CBI and OU-type processes in the literature. In fact, one may see that some parts of the proofs given above essentially follow the ideas of Li and Ma [Reference Li and Ma16] and Wang [Reference Wang29]; see also Wang [Reference Wang28]. For general affine processes on cones, Mayerhofer et al. [Reference Mayerhofer, Stelzer and Vestweber19] studied the exponential ergodicity in the total variation distance under certain irreducibility, aperiodicity, and finite second moment assumptions. Their techniques were based on the theory of stochastic stability of Markov processes; see Meyn and Tweedie [Reference Meyn and Tweedie20, Reference Meyn and Tweedie21] and the references therein. While the results of Mayerhofer et al. [Reference Mayerhofer, Stelzer and Vestweber19] were formulated in an abstract framework, it seems a delicate task to check their conditions for the process discussed here. Moreover, the finite second moment condition of Mayerhofer et al. [Reference Mayerhofer, Stelzer and Vestweber19] rules out some natural examples.

4. A weaker condition for ergodicity

Throughout this section, we assume $\beta_{11}< 0$ and $\beta_{22}< 0$ . We shall establish the ergodicity of the affine process under a condition on the Lévy measure $\nu$ weaker than Condition 3.4. The proof of the result is based on a coupling similar to those used in the last section. Suppose that $\nu(D)> 0$ and choose $0< \varepsilon< 1$ so that $0< \nu(D_\varepsilon)< \infty$ , where $D_\varepsilon=\mathbb{R}_+\times [\!-\varepsilon,\varepsilon]$ . Let $\nu_\varepsilon$ and $\hat{\nu}_\varepsilon$ be defined as in the last section. Let $\gamma^{\varepsilon}_z= \hat{\nu}_\varepsilon\land (\delta_{(0,z)}*\hat{\nu}_\varepsilon)$ for $z\in \mathbb{R}$ .

Condition 4.1. There are constants $\varepsilon\in (0,1]$ and $\delta> 0$ such that

\begin{align*}q\,{:\!=}\, \inf_{|z|\le \delta}\gamma^{\varepsilon}_z(D) =\inf_{|z|\le\delta}\hat{\nu}_\varepsilon\land (\delta_{(0,z)}*\hat{\nu}_\varepsilon)(D)> 0. \end{align*}

The above condition was introduced for general finite-dimensional OU-type processes by Schilling and Wang [Reference Schilling and Wang27] and Wang [Reference Wang29]; see also Wang [Reference Wang28]. As observed in Wang [Reference Wang29, p. 992], the condition is weaker than Condition 3.4.

Lemma 4.2. Suppose that Condition 4.1 is satisfied. For $|z|\le \delta$ , let $\hat{\gamma}^{\varepsilon}_z= \gamma^{\varepsilon}_z(D)^{-1} \gamma^{\varepsilon}_z$ , and let $(\eta,\rho_1,\zeta)$ be a random vector such that, for $A\in \mathscr{B}(D)$ ,

(4.1) \begin{align}\mathbb{P}\{(\eta,\rho_1,\zeta)\in A\times B\} =\left\{\begin{array}{l@{\quad}l} q\hat{\gamma}^{\varepsilon}_{-z}(A)/2, & B=\{z\}, \\ \\[-8pt] q\hat{\gamma}^{\varepsilon}_z(A)/2, & B=\{-z\}, \\ \\[-8pt] [\hat{\nu}_\varepsilon - q(\hat{\gamma}^{\varepsilon}_{-z}+\hat{\gamma}^{\varepsilon}_z)/2](A), & B=\{0\}.\end{array}\right. \end{align}

Let $\rho_2= \rho_1+\zeta$ . Then we have

(4.2) \begin{eqnarray}\mathbb{P}(\zeta= z)= \mathbb{P}(\zeta= -z)= q/2, \quad \mathbb{P}(\zeta=0)= 1-q \end{eqnarray}

and

(4.3) \begin{eqnarray}\mathbb{P}\{(\eta,\rho_1)\in A\}= \mathbb{P}\{(\eta,\rho_2)\in A\}= \hat{\nu}_\varepsilon(A), \quad A\in \mathscr{B}(D). \end{eqnarray}

