2. Main results

In terms of terminology, we mainly follow the coordinate-free notation used in Mayerhofer [Reference Mayerhofer42] and Keller-Ressel and Mayerhofer [Reference Keller-Ressel and Mayerhofer33].

Let $d\geq2$ and denote by $\mathbb{S}_{d}$ the space of symmetric $d\times d$ matrices equipped with the scalar product $\langle x,y\rangle=\text{tr}(xy)$ and the Frobenius norm $\Vert x\Vert\,:\!=\langle x,x\rangle^{1/2}$ , where tr(.) denotes the trace of a matrix. We list in Appendix A some properties of the trace and its induced norm which are repeatedly used in the remainder of the article. Denote by $\mathbb{S}_{d}^{+}$ (resp. $\mathbb{S}_{d}^{++}$ ) the cone of symmetric and positive semidefinite (resp. positive definite) real $d\times d$ matrices. We write $x\preceq y$ if $y-x\in\mathbb{S}_{d}^{+}$ and $x\prec y$ if $y-x\in\mathbb{S}_{d}^{++}$ for the natural partial and strict order relations introduced respectively by the cones $\mathbb{S}_{d}^{+}$ and $\mathbb{S}_{d}^{++}$ . Let $\mathcal{B}(\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace)$ be the Borel- $\sigma$ -algebra on $\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace$ . An $\mathbb{S}_{d}^{+}$ -valued measure $\eta$ on $\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace$ is a $d\times d$ matrix of signed measures on $\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace$ such that $\eta(A)\in\mathbb{S}_{d}^{+}$ whenever $A\in\mathcal{B}(\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace)$ with $0\not\in\overline{A}$ .

In the following we present the notion of admissible parameters first introduced in Cuchiero et al. [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Definition 2.3]. Here we mainly follow the definition given in Mayerhofer [Reference Mayerhofer42, Definition 3.1], with a slightly stronger condition on the linear jump coefficient.

According to our definition, a set of admissible parameters does not contain parameters corresponding to killing. In addition, compared with [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Definition 2.3], our definition involves an additional first-moment assumption on the linear jump coefficient $\mu$ . Thanks to this stronger assumption and [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Remark 2.5], the affine process we consider here is conservative.

Theorem 2.1. (Cuchiero et al. [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12].) Let $(\alpha,b,B,m,\mu)$ be admissible parameters in the sense of Definition 2.1. Then there exists a unique stochastically continuous transition kernel $p_{t}(x,\text{d}\xi)$ such that $p_{t}(x,\mathbb{S}_{d}^{+})=1$ and

(2) \begin{equation}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}p_{t}(x,\text{d}\xi)=\exp\!\left({-}\phi(t,u)-\langle\psi(t,u),x\rangle\right)\!,\quad t\geq0,\ \ x,u\in\mathbb{S}_{d}^{+},\end{equation}

where $\phi(t,u)$ and $\psi(t,u)$ in (2) are the unique solutions to the generalized Riccati differential equations; that is, for $u\in\mathbb{S}_{d}^{+}$ ,

(3) \begin{align}\frac{\partial\phi(t,u)}{\partial t} & =F\left(\psi(t,u)\right)\!,\quad\phi(0,u)=0,\end{align}

(4) \begin{align}\frac{\partial\psi(t,u)}{\partial t} & =R\left(\psi(t,u)\right)\!,\quad\psi(0,u)=u,\end{align}

where the functions F and R are given by

\begin{align*}F(u) & =\langle b,u\rangle-\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}(\text{e}^{-\langle u,\xi\rangle}-1)m\left(\text{d}\xi\right)\!,\\[5pt] R(u) & =-2u\alpha u+B{}^{\top}\left(u\right)-\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}(\text{e}^{-\langle u,\xi\rangle}-1)\mu\left(\text{d}\xi\right)\!.\end{align*}

Here, $B^{\top}$ denotes the adjoint operator on $\mathbb{S}_{d}$ defined by the relation $\langle u,B(\xi)\rangle=\langle B^{\top}(u),\xi\rangle$ for $u,\thinspace\xi\in\mathbb{S}_{d}$ . Under the additional moment condition (iv) of Definition 2.1, we will show in Lemma 4.2 below that R(u) is continuously differentiable and thus locally Lipschitz continuous on $\mathbb{S}_{d}^{+}$ . This fact, together with the absence of parameters according to killing, implies that the affine process under consideration is indeed conservative (see [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Remark 2.5]) and, moreover, the Riccati equations have unique solutions for all $u \in \mathbb{S}_d^+$ . Also, following [Reference Mayerhofer42], each affine process on $\mathbb{S}_d^+$ , when $d \geq 2$ , has jumps of finite variation, which is why we omitted, compared with [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12], additional compensation in the integral against $\mu$ .

2.1. First moment

Our first result provides existence and a precise formula for the first moment of conservative affine processes on $\mathbb{S}_{d}^{+}$ . For this purpose, we define the effective drift

(5) \begin{equation}\widetilde{B}(u)\,:\!=B(u)+\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}\xi\langle u,\mu(\text{d}\xi)\rangle,\quad\text{for }u\in\mathbb{S}_{d}.\end{equation}

Then note that $\widetilde{B}\,:\,\mathbb{S}_{d}\to\mathbb{S}_{d}$ is a linear map. We define the corresponding semigroup $(\exp(t\widetilde{B}))_{t\geq0}$ by its Taylor series $\exp(t\widetilde{B})(u)=\sum_{n=0}^{\infty}t^{n}/n!\widetilde{B}^{\circ n}(u)$ , where $\widetilde{B}^{\circ n}$ denotes the n-times composition of $\widetilde{B}$ . For the remainder of the article we write $\mathbbm{1}$ without an index for the $d\times d$ -identity matrix, while $\mathbbm{1}_{A}$ denotes the standard indicator function of a set A.

Theorem 2.2. Let $p_{t}(x,\text{d}\xi)$ be the transition kernel of an affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ satisfying

(6) \begin{equation}\int_{\{\Vert\xi\Vert>1\}}\Vert\xi\Vert m\left(\text{d}\xi\right)<\infty.\end{equation}

Then for each $t\geq0$ and $x\in\mathbb{S}_{d}^{+}$ , the first moment of $p_{t}(x,\text{d}\xi)$ exists and equals

(7) \begin{equation}\int_{\mathbb{S}_{d}^{+}}\xi p_{t}(x,\text{d}\xi)=\text{e}^{t\widetilde{B}}x+\int_{0}^{t}\text{e}^{s\widetilde{B}}\left(b+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\xi m(\text{d}\xi)\right)\text{d}s.\end{equation}

Although this result is not very surprising, the proof of our main result (Theorem 2.3) crucially relies on (7). For the sake of completeness we therefore provide a full and detailed proof of Theorem 2.2 in Section 4.

Based on methods of stochastic calculus, similar results were obtained for affine processes with state-space $\mathbb{R}_{\geq0}^{m}$ in [Reference Barczy, Li and Pap5, Lemma 3.4] and on the canonical state space $\mathbb{R}_{\geq0}^{m}\times\mathbb{R}^{n}$ in [Reference Friesen, Jin and Rüdiger19, Lemma 5.2]. For affine processes on $\mathbb{R}_{\geq0}$ , i.e., continuous-state branching processes with immigration, and also for the more general class of Dawson–Watanabe superprocesses, an alternative approach based on a fine analysis of the Laplace transform is provided in [Reference Li39]. The latter approach clearly has the advantage that it is purely analytical and does not rely on the use of stochastic equations and semimartingale representations for these processes. Our proof provided in Section 4 is also purely analytical and uses some ideas taken from [Reference Li39].

Remark 2.1. Note that the transition kernel $p_{t}(x,\cdot)$ with admissible parameters $(\alpha,b,B,m,\mu)$ is Feller by virtue of [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Theorem 2.4]. Therefore, there exists a canonical realization $(X,(\mathbb{P}_{x})_{x\in\mathbb{S}_{d}^{+}})$ of the corresponding Markov process on the filtered space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ , where $\Omega=\mathbb{D}(\mathbb{S}_{d}^{+})$ is the set of all càdlàg paths $\omega\,:\,\mathbb{R}_{\geq0}\to\mathbb{S}_{d}^{+}$ , and $X_{t}(\omega)=\omega(t)$ for $\omega\in\Omega$ . Here $(\mathcal{F}_{t})_{t\geq0}$ is the natural filtration generated by X, and $\mathcal{F}=\bigvee_{t\geq0}\mathcal{F}_{t}$ . For $x\in\mathbb{S}_{d}^{+}$ , the probability measure $\mathbb{P}_{x}$ on $\Omega$ represents the law of the Markov process X given $X_{0}=x$ . With this notation, under the conditions of Theorem 2.2, the formula (7) reads

\begin{equation*}\mathbb{E}_{x}\big[X_{t}\big]=\text{e}^{t\widetilde{B}}x+\int_{0}^{t}\text{e}^{s\widetilde{B}}\left(b+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\xi m(\text{d}\xi)\right)\text{d}s,\end{equation*}

where $\mathbb{E}_{x}$ denotes the expectation with respect to $\mathbb{P}_{x}$ .

2.2. Existence and convergence to the invariant distribution

In this subsection we formulate our main result. Let $p_{t}(x,\cdot)$ be the transition kernel of an affine process on $\mathbb{S}_{d}^{+}$ . Motivated by Theorem 2.2 it is reasonable to relate the long-time behavior of the process to the spectrum $\sigma(\widetilde{B})$ of $\widetilde{B}$ . More precisely, an affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ is said to be subcritical if

(8) \begin{equation}\sup\left\lbrace \text{Re}\thinspace\lambda\in\mathbb{C}\thinspace:\thinspace\lambda\in\sigma\left(\widetilde{B}\right)\right\rbrace <0.\end{equation}

Under the condition (8), it is well-known that there exist constants $M\ge 1$ and $\delta>0$ such that

(9) \begin{equation}\left\Vert \text{e}^{t\widetilde{B}}\right\Vert \leq M\text{e}^{-\delta t},\quad t\geq0.\end{equation}

The next remark provides a sufficient condition for (9).

Remark 2.2. According to [Reference Mayerhofer, Stelzer and Vestweber44, Theorem 2.7], (9) is satisfied if and only if there exists a $v\in\mathbb{S}_{d}^{++}$ such that $-\widetilde{B}^{\top}(v)\in\mathbb{S}_{d}^{++}$ . However, in many application the linear drift is of the form $\widetilde{B}(x)=\beta x+x\beta^{\top}$ , where $\beta$ is a real-valued $d\times d$ matrix; see Section 3. In this case, it follows from [Reference Mayerhofer, Stelzer and Vestweber44, Corollary 5.1] that (9) is satisfied if and only if

\begin{equation*}\sup\left\lbrace \text{Re}\thinspace\lambda\in\mathbb{C}\thinspace:\thinspace\lambda\in\sigma\left(\beta\right)\right\rbrace <0,\end{equation*}

which in turn holds true if and only if there exists a $v\in\mathbb{S}_{d}^{++}$ such that $-(\beta^{\top}v+v\beta)\in\mathbb{S}_{d}^{++}$ .

Let $\mathcal{P}(\mathbb{S}_{d}^{+})$ be the space of all Borel probability measures on $\mathbb{S}_{d}^{+}$ . We call $\pi\in\mathcal{P}(\mathbb{S}_{d}^{+})$ an invariant distribution if

\begin{equation*}\int_{\mathbb{S}_{d}^{+}}p_{t}(x,\text{d}\xi)\pi(\text{d}x)=\pi(\text{d}\xi),\qquad t\geq0.\end{equation*}

The following is our main result.

Theorem 2.3. Let $p_{t}(x,\text{d}\xi)$ be the transition kernel of a subcritical affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ . Suppose that the measure m satisfies

(10) \begin{align}\int_{\{\Vert\xi\Vert>1\}}\log\left\Vert \xi\right\Vert m\left(\text{d}\xi\right)<\infty.\end{align}

Then there exists a unique invariant distribution $\pi$ . Moreover, for each $x\in\mathbb{S}_{d}^{+}$ , one has $p_{t}(x,\cdot)\to\pi$ weakly as $t\to\infty$ , and $\pi$ has Laplace transform

(11) \begin{equation}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,x\rangle}\pi\left(\text{d}x\right)=\exp\!\left({-}\int_{0}^{\infty}F\left(\psi(s,u)\right)\text{d}s\right)\!,\quad u\in\mathbb{S}_{d}^{+}.\end{equation}

The proof of Theorem 2.3 is postponed to Section 6. Let us make a few comments. Note that in dimension $d=1$ it holds that $\mathbb{S}_{1}^{+}=\mathbb{R}_{\geq0}$ , and affine processes on this state space coincide with the class of continuous-state branching processes with immigration introduced by Kawazu and Watanabe [Reference Kawazu and Watanabe31]. In this case, the long-time behavior has been extensively studied in the articles [Reference Keller-Ressel and Steiner36, Theorem 3.16] and [Reference Keller-Ressel and Mijatović34, Theorem 2.6] and in the monograph [Reference Li39, Theorem 3.20 and Corollary 3.21]. This is why we restrict ourselves to the case $d\geq2$ .

