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\begin{document}




\noindent{\small Published in: {\it The Annals of Probability}
\textbf{34} (2006), 3: 1103--1142}


\vskip1.0cm

\centerline{\large\textbf{SKEW CONVOLUTION SEMIGROUPS AND}}

\smallskip

\centerline{\large\textbf{AFFINE MARKOV PROCESSES}}

\bigskip

\centerline{By D.\,A.~Dawson\footnote{Supported by an NSERC
Research Grant.} and Zenghu Li\footnote{Supported by NSFC and NCET
Research Grants.}}

\medskip

\centerline{\it Carleton University and Beijing Normal University}

\bigskip


{\narrower{\narrower


A general affine Markov semigroup is formulated as the convolution
of a homogeneous one with a skew convolution semigroup. We provide
some sufficient conditions for the regularities of the homogeneous
affine semigroup and the skew convolution semigroup. The
corresponding affine Markov process is constructed as the strong
solution of a system of stochastic equations with non-Lipschitz
coefficients and Poisson-type integrals over some random sets.
Based on this characterization, it is proved that the affine
process arises naturally in a limit theorem for the difference of
a pair of reactant processes in a catalytic branching system with
immigration.

\smallskip

\noindent\textit{AMS 2000 subject classifications.} Primary 60J35;
secondary 60J80, 60H20, 60K37.

\smallskip

\noindent\textit{Key words and phrases.} Skew convolution
semigroup, affine process, continuous state branching process,
catalytic branching process, immigration, Ornstein-Uhlenbeck
process, stochastic integral equation, Poisson random measure.

\smallskip

\noindent\textbf{Abbreviated Title:} Affine Markov processes


\par}\par}


%%%%%%%%%%%%%%%%%%%% (Section 1) %%%%%%%%%%%%%%%%%%%%%


\section{Introduction}

\setcounter{equation}{0}

The concept of \textit{affine processes} unifies a wide class of
Markov processes including Ornstein-Uhlenbeck processes
(OU-processes) and continuous state branching processes with
immigration (CBI-processes). Those processes involve rich common
mathematical structures and the unified treatment of them develops
interesting connections between several areas in the theory of
probability. The ``affine property'' is roughly that the logarithm
of the characteristic function of the transition semigroup is
given by an affine transformation of the initial state $x \mapsto
\<x,\psi(t,u)\> + \phi(t,u)$; see Section~3. An important special
case is where the affine transformation is homogeneous, that is,
it only contains a non-trivial linear part $\<x,\psi(t,u)\>$. In
this case, we refer to the affine semigroup as
\textit{homogeneous}. A general affine semigroup can be
constructed as the convolution of a homogeneous one with an
associated \textit{skew convolution semigroup}, which corresponds
to the constant term $\phi(t,u)$ and gives the one-dimensional
distributions of the affine process started with the trivial
initial state. A complete characterization of general
finite-dimensional affine processes was recently given by Duffie
\textit{et al.}\ (2003) under a regularity assumption, which
requires that the coefficients $\psi(t,u)$ and $\phi(t,u)$ are
both differentiable at $t=0$. Based on this characterization, they
discussed a wide range of applications of affine processes in
mathematical finance.

The problems of characterizing different particular classes of
affine processes have also been studied by some other authors. In
particular, Watanabe (1969) gave a complete description of regular
two-dimensional continuous state branching processes. He also
proved that the regularity property of such processes is implied
by the stochastic continuity. A similar characterization of
finite-dimensional continuous state branching processes was given
in Rhyzhov and Skorokhod (1970). The same problem for
measure-valued branching processes was investigated in Dynkin
\textit{et al.}\ (1994). In those cases, the processes are defined
by homogeneous affine semigroups. On the other hand, a complete
characterization for stochastically continuous one-dimensional
CBI-processes was given in Kawazu and Watanabe (1971); see also
Shiga and Watanabe (1973). In the setting of measure-valued
processes, Li (1996a) gave a formulation of immigration structures
in terms of skew convolution semigroups. It was proved in Li
(1996a) that the skew convolution semigroups associated with a
measure-valued branching process are in 1-1 correspondence with a
class of infinitely divisible probability entrance laws; see also
Li (1996b, 1998). A construction of trajectories of the
corresponding immigration processes was given in Li (2002) by
summing up measure-valued paths in some Kuznetsov processes. Skew
convolution semigroups and OU-processes on real separable Hilbert
spaces were studied in Bogachev \textit{et al.}\ (1996), Dawson
and Li (2004), Dawson \textit{et al.}\ (2004), Fuhrman and
R\"ockner (2000), van Neerven (2000) and Schmuland and Sun (2001).
Roughly speaking, a skew convolution semigroup is regular if and
only if it is determined by a closable entrance law. For both the
Hilbert spaces and the spaces of measures, a stochastically
continuous skew convolution semigroup can be irregular. A number
of such examples arising in applications were discussed in Dawson
and Li (2004), Dawson \textit{et al.}\ (2004) and Li (1996b,
1998).