Proof. From (4.1) it is easy to see that $\zeta$ has distribution given by (4.2). Moreover, for any $A\in \mathscr{B}(D)$ we have

\begin{align*}\mathbb{P}\{(\eta,\rho_1)\in A\} &= \mathbb{P}\{(\eta,\rho_1)\in A, \zeta= z\} + \mathbb{P}\{(\eta,\rho_1)\in A, \zeta= -z\} \nonumber\\[3pt] &\quad +\, \mathbb{P}\{(\eta,\rho_1)\in A, \zeta= 0\} \nonumber\\[3pt] &= q\hat{\gamma}^{\varepsilon}_{-z}(A)/2 + q\hat{\gamma}^{\varepsilon}_z(A)/2 + [\hat{\nu}_\varepsilon - q(\hat{\gamma}^{\varepsilon}_{-z} + \hat{\gamma}^{\varepsilon}_z)/2](A) \nonumber\\[3pt] & = \hat{\nu}_\varepsilon(A) \end{align*}

and

\begin{align*}\mathbb{P}\{(\eta,\rho_2)\in A\} &= \mathbb{P}\{(\eta,\rho_2)\in A, \zeta= z\} + \mathbb{P}\{(\eta,\rho_2)\in A, \zeta= -z\} \nonumber\\[3pt] &\quad +\, \mathbb{P}\{(\eta,\rho_2)\in A, \zeta= 0\} \nonumber\\[3pt] &= \mathbb{P}\{(\eta,\rho_1)\in A-(0,z), \zeta= z\} + \mathbb{P}\{(\eta,\rho_1)\in A+(0,z), \zeta= -z\} \nonumber\\[3pt] &\quad +\, \mathbb{P}\{(\eta,\rho_1)\in A, \zeta= 0\} \nonumber\\[3pt] &= q\hat{\gamma}^{\varepsilon}_{-z}(A-(0,z))/2 + q\hat{\gamma}^{\varepsilon}_z(A+(0,z))/2 + [\hat{\nu}_\varepsilon - q(\hat{\gamma}^{\varepsilon}_{-z} + \hat{\gamma}^{\varepsilon}_z)/2](A) \nonumber\\[3pt] & = q\hat{\gamma}^{\varepsilon}_z(A)/2 + q\hat{\gamma}^{\varepsilon}_{-z}(A)/2 + [\hat{\nu}_\varepsilon - q(\hat{\gamma}^{\varepsilon}_{-z} + \hat{\gamma}^{\varepsilon}_z)/2](A) \nonumber\\[3pt] & = \hat{\nu}_\varepsilon(A). \end{align*}

Then (4.3) holds.

Proof. It is easy to see that $z\mapsto \delta_{(0,z)}*\hat{\nu}_\varepsilon$ is a Borel probability kernel from $[\!-\delta,\delta]$ to D. Let

\begin{align*}m^{\varepsilon}(z,\cdot)= (\hat{\nu}_\varepsilon - \delta_{(0,z)}*\hat{\nu}_\varepsilon)^+ + (\hat{\nu}_\varepsilon - \delta_{(0,z)}*\hat{\nu}_\varepsilon)^- \end{align*}

denote the total variation of the signed measure $\hat{\nu}_\varepsilon - \delta_{(0,z)}*\hat{\nu}_\varepsilon$ . By the regularity of the measures, for any bounded positive continuous function f on D we have

\begin{align*}\int_{D} f(v)m^{\varepsilon}(z,\textrm{d} v) =\sup_{g\in \mathscr{C}_1}\int_{D} f(v)g(v) \big[\hat{\nu}_\varepsilon(\textrm{d} v) - (\delta_{z}*\hat{\nu}_\varepsilon)(\textrm{d} v)\big], \end{align*}

where $\mathscr{C}_1$ is the set of continuous functions g on D satisfying $|g|\le 1$ . Then the mapping

\begin{align*}z\mapsto \int_{D} f(v)m^{\varepsilon}(z,\textrm{d} v) \end{align*}

is lower semicontinuous, so it is a Borel function on $[\!-\delta,\delta]$ . It follows that $m^{\varepsilon}(z,\cdot)$ and $\gamma^{\varepsilon}_z= \hat{\nu}_\varepsilon + \delta_{z}*\hat{\nu}_\varepsilon - m^{\varepsilon}(z,\cdot)$ are kernels from $[\!-\delta,\delta]$ to D. From (4.1) we see that $Q(z,\cdot)$ is a Borel probability kernel from $[\!-\delta,\delta]\setminus \{0\}$ to $\mathbb{R}_+\times \mathbb{R}^2$ .