Theorem 2.3 establishes sufficient conditions for the existence, uniqueness, and convergence to the invariant distribution. For affine processes on the canonical state space $\mathbb{R}_{\geq0}^{m}\times\mathbb{R}^{n}$ a similar statement was recently shown in [Reference Jin, Kremer and Rüdiger30]. However, the method of [Reference Jin, Kremer and Rüdiger30] cannot be applied to the state space $\mathbb{S}_{d}^{+}$ . The reason is as follows. To apply the arguments of [Reference Jin, Kremer and Rüdiger30] to $\mathbb{S}_{d}^{+}$ -valued affine processes requires the decomposition

(12) \begin{equation} p_{t}(x,\cdot)=r_{t}(x,\cdot)\ast p_{t}(0,\cdot),\end{equation}

where $r_{t}(x,\cdot)$ is the transition kernel of an affine process on $\mathbb{S}_{d}^{+}$ whose Laplace transform is given by

(13) \begin{align}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}r_{t}(x,\text{d}\xi)=\exp\!\left({-}\langle\psi(t,u),x\rangle\right);\end{align}

that is, $r_{t}(x,\cdot)$ should have admissible parameters $(\alpha,b=0,B,m=0,\mu)$ . Unfortunately, such a transition kernel $r_{t}(x,\cdot)$ is well-defined if and only if $(\alpha,b=0,B,m=0,\mu)$ are admissible parameters in the sense of Definition 2.1. This in turn is true if and only if $\alpha=0$ , which is a consequence of the particular structure of the boundary $\partial \mathbb{S}_{d}^{+}$ . To overcome this difficulty, we use the cone structure to show that $\psi(t,u) \preceq \exp\{t \widetilde{B}^{\top}\}(u)$ , which requires us to study the first moment of the affine process with admissible parameters $(\alpha,b,B, m = 0, \mu)$ ; see (29). Note that the first moment of affine processes is already studied in Theorem 2.2, where the first moment condition for $\mu$ is also explicitly used. Uniform exponential stability for $\psi(t,u)$ is then an immediate consequence of the assumption that the affine process is subcritical, i.e. that (9) holds. Compared with [Reference Jin, Kremer and Rüdiger30], our proof does not rely on stability results for ordinary differential equations; moreover, it has the advantage that it also applies to more general convex cones other than $\mathbb{S}_d^+$ (see Section 9). On the other hand, since this argument is crucially based on the proper closed convex cone structure, it cannot be applied to affine processes on the canonical state space $\mathbb{R}_{\geq0}^m \times \mathbb{R}^n$ .

Based on strong solutions to the stochastic equation and the construction of monotone couplings, an alternative approach for the study of the long-time behavior of affine processes on the canonical state space $\mathbb{R}_{\geq0}^m \times \mathbb{R}^n$ was recently given in [Reference Friesen, Jin and Rüdiger19]. This approach certainly does not apply here since it is still an open problem whether each affine process on $\mathbb{S}_{d}^{+}$ can be obtained as a strong solution to a certain stochastic equation driven by Brownian motions and Poisson random measures (it is actually not even known what such a stochastic equation in the general case should look like). We refer the reader to [Reference Mayerhofer, Pfaffel and Stelzer43] for some related results. In addition, we do not know if a comparison principle for such processes would be available.

For dimension $d=1$ it is known that (10) is not only sufficient, but also necessary for the convergence to some limiting distribution; see, e.g., [Reference Li39, Theorem 3.20 and Corollary 3.21]. To our knowledge, extensions of this result to higher-dimensional state spaces have not yet been obtained. In this context, we have the following partial result for subcritical affine processes on $\mathbb{S}_{d}^{+}$ .

Proposition 2.1. Let $p_{t}(x,\text{d}\xi)$ be the transition kernel of a subcritical affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ . Suppose that there exist $x\in\mathbb{S}_{d}^{+}$ and $\pi\in\mathcal{P}(\mathbb{S}_{d}^{+})$ such that $p_{t}(x,\cdot)\to\pi$ weakly as $t\to\infty$ . If $\alpha=0$ and if there exists a constant $K>0$ satisfying

(14) \begin{align}K\xi+B(\xi)\succeq0,\quad\xi\in\mathbb{S}_{d}^{+},\end{align}

then (10) holds.

In order to prove Proposition 2.1 we first establish in Section 5 a precise lower bound for $\psi(t,u)$ . Since in dimension $d\geq2$ different components of the process interact through the drift B in a nontrivial manner on $\mathbb{S}_{d}^{+}$ , the proof of the lower bound is deduced from the additional conditions $\alpha=0$ and (14), which guarantee that these components are coupled in a well-behaved way. Let us remark that, in contrast to the proof of Theorem 2.3, the proof of Proposition 2.1 does not explicitly use the first moment condition for $\mu$ . Indeed, it suffices to assume that $\mu$ satisfies the condition $\int_{\mathbb{S}_d^+ \backslash \{0\}} ( 1 \wedge \| \xi \| ) \text{tr}(\mu)(\text{d}\xi) < \infty$ , which is weaker than (1), and, in addition, that the affine process is conservative.

We note that any linear map $B\,:\,\mathbb{S}_d\to\mathbb{S}_d$ with $B(\mathbb{S}_d^+) \subset \mathbb{S}_d^+$ satisfies the condition (14) for each $K>0$ . As an example of such a map, let $B(x)=\beta x\beta^{\top}$ for $x\in\mathbb{S}_d$ , where $\beta$ is a real-valued $d\times d$ matrix. Obviously, B defined in this way is admissible in the sense of Definition 2, and $B(\mathbb{S}_d^+) \subset \mathbb{S}_d^+$ . However, such an example will not lead to a subcritical affine process. Just to understand this point, we can think of the simplest case when $d=1$ (although we always assume $d\ge 2$ in the other parts of the paper): $B(x)=\beta^2 x$ and it is impossible for B to fulfill the subcritical condition (8).

The simplest example of an admissible B that produces subcriticality and at the same time satisfies (14) is $B(x)=-x$ , $x\in S_{d}$ . However, there are also other examples like this one. For instance, consider $d=2$ and the linear map B on $S_{2}$ defined by

\begin{equation*}B\left(\left(\begin{array}{c@{\quad}c}x_{1} & x_{2}\\[3pt] x_{2} & x_{3}\end{array}\right)\right)=\left(\begin{array}{c@{\quad}c}-x_{1} & -\kappa x_{2}\\[3pt] -\kappa x_{2} & -x_{3}\end{array}\right)\!,\end{equation*}

where $\kappa>1$ is a constant. First, such a B is admissible in the sense of Definition 2.1, since for $u,x\in S_{2}^{+}$ with $\left\langle x,u\right\rangle =0$ ,

\begin{equation*}\left\langle B(x),u\right\rangle =-\kappa\left\langle x,u\right\rangle +(\kappa-1)\left\langle \left(\begin{array}{c@{\quad}c}x_{1} & 0\\[3pt] 0 & x_{3}\end{array}\right)\!,u\right\rangle =(\kappa-1)\left\langle \left(\begin{array}{c@{\quad}c}x_{1} & 0\\[3pt] 0 & x_{3}\end{array}\right)\!,u\right\rangle \ge0.\end{equation*}

It is easy to see that the spectrum $\sigma(B)=\left\{ -1,-\kappa\right\} $ , so that we can choose a relatively small $\mu$ to make $\tilde{B}$ defined in (5) subcritical; moreover, (14) holds for $K=\kappa$ .

We close this section with a useful moment result regarding the invariant distribution.

Corollary 2.1. Let $p_{t}(x,\text{d}\xi)$ be the transition kernel of a subcritical affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ satisfying (6). Let $\pi$ be the unique invariant distribution. Then

\begin{equation*}\lim_{t\to\infty}\int_{\mathbb{S}_{d}^{+}}yp_{t}(x,\text{d}y)=\int_{\mathbb{S}_{d}^{+}}y\pi(\text{d}y)= -\widetilde{B}^{-1}\left( b + \int_{\mathbb{S}_d^+ \backslash \{0\}} \xi m(\text{d}\xi) \right)\!.\end{equation*}

2.3. Study of the convergence rate

Consider a subcritical affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ and let $\delta$ be defined by (9). In particular, $\delta$ is strictly positive, and we will see that it appears naturally in the convergence rate towards the invariant distribution. In order to measure this rate of convergence we introduce

\begin{equation*}d_{L}(\eta,\nu)\,:\!=\sup_{u\in\mathbb{S}_{d}^{+}\setminus\{0\}}\frac{1}{\Vert u\Vert}\left|\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,x\rangle}\eta(\text{d}x)-\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,x\rangle}\nu(\text{d}x)\right|,\quad\eta,\thinspace\nu\in\mathcal{P}(\mathbb{S}_{d}^{+}).\end{equation*}

Note that this supremum is not necessarily finite. However, it is finite for elements from

\begin{equation*}\mathcal{P}_{1}(\mathbb{S}_{d}^{+})=\left\lbrace \varrho\in\mathcal{P}(\mathbb{S}_{d}^{+})\thinspace:\thinspace\int_{\mathbb{S}_{d}^{+}}\Vert x\Vert\varrho(\text{d}x)<\infty\right\rbrace,\end{equation*}

which essentially follows from

(15) \begin{align} \left|\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,x\rangle}\eta(\text{d}x)-\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,x\rangle}\nu(\text{d}x)\right| &= \left|\int_{\mathbb{S}_{d}^{+} \times \mathbb{S}_d^+}\left(\text{e}^{-\langle u,x\rangle} - \text{e}^{- \langle u,y \rangle} \right) \eta(\text{d}x)\nu(\text{d}y) \right|\\ &\leq \| u \| \int_{\mathbb{S}_d^+ \times \mathbb{S}_d^+} \| x -y\| \eta(\text{d}x) \nu(\text{d}y)\nonumber \\ & \leq \| u \| \left(\int_{\mathbb{S}_d^+} \| x \| \eta(\text{d}x)+ \int_{\mathbb{S}_d^+} \| y\| \nu(\text{d}y)\right)\!.\nonumber \end{align}

It is easy to see that $d_L$ is a metric on $\mathcal{P}_1(\mathbb{S}_{d}^{+})$ ; moreover, $\big(\mathcal{P}_1(\mathbb{S}_{d}^{+}),d_L\big)$ is complete. Using well-known properties of Laplace transforms, it can be shown that convergence with respect to $d_{L}$ implies weak convergence. The next result provides an exponential rate in $d_{L}$ distance.

Theorem 2.4. Let $p_{t}(x,\text{d}\xi)$ be the transition kernel of a subcritical affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ . Suppose that (10) holds and denote by $\pi$ the unique invariant distribution. Then there exists a constant $C>0$ such that

(16) \begin{equation}d_{L}\left(p_{t}(x,\cdot),\pi\right)\leq C\left(1+\Vert x\Vert\right)\text{e}^{-\delta t},\quad t\geq0,\ \ x\in\mathbb{S}_{d}^{+}.\end{equation}

The proof of this result is given in Section 7. Although under the given conditions $p_t(x,\cdot)$ and $\pi$ do not necessarily belong to $\mathcal{P}_1(\mathbb{S}_d^+)$ , the proof of (16) reveals that $d_L(p_t(x,\cdot), \pi)$ is well-defined. Let us mention that the main purpose of this work is dedicated to Theorem 2.3. The above convergence rate in the $d_L$ distance is a simple byproduct of the estimates derived in the process of proving Theorem 2.3.