The main purpose of this paper is to investigate the basic
characterizations and regularities of affine Markov semigroups and
processes. For simplicity of the presentation we shall confine
ourselves to the non-degenerate two-dimensional case, but most of
the arguments can be generalized to higher dimensions. {From} the
results of Duffie \textit{et al.}\ (2003) we know that the
constant part in the affine structure is usually smoother than the
linear part. Therefore, we discuss \textit{separately} the
regularities of homogeneous affine semigroups and those of skew
convolution semigroups. It turns out that a skew convolution
semigroup always consists of infinitely divisible probability
measures. We prove that such a semigroup is regular if and only if
the linear part of the logarithm of its characteristic function is
absolutely continuous. Some sufficient conditions for the
regularities of homogeneous affine semigroups and skew convolution
semigroups are given in terms of their first moments. Those
results give a partial solution of the problem of characterizing
all affine semigroups without regularity assumption; see Duffie
\textit{et al.}\ (2003, Remark~2.11). We then give a construction
of the affine process as the strong solution of a system of
stochastic integral equations with random and non-Lipschitz
coefficients and jumps of Poisson type selected from some random
sets. A similar equation system is used to construct a class of
catalytic CBI-processes. The concept of \textit{catalytic
branching processes} was first introduced by Dawson and
Fleischmann (1997) in the setting of measure-valued diffusions;
see, e.g., Dawson \textit{et al.}\ (2002) and Dawson and
Fleischmann (2000) for some related developments. As an
application of the characterization by stochastic equations, we
show that an affine process arises naturally in a limit theorem
for the difference of a pair of reactant processes in a catalytic
CBI-process. This result is of interest since it seems that the
connection between affine processes and catalytic branching
processes has not been noticed before. The studies of those two
classes of processes have been undergoing rapid developments in
recent years with different motivations. The interplay between
them provides new motivations for both sides and might stimulate
some further studies on related topics.



\noindent\textbf{Notation.} Let $\mbb{R}_+ = [0, \infty)$ and
$\mbb{R}_- = (-\infty, 0]$. Let $\lambda$ denote the Lebesgue
measure on $\mbb{R}$. For a Borel measure $\nu$ and a Borel
function $f$ on $E \subseteq \mbb{R}$ or $\mbb{R}^2$, we write
$\nu(f)$ for $\int_E f d\nu$ if the integral exists. Write
$\Delta_\xi f(x) = f(x+\xi) - f(x)$ if the right-hand side is
well-defined. Let $\hat\nu$ denote the \textit{characteristic
function} of $\nu$ defined by
 \begin{eqnarray*}
\hat\nu(u)
 :=
\int_{E} \exp\{\<u,\xi\>\} \nu(d\xi), \qquad u\in U,
 \end{eqnarray*}
where $U \subseteq \mbb{C}$ or $\mbb{C}^2$ is to be specified. For
$x\in \mbb{R}$ set $l_1(x) = |x|$ and $l_{12}(x) = |x| \land
|x|^2$. Let
 \begin{eqnarray*}
\chi(x) =\left\{\begin{array}{ll}
x   \ar\mbox{ if $x \in [-1,1]$},  \\
1   \ar\mbox{ if $x \in (1,\infty)$},  \\
-1  \ar\mbox{ if $x \in (-\infty,1)$}.
\end{array}\right.
 \end{eqnarray*}
For $x=(x_1,x_2) \in \mbb{R}^2$ define $\chi(x) = (\chi(x_1),
\chi(x_2))$. We make the convention that
 \begin{eqnarray*}
\int_r^t = - \int_t^r = \int_{(r,t]}
 \qquad\mbox{and}\qquad
\int_r^\infty = \int_{(r,\infty)}
 \end{eqnarray*}
for $r\le t \in \mbb{R}$.








%%%%%%%%%%%%%%%%%%%% (Section 2) %%%%%%%%%%%%%%%%%%%%%


\section{Homogeneous affine semigroups}

\setcounter{equation}{0}

In this section, we give the definition and prove some simple
properties of homogeneous affine semigroups. Let $D = \mbb{R}_+
\times \mbb{R}$ and $U = \mbb{C}_- \times (i\mbb{R})$, where
$\mbb{C}_- = \{a+ib: a\in \mbb{R}_-, b\in \mbb{R}\}$ and $i\mbb{R}
= \{ib: b\in \mbb{R}\}$. Note that the word ``homogeneous'' in the
following definition has a meaning different from the one of
``time-homogeneous''.