Now let us define the first coupling in this section. The basic idea follows that of Schilling and Wang [Reference Schilling and Wang27] and Wang [Reference Wang29]. Here we give a pathwise construction of the coupling in terms of stochastic integrals. Let $x_1= (y,z_1)\in D$ and $x_2= (y,z_2)\in D$ , where $y\in \mathbb{R}_+$ and $z_1, z_2\in \mathbb{R}$ satisfy $z_1\neq z_2$ . Let $k= \lfloor\delta^{-1}|z_1-z_2|\rfloor+1$ . Then $k^{-1}|z_1-z_2|\le \delta$ . Let $G= \mathbb{R}_{+}\times \mathbb{R}^2$ be endowed with its Borel $\sigma$ -algebra. In addition to the noises in (2.14) and (2.15), let $N_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2)$ be an $(\mathscr{F}_t)$ -Poisson random measure on $(0,\infty)\times G$ with intensity

(4.4) \begin{eqnarray}\nu_\varepsilon(D) \textrm{d} sQ(k^{-1}(z_1-z_2)\textrm{e}^{\beta_{22}s},\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2). \end{eqnarray}

We assume all of these noises are independent of each other. For $t\ge 0$ let

(4.5) \begin{eqnarray}\xi(t)= (z_1-z_2) + \int_0^t\int_G (v_1-v_2)\textrm{e}^{-\beta_{22}s} N_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2). \end{eqnarray}

Then we have

(4.6) \begin{eqnarray}\xi(t)= (z_1-z_2)[1 + L(t)], \quad t\ge 0, \end{eqnarray}

where

\begin{align*}L(t)= (z_1-z_2)^{-1}\int_0^t\int_G (v_1-v_2)\textrm{e}^{-\beta_{22}s} N_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2). \end{align*}

Let $N_0(\textrm{d} s,\textrm{d} r)$ be the image of $N_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2)$ under the mapping

\begin{align*}(s,u,v_1,v_2)\mapsto (s,r)= (s,(z_1-z_2)^{-1}(v_1-v_2)\textrm{e}^{\beta_{11}s}). \end{align*}

We easily see that $N_0(\textrm{d} s,\textrm{d} r)$ is a Poisson random measure on $(0,\infty)\times \mathbb{R}$ with intensity $\nu_\varepsilon(D)\textrm{d} s\pi(s,\textrm{d} r)$ , where $\pi(s,\textrm{d} r)$ is the probability measure on $\mathbb{R}$ defined by

\begin{align*}\pi(s,\{1/k\}) =\pi(s,\{-1/k\})= q/2, \quad\pi(s,\{0\})= 1-q. \end{align*}

It follows that

\begin{align*}N(t)\,{:\!=}\, \int_0^t\int_{\mathbb{R}_+}\int_{\mathbb{R}^2} 1_{\{v_1\neq v_2\}} N_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2) \end{align*}

is a Poisson process with parameter $q\nu_\varepsilon(D)$ . For $i\ge 1$ , let $\xi_i$ denote the size of the ith jump of the process $\{L(t)\,{:}\,t\ge 0\}$ . Then $\{\xi_i\,{:}\,i\ge 1\}$ are independent and identically distributed (i.i.d.) random variables with

(4.7) \begin{eqnarray}\mathbb{P}(\xi_i= 1/k)= \mathbb{P}(\xi_i= -1/k)= 1/2. \end{eqnarray}

Let $S(n)= \sum_{i=1}^n \xi_i$ . Then the processes $\{N(t)\,{:}\,t\ge 0\}$ and $\{S(n)\,{:}\, n\ge 0\}$ are independent and

\begin{align*}L(t)= S(N(t)), \quad t\ge 0. \end{align*}

This proves the following.