We turn to investigate the convergence rate from the affine transition kernel to the invariant distribution in the Wasserstein-1-distance introduced below. Given $\varrho,\thinspace\widetilde{\varrho}\in\mathcal{P}_{1}(\mathbb{S}_{d}^{+})$ , a coupling H of $(\varrho,\widetilde{\varrho})$ is a Borel probability measure on $\mathbb{S}_{d}^{+}\times\mathbb{S}_{d}^{+}$ which has marginals $\varrho$ and $\widetilde{\varrho}$ , respectively. We denote by $\mathcal{H}(\varrho,\widetilde{\varrho})$ the collection of all such couplings. We define the Wasserstein distance on $\mathcal{P}_{1}(\mathbb{S}_{d}^{+})$ by

\begin{equation*}W_{1}\left(\varrho,\widetilde{\varrho}\right)=\inf\left\lbrace \int_{\mathbb{S}_{d}^{+}\times\mathbb{S}_{d}^{+}}\Vert x-y\Vert H\left(\text{d}x,\text{d}y\right)\thinspace:\thinspace H\in\mathcal{H}\left(\varrho,\widetilde{\varrho}\right)\right\rbrace .\end{equation*}

Since $\varrho$ and $\widetilde{\varrho}$ belong to $\mathcal{P}_{1}(\mathbb{S}_{d}^{+})$ , it holds that $W_{1}(\varrho,\widetilde{\varrho})$ is finite. According to [Reference Villani48, Theorem 6.16], we have that $(\mathcal{P}_1(\mathbb{S}_{d}^{+}),W_{1})$ is a complete separable metric space. Note that, by an argument similar to (15), one easily finds that $d_{L}(\eta,\nu) \leq W_1(\eta,\nu)$ for all $\eta,\nu\in \mathcal{P}_1(\mathbb{S}_{d}^{+})$ . Exponential ergodicity in different Wasserstein distances for affine processes on the canonical state space $\mathbb{R}_{\geq0}^{m}\times\mathbb{R}^{n}$ was very recently studied in [Reference Friesen, Jin and Rüdiger19]. Below we provide a corresponding result for affine processes on $\mathbb{S}_{d}^{+}$ .

Theorem 2.5. Let $p_{t}(x,\text{d}\xi)$ be the transition kernel of a subcritical affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ satisfying (6). If $\alpha=0$ , then

(17) \begin{equation}W_{1}\left(p_{t}(x,\cdot),\pi\right)\leq\sqrt{d}Me^{-\delta t}\left(\|x\|+\int_{\mathbb{S}_{d}^{+}}\|y\|\pi(\text{d}y)\right)\!,\quad t\geq0,\ \ x\in\mathbb{S}_{d}^{+}.\end{equation}

The proof of Theorem 2.5, which is given in Section 8, largely follows some ideas of [Reference Friesen, Jin and Rüdiger19]. In contrast to the latter work, for the study of affine processes on $\mathbb{S}_{d}^{+}$ we encounter similar difficulties as in the proof of Theorem 2.3. Namely, the affine property allows us only to use the convolution argument (12) when $\alpha = 0$ . Moreover, since it is still an open problem whether each affine process on $\mathbb{S}_{d}^{+}$ can be obtained as a strong solution to a certain stochastic equation driven by Brownian motions and Poisson random measures, we are not able to improve our Theorem 2.5 to other variants of Wasserstein distances as used in [Reference Friesen, Jin and Rüdiger19].

3. Applications and examples

Let $(W_{t})_{t\geq0}$ be a $d\times d$ matrix of independent standard Brownian motions. Denote by $(J_{t})_{t\geq0}$ an $\mathbb{S}_{d}^{+}$ -valued Lévy subordinator with Lévy measure m. Suppose that these two processes are independent of each other. Following [Reference Mayerhofer, Pfaffel and Stelzer43], the stochastic differential equation

(18) \begin{equation}\begin{cases}\text{d}X_{t}=\left(b+\beta X_{t}+X_{t}\beta^{\top}\right)\text{d}t+\sqrt{X_{t}}\text{d}W_{t}\Sigma+\Sigma^{\top}\text{d}W_{t}^{\top}\sqrt{X_{t}}+\text{d}J_{t}, & t\geq0,\\[5pt] X_{0}=x\in\mathbb{S}_{d}^{+},\end{cases}\end{equation}

has a unique weak solution if $b\succeq(d-1)\Sigma^{\top}\Sigma$ and $\Sigma,\thinspace\beta$ are real-valued $d\times d$ matrices. Moreover, according to [Reference Mayerhofer, Pfaffel and Stelzer43, Corollary 3.2], if $b\succeq(d+1)\Sigma^{\top}\Sigma$ and $x \in \mathbb{S}_d^{++}$ , then there also exists an $\mathbb{S}_d^{++}$ -valued strong solution. The corresponding Markov process $X=(X_{t})_{t\geq0}$ is a conservative affine process with admissible parameters $(\alpha,b,B,m,0)$ with diffusion $\alpha=\Sigma^{\top}\Sigma$ and linear drift $B(x)=\beta x+x\beta^{\top}$ . The functions F and R are given by

\begin{equation*}F(u)=\langle b,u\rangle+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}(1-\text{e}^{-\langle u,\xi\rangle})\,m(\text{d}\xi)\end{equation*}

and

\begin{equation*}R(u)=-2u\alpha u+u\beta+\beta^{\top}u.\end{equation*}

The generalized Riccati equations are now given by

\begin{align*}\partial_{t}\phi(t,u) & =\langle b,\psi(t,u)\rangle+\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}(1-\text{e}^{-\langle\psi(t,u),\xi\rangle})\,m(\text{d}\xi),\\[4pt] \partial_{t}\psi(t,u) & =-2\psi(t,u)\alpha\psi(t,u)+\psi(t,u)\beta+\beta^{\top}\psi(t,u),\end{align*}

with initial conditions $\phi(0,u)=0$ and $\psi(0,u)=u$ . Let $\sigma_{t}^{\beta}\,:\,\mathbb{S}_{d}^{+}\to\mathbb{S}_{d}^{+}$ be given by

\begin{equation*}\sigma_{t}^{\beta}(x)\,:\!=2\int_{0}^{t}\text{e}^{\beta s}x\text{e}^{\beta^{\top}s}\text{d}s,\quad t\geq0.\end{equation*}

According to [Reference Mayerhofer42, Section 4.3], we have

\begin{align*}\phi(t,u) & =\langle b,\int_{0}^{t}\psi(s,u)\text{d}s \rangle +\int_{0}^{t}\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}(1-\text{e}^{-\langle\psi(s,u),\xi\rangle})\,m(\text{d}\xi)\text{d}s,\\[3pt] \psi(t,u) & =\text{e}^{\beta^{\top}t}(u^{-1}+\sigma_{t}^{\beta}(\alpha))^{-1}\text{e}^{\beta t}.\end{align*}

Since $\widetilde{B}(x)=B(x)$ , Remark 2.2 implies that X is subcritical, provided $\beta$ has only eigenvalues with negative real parts. If the Lévy measure m satisfies (10), then Theorem 2.3 implies existence, uniqueness, and convergence to the invariant distribution $\pi$ whose Laplace transform satisfies

\begin{equation*}\int_{0}^{\infty}\text{e}^{-\langle u,x\rangle}\pi\left(\text{d}x\right) =\langle b,\int_{0}^{\infty}\psi(s,u)\text{d}s\rangle \exp\bigg({-}\int_{0}^{\infty}\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}(1-\text{e}^{-\langle\psi(s,u),\xi\rangle})\,m(\text{d}\xi)\text{d}s\bigg).\end{equation*}

Moreover, if in addition $\int_{\lbrace\Vert\xi\Vert\geq1\rbrace}\Vert\xi\Vert m(\text{d}\xi)<\infty$ , then we infer from Corollary 2.1 that

\begin{equation*}\lim_{t\to\infty}\mathbb{E}_{x}\!\left[X_{t}\right]=\int_{0}^{\infty}\text{e}^{s\beta^{\top}}\left(b+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\xi m(\text{d}\xi)\right)\text{e}^{s\beta}\text{d}s=\int_{\mathbb{S}_{d}^{+}}y\pi(\text{d}y).\end{equation*}

We end this section by considering the following examples.

Example 3.1. (The matrix-variate basic affine jump-diffusion and Wishart process.) Take $b=2k\Sigma^{\top}\Sigma$ with $k\geq d-1$ in (18). This process is called matrix-variate basic affine jump-diffusion on $\mathbb{S}_{d}^{+}$ (MBAJD for short); see [Reference Mayerhofer42, Section 4]. Following [Reference Mayerhofer42, Section 4.3], $\phi(t,u)$ is precisely given by

\begin{equation*}\phi(t,u)=k\log\det\left(\mathbbm{1}+u\sigma_{t}^{\beta}(\alpha)\right)\int_{0}^{t}\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}\big(1-\text{e}^{-\langle\psi(s,u),\xi\rangle}\big)\,m(\text{d}\xi)\text{d}s,\end{equation*}

and Theorem 2.3 implies that the unique invariant distribution is given by

\begin{align*}\int_{0}^{\infty}\text{e}^{-\langle u,x\rangle}\pi\left(\text{d}x\right) & =\left(\det\left(\mathbbm{1}+\sigma_{\infty}^{\beta}(\alpha)u\right)\right)^{-k}\exp\!\left({-}\int_{0}^{\infty}\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}(1-\text{e}^{-\langle\psi(s,u),\xi\rangle})\,m(\text{d}\xi)\text{d}s\right)\!,\end{align*}

where $\sigma_{\infty}^{\beta}(\alpha)=\int_{0}^{\infty}\exp(s\beta)\alpha\exp(s\beta^{\top})\text{d}s$ .

The well-known Wishart process, introduced by Bru [Reference Bru10], is a special case of the MBAJD with $m=0$ . Existence of a unique distribution was then obtained in [Reference Alfonsi, Kebaier and Rey1, Lemma C.1]. In this case $\pi$ is a Wishart distribution with shape parameter k and scale parameter $\sigma_{\infty}^{\beta}(\alpha)$ .

Example 3.2. (Matrix-variate Ornstein–Uhlenbeck-type processes.) For $b=0$ and $\Sigma=0$ , we call the solutions to the stochastic differential equation (18) matrix-variate Ornstein–Uhlenbeck-type (OU-type) processes; see [Reference Barndorff-Nielsen and Stelzer7]. Properties of the stationary matrix-variate OU-type processes were investigated in [Reference Pigorsch and Stelzer45]. Provided that

\begin{equation*}\int_{\lbrace\Vert\xi\Vert\geq1\rbrace}\Vert\xi\Vert m(\text{d}\xi)<\infty,\end{equation*}

Theorem 2.5 implies that the matrix-variate OU-type process is also exponentially ergodic in the Wasserstein-1-distance.

Note that in [Reference Masuda41] ergodicity in total variation was studied for matrix-variate OU-type processes. The proof crucially relies on particular properties of Ornstein–Uhlenbeck processes and hence cannot be extended to cases where the parameter $\mu$ for state-dependent jumps does not vanish. In contrast, our method would also apply for non-vanishing $\mu$ .

4. Proof of Theorem 2.2

In this section we study the first moment of a conservative affine process on $\mathbb{S}_{d}^{+}$ . In particular, we prove Theorem 2.2. Essential to the proof is the space-differentiability of the functions F and R as well as of $\phi$ and $\psi$ . To simplify the notation we introduce $L(\mathbb{S}_{d},\mathbb{S}_{d})$ as the space of all linear operators $\mathbb{S}_{d}\to\mathbb{S}_{d}$ , and similarly $L(\mathbb{S}_{d},\mathbb{R})$ stands for the space of all linear functionals $\mathbb{S}_{d}\to\mathbb{R}$ . For a function $G\,:\,\mathbb{S}_{d}\to\mathbb{S}_{d}$ we denote its derivative at $u\in\mathbb{S}_{d}$ , if it exists, by $DG(u)\in L(\mathbb{S}_{d},\mathbb{S}_{d})$ . Similarly, we denote the derivative of $H\,:\,\mathbb{S}_{d}\to\mathbb{R}$ by $DH(u)\in L(\mathbb{S}_{d},\mathbb{R})$ . We equip $L(\mathbb{S}_{d},\mathbb{S}_{d})$ and $L(\mathbb{S}_{d},\mathbb{R})$ with the corresponding norms

\begin{equation*}\left\Vert DG(u)\right\Vert =\sup_{\Vert x\Vert=1}\left\Vert DG(u)(x)\right\Vert \quad\text{and}\quad\left\Vert DH(u)\right\Vert =\sup_{\Vert x\Vert=1}\left\Vert DH(u)(x)\right\Vert .\end{equation*}

Let F and R be as in Theorem 2.1 According to [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Lemma 5.1] the function R is analytic on $\mathbb{S}_{d}^{++}$ . Below we study the differentiability of F and R on the entire cone $\mathbb{S}_{d}^{+}$ .

We first give a lemma that slightly extends [Reference Mayerhofer42, Lemma 3.3].