\bdefinition\label{d2.1} A transition semigroup $(Q(t))_{t\ge0}$
with state space $D$ is called a \textit{homogeneous affine
semigroup (HA-semigroup)} if for each $t\ge0$ there exists a
continuous two-dimensional complex function $\psi(t,\cdot) :=
(\psi_1(t,\cdot), \psi_2(t,\cdot))$ on $U$ such that
 \begin{eqnarray}\label{2.1}
\int_D \exp\{\<u,\xi\>\} Q(t,x,d\xi)
 =
\exp\{\<x,\psi(t,u)\>\}, \qquad x \in D, u \in U.
 \end{eqnarray}
The HA-semigroup $(Q(t))_{t\ge0}$ given by (\ref{2.1}) is called
\textit{regular} if it is stochastically continuous and the
derivative $\psi^\prime_t (0,u)$ exists for all $u\in U$ and is
continuous at $u=0$. \edefinition


\bproposition\label{p2.1} Let $(Q(t))_{t\ge0}$ be a HA-semigroup
defined by (\ref{2.1}). Then $\psi(t,u) \in U$ and
 \begin{eqnarray}\label{2.2}
\psi(r+t,u) = \psi(r,\psi(t,u)), \qquad r,t\ge0, u \in U.
 \end{eqnarray}
Moreover, $\psi_2(t,u)$ has the form
 \begin{eqnarray}\label{2.3}
\psi_2(t,u)
 =
\beta_{22}(t)u_2, \qquad t\ge0, u\in U,
 \end{eqnarray}
where $\beta_{22}(\cdot)$ is a function on $[0,\infty)$ satisfying
 \begin{eqnarray}\label{2.4}
\beta_{22}(r+t) = \beta_{22}(r)\beta_{22}(t), \qquad r,t\ge0.
 \end{eqnarray}
\eproposition

\proof {From} (\ref{2.1}) we see that $\<x,\psi(t,u)\> \in
\mbb{C}_-$ for all $x\in D$. This implies that $\psi(t,u) \in U$
for all $u\in U$. For any $x_2\in \mbb{R}$ we have
 \begin{eqnarray*}
Q(t,(0,x_2),\cdot)*Q(t,(0,-x_2),\cdot) = \delta_{(0,0)}.
 \end{eqnarray*}
Then $Q(t,(0,x_2),\cdot)$ must be degenerate. Let $Q(t, (0,x_2),
\cdot) = \delta_{\beta(t,x_2)}$ for $\beta(t,x_2) \in D$. It
follows that
 \begin{eqnarray}\label{2.5}
\exp\{x_2\psi_2(t,u)\}
 =
\int_{D} \exp\{\<u,\xi\>\} Q(t,(0,x_2),d\xi)
 =
\exp\{\<u,\beta(t,x_2)\>\}.
 \end{eqnarray}
Then $x_2\psi_2(t,u) = \<u,\beta(t,x_2)\>$ and so $\beta(t,x_2) =
\beta(t,1)x_2$. Since $\beta(t,x_2) \in D$ for all $x_2 \in
\mbb{R}$, we must have $\beta_1(t,1) =0$ and hence $\psi_2(t,u) =
\beta_2(t,1)u_2$ for all $u\in U$. That is, (\ref{2.3}) holds with
$\beta_{22}(t) = \beta_2(t,1)$. The relation (\ref{2.2}) follows
from (\ref{2.1}) and the Chapman-Kolmogorov equation for
$(Q(t))_{t\ge0}$. By (\ref{2.2}) and (\ref{2.3}) we obtain
 \begin{eqnarray*}
\beta_{22}(r+t)u_2 = \psi_2(r,\psi(t,u)) = \beta_{22}(r)
\psi_2(t,u) = \beta_{22}(r)\beta_{22}(t)u_2,
 \end{eqnarray*}
which implies (\ref{2.4}).  \qed



\bcorollary\label{c2.1} If (\ref{2.1}) defines a stochastically
continuous HA-semigroup $(Q(t))_{t\ge0}$, then there is a constant
$\beta_{22} \in \mbb{R}$ such that $\beta_{22}(t) =
\exp\{\beta_{22}t\}$ for all $t\ge0$. \ecorollary