Let $\tau= \inf\{t\ge 0\,{:}\,\xi(t)= 0\}= \inf\{t\ge 0\,{:}\,L(t)= -1\}$ , and let $\{Y'(t)\,{:}\,t\ge 0\}$ be the solution of

(4.8) \begin{align}Y'(t) &= y + \int_0^t [b_1+\beta_{11}Y'(s)] \textrm{d} s + \sqrt{2}\sigma_{11}\int_0^t\int_0^{Y'(s)} W_1(\textrm{d} s,\textrm{d} u) \nonumber\\[3pt] &\quad + \sqrt{2}\sigma_{12}\int_0^t\int_0^{Y'(s)} W_2(\textrm{d} s,\textrm{d} u) + \int_0^t\int_0^{Y'(s-\!)}\int_D v \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v,\textrm{d} r) \nonumber\\[3pt] &\quad + \int_{t\land \tau}^t\int_D v_1 N(\textrm{d} s,\textrm{d} v) + \int_0^{t\land \tau}\int_G u \tilde{N}_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2). \end{align}

Let $\{Z_1'(t)\,{:}\,t\ge 0\}$ and $\{Z_2'(t)\,{:}\,t\ge 0\}$ be defined by

(4.9) \begin{align}Z_1'(t) &= \textrm{e}^{\beta_{22}t}\Big[z_1 + \int_0^t \textrm{e}^{-\beta_{22}s}[b_2+\beta_{21}Y'(s)] \textrm{d} s + \sqrt{2}\sigma_0\int_0^t \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(s) \nonumber \\[3pt]&\quad + \sqrt{2}\sigma_{21}\int_0^t\int_0^{Y'(s)}\textrm{e}^{-\beta_{22}s} W_1(\textrm{d} s,\textrm{d} u) + \sqrt{2}\sigma_{22} \int_0^t\int_0^{Y'(s)} \textrm{e}^{-\beta_{22}s} W_2(\textrm{d} s,\textrm{d} u) \nonumber \\[3pt] &\quad + \int_0^t\int_0^{Y'(s-\!)}\int_{D} \textrm{e}^{-\beta_{22}s}r \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v,\textrm{d} r) + \int_{t\land \tau}^t\int_{D} \textrm{e}^{-\beta_{22}s}v_2 \tilde{N}(\textrm{d} s,\textrm{d} v) \nonumber \\[3pt] &\quad + \int_0^{t\land \tau}\int_G \textrm{e}^{-\beta_{22}s}v_1 \tilde{N}_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2)\Big] \end{align}

and

(4.10) \begin{align}Z_2'(t) & = \textrm{e}^{\beta_{22}t}\Big[z_2 + \int_0^t \textrm{e}^{-\beta_{22}s}[b_2+\beta_{21}Y'(s)] \textrm{d} s + \sqrt{2}\sigma_0\int_0^t \textrm{e}^{-\beta_{22}s} \textrm{d} W_0(s) \nonumber\\[3pt] &\quad + \sqrt{2}\sigma_{21}\int_0^t\int_0^{Y'(s)}\textrm{e}^{-\beta_{22}s} W_1(\textrm{d} s,\textrm{d} u) + \sqrt{2}\sigma_{22}\int_0^t\int_0^{Y'(s)} \textrm{e}^{-\beta_{22}s} W_2(\textrm{d} s,\textrm{d} u) \nonumber\\[3pt] &\quad + \int_0^t\int_0^{Y'(s-\!)}\int_{D} \textrm{e}^{-\beta_{22}s}r \tilde{M}(\textrm{d} s,\textrm{d} u,\textrm{d} v,\textrm{d} r) + \int_{t\land \tau}^t\int_{D} \textrm{e}^{-\beta_{22}s}v_2 \tilde{N}(\textrm{d} s,\textrm{d} v) \nonumber\\[3pt] &\quad + \int_0^{t\land \tau}\int_G \textrm{e}^{-\beta_{22}s}v_2 \tilde{N}_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2)\Big]. \end{align}

Let $N_1(\textrm{d} s,\textrm{d} u,\textrm{d} v_1)$ and $N_2(\textrm{d} s,\textrm{d} u,\textrm{d} v_2)$ respectively denote the images of the random measure $N_0(\textrm{d} s,\textrm{d} u,\textrm{d} v_1,\textrm{d} v_2)$ under the mappings

\begin{align*}(s,u,v_1,v_2)\mapsto (s,u,v_1), \quad(s,u,v_1,v_2)\mapsto (s,u,v_2). \end{align*}