Lemma 4.1. Let g be a measurable function on $\mathbb{S}_{d}^{+}$ with $\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\left\vert g(\xi)\right\vert \text{tr}(\mu)(\text{d}\xi)<\infty$ . Then $\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g(\xi)\mu(\text{d}\xi)$ is finite and

\begin{equation*}\left\Vert \int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g(\xi)\mu(\text{d}\xi)\right\Vert \le\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\left\vert g(\xi)\right\vert \text{tr}(\mu)(\text{d}\xi).\end{equation*}

Proof. Let $\mu=(\mu_{ij})$ and $\mu_{ij}=\mu_{ij}^{+}-\mu_{ij}^{-}$ be the Jordan decomposition of $\mu_{ij}$ . Suppose $\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\left\vert g(\xi)\right\vert \text{tr}(\mu)(\text{d}\xi)<\infty$ . Then [Reference Mayerhofer42, Lemma 3.3] implies that $\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}|g(\xi)|\mu(\text{d}\xi)$ is finite and

\begin{equation*}\left\Vert \int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}|g(\xi)|\mu(\text{d}\xi)\right\Vert \le\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\left\vert g(\xi)\right\vert \text{tr}(\mu)(\text{d}\xi).\end{equation*}

Since the ijth entry of $\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}|g(\xi)|\mu(\text{d}\xi)$ is given by

\begin{equation*}\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}|g(\xi)|\mu_{ij}^{+}(\text{d}\xi)-\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}|g(\xi)|\mu_{ij}^{-}(\text{d}\xi),\end{equation*}

which is finite, we must have

\begin{equation*}\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}|g(\xi)|\mu_{ij}^{+}(\text{d}\xi)<\infty\qquad\mbox{and\ensuremath{\qquad}}\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}|g(\xi)|\mu_{ij}^{-}(\text{d}\xi)<\infty,\quad\forall i,j\in\left\{ 1,\ldots d\right\} .\end{equation*}

So $\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g(\xi)\mu(\text{d}\xi)$ is finite. Again by [Reference Mayerhofer42, Lemma 3.3],

\begin{align*}\left\Vert \int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g(\xi)\mu(\text{d}\xi)\right\Vert & =\left\Vert \int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g^{+}(\xi)\mu(\text{d}\xi)-\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g^{-}(\xi)\mu(\text{d}\xi)\right\Vert \\[3pt] & \le\left\Vert \int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g^{+}(\xi)\mu(\text{d}\xi)\right\Vert +\left\Vert \int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g^{-}(\xi)\mu(\text{d}\xi)\right\Vert \\[3pt] & \le\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g^{+}(\xi)\text{tr}(\mu)(\text{d}\xi)+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}g^{-}(\xi)\text{tr}(\mu)(\text{d}\xi)\\[3pt] & \le\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\left\vert g(\xi)\right\vert \text{tr}(\mu)(\text{d}\xi).\end{align*}

The lemma is proved.

Proof. (a) Let $u\in\mathbb{S}_{d}^{++}$ . Consider $h\in\mathbb{S}_{d}$ with sufficiently small $\|h\|$ such that $u+h\in\mathbb{S}_{d}^{+}$ . An easy calculation shows that

\begin{equation*}R(u+h)-R(u)=DR(u)(h)+r(u,h),\end{equation*}

where

\begin{equation*}r(u,h)\,:\!=-2h\alpha h+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\text{e}^{-\langle u,\xi\rangle}\left(1-\text{e}^{-\langle h,\xi\rangle}-\langle h,\xi\rangle\right)\mu(\text{d}\xi).\end{equation*}

Let us prove that $\lim_{0\not=\Vert h\Vert\to0}\Vert r(u,h)\Vert/\Vert h\Vert=0$ . Assume $\Vert h\Vert\not=0$ . First, note that

\begin{equation*}\frac{\Vert2h\alpha h\Vert}{\Vert h\Vert}\leq2\Vert\alpha\Vert\frac{\Vert h\Vert^{2}}{\Vert h\Vert}\leq2\Vert\alpha\Vert\Vert h\Vert.\end{equation*}

Let $M>0$ . For $\Vert\xi\Vert\leq M$ , we have

(21) \begin{align}\left|\text{e}^{-\langle u,\xi\rangle}\left(1-\text{e}^{-\langle h,\xi\rangle}-\langle h,\xi\rangle\right)\right| & =\left|\langle h,\xi\rangle\left(\int_{0}^{1}\text{e}^{-\langle u+sh,\xi\rangle}\text{d}s-\text{e}^{-\langle u,\xi\rangle}\right)\right|\nonumber \\[4pt] & =\left|\langle h,\xi\rangle\right|\cdot\left|\int_{0}^{1}\left(\text{e}^{-\langle u+sh,\xi\rangle}-\text{e}^{-\langle u,\xi\rangle}\right)\text{d}s\right|\nonumber \\[4pt] & \le\left|\langle h,\xi\rangle\right|^{2},\end{align}

where we used the fact that $\langle u+sh,\xi\rangle\geq0$ and the Lipschitz continuity of $[0,\infty)\in x\mapsto\exp({-}x)$ to get the last inequality. Similarly, for $\Vert\xi\Vert>M$ ,

(22) \begin{equation}\left|\text{e}^{-\langle u,\xi\rangle}\left(1-\text{e}^{-\langle h,\xi\rangle}-\langle h,\xi\rangle\right)\right|\leq \left|\text{e}^{-\langle u,\xi\rangle}-\text{e}^{-\langle u+h,\xi\rangle}\right|+\left|\text{e}^{-\langle u,\xi\rangle}\langle h,\xi\rangle\right|\leq 2\left|\langle h,\xi\rangle\right|.\end{equation}

Combining (21) and (22) and applying Lemma 4.1, we get

\begin{align*}\frac{1}{\Vert h\Vert} & \left\Vert \int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\text{e}^{-\langle u,\xi\rangle}\left(1-\text{e}^{-\langle h,\xi\rangle}-\langle h,\xi\rangle\right)\mu(\text{d}\xi)\right\Vert \\[4pt] & \quad\leq\frac{1}{\Vert h\Vert}\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\left\vert \text{e}^{-\langle u,\xi\rangle}\left(1-\text{e}^{-\langle h,\xi\rangle}-\langle h,\xi\rangle\right)\right\vert \text{tr}(\mu)(\text{d}\xi)\\[4pt] & \quad\leq\Vert h\Vert\int_{\lbrace\Vert\xi\Vert\leq M\rbrace}\Vert\xi\Vert^{2}\text{tr}(\mu)(\text{d}\xi)+2\int_{\lbrace\Vert\xi\Vert>M\rbrace}\Vert\xi\Vert\text{tr}(\mu)(\text{d}\xi).\end{align*}

So

\begin{equation*}\frac{\Vert r(u,h)\Vert}{\Vert h\Vert}\leq\left(2\Vert\alpha\Vert+\int_{\lbrace\Vert\xi\Vert\leq M\rbrace}\Vert\xi\Vert^{2}\text{tr}(\mu)(\text{d}\xi)\right)\Vert h\Vert+2\int_{\lbrace\Vert\xi\Vert>M\rbrace}\Vert\xi\Vert\text{tr}(\mu)(\text{d}\xi).\end{equation*}

Note that $\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\Vert\xi\Vert\text{tr}(\mu)(\text{d}\xi)<\infty$ by virtue of Definition 2.1(iv). Let $\varepsilon>0$ be arbitrary and fix some $M=M(\varepsilon)>0$ large enough so that $\int_{\lbrace\Vert\xi\Vert>M\rbrace}\Vert\xi\Vert\text{tr}(\mu)(\text{d}\xi)<\varepsilon/4$ . Define

\begin{equation*}\delta=\delta(\varepsilon)\,:\!=\left(1+2\Vert\alpha\Vert+\int_{\lbrace\Vert\xi\Vert\leq M\rbrace}\Vert\xi\Vert^{2}\text{tr}(\mu)(\text{d}\xi)\right)^{-1}\frac{\varepsilon}{2}.\end{equation*}

Then, for $\Vert h\Vert\leq\delta$ , we see that

\begin{equation*}\frac{\Vert r(u,h)\Vert}{\Vert h\Vert}\leq\left(2\Vert\alpha\Vert+\int_{\lbrace\Vert\xi\Vert\leq M\rbrace}\Vert\xi\Vert^{2}\text{tr}(\mu)(\text{d}\xi)\right)\delta+\frac{\varepsilon}{2}\leq\varepsilon.\end{equation*}

This proves (19) for $u\in\mathbb{S}_{d}^{++}$ . Finally, the continuity of $u\mapsto$ DR(u) in $\mathbb{S}_{d}^{+}$ can be easily obtained from the dominated convergence theorem.

(b) Similarly as before, we derive $F(u+h)-F(u)=DF(u)(h)+r(u,h)$ with

\begin{equation*}r(u,h)\,:\!=\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\exp({-}\langle u,\xi\rangle)(1-\exp(\langle h,\xi\rangle)-\langle h,\xi\rangle)m(\text{d}\xi).\end{equation*}

Let $\Vert h\Vert\not=0$ . By essentially the same reasoning as in (a), we obtain that

\begin{equation*}\frac{\Vert r(u,h)\Vert}{\Vert h\Vert}\leq\Vert h\Vert\int_{\lbrace\Vert\xi\Vert\leq M\rbrace}\Vert\xi\Vert^{2}m(\text{d}\xi)+2\int_{\lbrace\Vert\xi\Vert>M\rbrace}\Vert\xi\Vert m(\text{d}\xi),\end{equation*}

and the second integral on the right-hand side is now finite by (6). Hence, we may follow the same steps as in (a) to see that $\Vert r(u,h)\Vert/\Vert h\Vert\to0$ as $\Vert h\Vert\to0$ and to obtain the continuity of DF(u) in $\mathbb{S}_{d}^{+}$ .

Let $\phi$ and $\psi$ be as in Theorem 2.1. We know from [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Lemma 3.2 (iii)] that $\phi(t,u)$ and $\psi(t,u)$ are jointly continuous on $\mathbb{R}_{\geq0}\times\mathbb{S}_{d}^{+}$ and, moreover, $u\mapsto\phi(t,u)$ and $u\mapsto\psi(t,u)$ are analytic on $\mathbb{S}_{d}^{++}$ for $t\geq0$ .

Proof. (a) Noting that $s\mapsto DR(\psi(s,u))\in L(\mathbb{S}_{d},\mathbb{S}_{d})$ is continuous, we may define $f_{u}(t)$ as the unique solution in $L(\mathbb{S}_{d},\mathbb{S}_{d})$ to

\begin{equation*}f_{u}(t)=\mathbbm{1}+\int_{0}^{t}DR\left(\psi(s,u)\right)f_{u}(s)\text{d}s.\end{equation*}

Further, we then define the extension of $D\psi$ onto $\mathbb{R}_{\geq0}\times\partial\mathbb{S}_{d}^{+}$ simply by

\begin{equation*}D\psi(t,u)=f_{u}(t),\quad(t,u)\in\mathbb{R}_{\geq0}\times\partial\mathbb{S}_{d}^{+}.\end{equation*}

It remains to verify the joint continuity of $D\psi(t,u)$ on $\mathbb{R}_{\geq0}\times\mathbb{S}_{d}^{+}$ extended in this way. By the Riccati differential equation (4) we have

\begin{equation*}D\psi(t,u)=\mathbbm{1}+\int_{0}^{t}DR\left(\psi(s,u)\right)D\psi(s,u)\text{d}s,\quad t\geq0,\thinspace u\in\mathbb{S}_{d}^{+}.\end{equation*}

Using that $u\mapsto R(u)$ is continuous on $\mathbb{S}_{d}^{+}$ and $\psi$ is jointly continuous on $\mathbb{R}_{\geq0}\times\mathbb{S}_{d}^{+}$ , we have that for all $T>0$ and $M>0$ , there exists a constant $C(T,M)>0$ such that

\begin{equation*}\sup_{s\in[0,T],\thinspace u\in\mathbb{S}_{d}^{+},\thinspace\Vert u\Vert\leq M}\left\Vert DR\left(\psi(s,u)\right)\right\Vert =\!:\,C(T,M)<\infty.\end{equation*}

Hence, for each $u\in\mathbb{S}_{d}^{+}$ with $\Vert u\Vert\leq M$ , we obtain

\begin{equation*}\left\Vert D\psi(t,u)\right\Vert \leq1+C(T,M)\int_{0}^{t}\Vert D\psi(s,u)\Vert\text{d}s.\end{equation*}

Applying Gronwall’s inequality yields

\begin{equation*}\left\Vert D\psi(t,u)\right\Vert \leq\text{e}^{C(T,M)T}=\!:\,K(T,M)<\infty\end{equation*}

for all $t\in[0,T]$ and $u\in\mathbb{S}_{d}^{+}$ with $\Vert u\Vert\leq M$ . Because $D\psi$ is jointly continuous in $\mathbb{R}_{\geq0}\times\mathbb{S}_{d}^{++}$ , it is enough to prove continuity at some fixed point $(t,u)\in\mathbb{R}_{\geq0}\times\partial\mathbb{S}_{d}^{+}$ , where $\partial \mathbb{S}_d^+:=\mathbb{S}_d^+ \backslash \mathbb{S}_d^{++}$ .