\bproposition\label{p2.2} Let $(Q(t))_{t\ge0}$ be a HA-semigroup
defined by (\ref{2.1}). Then $\psi_1(t,u)$ has the representation
 \begin{eqnarray}\label{2.6}
\psi_1(t,u) = \beta_{11}(t)u_1 + \beta_{12}(t)u_2 + \alpha(t)u_2^2
+ \int_D (e^{\<u,\xi\>} - 1 - u_2\chi(\xi_2)) \mu(t,d\xi),
 \end{eqnarray}
where $\alpha(t) \in \mbb{R}_+$, $(\beta_{11}(t), \beta_{12}(t))
\in D$ and $\mu(t,d\xi)$ is a $\sigma$-finite measure on $D$
supported by $D \setminus \{0\}$ such that
 \begin{eqnarray*}
\int_D [\chi(\xi_1) + \chi^2(\xi_2)] \mu(t,d\xi) < \infty.
 \end{eqnarray*}
Moreover, for any $r,t\ge0$ we have
 \begin{eqnarray}
\beta_{11}(r+t) \ar=\ar \beta_{11}(r)\beta_{11}(t), \label{2.7} \\
\beta_{12}(r+t) \ar=\ar \beta_{11}(r)\beta_{12}(t) + \beta_{12}(r)
\beta_{22}(t)  \nnm \\
 \ar \ar + \int_D[Q(t)\chi_2(\xi) - \beta_{22}(t)\chi(\xi_2)]
\mu(r,d\xi), \label{2.8} \\
\alpha(r+t) \ar=\ar \beta_{11}(r)\alpha(t) +
\alpha(r)\beta_{22}^2(t),
\label{2.9} \\
\mu(r+t,\cdot) \ar=\ar \int_D \mu(r,d\xi)Q(t,\xi,\cdot) +
\beta_{11}(r)\mu(t,\cdot), \label{2.10}
 \end{eqnarray}
where
 \begin{eqnarray}\label{2.11}
Q(t)\chi_2(\xi) = \int_D \chi(\eta_2)Q(t,\xi,d\eta)
 \end{eqnarray}
and $\beta_{22}(t)$ is given by Proposition~\ref{p2.1}.
\eproposition

\proof In view of (\ref{2.1}) it is easy to see that each
$Q(t,(x_1,0),\cdot)$ is an infinitely divisible probability
measure on $D$. Then (\ref{2.6}) follows by the special structure
of $D$ and the L\'evy-Khintchine representation for the
characteristic function of an infinitely divisible distribution;
see, e.g., Laha and Rohatgi (1979, pp.499-500). {From} (\ref{2.6})
and the results of Proposition~\ref{p2.1} we get
 \begin{eqnarray*}
\psi_1(r+t,u)
 \ar=\ar
\beta_{11}(r)\psi_1(t,u) + \beta_{12}(r)\beta_{22}(t)u_2 +
\alpha(r)\beta_{22}^2(t)u_2^2   \\
 \ar \ar
+ \int_D [e^{\<\psi(t,u),\xi\>} - 1 - \beta_{22}(t)u_2
\chi(\xi_2)] \mu(r,d\xi)  \\
 \ar=\ar
\beta_{11}(r)\beta_{11}(t)u_1 + \beta_{11}(r)\beta_{12}(t)u_2 \\
 \ar \ar
+\, \beta_{11}(r)\alpha(t)u_2^2 + \beta_{12}(r)\beta_{22}(t)u_2  \\
 \ar \ar
+\, \alpha(r)\beta_{22}^2(t)u_2^2 + \beta_{11}(r)\int_D
[e^{\<u,\xi\>} - 1 - u_2\chi(\xi_2)] \mu(t,d\xi)  \\
 \ar \ar
+ \int_D [e^{\<\psi(t,u),\xi\>} - 1 - u_2Q(t)\chi_2(\xi)]
\mu(r,d\xi)  \\
 \ar \ar
+\, u_2\int_D [Q(t)\chi_2(\xi) - \beta_{22}(t) \chi(\xi_2)]
\mu(r,d\xi).
 \end{eqnarray*}
Then (\ref{2.7}) -- (\ref{2.10}) follow by a comparison of the
above expression with (\ref{2.6}). \qed



\blemma\label{l2.1} If (\ref{2.1}) defines a stochastically
continuous HA-semigroup and $\psi_1(t,u)$ is given by (\ref{2.6}),
then $t \mapsto \beta_{12}(t)$ is continuous. \elemma