Clearly, both $N_1(\textrm{d} s,\textrm{d} u,\textrm{d} w)$ and $N_2(\textrm{d} s,\textrm{d} u,\textrm{d} w)$ are Poisson random measures on $(0,\infty)\times D$ with intensity $\nu_\varepsilon(D)\textrm{d} s\hat{\nu}_\varepsilon(\textrm{d} u,\textrm{d} w)= \textrm{d} s\nu_\varepsilon(\textrm{d} u,\textrm{d} w)$ . It follows that both $\{(Y'(t),Z_1'(t))\,{:}\,t\ge 0\}$ and $\{(Y'(t),Z_2'(t))\,{:}\,t\ge 0\}$ are affine processes with transition semigroup $(P_t)_{t\ge 0}$ . From (4.8), (4.9), and (4.10) we see that

(4.11) \begin{eqnarray}\zeta(t)\,{:\!=}\, Z_1'(t)-Z_2'(t) =\textrm{e}^{\beta_{22}t}\xi(t\land \tau). \end{eqnarray}

Then $\{(Y'(t),Z_1'(t)$ , $Y'(t),Z_2'(t))\,{:}\,t\ge 0\}$ is a coupling of the affine process with coupling time

\begin{align*}\tau= \inf\{t\ge 0\,{:}\,\zeta(t)= 0\}= \inf\{t\ge 0\,{:}\,X_1'(t)= X_2'(t)\}. \end{align*}

Lemma 4.6. Suppose that Condition 4.1 is satisfied. Then there exists a constant $C_\varepsilon> 0$ such that

(4.12) \begin{eqnarray}\mathbb{P}\{Z_1'(t)\neq Z_2'(t)\}= \mathbb{P}(\tau>t) \le C_\varepsilon\big(1+|x_2-x_1|\big)\frac{1}{\sqrt{t}}, \quad t>0. \end{eqnarray}

Proof. By the reflection principle for the symmetric simple random walk, we have

\begin{align*}\mathbb{P}\Big(\min_{k\le n}S(k)> -1\Big) \le\mathbb{P}\big(|S(n)|\le 1\big) =\mathbb{P}\Big(\frac{k|S(n)|}{\sqrt{n}}\le \frac{k}{\sqrt{n}}\Big); \end{align*}

see, e.g., Lemma 2.3 in Schilling and Wang [Reference Schilling and Wang27]. By the Berry–Esseen inequality, there is a universal constant $C_0> 0$ such that

\begin{align*}\Big|\mathbb{P}\Big(x\le \frac{kS(n)}{\sqrt{n}}\le y\Big) - \frac{1}{\sqrt{2\pi}}\int_x^y\textrm{e}^{-z^2/2}\textrm{d} z\Big| \le\frac{C_0}{\sqrt{n}}, \quad x\le y\in \mathbb{R}. \end{align*}

Let $T= \inf\{n\ge 0\,{:}\,S(n)= -1\}$ . Then we have

\begin{align*}\mathbb{P}(T> n) &= \mathbb{P}\Big(\min_{k\le n}S(k)> -1\Big) \\[4pt] &\le \frac{C_0}{\sqrt{n}} + \frac{1}{\sqrt{2\pi}}\int_{-k/\sqrt{n}}^{k/\sqrt{n}}\textrm{e}^{-z^2/2}\textrm{d} z \\[4pt] &\le \frac{C_0}{\sqrt{n}} + \frac{\sqrt{2}k}{\sqrt{n\pi}} \le C_1(k+1)\frac{1}{\sqrt{n}}, \end{align*}

where $C_1= 1\vee C_0$ . By the total probability formula and the independence of $\{S(n)\,{:}\,n\ge 0\}$ and $\{N(t)\,{:}\,t\ge 0\}$ , it follows that