Without loss of generality we assume $t\in[0,T]$ and $u\in\partial\mathbb{S}_{d}^{+}$ with $\|u\|\le M$ . Let $s\in\mathbb{R}_{\geq0}$ and $v\in\mathbb{S}_{d}^{+}$ with $s\in[0,T]$ and $\Vert v\Vert\leq M$ . We have

(23) \begin{equation}\left\Vert D\psi(t,u)-D\psi(s,v)\right\Vert \leq\left\Vert D\psi(t,u)-D\psi(s,u)\right\Vert +\left\Vert D\psi(s,u)-D\psi(s,v)\right\Vert .\end{equation}

We estimate the first term on the right-hand side of (23) by

(24) \begin{align}\left\Vert D\psi(t,u)-D\psi(s,u)\right\Vert & \leq\left\Vert \int_{0}^{t}DR\left(\psi(r,u)\right)D\psi(r,u)\text{d}r-\int_{0}^{s}DR\left(\psi(r,u)\right)D\psi(r,u)\text{d}r\right\Vert \nonumber \\[3pt] & \leq C(T,M)\int_{[s,t]\cup[t,s]}\left\Vert D\psi(r,u)\right\Vert \text{d}r\nonumber \\[3pt] & \leq C(T,M)K(T,M)\vert t-s\vert.\end{align}

Turning to the second term, for $v\in\mathbb{S}_{d}^{++}$ with $\Vert v\Vert\leq M$ , $D\psi(s,u)=f_{u}(s)$ , and $D\psi(r,u)=f_{u}(r)$ , we obtain

\begin{align*}\left\Vert D\psi(s,u)-D\psi(s,v)\right\Vert & \leq\int_{0}^{s}\left\Vert DR\left(\psi(r,u)\right)D\psi(r,u)-DR\left(\psi(r,v)\right)D\psi(r,v)\right\Vert \text{d}r\\[3pt] & \leq\int_{0}^{s}\left\Vert DR(\psi(r,u))-DR\left(\psi(r,v)\right)\right\Vert \left\Vert D\psi(r,v)\right\Vert \text{d}r\\[3pt] & \quad+\int_{0}^{s}\left\Vert DR\left(\psi(r,u)\right)\right\Vert \left\Vert D\psi(r,u)-D\psi(r,v)\right\Vert \text{d}r\\[3pt] & \leq K(T,M)\int_{0}^{T}\left\Vert DR(\psi(r,u))-DR\left(\psi(r,v)\right)\right\Vert \text{d}r\\[3pt] & \quad+C(T,M)\int_{0}^{s}\left\Vert D\psi(r,u)-D\psi(r,v)\right\Vert \text{d}r\\[3pt] & =K(T,M)a_{T}(v,u)+C(T,M)\int_{0}^{s}\left\Vert D\psi(r,u)-D\psi(r,v)\right\Vert \text{d}r,\end{align*}

where $a_{T}(v,u)\,:\!=\int_{0}^{T}\Vert DR(\psi(r,u))-DR(\psi(r,v))\Vert\text{d}r$ . Using once again Gronwall’s inequality, we deduce

(25) \begin{equation}\left\Vert D\psi(s,u)-D\psi(s,v)\right\Vert \leq K(T,M)a_{T}(v,u)\text{e}^{C(T,M)T}.\end{equation}

Noting that $R\in C^{1}(\mathbb{S}_{d}^{+})$ and $\psi(r,0)=0$ by [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Remark 2.5], by the dominated convergence theorem we see that $a_{T}(v,u)$ tends to zero as $v\to u$ . Consequently, the right-hand side of (25) tends to zero as $v\to u$ . Combining (23) with (24) and (25), we conclude that $D\psi$ extended in this way is jointly continuous in $(t,u)\in\mathbb{R}_{\geq0}\times\mathbb{S}_{d}^{+}$ .

(b) We know from the generalized Riccati equation (3) that $\phi(t,u)=\int_{0}^{t}F(\psi(s,u))\text{d}s$ . Noting that $F\in C^{1}(\mathbb{S}_{d}^{+})$ thanks to (6), the chain rule combined with the dominated convergence theorem implies the assertion.

We are ready to prove Theorem 2.2.

Proof of Theorem 2.2. Let $\varepsilon>0$ . We have

\begin{align*}\frac{\partial}{\partial\varepsilon}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle\varepsilon u,\xi\rangle}p_{t}(x,\text{d}\xi) & =\frac{\partial}{\partial\varepsilon}\text{e}^{-\phi(t,\varepsilon u)-\langle x,\psi(t,\varepsilon u)\rangle}\\ & =-\left(D\phi(t,\varepsilon u)(u)+\langle x,D\psi(t,\varepsilon u)(u)\rangle\right)\text{e}^{-\phi(t,\varepsilon u)-\langle x,\psi(t,\varepsilon u)\rangle}\\ & \to-\left(D\phi(t,0)(u)+\langle x,D\psi(t,0)(u)\rangle\right)\quad\text{as }\varepsilon\to0,\end{align*}

where we used that the functions $D\phi$ and $D\psi$ have a jointly continuous extension on $\mathbb{R}_{\geq0}\times\mathbb{S}_{d}^{+}$ in accordance with Proposition 4.1. On the other hand, noting that

\begin{equation*}\left|\langle u,\xi\rangle\exp({-}\langle\varepsilon u,\xi\rangle)\right|\le\varepsilon^{-1}\text{e}^{-1}\end{equation*}

and applying the dominated convergence theorem, we get

\begin{equation*}\frac{\partial}{\partial\varepsilon}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle\varepsilon u,\xi\rangle}p_{t}(x,\text{d}\xi)=-\int_{\mathbb{S}_{d}^{+}}\langle u,\xi\rangle\text{e}^{-\langle\varepsilon u,\xi\rangle}p_{t}(x,\text{d}\xi)\to-\int_{\mathbb{S}_{d}^{+}}\langle u,\xi\rangle p_{t}(x,\text{d}\xi)\quad\text{as }\varepsilon\to0.\end{equation*}

Note that the limit on the right-hand side is finite. Indeed, using Fatou’s lemma, we obtain

\begin{equation*}\int_{\mathbb{S}_{d}^{+}}\langle u,\xi\rangle p_{t}(x,\text{d}\xi)\leq\liminf_{\varepsilon\to0}\int_{\mathbb{S}_{d}^{+}}\langle u,\xi\rangle\text{e}^{-\langle\varepsilon u,\xi\rangle}p_{t}(x,\text{d}\xi)=D\phi(t,0)(u)+\langle x,D\psi(t,0)(u)\rangle<\infty\end{equation*}

for all $u\in\mathbb{S}_{d}^{+}$ . So

(26) \begin{equation}\int_{\mathbb{S}_{d}^{+}}\langle u,\xi\rangle p_{t}(x,\text{d}\xi)=D\phi(t,0)(u)+\langle x,D\psi(t,0)(u)\rangle.\end{equation}

In what follows, we compute the derivatives $D\phi(t,0)$ and $D\psi(t,0)$ explicitly. By means of the generalized Riccati equation (4), we have

\begin{equation*}\psi(t,u)-u=\int_{0}^{t}R\left(\psi(s,u)\right)ds,\quad t\geq0,\thinspace u\in\mathbb{S}_{d}^{+}.\end{equation*}

According to Lemma 4.2 and Proposition 4.1 we are allowed to differentiate both sides of the latter equation with respect to $u\in\mathbb{S}_{d}^{+}$ and evaluate at $u=0$ ; thus, using the dominated convergence theorem,

\begin{equation*}\left.D\psi(t,u)\right|_{u=0}-\text{Id}=\int_{0}^{t}\left.DR\left(\psi(s,u)\right)D\psi(s,u)\right\vert _{u=0}\text{d}s,\quad t\geq0,\end{equation*}

where Id denotes the identity map on $\mathbb{S}_{d}^{+}$ . From [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Lemma 3.2(iii)] we know that $\psi(t,u)$ is continuous in $\mathbb{R}_{\geq0}\times\mathbb{S}_{d}^{+}$ , and noting that $\psi(s,0)=0$ (see [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Remark 2.5]), we get

\begin{equation*}D\psi(t,0)-\text{Id}=\int_{0}^{t}DR\left(0\right)D\psi(s,0)\text{d}s,\quad t\geq0.\end{equation*}

From this and the precise formula for $\phi(t,h)$ we deduce that

\begin{equation*}D\psi(t,0)=\text{e}^{tDR(0)}\quad\text{and}\quad\quad D\phi(t,0)=\int_{0}^{t}DF(0)\text{e}^{sDR(0)}\text{d}s.\end{equation*}

We use Lemma 4.2 to get that

\begin{equation*}DR(0)(u)=\widetilde{B}^{\top}(u)\quad\text{and}\quad DF(0)(u)=\langle b+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\xi m(\text{d}\xi),u\rangle.\end{equation*}

Finally, combining this with (26) yields

\begin{align*}\int_{\mathbb{S}_{d}^{+}}\langle u,\xi\rangle p_{t}(x,\text{d}\xi) & =\int_{0}^{t}\left(DF(0)\right)\text{e}^{sDR(0)}(u)\text{d}s+\langle x,\text{e}^{tDR(0)}(u)\rangle\\ & =\int_{0}^{t}\langle\text{e}^{s\widetilde{B}}\left(b+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\xi m(\text{d}\xi)\right)\!,u\rangle\text{d}s+\langle\text{e}^{t\widetilde{B}}x,u\rangle.\end{align*}

Since the equality holds for each $u\in\mathbb{S}_{d}^{+}$ , the assertion is proved.

5. Estimates on $\psi(t,u)$

We fix an admissible parameter set $(\alpha,b,B,m,\mu)$ and let $\psi$ be the unique solution to (4). In this section we study upper and lower bounds for $\psi$ . Let us start with an upper bound for $\psi(t,u)$ .

Propostion 5.1. Let $\psi$ be the unique solution to (4). Then

(27) \begin{equation}\left\Vert \psi\left(t,u\right)\right\Vert \leq M\Vert u\Vert\text{e}^{-t\delta},\quad t\geq0,\end{equation}

where M and $\delta$ are given by (9).

Proof. The proof is divided into three steps.

Step 1: Denote by $q_{t}(x,\text{d}\xi)$ the unique transition kernel of an affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m=0,\mu)$ ; that is, for each $u,\thinspace x\in\mathbb{S}_{d}^{+}$ , we have

(28) \begin{align}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}q_{t}(x,\text{d}\xi)=\exp\left({-}\int_{0}^{t}\langle b,\psi(s,u)\rangle\text{d}s-\langle x,\psi(t,u)\rangle\right)\!,\quad t\geq0.\end{align}

Applying Jensen’s inequality to the convex function $t\mapsto\exp({-}t)$ yields

(29) \begin{align}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}q_{t}(x,\text{d}\xi) & \geq\exp\!\left({-}\int_{\mathbb{S}_{d}^{+}}\langle u,\xi\rangle q_{t}(x,\text{d}\xi)\right) \notag\\ & =\exp\!\left({-}\int_{0}^{t}\langle\text{e}^{s\widetilde{B}}b,u\rangle\text{d}s-\langle\text{e}^{t\widetilde{B}}x,u\rangle\right)\!, \end{align}

where the last identity is a special case of Theorem 2.2. Using (28) we obtain

(30) \begin{equation}\langle x,\psi(t,u)\rangle+\int_{0}^{t}\langle b,\psi(t,u)\rangle\text{d}s\leq\langle\text{e}^{t\widetilde{B}}x,u\rangle+\int_{0}^{t}\langle\text{e}^{s\widetilde{B}}b,u\rangle\text{d}s,\quad\text{for all }u,\thinspace x\in\mathbb{S}_{d}^{+},\ t\ge 0.\end{equation}

Step 2: Let $\alpha \in \mathbb{S}_{d}^{+}$ be fixed. We claim that (30) holds not only for $b\succeq(d-1)\alpha$ but also for any $b\in\mathbb{S}_{d}^{+}$ . Aiming for a contradiction, suppose that there exist $t_{0}>0$ and $\xi,\thinspace x_{0},\thinspace u_{0}\in\mathbb{S}_{d}^{+}$ such that

\begin{equation*}I\,:\!=\langle x_{0},\psi(t_{0},u_{0})\rangle+\int_{0}^{t_{0}}\langle\xi,\psi(s,u_{0})\rangle\text{d}s-\langle x_{0},\text{e}^{t_{0}\widetilde{B}^{\top}}u_{0}\rangle-\int_{0}^{t_{0}}\langle\xi,\text{e}^{s\widetilde{B}^{\top}}u_{0}\rangle\text{d}s>0.\end{equation*}

We now take an arbitrary but fixed $b_{0}\succeq(d-1)\alpha$ . Noting that

\begin{equation*}\Delta\,:\!=\int_{0}^{t_{0}}\langle b_{0},\psi(s,u_{0})\rangle\text{d}s-\int_{0}^{t_{0}}\langle b_{0},\text{e}^{s\widetilde{B}^{\top}}u_{0}\rangle\text{d}s\end{equation*}

is finite, we find a constant $K>0$ large enough so that $KI+\Delta>0$ , i.e.,

(31) \begin{equation}\langle Kx_{0},\psi(t_{0},u_{0})\rangle+\int_{0}^{t_{0}}\langle b_{0}+K\xi,\psi(s,u_{0})\rangle\text{d}s>\langle Kx_{0},\text{e}^{t_{0}\widetilde{B}^{\top}}u_{0}\rangle+\int_{0}^{t_{0}}\langle b_{0}+K\xi,\text{e}^{s\widetilde{B}^{\top}}u_{0}\rangle\text{d}s.\end{equation}

Now, since $b_{0}+K\xi\succeq(d-1)\alpha$ , we see that (31) contradicts (30) if we choose $b=b_{0}+K\xi$ , $x=Kx_{0}$ , $u=u_{0}$ , and $t=t_{0}$ . Hence (30) holds for all $b\in\mathbb{S}_{d}^{+}$ .