\proof The proof of Duffie \textit{et al.}\ (2003, Lemma~3.1)
shows that $\psi(t,u)$ is jointly continuous in $(t,u)$. Let
$\mu_2 (t,d\xi_2)$ denote the projection of $\mu(t,d\xi)$ to
$\mbb{R}$. Then
 \begin{eqnarray*}
\psi_1(t,(0,iz))
 =
i\beta_{12}(t)z - \alpha(t)z^2 + \int_\mbb{R} (e^{iz\xi_2} - 1 -
iz\chi(\xi_2)) \mu_2(t,d\xi_2)
 \end{eqnarray*}
is a continuous function of $(t,z) \in [0,\infty) \times \mbb{R}$.
It is not hard to find universal constants $0<c_1<c_2<\infty$ so
that
 \begin{eqnarray}\label{2.12}
c_1\le \Big(1 - \frac{\sin\xi_2} {\xi_2}\Big)\chi(\xi_2)^{-2} \le
c_2
 \end{eqnarray}
for all $\xi_2\in\mbb{R} \setminus \{0\}$. Then for each $t\ge0$
we can define the Borel measure $G(t,d\xi_2)$ on $\mbb{R}$ by
setting $G(t,\{0\}) = \alpha(t)/3$ and
 \begin{eqnarray*}
G(t,d\xi_2)
 =
\Big(1 - \frac{\sin\xi_2} {\xi_2}\Big) \mu_2(t,d\xi_2), \qquad
\xi_2 \in \mbb{R} \setminus \{0\}.
 \end{eqnarray*}
It follows that
 \begin{eqnarray}\label{2.13}
\psi_1(t,(0,iz))
 =
i\beta_{12}(t)z + \int_{\mbb{R}} \big(e^{iz\xi_2} - 1 -
iz\chi(\xi_2)\big)\Big(1 - \frac{\sin\xi_2} {\xi_2}\Big)^{-1}
G(t,d\xi_2)
 \end{eqnarray}
where the integrand is defined at $\xi_2=0$ by continuity as
$-3z^2$. By dominated convergence,
 \begin{eqnarray*}
t \mapsto v(t,\lambda) := 2\psi_1(t,(0,i\lambda)) -
\int_{\lambda-1}^{\lambda+1} \psi_1(t,(0,iz))dz
 \end{eqnarray*}
is continuous for each $\lambda \in \mbb{R}$. One may check easily
that $v(t,\lambda)$ is the characteristic function of
$G(t,d\xi_2)$. Then L\'evy's continuity theorem implies that $t
\mapsto G(t,d\xi_2)$ is continuous by weak convergence. For any
fixed $z \in \mbb{R}$, the integrand in (\ref{2.13}) is bounded
and continuous in $\xi_2$, so the integral term is continuous in
$t\ge0$. By the continuity of $\psi_1(t, (0,iz))$ we find that $t
\mapsto \beta_{12}(t)$ is continuous. \qed


\bproposition\label{p2.3} Suppose that (\ref{2.1}) defines a
stochastically continuous HA-semigroup and $\psi_1(t,u)$ is given
by (\ref{2.6}). Then
 \begin{eqnarray}\label{2.14}
B(t) := \beta_{11}(t) + \beta_{12}^2(t) + \alpha(t) + \int_D
[\chi(\xi_1) + \chi^2(\xi_2)] \mu(t,d\xi)
 \end{eqnarray}
is a locally bounded function of $t\ge0$. \eproposition


\proof Let $\mu_1(t,d\xi_1)$ denote the projection of
$\mu(t,d\xi)$ to $\mbb{R}_+$. Then
 \begin{eqnarray}\label{2.15}
\psi_1(t,(-z,0))
 =
- \beta_{11}(t)z + \int_0^\infty (e^{-z\xi_1} - 1) \mu_1(t,d\xi_1)
 \end{eqnarray}
is continuous in $(t,z) \in [0,\infty) \times \mbb{R}_+$. In
particular, (\ref{2.15}) is locally bounded in $t\ge0$ for any
fixed $z \in \mbb{R}_+$. Taking $\lambda=1$ one finds that
 \begin{eqnarray*}
\beta_{11}(t) + \int_0^\infty \chi(\xi_1) \mu_1(t,d\xi_1)
 \end{eqnarray*}
is locally bounded in $t\ge 0$. By the proof of Lemma~\ref{l2.1},
 \begin{eqnarray*}
v(t,0) = \frac{1}{3}\alpha(t) + \int_\mbb{R} \Big(1 -
\frac{\sin\xi_2}{\xi_2}\Big)\mu_2(t,d\xi_2)
 \end{eqnarray*}
is continuous and hence locally bounded in $t\ge0$. In view of
(\ref{2.12}) we find that
 \begin{eqnarray}\label{2.16}
\alpha(t) + \int_\mbb{R} \chi^2(\xi_2) \mu_2(t,d\xi_2)
 \end{eqnarray}
is also locally bounded in $t\ge0$. By Lemma~\ref{l2.1},
$\beta_{12}(t)$ is locally bounded in $t\ge0$. Then we have the
desired result. \qed