\begin{align*}\mathbb{P}(\tau>t) &= \sum_{n=0}^\infty \mathbb{P}(N(t)=n) \mathbb{P}(T> n|N(t)=n) \\ & = \textrm{e}^{-q\nu_\varepsilon(D)t}\Big[1 + \sum_{n=1}^\infty \frac{(q\nu_\varepsilon(D)t)^n}{n!} \mathbb{P}(T> n)\Big] \\ & \le \textrm{e}^{-q\nu_\varepsilon(D)t}\Big[1 + C_1(k+1)\sum_{n=1}^\infty \frac{(q\nu_\varepsilon(D)t)^n}{n!} \frac{1}{\sqrt{n}}\Big] \\ & \le \textrm{e}^{-q\nu_\varepsilon(D)t}\Big[1 + C_1(k+1)(\textrm{e}^{q\nu_\varepsilon(D)t}-1)^{1/2} \Big(\sum_{n=1}^\infty \frac{(q\nu_\varepsilon(D)t)^n}{n\cdot n!}\Big)^{1/2}\Big] \\ & \le \textrm{e}^{-q\nu_\varepsilon(D)t}\Big[1 + C_1(k+1) \Big(\frac{2(\textrm{e}^{q\nu_\varepsilon(D)t}-1)} {q\nu_\varepsilon(D)t} \sum_{n=1}^\infty \frac{(q\nu_\varepsilon(D)t)^{n+1}}{(n+1)\cdot n!}\Big)^{1/2}\Big] \\ & \le \textrm{e}^{-q\nu_\varepsilon(D)t}\Big[1 + \sqrt{2}C_1(k+1) (\textrm{e}^{q\nu_\varepsilon(D)t}-1) \frac{1}{\sqrt{q\nu_\varepsilon(D)t}}\Big] \\ & \le \textrm{e}^{-q\nu_\varepsilon(D)t} + \sqrt{2}C_1(|x_1-x_2|+2)(1-\textrm{e}^{-q\nu_\varepsilon(D)t}) \frac{1}{\sqrt{q\nu_\varepsilon(D)t}}. \end{align*}

Then (4.12) holds for some constant $C_\varepsilon\ge 0$ .

We next construct the main coupling of this section, through a concatenation of two couplings. Let $D([0,\infty),D^2)$ denote the space of càdlàg paths from $[0,\infty)$ to D. Let $\{w(t)\,{:}\,t\ge 0\}= \{(w_1(t),w_2(t)$ , $w_3(t),w_4(t))\,{:}\,t\ge 0\}$ denote the coordinate process of this space, and let $(\mathscr{F}_t\,{:}\,t\ge 0)$ be its natural filtration generated by the coordinate process. Let $\tau_0^w= \{t\ge 0\,{:}\,w_1(t)= w_3(t)\}$ and $\tau^w= \{t\ge \tau_0(w)\,{:}\,w_2(t)= w_4(t)\}$ . For $s\ge 0$ , let $\theta_s$ be the shifting operator on $D([0,\infty),D^2)$ defined by $\theta_sw(t)= w(s+t)$ , $t\ge 0$ . For $s\ge 0$ and $w\in D([0,\infty),D^2)$ , the stopped path $w^s\in D([0,\infty),D^2)$ is defined by $w^s(t)= w(s\land t)$ , $t\ge 0$ .

Let $x_1= (y_1,z_1)\in D$ and $x_2= (y_2,z_2)\in D$ , where $y_1,y_2\in \mathbb{R}_+$ and $z_1, z_2\in \mathbb{R}$ . For $i=1,2$ , let $\{Y_i(t)\,{:}\,t\ge 0\}$ be the solution of (2.14) with $Y_i(0)= y_i$ , and let $\{Z_i(t)\,{:}\,t\ge 0\}$ be defined by (2.16) with $Z_i(0)= z_i$ . Let $\mathbb{P}^{2,2}_{(x_1,x_2)}$ be the distribution on $D([0,\infty),D^2)$ of $\{(Y_1(t),Z_1(t)$ , $Y_2(t),Z_2(t))\,{:}\,t\ge 0\}$ . Let $\mathbb{P}^{1,2}_{(y,z_1,z_2)}$ be the distribution on $D([0,\infty),D^2)$ of the process $\{(Y'(t),Z_1'(t)$ , $Y'(t),Z_2'(t))\,{:}\,t\ge 0\}$ defined by (4.8)–(4.10). Let $\mathbb{P}_{(x_1,x_2)}$ be the probability measure on $D([0,\infty),D^2)$ defined by