Step 3: According to Step 2, we are allowed to choose $b=0$ in (30), which implies

\begin{equation*}\langle x,\psi(t,u)\rangle\leq\langle x,\text{e}^{t\widetilde{B}^{\top}}u\rangle\end{equation*}

for all $t\geq0$ and $x,\thinspace u\in\mathbb{S}_{d}^{+}$ . This completes the proof.

We continue with a lower bound for $\psi(t,u)$ .

Proposition 5.2. Let $\psi$ be the unique solution to (4) and suppose that $\alpha=0$ and (14) is satisfied. Then, for each $u,\thinspace\xi\in\mathbb{S}_{d}^{+}$ ,

(32) \begin{equation}\langle\xi,\psi(t,u)\rangle\geq\text{e}^{-Kt}\langle\xi,u\rangle,\quad t\geq0.\end{equation}

Proof. Fix $u\in\mathbb{S}_{d}^{+}$ and define $W_{t}(u)\,:\!=\psi(t,u)-\exp({-}Kt)u$ . Using that $\exp({-}Kt)u=\psi(t,u)-W_{t}(u)$ we obtain

\begin{align*}\frac{\partial W_{t}(u)}{\partial t}=R(\psi(t,u))+K\psi(t,u)-KW_{t}(u).\end{align*}

Since $W_{0}(u)=0$ , the latter implies

\begin{equation*}W_{t}(u)=\int_{0}^{t}\text{e}^{-K(t-s)}\left(K\psi(s,u)+R(\psi(s,u))\right)\text{d}s.\end{equation*}

Fix $\xi\in\mathbb{S}_{d}^{+}$ ; then

(33) \begin{align}\langle\xi,W_{t}(u)\rangle=\int_{0}^{t}\text{e}^{-K(t-s)}\left(K\langle\xi,\psi(s,u)\rangle+\langle\xi,R(\psi(s,u))\rangle\right)\text{d}s.\end{align}

In the following we estimate the integrand. For this, we write $\langle\xi,R(\psi(s,u))\rangle=I_{1}+I_{2}$ , where

\begin{equation*}I_{1}=\langle\xi,B{}^{\top}\left(\psi(s,u)\right)\rangle\quad\text{and}\quad I_{2}=-\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}\big(\text{e}^{-\langle\psi(s,u),\zeta\rangle}-1\big)\langle\xi,\mu\left(\text{d}\zeta\right)\rangle,\end{equation*}

and estimate $I_{1}$ and $I_{2}$ separately. For $I_{1}$ , by (14) we get

\begin{equation*}I_{1}=\langle B(\xi),\psi(s,u)\rangle\geq-K\langle\xi,\psi(s,u)\rangle,\end{equation*}

where we used the self-duality of the cone $\mathbb{S}_d^+$ (see [Reference Horn and Johnson27, Theorem 7.5.4]). Turning to $I_{2}$ , we simply have

\begin{align*}I_{2} & =\int_{\mathbb{S}_{d}^{+}\backslash\{0\}}\left(1-\text{e}^{-\langle\psi(s,u),\zeta\rangle}\right)\langle\xi,\mu\left(\text{d}\zeta\right)\rangle\geq0.\end{align*}

Collecting now the estimates for $I_{1}$ and $I_{2}$ , we see that

\begin{equation*}\left(K\langle\xi,\psi(s,u)\rangle+\langle\xi,R(\psi(s,u))\rangle\right)\geq0,\end{equation*}

and thus $\langle\xi,W_{t}(u)\rangle\geq0$ by (33). This proves the assertion.

6. Proofs of the main results

In this section we will prove Theorem 2.2, Proposition 2.1, and Corollary 2.1. Let $p_{t}(x,\text{d}\xi)$ be the transition kernel of a subcritical affine process on $\mathbb{S}_{d}^{+}$ with admissible parameters $(\alpha,b,B,m,\mu)$ , and let $\delta>0$ be given by (9).

We note that $F(u)\geq0$ for all $u\in\mathbb{S}_{d}^{+}$ . Based on the estimates on $\psi(t,u)$ that we derived in the previous section, we easily obtain the following lemma.

Lemma 6.1. Suppose that (10) holds. Then there exists a constant $C>0$ such that

(34) \begin{equation}F(\psi(s,u))\leq C\Vert u\Vert\text{e}^{-s\delta},\quad s\geq0,\ \ u\in\mathbb{S}_{d}^{+}.\end{equation}

Consequently,

(35) \begin{align}\int_{0}^{\infty}F(\psi(s,u))\text{d}s\leq\frac{C}{\delta}\|u\|,\quad u\in\mathbb{S}_{d}^{+}.\end{align}

Proof. We know that

\begin{align*}F(\psi(s,u)) & =\langle b,\psi(s,u)\rangle+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\big(1-\text{e}^{-\langle\psi(s,u),\xi\rangle}\big)\,m(\text{d}\xi)\\ & =\!:\,\langle b,\psi(s,u)\rangle+I(u).\end{align*}

Now, first note that, by (27),

(36) \begin{equation}\langle b,\psi(s,u)\rangle\leq\Vert b\Vert\Vert\psi(s,u)\Vert\leq\Vert b\Vert\Vert u\Vert\text{e}^{-s\delta}.\end{equation}

We turn to the estimate of I(u). Once again using (27), we obtain

\begin{align*}I(u) & =\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\big(1-\text{e}^{-\langle\psi(s,u),\xi\rangle}\big)\,m(\text{d}\xi)\\ & \leq\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\min\left\lbrace 1,\langle\psi(s,u),\xi\rangle\right\rbrace m(\text{d}\xi)\\ & \leq\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\min\left\lbrace 1,\Vert\xi\Vert\Vert u\Vert\text{e}^{-s\delta}\right\rbrace m(\text{d}\xi).\end{align*}

For all $a\geq0$ it holds that $1\wedge a\leq\log(2)^{-1}\log(1+a)$ ; hence

\begin{align*}I(u) & \leq\frac{1}{\log(2)}\int_{\lbrace\Vert\xi\Vert\leq1\rbrace}\Vert\xi\Vert\Vert u\Vert\text{e}^{-s\delta}m(\text{d}\xi)+\frac{1}{\log(2)}\int_{\lbrace\Vert\xi\Vert>1\rbrace}\log\left(1+\Vert\xi\Vert\Vert u\Vert\text{e}^{-s\delta}\right)m(\text{d}\xi)\\ & =\!:\,J_{1}(u)+J_{2}(u).\end{align*}

Let $C>0$ be a generic constant which may vary from line to line. Since $m(\text{d}\xi)$ integrates $\Vert\xi\Vert\mathbbm{1}_{\lbrace\Vert\xi\Vert\leq1\rbrace}$ by definition, we have

\begin{equation*}J_{1}(u)\leq C\Vert u\Vert\text{e}^{-s\delta}.\end{equation*}

Moreover, noting that $m(\text{d}\xi)$ integrates $\log\Vert\xi\Vert\mathbbm{1}_{\lbrace\Vert\xi\Vert>1\rbrace}$ by assumption, for $J_{2}(u)$ we use the elementary inequality (see [Reference Friesen, Jin and Rüdiger19, Lemma 8.5])

\begin{align*}\log(1+a\cdot c) & \leq C\min\left\lbrace \log(1+a),\log(1+c)\right\rbrace +C\log(1+a)\log(1+c)\\ & \leq C\log(1+a)+Ca\log(1+c)\\ & \leq Ca\left(1+\log(1+c)\right)\end{align*}

for $a=\Vert u\Vert\exp({-}s\delta)$ and $c=\Vert\xi\Vert$ to get

\begin{equation*}J_{2}(u)\leq C\Vert u\Vert\text{e}^{-s\delta}\int_{\lbrace\Vert\xi\Vert>1\rbrace}\left(1+\log\left(1+\Vert\xi\Vert\right)\right)m(\text{d}\xi)\leq C\Vert u\Vert\text{e}^{-s\delta}.\end{equation*}

Combining the estimates for $J_{1}(u)$ and $J_{2}(u)$ yields

(37) \begin{equation}I(u)=J_{1}(u)+J_{2}(u)\leq C\Vert u\Vert\text{e}^{-s\delta}.\end{equation}

So, by (36) and (37), we have (34), which proves the assertion.

We are now able to prove Theorem 2.3.

Proof of Theorem 2.3. Fix $x\in\mathbb{S}_{d}^{+}$ . By means of Proposition 5.1, we see that

\begin{align*}\lim_{t\to\infty}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}p_{t}(x,\text{d}\xi) & =\lim_{t\to\infty}\exp\!\left({-}\int_{0}^{t}F(\psi(s,u))\text{d}s-\langle x,\psi(t,u)\rangle\right)\\[3pt] & =\exp\!\left({-}\int_{0}^{\infty}F(\psi(s,u))\text{d}s\right)\!,\end{align*}

and the limit on the right-hand side is finite by Lemma 6. Clearly, by (35), we also have that $u\mapsto\int_{0}^{\infty}F(\psi(s,u)))\text{d}s$ is continuous at $u=0$ . Now, Lévy’s continuity theorem (cf. [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Lemma 4.5]) implies that $p_{t}(x,\cdot)\rightarrow\pi$ weakly as $t\to\infty$ . Moreover, $\pi$ has Laplace transform (11). It remains to verify that $\pi$ is the unique invariant distribution.

Invariance. Fix $u\in\mathbb{S}_{d}^{+}$ and let $t\geq0$ be arbitrary. Then

\begin{align*}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}\left(\int_{\mathbb{S}_{d}^{+}}p_{t}(x,\text{d}\xi)\pi(\text{d}x)\right) & =\int_{\mathbb{S}_{d}^{+}}\left(\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}p_{t}(x,\text{d}\xi)\right)\pi(\text{d}x)\\[3pt] & =\text{e}^{-\phi(t,u)}\int_{\mathbb{S}_{d}^{+}}\exp\!\left({-}\langle x,\psi(t,u)\rangle\right)\pi(\text{d}x).\end{align*}

Note that $\psi$ satisfies the semi-flow equation by [Reference Cuchiero, Filipović, Mayerhofer and Teichmann12, Lemma 3.2]; that is, $\psi(t+s,u)=\psi\left(s,\psi(t,u)\right)$ for all $t,\thinspace s\geq0$ . Using that the Laplace transform of $\pi$ is given by (11), for each $u\in\mathbb{S}_{d}^{+}$ we obtain

\begin{align*}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}\left(\int_{\mathbb{S}_{d}^{+}}p_{t}(x,\text{d}\xi)\pi(\text{d}x)\right) & =\text{e}^{-\phi(t,u)}\exp\!\left({-}\int_{0}^{\infty}F\left(\psi\left(s,\psi(t,u)\right)\right)\text{d}s\right)\\[3pt] & =\text{e}^{-\phi(t,u)}\exp\!\left({-}\int_{0}^{\infty}F\left(\psi(t+s,u)\right)\text{d}s\right)\\[3pt] & =\text{e}^{-\phi(t,u)}\exp\!\left({-}\int_{t}^{\infty}F\left(\psi(s,u)\right)\text{ds}\right)\\[3pt] & =\exp\!\left({-}\int_{0}^{\infty}F\left(\psi(s,u)\right)\text{d}s\right)\\[3pt] & =\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle x,u\rangle}\pi\left(\text{d}x\right)\!.\end{align*}

Consequently, $\pi$ is invariant.