\bproposition\label{p2.4} Let $(Q(t))_{t\ge0}$ be a stochastically
continuous HA-semigroup defined by (\ref{2.1}). Then there is a
locally bounded non-negative function $c_0(\cdot)$ on $[0,\infty)$
such that
 \begin{eqnarray}\label{2.17}
\int_D\chi(\xi_1)Q(t,x,d\xi) \le c_0(t)\chi(x_1)
 \end{eqnarray}
and
 \begin{eqnarray}\label{2.18}
\int_D\chi^2(\xi_2)Q(t,x,d\xi) \le c_0(t)[\chi(x_1) +
\chi^2(x_2)].
 \end{eqnarray}
\eproposition

\proof In view of (\ref{2.15}), we have $\psi_1(t,(-z,0))\le 0$.
{From} (\ref{2.1}) it follows that
 \begin{eqnarray*}
\int_D (1 - e^{-z\xi_1}) Q(t,x,d\xi)
 =
1 - \exp\{-x_1|\psi_1(t,(-z,0))|\}
 \le
c_1(t,z)(1 - e^{-x_1}),
 \end{eqnarray*}
where $c_1(t,z) := 1\vee |\psi_1(t,(-z,0))|$ is locally bounded in
$t\ge0$. Then we get the first inequality by letting $\lambda=1$.
The second inequality is obvious if $x_1\ge 1$ or $|x_2|\ge 1$. On
the other hand, we have
 \begin{eqnarray*}
\int_D \Big(1 - \frac{\sin\xi_2}{\xi_2}\Big) Q(t,x,d\xi)
 \ar=\ar
\frac{1}{2}\int_{-1}^1dz\int_D (1-e^{iz\xi_2}) Q(t,x,d\xi) \\
 \ar=\ar
\frac{1}{2}\int_{-1}^1 (1 - \exp\{x_1\psi_1(t,(0,iz))
+ ix_2\beta_{22}(t)z\}) dz \\
 \ar\le\ar
\frac{1}{2}\int_{-1}^1[x_1|\psi_1(t,(0,iz))| + h(t,x_1,x_2,z)]dz,
 \end{eqnarray*}
where
 \begin{eqnarray*}
h(t,x_1,x_2,z)
 \ar=\ar
\sum_{k=2}^\infty \frac{1}{k!}(x_1|\psi_1(t,(0,iz))|
+ |x_2\beta_{22}(t)z|)^k   \\
 \ar\le\ar
\sum_{k=2}^\infty \frac{1}{k!}(x_1 + |x_2|)^k
(|\psi_1(t,(0,iz))| + |\beta_{22}(t)z|)^k   \\
 \ar\le\ar
2(x_1^2 + x_2^2)q(t,x_1,x_2,z)
 \end{eqnarray*}
with
 \begin{eqnarray*}
q(t,x_1,x_2,z)
 =
\sum_{k=2}^\infty \frac{1}{k!}(x_1 + |x_2|)^{k-2}
(|\psi_1(t,(0,iz))| + |\beta_{22}(t)z|)^k.
 \end{eqnarray*}
It follows that
 \begin{eqnarray*}
\int_D \Big(1 - \frac{\sin\xi_2}{\xi_2}\Big) Q(t,x,d\xi)
 \ar\le\ar
\frac{1}{2}x_1\int_{-1}^1|\psi_1(t,(0,iz))|dz   \\
 \ar \ar
+\, (x_1^2 + x_2^2)\int_{-1}^1 q(t,x_1,x_2,z)dz,
 \end{eqnarray*}
which implies (\ref{2.18}) for $0\le x_1\le 1$ and $|x_2|\le 1$.
\qed

As an immediate consequence of the above proposition we have the
following

\bcorollary\label{c2.2} Suppose that $(Q(t))_{t\ge0}$ is a
stochastically continuous HA-semigroup defined by (\ref{2.1}). Let
$U_\varepsilon = [0,\varepsilon) \times (-\varepsilon,
\varepsilon)$ for $\varepsilon >0$. Then for each $T\ge0$ we have
 \begin{eqnarray}\label{2.19}
\lim_{|x|\to0} \sup_{0\le t\le T}Q(t,x,D\setminus U_\varepsilon) =
0.
 \end{eqnarray}
\ecorollary