\begin{align*} &\mathbb{P}_{(x_1,x_2)}\big[F\big((w_1,w_2,w_3,w_4)^{\tau_0^w}\big) G\big(\theta_{\tau_0^w}(w_1,w_2,w_3,w_4)\big)\big] \nonumber\\[3pt] &\qquad=\, \mathbb{P}^{2,2}_{(x_1,x_2)}\big[F\big((w_1,w_2,w_3,w_4)^{\tau_0^w}\big) \mathbb{P}^{1,2}_{(w_1(\tau_0^w),w_3(\tau_0^w),w_4(\tau_0^w))}G(w_1,w_2,w_3,w_4)\big], \end{align*}

where F and G are Borel functions on $D([0,\infty),D^2)$ , and the probability symbols are used to denote the corresponding expectations. Then under $\mathbb{P}_{(x_1,x_2)}$ the coordinate process $\{(w_1(t),w_2(t),$ $w_3(t),w_4(t))\,{:}\,t\ge 0\}$ evolves according to the transition law of $\{(Y_1(t),Z_1(t),$ $Y_2(t),Z_2(t))\,{:}\,t\ge 0\}$ up to time $\tau_0^w$ , after which it evolves according to the transition law of $\{(Y'(t),Z_1'(t),$ $Y'(t),Z_2'(t))\,{:}\,t\ge 0\}$ . It is clear that both $\{(w_1(t),w_2(t))\,{:}\,t\ge 0\}$ and $\{(w_3(t),w_4(t))\,{:}\,t\ge 0\}$ are affine processes with transition semigroup $(P_t)_{t\ge 0}$ . Thus they form a coupling of the affine process with coupling time $\tau^w$ .

Theorem 4.8. Suppose that Condition 4.1 is satisfied. Then there is a constant $C_\varepsilon\ge 0$ such that

(4.13) \begin{eqnarray}\|P_t(x_1,\cdot)-P_t(x_2,\cdot)\|_{\textrm{var}} \le C_\varepsilon(1+|x_1-x_2|)t^{-1/2}, \quad t>0,\ x_1, x_2\in D. \end{eqnarray}

Proof. There is no loss of generality in assuming $y_1\ge y_2$ . Using the coupling of the affine process constructed above, we have

\begin{align*}\mathbb{P}_{(x_1,x_2)}(\tau_1^w> t) \le\mathbb{P}_{(x_1,x_2)}(\tau_0^w> t/2) + \mathbb{P}_{(x_1,x_2)}(\tau_0^w\le t/2,\tau_1^w> t), \end{align*}

where $\mathbb{P}_{(x_1,x_2)}(\tau_0^w> t/2)\le |y_1-y_2|\bar{v}_{t/2}$ by (3.1). Since $t-\tau_0^w$ is measurable relative to $\mathscr{F}_{\tau_0^w}$ , we can use Lemma 4.6 to see

\begin{align*} &\mathbb{P}_{(x_1,x_2)}\big(\tau_0^w\le t/2,\tau_1^w> t\big) \nonumber\\[3pt] &\qquad=\, \mathbb{E}_{(x_1,x_2)}\big[1_{\{\tau_0^w\le t/2\}} \mathbb{P}_{(x_1,x_2)}\big(\tau_1^w> t|\mathscr{F}_{\tau_0^w}\big)\big] \nonumber\\[3pt] &\qquad\le\, C_\varepsilon\mathbb{E}_{(x_1,x_2)}\big[1_{\{\tau_0^w\le t/2\}} \big(1 + |w_3(\tau_0^w)-w_4(\tau_0^w)|\big) (t-\tau_0^w)^{-1/2}\big] \nonumber\\[3pt] &\qquad\le C_\varepsilon t^{-1/2}\mathbb{E}\Big(1 + \sup_{0\le s\le t/2}|Z_1(s)-Z_2(s)|\Big). \end{align*}

Then the result follows by Proposition 2.6 and (2.21).

Corollary 4.9. Suppose that Condition 4.1 is satisfied. Then there is a constant $C_\varepsilon\ge 0$ such that

(4.14) \begin{eqnarray}\|P_t(x,\cdot)-\pi\|_{\textrm{var}}\le C_\varepsilon(1+|x|)t^{-1/2}, \quad t> 0,\ x\in D. \end{eqnarray}

The above corollary gives an extension of Theorem 1.1 of Schilling and Wang [Reference Schilling and Wang27]; see also Theorem 2(i) of Wang [Reference Wang29].