Uniqueness. Let $\pi'$ be another invariant distribution. For fixed $u\in\mathbb{S}_{d}^{+}$ and $t\geq0$ we have

\begin{align*}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle x,u\rangle}\pi'(\text{d}x) & =\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}\left(\int_{\mathbb{S}_{d}^{+}}p_{t}(x,\text{d}\xi)\pi'(\text{d}x)\right)\\[3pt] & =\int_{\mathbb{S}_{d}^{+}}\exp\!\left({-}\phi(t,u)-\langle x,\psi(t,u)\rangle\right)\pi'(\text{d}x).\end{align*}

Letting $t\to\infty$ shows that $\pi'$ also satisfies (11). By uniqueness of the Laplace transforms, it holds that $\pi'=\pi$ .

Proof of Proposition 2.1. Let $x\in\mathbb{S}_{d}^{+}$ and $\pi\in\mathcal{P}(\mathbb{S}_{d}^{+})$ be such that $p_{t}(x,\cdot)\to\pi$ weakly as $t\to\infty$ . It follows that

\begin{equation*}\lim_{t\to\infty}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}p_{t}(x,\text{d}\xi)=\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,y\rangle}\pi(\text{d}y),\quad u\in\mathbb{S}_{d}^{+},\end{equation*}

and we obtain from (2) that

\begin{equation*}\lim_{t\to\infty}\exp\!\left({-}\int_{0}^{t}F(\psi(s,u))\text{d}s\right)=\lim_{t\to\infty}\text{e}^{\langle x,\psi(t,u)\rangle}\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}p_{t}(x,\text{d}\xi)=\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,y\rangle}\pi(\text{d}y).\end{equation*}

In particular, this implies

\begin{align*}\int_{0}^{\infty}F(\psi(s,u))\text{d}s<\infty,\quad u\in\mathbb{S}_{d}^{+}.\end{align*}

Fix $u\in\mathbb{S}_{d}^{++}$ . Assume that $\alpha=0$ and (14) holds. By definition of F we have $F(u)\geq\int_{\mathbb{S}_{d}^{+}}(1-\exp({-}\langle u,\xi\rangle))m(\text{d}\xi)$ and thereby

\begin{align*}F(\psi(s,u))\geq\int_{\lbrace\langle\xi,u\rangle>1\rbrace}\left(1-\text{e}^{-\text{e}^{-Ks}\langle\xi,u\rangle}\right)m(\text{d}\xi),\end{align*}

where we used (32). Integrating over $[0,\infty)$ and using a change of variable $r\,:\!=\exp({-}Ks)\langle\xi,u\rangle$ with $\text{d}s=-1/K\cdot\text{d}r/r$ yields

\begin{align*}\int_{0}^{\infty}F(\psi(s,u))\text{d}s & \geq\frac{1}{K}\int_{\lbrace\langle\xi,u\rangle>1\rbrace}\int_{0}^{\langle\xi,u\rangle}\frac{1-\text{e}^{-r}}{r}\text{d}rm(\text{d}\xi)\\[3pt] & \geq\frac{1}{K}\int_{\lbrace\langle\xi,u\rangle>1\rbrace}\int_{1}^{\langle\xi,u\rangle}\frac{1-\text{e}^{-r}}{r}\text{d}rm(\text{d}\xi)\\[3pt] & \geq\frac{1-\text{e}^{-1}}{K}\int_{\lbrace\langle\xi,u\rangle>1\rbrace}\log(\langle\xi,u\rangle)m(\text{d}\xi),\end{align*}

where we used in the last inequality that $1-\exp({-}r)\geq1-\exp({-}1)>0$ for $r\ge1$ . This leads to the estimate

\begin{equation*}\int_{\lbrace\langle\xi,u\rangle>1\rbrace}\log(\langle\xi,u\rangle)m(\text{d}\xi)\leq\frac{K}{1-\text{e}^{-1}}\int_{0}^{\infty}F(\psi(s,u))\text{d}s<\infty.\end{equation*}

Letting $u=\mathbbm{1}\in\mathbb{S}_{d}^{++}$ gives $\langle\xi,\mathbbm{1}\rangle=\text{tr}(\xi)\geq\Vert\xi\Vert$ , so that

\begin{equation*}\int_{\lbrace\Vert\xi\Vert>1\rbrace}\log\left(\Vert\xi\Vert\right)m(\text{d}\xi)\leq\int_{\lbrace\langle\xi,\mathbbm{1}\rangle>1\rbrace}\log\left(\langle\xi,\mathbbm{1}\rangle\right)m(\text{d}\xi)<\infty.\end{equation*}

This completes the proof.

Proof of Corollary 2.1. Using that $\Vert\exp(t\widetilde{B})\Vert\leq M\exp({-}\delta t)$ , where $\delta$ is given by (9), we have

\begin{align*}\lim_{t\to\infty}\int_{\mathbb{S}_{d}^{+}}yp_{t}(x,\text{d}y) & =\int_{0}^{\infty}\text{e}^{s\widetilde{B}}\left(b+\int_{\mathbb{S}_{d}^{+}}\xi m(\text{d}\xi)\right)\text{d}s\in\mathbb{S}_{d}^{+}.\end{align*}

By direct computation we find that

\begin{equation*} \int_{0}^{\infty}\text{e}^{s\widetilde{B}}\left(b+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\xi m(\text{d}\xi)\right)\text{d}s = - \widetilde{B}^{-1}\left( b + \int_{\mathbb{S}_d^+ \backslash \{0\}} \xi m(\text{d}\xi) \right)\!,\end{equation*}

and hence it suffices to prove that $\lim_{t\to\infty}\int_{\mathbb{S}_{d}^{+}}yp_{t}(x,\text{d}y)=\int_{\mathbb{S}_{d}^{+}}y\pi(\text{d}y)$ . To do so, we can proceed as in the proof of Theorem 2.2. Indeed, by Lemma A.1, we estimate

\begin{equation*}\sup_{t\geq0}\int_{\mathbb{S}_{d}^{+}}\Vert y\Vert p_{t}(x,\text{d}y)\leq\sup_{t\geq0}\text{tr}\left(\int_{\mathbb{S}_{d}^{+}}yp_{t}(x,\text{d}y)\right)\leq\sqrt{d}\sup_{t\geq0}\left\Vert \int_{\mathbb{S}_{d}^{+}}yp_{t}(x,\text{d}y)\right\Vert <\infty.\end{equation*}

Therefore, applying Fatou’s lemma yields

\begin{equation*}\int_{\mathbb{S}_{d}^{+}}\Vert y\Vert\pi(\text{d}y)\leq\sup_{t\geq0}\int_{\mathbb{S}_{d}^{+}}\Vert y\Vert p_{t}(x,\text{d}y)<\infty.\end{equation*}

So $\pi\in\mathcal{P}_{1}(\mathbb{S}_{d}^{+})$ . Now, let $\varepsilon>0$ . By the dominated convergence theorem, we see that

\begin{equation*}\lim_{\varepsilon\searrow0}\int_{\mathbb{S}_{d}^{+}}\frac{1-\text{e}^{-\langle\varepsilon u,y\rangle}}{\varepsilon}\pi(\text{d}y)=\int_{\mathbb{S}_{d}^{+}}\langle u,y\rangle\pi(\text{d}y).\end{equation*}

Moreover, noting that, by Proposition 5.1,

\begin{equation*}1-\text{e}^{-\langle\psi(s,\varepsilon u),\xi\rangle}\leq\langle\psi(s,\varepsilon u),\xi\rangle\leq\Vert\xi\Vert\Vert\varepsilon u\Vert\text{e}^{-\delta t},\end{equation*}

we can use once again the dominated convergence theorem to obtain

\begin{align*}\lim_{\varepsilon\searrow0}\int_{\mathbb{S}_{d}^{+}}\frac{1-\text{e}^{-\langle\varepsilon u,y\rangle}}{\varepsilon}\pi(\text{d}y) & =\lim_{\varepsilon\searrow0}\frac{1}{\varepsilon}\int_{0}^{\infty}F(\psi(s,\varepsilon u))\text{d}s\\[4pt] & =\lim_{\varepsilon\searrow0}\int_{0}^{\infty}\left(\langle b,\frac{\psi(s,\varepsilon u)}{\varepsilon}\rangle\text{d}s+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\frac{1-\text{e}^{-\langle\psi(s,\varepsilon u),\xi\rangle}}{\varepsilon}m(\text{d}\xi)\right)\text{d}s\\[4pt] & =\int_{0}^{\infty}\langle b,D\psi(s,0)(u)\rangle\text{d}s+\int_{0}^{\infty}\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\langle D\psi(s,0)(u),\xi\rangle m(\text{d}\xi)\text{d}s\\[4pt] & =\int_{0}^{\infty}\langle\text{e}^{s\widetilde{B}}\left(b+\int_{\mathbb{S}_{d}^{+}\backslash\lbrace0\rbrace}\xi m(\text{d}\xi)\right)\!,u\rangle\text{d}s,\end{align*}

where we used that $D\psi(s,0)(u)=\exp(s\widetilde{B}^{\top})u$ (see the proof of Theorem 2.2). Since the latter identity holds for all $u\in\mathbb{S}_{d}^{+}$ , this concludes the proof.

7. Proof of Theorem 2.4

Proof of Theorem 2.4. Suppose that (10) holds. By definition of $d_{L}$ , we have

(38) \begin{align} & d_{L}\left(p_{t}(x,\text{d}\xi),\pi(\text{d}\xi)\right)\nonumber \\ & =\sup_{u\in\mathbb{S}_{d}^{+}\setminus\{0\}}\frac{1}{\Vert u\Vert}\left\vert \int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}p_{t}(x,\text{d}\xi)-\int_{\mathbb{S}_{d}^{+}}\text{e}^{-\langle u,\xi\rangle}\pi(\text{d}\xi)\right\vert \nonumber \\ & =\sup_{u\in\mathbb{S}_{d}^{+}\setminus\{0\}}\frac{1}{\Vert u\Vert}\left\vert \exp\!\left({-}\int_{0}^{t}F\left(\psi(s,u)\right)\text{d}s-\langle x,\psi(t,u)\rangle\right)-\exp\!\left({-}\int_{0}^{\infty}F\left(\psi(s,u)\right)\text{d}s\right)\right\vert .\end{align}

Let $C>0$ be a generic constant that may vary from line to line. Then, using (34), for each $t\geq0$ we have

\begin{align*} & \left\vert \exp\!\left({-}\int_{0}^{t}F\left(\psi(s,u)\right)\text{d}s-\langle x,\psi(t,u)\rangle\right)-\exp\!\left({-}\int_{0}^{\infty}F\left(\psi(s,u)\right)\text{d}s\right)\right\vert \\ & \qquad\qquad\qquad\leq\left\vert \exp\!\left({-}\langle x\psi(t,u)\rangle\right)-1\right\vert \cdot\left|\exp\!\left({-}\int_{0}^{t}F\left(\psi(s,u)\right)\text{d}s\right)\right\vert \\ & \qquad\qquad\qquad\quad+\left\vert \exp\!\left({-}\int_{0}^{t}F\left(\psi(s,u)\right)\text{d}s\right)-\exp\!\left({-}\int_{0}^{\infty}F\left(\psi(s,u)\right)\text{d}s\right)\right\vert \\ & \qquad\qquad\qquad\leq\left\vert \langle x,\psi(s,u)\rangle\right|+\left|\int_{t}^{\infty}F\left(\psi(s,u)\right)\text{d}s\right\vert \\ & \qquad\qquad\qquad\leq M\Vert x\Vert\Vert u\Vert\text{e}^{-t\delta}+C\Vert u\Vert\int_{t}^{\infty}\text{e}^{-s\delta}\text{d}s\\ & \qquad\qquad\qquad\leq C\left(1+\Vert x\Vert\right)\Vert u\Vert\text{e}^{-t\delta},\end{align*}

which when plugged back into (38) implies (16).