%%%%%%%%%%%%%%%%%%%% (Section 3) %%%%%%%%%%%%%%%%%%%%%


\section{Skew convolution semigroups}

\setcounter{equation}{0}


In this section, we give a formulation of the general affine
Markov semigroup in terms of a homogeneous one and a skew
convolution semigroup. It turns out that a skew convolution
semigroup always consists of infinitely divisible probability
measures. We prove that such a semigroup is regular if and only if
the linear part of the logarithm of its characteristic function is
absolutely continuous. We shall fix a regular HA-semigroup
$(Q(t))_{t\ge0}$ on $D$ defined by (\ref{2.1}), where $\psi(t,u) =
(\psi_1(t,u), \beta_{22}(t)u_2)$ is given by Corollary~\ref{c2.1}
and Proposition~\ref{p2.2}.


\bdefinition\label{d3.1} A family of probability measures
$(\gamma(t))_{t\ge0}$ on $D$ is called a \textit{skew convolution
semigroup (SC-semigroup)} associated with $(Q(t))_{t\ge0}$ if
 \begin{eqnarray}\label{3.1}
\gamma(r+t)
 =
(\gamma(r)Q(t))*\gamma(t), \qquad r,t\ge0,
 \end{eqnarray}
where ``$*$'' denotes the convolution operation and $\gamma(r)
Q(t)$ is the probability measure on $D$ defined by
 \begin{eqnarray}\label{3.2}
\gamma(r)Q(t)(B) = \int_DQ(t,\xi,B)\gamma(r,d\xi),
 \qquad B\in{\mcr B}(D).
 \end{eqnarray}
\edefinition


The concept of SC-semigroup generalizes that of the usual
convolution semigroup; see also Dawson and Li (2004) and Li
(1996a, 1996b, 1998, 2002). We refer the reader to Bertoin (1996)
and Sato (1999) for the general theory of convolution semigroups
and L\'evy processes.


\bproposition\label{p3.1} Let $(\gamma(t))_{t\ge0}$ be a
stochastically continuous SC-semigroup associated with
$(Q(t))_{t\ge0}$. Then each $\gamma(t)$ is an infinitely divisible
probability measure, so we have the representation
 \begin{eqnarray}\label{3.3}
\int_{D} \exp\{\<u,\xi\>\} \gamma(t,d\xi)
 =
\exp\{\phi(t,u)\},
 \qquad u\in U,
 \end{eqnarray}
with
 \begin{eqnarray}\label{3.4}
\phi(t,u) \ar=\ar b_1(t)u_1 + b_2(t)u_2 \nnm \\
 \ar \ar
+\, a(t)u_2^2 + \int_D (e^{\<u,\xi\>} - 1 - u_2\chi(\xi_2))
m(t,d\xi),
 \end{eqnarray}
where $a(t) \in \mbb{R}_+$, $(b_1(t), b_2(t)) \in D$ and
$m(t,d\xi)$ is a $\sigma$-finite measure on $D$ supported by $D
\setminus \{0\}$ such that
 \begin{eqnarray*}
\int_D [\chi(\xi_1) + \chi^2(\xi_2)] m(t,d\xi) < \infty.
 \end{eqnarray*}
Moreover, for any $r,t\ge0$ we have
 \begin{eqnarray}
b_1(r+t) \ar=\ar b_1(r)\beta_{11}(t) + b_1(t), \label{3.5} \\
b_2(r+t) \ar=\ar b_1(r)\beta_{12}(t) + b_2(r) \beta_{22}(t) +
b_2(t) \nnm \\
 \ar \ar + \int_D[Q(t)\chi_2(\xi) - \beta_{22}(t)\chi(\xi_2)]
m(r,d\xi), \label{3.6} \\
a(r+t) \ar=\ar b_1(r)\alpha(t) + a(r)\beta_{22}^2(t) + a(t), \label{3.7} \\
m(r+t,\cdot) \ar=\ar \int_D m(r,d\xi)Q(t,\xi,\cdot) +
b_1(r)\mu(t,\cdot) + m(t,\cdot). \label{3.8}
 \end{eqnarray}
\eproposition