8. Proof of Theorem 2.5

Proof of Theorem 2.5. Note that $\pi\in\mathcal{P}_{1}(\mathbb{S}_{d}^{+})$ by Corollary 2.1. Let $q_{t}(x,\text{d}\xi)$ be a transition kernel for the conservative, subcritical affine process with admissible parameters $(\alpha=0,b=0,B,m=0,\mu)$ . Using the particular form of the Laplace transform for $p_{t}(x,\cdot)$ (see (2)), it is not difficult to see that $p_{t}(x,\cdot)=q_{t}(x,\cdot)\ast p_{t}(0,\cdot)$ , where ‘ $\ast$ ’ denotes the convolution of measures. Let H be any coupling with marginals $\delta_{x}$ and $\pi$ , i.e., $H\in\mathcal{H}(\delta_{x},\pi)$ . Using the invariance of $\pi$ , together with the convexity of $W_{1}$ (see [Reference Villani48, Theorem 4.8] and [Reference Friesen, Jin, Kremer and Rüdiger18, Lemma 2.3]), we find that

\begin{align*}W_{1}\left(p_{t}(x,\cdot),\pi\right) & =W_{1}\left(\int_{\mathbb{S}_{d}^{+}}p_{t}(y,\cdot)\delta_{x}(\text{d}y),\int_{\mathbb{S}_{d}^{+}}p_{t}(y',\cdot)\pi(\text{d}y')\right)\\[3pt] & \leq\int_{\mathbb{S}_{d}^{+}\times\mathbb{S}_{d}^{+}}W_{1}\left(p_{t}(y,\cdot),p_{t}(y',\cdot)\right)H(\text{d}y,\text{d}y')\\[3pt] & \leq\int_{\mathbb{S}_{d}^{+}\times\mathbb{S}_{d}^{+}}W_{1}\left(q_{t}(y,\cdot),q_{t}(y',\cdot)\right)H(\text{d}y,\text{d}y').\end{align*}

The integrand can now be estimated as follows:

\begin{align*}W_{1}\left(q_{t}(y,\cdot),q_{t}(y',\cdot)\right) & \leq\int_{\mathbb{S}_{d}^{+}\times\mathbb{S}_{d}^{+}}\|z-z'\|G(\text{d}z,\text{d}z')\\[3pt] & \leq\int_{\mathbb{S}_{d}^{+}}\|z\|q_{t}(y,\text{d}z)+\int_{\mathbb{S}_{d}^{+}}\|z'\|q_{t}(y',\text{d}z')\\[3pt] & \leq M\sqrt{d}\text{e}^{-t\delta}\left(\|y\|+\|y'\|\right)\!,\end{align*}

where G is any coupling of $(q_{t}(y,\cdot),q_{t}(y',\cdot))$ and we have used Lemma A.1 to obtain

\begin{align*}\int_{\mathbb{S}_{d}^{+}}\Vert z\Vert q_{t}(y,\text{d}z) & \leq\text{tr}\left(\int_{\mathbb{S}_{d}^{+}}zq_{t}(y,\text{d}z)\right)\\[3pt] & =\text{tr}\left(\text{e}^{t\widetilde{B}}y\right)\\[3pt] & \leq\sqrt{d}\left\Vert \text{e}^{t\widetilde{B}}y\right\Vert \\[3pt] & \leq M\sqrt{d}\text{e}^{-t\delta}\Vert y\Vert.\end{align*}

Combining these estimates, we obtain

\begin{align*}W_{1}(p_{t}(x,\cdot),\pi) & \leq M\sqrt{d}\text{e}^{-t\delta}\int_{\mathbb{S}_{d}^{+}\times\mathbb{S}_{d}^{+}}\left(\|y\|+\|y'\|\right)H(\text{d}y,\text{d}y')\\ & \leq M\sqrt{d}\text{e}^{-t\delta}\left(\|x\|+\int_{\mathbb{S}_{d}^{+}}\Vert y\Vert\pi(\text{d}y)\right)\!,\end{align*}

which yields (17).

9. Extension to affine processes on finite-dimensional convex cones

In this section we outline how our main result, Theorem 2.3, can be extended to the more general framework of affine processes on convex cones. Below we briefly introduce the necessary concepts, while additional details, proofs, and references can be found in [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13].

Let V be a finite-dimensional Euclidean space with scalar product $\langle\cdot,\cdot\rangle$ and associated norm $\|\cdot\|$ . Let $K\subset V$ be a closed convex cone and suppose that it is proper, i.e. $K\cap({-}K)=\{0\}$ , and generating, i.e. $V=K-K$ . Note that its closed dual cone

\begin{equation*}K^{*}=\{u\in V\ |\ \langle x,u\rangle\geq0,\ \ \forall x\in K\}\end{equation*}

is then also generating and proper. The partial order on V is, for $u,v\in V$ , defined by $u\preceq v\Longleftrightarrow v-u\in K^{*}$ . The following is the definition of affine transition semigroups due to [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Definition 2.1].

Note that this definition allows non-conservative affine transition kernels. However, having in mind that we investigate existence, uniqueness, and convergence of $p_{t}(x,\cdot)$ to the invariant distribution, we restrict our study to conservative kernels; that is, $p_{t}(x,K)=1$ for all $t\geq0$ and $x\in K$ . The following is a summary of the main results obtained in [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13] on the existence and structure of conservative affine transition kernels. We emphasize that these results hold in a more general setting that allows killing and explosion of the process; see [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13] for more details.

Note that, for a given parameter set $(Q,b,B,m,\mu)$ , existence of an affine transition kernel is only shown for the case $Q=0$ , while conservativeness is a consequence of the first moment condition (41) (see [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, p. 373] and follow the argument given in [Reference Duffie, Filipović and Schachermayer16, Section 9]).

Assuming, in addition, that K is an irreducible symmetric cone (see [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Definition 2.6]), there exists a conservative affine transition semigroup for any parameter set $(Q,b,B,m,\mu)$ fulfilling the above conditions (i)–(v) and (41); see [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Theorem 2.19]. Moreover, for dimensions larger than 2 it has been shown that the jumps are of finite variation and the drift satisfies $b\succeq4^{-1}d(r-1)Q(e,e)$ , where e denotes the identity element for the multiplication on V, and r denotes the rank and d the Peirce invariant of V; see [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Theorem 2.13] and [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Appendix] for additional details. Note that if we let $V=S_{d}$ and $K=S_{d}^{+}$ , we obtain the case of positive semidefinite matrices as a particular case of these results.

Below we provide an extension of Theorem 2.3 for a conservative affine transition kernel on a proper closed convex cone K, which is generating. Since we do not suppose that this cone is symmetric, existence of such a kernel is a priori not clear and should be established separately.

Theorem 9.2. Suppose that there exists an affine transition kernel $p_{t}(x,\text{d}\xi)$ with parameters $(Q,b,B,m,\mu)$ satisfying (41) (which makes the kernel $p_{t}$ necessarily conservative). Define $\widetilde{B}:V\longrightarrow V$ by

\begin{equation*}\widetilde{B}(u)\,:\!=B(u)+\int_{K\backslash\{0\}}\mathbbm{1}_{\{\|\xi\|>1\}}\xi\langle u,\mu(\text{d}\xi)\rangle,\qquad u\in V,\end{equation*}

and suppose that $\sigma(\widetilde{B})\subset\{\lambda\in\mathbb{C}\ |\ \text{Re}(\lambda)<0\}$ . Moreover, assume that

\begin{equation*}\int_{K\backslash\{0\}}\mathbbm{1}_{\{\|\xi\|>1\}}\log(\|\xi\|)m(\text{d}\xi)<\infty.\end{equation*}

Then the conservative affine transition kernel $p_{t}(x,\text{d}\xi)$ has a unique invariant distribution $\pi$ . Moreover, for each $x\in K$ , one has $p_{t}(x,\cdot)\longrightarrow\pi$ weakly as $t\to\infty$ , and $\pi$ has Laplace transform

\begin{equation*}\int_{K}\text{e}^{-\langle u,\xi\rangle}\pi(\text{d}\xi)=\exp\!\left({-}\int_{0}^{\infty}F(\psi(s,u))\text{d}s\right)\!,\qquad u\in K^{*},\end{equation*}

where the latter integral is absolutely convergent.

Proof. We follow the proof of our main result, Theorem 2.3.

Step 1. Let us prove that there exist constants $M\geq1$ and $\delta>0$ such that

(42) \begin{align}\|\psi(t,u)\|\leq M\|u\|\text{e}^{-\delta t},\qquad t\geq0,\ \ u\in K^{*}.\end{align}

Indeed, one could check that Theorem 2.2 also holds in this situation (provided that m has finite first moment). One may think of deducing the assertion by arguments similar to those for Proposition 5.1. Unfortunately, the latter is built on a convolution argument, and it requires the existence of a conservative affine transition kernel with parameters $(Q,b,B,m=0,\mu)$ , which is currently not known. So we provide below an alternative and direct proof which avoids the convolution trick.

First observe that, for $u\in K^{*}$ , it holds that

\begin{align*}R(u)=-\frac{1}{2}Q(u,u)+\widetilde{B}^{\top}(u)-\int_{K\backslash\{0\}}\left(\text{e}^{-\langle u,x\rangle}-1+\langle\xi,u\rangle\right)\mu(\text{d}\xi)\preceq\widetilde{B}^{\top}(u).\end{align*}

Let $y(t,u)=e^{t\widetilde{B}^{\top}}(u)$ be the unique solution to

\begin{equation*}\partial_{t}y(t,u)=\widetilde{B}^{\top}(y(t,u)),\ \ y(0,u)=u\in K^{*}.\end{equation*}

Note that R is quasi-monotone increasing on $K^{*}$ by [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Proposition 3.12]. Then

\begin{equation*}0=\partial_{t}\psi(t,u)-R(\psi(t,u))=\partial_{t}y(t,u)-\widetilde{B}^{\top}(y(t,u))\preceq\partial_{t}y(t,u)-R(y(t,u)),\end{equation*}

and hence by Volkmann’s comparison theorem for ordinary differential equations (see [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Theorem 3.13]) we conclude that

\begin{equation*}\psi(t,u)\preceq y(t,u)=\text{e}^{t\widetilde{B}^{\top}}(u);\end{equation*}

that is,

\begin{equation*} \langle\psi(t,u),x\rangle\leq\langle \text{e}^{t\widetilde{B}^{\top}}(u),x\rangle,\quad x\in K.\end{equation*}

Now we can repeat the argument in [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, p. 387]: since $K^{*}$ is a finite-dimensional proper closed convex cone, it is normal; i.e., there exists a constant $\gamma_{K^{*}}$ such that for $x,y\in K^{*}$ ,

\begin{equation*}0\preceq x\preceq y\implies\|x\|\le\gamma_{K^{*}}\|y\|.\end{equation*}

We thus have

\begin{equation*}\|\psi(t,u)\|\le\gamma_{K^{*}}\left\Vert \text{e}^{t\widetilde{B}^{\top}}(u)\right\Vert ,\quad u\in K^{*}.\end{equation*}

The property (42) is now an immediate consequence of the assumption that the spectrum of $\widetilde{B}$ is contained in the left half-plane of $\mathbb{C}$ .

Step 2. Following exactly the same arguments as in Lemma 6.1, we find a constant $C>0$ such that

\begin{equation*}\int_{0}^{\infty}F(\psi(t,u))\text{d}t\leq C\|u\|,\qquad u\in K^{*}.\end{equation*}

Step 3. Using [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Proposition 3.1(i)–(ii)] combined with a version of Lévy’s continuity theorem [Reference Cuchiero, Keller-Ressel, Mayerhofer and Teichmann13, Lemma 3.7], the assertion can be deduced by literally the same arguments as given in the proof of Theorem 2.3.

Appendix A. Matrix Calculus

For a $d\times d$ square matrix x, recall that $\text{tr}(x)=\sum_{i=1}^{d}x_{ii}$ . The Frobenius norm of x is given by $\Vert x\Vert=\text{tr}(xx)^{1/2}=(\sum_{i,j=1}^{d}\vert x_{ij}\vert^{2})^{1/2}$ . Let us recollect one property of this norm.

Lemma A.1 Let $x\in\mathbb{S}_{d}^{+}$ ; then

\begin{equation*}\Vert x\Vert\leq\text{tr}(x)\leq\sqrt{d}\Vert x\Vert.\end{equation*}

Proof. Write $x=u^{\top}\kappa u$ , where u is orthogonal and $\kappa$ is diagonal with its entries given by $\lambda_{i}(x)$ , $i = 1,\ldots,d$ , the eigenvalues of x. We have

\begin{equation*}\Vert x\Vert^{2}=\text{tr}\left(u^{\top}\kappa^{2}u\right)=\sum_{i=1}^{d}\lambda_{i}(x)^{2}.\end{equation*}

Since $x\in\mathbb{S}_{d}^{+}$ , it holds that $\lambda_{i}(x)\geq0$ , $i=1,\ldots,d$ . Then

\begin{equation*}\left\Vert x\right\Vert =\left(\sum_{i=1}^{d}\lambda_{i}^{2}(x)\right)^{1/2}\leq\sum_{i=1}^{d}\lambda_{i}(x)\leq\sqrt{d}\left(\sum_{i=1}^{d}\lambda_{i}^{2}(x)\right)^{1/2}=\sqrt{d}\Vert x\Vert.\end{equation*}