\proof Based on Corollary~\ref{c2.2}, the proof is a
simplification of the arguments of Schmuland and Sun (2001). Let
$t\ge0$ be fixed. For each integer $n\ge1$ we may use (\ref{3.1})
inductively to obtain
 \begin{eqnarray*}
\gamma(t) = \prod_{j=1}^n* \gamma(t/n)Q((j-1)t/n).
 \end{eqnarray*}
{From} the stochastic continuity of the SC-semigroup, we have
$\lim_{n\to0} \gamma(t/n) = \delta_0$. By virtue of
Corollary~\ref{c2.2}, it is easy to show that $\{\gamma(t/n)
Q((j-1)t/n): j=1,\cdots,n; n=1,2,\cdots\}$ form an infinitesimal
triangular array. It follows that $\gamma(t)$ is infinitely
divisible. Then we have representation (\ref{3.3}) with
$\phi(t,u)$ given by (\ref{3.4}); see, e.g., Laha and Rohatgi
(1979, pp.499-500 and pp.515-519). {From} (\ref{3.1}) we have
 \begin{eqnarray}\label{3.9}
\phi(r+t,u) = \phi(r,\psi(t,u)) + \phi(t,u), \qquad r,t\ge0, u\in
U.
 \end{eqnarray}
Then relations (\ref{3.5})--(\ref{3.8}) follow as in the proof of
Proposition~\ref{p2.2}. \qed


\bdefinition\label{d3.2} A SC-semigroup $(\gamma(t))_{t\ge0}$
associated with $(Q(t))_{t\ge0}$ is called \textit{regular} if
$\phi(t,u) = \log\hat \gamma(t,u)$ has representation
 \begin{eqnarray}\label{3.10}
\phi(t,u)
 =
\int_0^t F(\psi(s,u)) ds, \qquad t\ge0, u\in U,
 \end{eqnarray}
where $F = \log\hat\nu$ for an infinitely divisible probability
measure $\nu$ on $D$. \edefinition


We remark that if $\nu$ is an infinitely divisible probability
measure on $D$, the function $F$ is well-defined and (\ref{3.10})
really determines a SC-semigroup. A simple but irregular
SC-semigroup can be constructed by letting $Q(t)$ be the identity
and letting $\gamma(t) = \delta_{(0,b_2(t))}$, where $b_2(t)$ is a
discontinuous solution of $b_2(r+t) = b_2(r) + b_2(t)$; see, e.g.,
Sato (1999, p.37). This example shows that some condition on the
function $t \mapsto b_2(t)$ has to be imposed to get the
regularity of the SC-semigroup $(\gamma(t))_{t\ge0}$ given by
(\ref{3.3}) and (\ref{3.4}).



\bdefinition\label{d3.3} A transition semigroup $(P(t))_{t\ge0}$
on $D$ is called a (general) \textit{affine semigroup} associated
with the HA-semigroup $(Q(t))_{t\ge0}$ if its characteristic
function has representation
 \begin{eqnarray}\label{3.11}
\int_{D} \exp\{\<u,\xi\>\} P(t,x,d\xi)
 =
\exp\{\<x,\psi(t,u)\>+\phi(t,u)\},
 \end{eqnarray}
where $\phi(t,\cdot)$ is a continuous function on $U$ satisfying
$\phi(t,0) = 0$. The affine semigroup $(P(t))_{t\ge0}$ defined by
(\ref{3.11}) is called \textit{regular} if it is stochastically
continuous and the derivatives $\psi^\prime_t (0,u)$ and
$\phi^\prime_t (0,u)$ exist for all $u\in U$ and are continuous at
$u=0$. \edefinition



\bproposition\label{p3.2} Let $(\gamma(t))_{t\ge0}$ be a
stochastically continuous SC-semigroup associated with the
HA-semigroup $(Q(t))_{t\ge0}$. Then $P(t,x,\cdot) = Q(t,x,\cdot)
* \gamma(t,\cdot)$ defines a Feller affine semigroup $(P(t))_{t\ge0}$.
\eproposition

\proof It is easy to show that the kernels $P(t,x,\cdot)$ satisfy
the Chapman-Kolmogorov equation. {From} Duffie \textit{et al.}\
(2003, Proposition~8.2) we know that $(Q(t))_{t\ge0}$ is a Feller
semigroup. For any $f\in C_0(D)$ we have
 \begin{eqnarray*}
P(t)f(x) = \int_D \gamma(t,dy)\int_D f(\xi+y)Q(t,x,d\xi).
 \end{eqnarray*}
Then we can use dominated convergence to find that $P(t)f\in
C_0(D)$. Since both $(Q(t))_{t\ge0}$ and $(\gamma(t))_{t\ge0}$ are
stochastically continuous, so is $(P(t))_{t\ge0}$. It follows that
$(P(t))_{t\ge0}$ is a Feller semigroup. \qed


Clearly, if $(\gamma(t))_{t\ge0}$ is a regular SC-semigroup, then
the general affine semigroup $(P(t))_{t\ge0}$ defined in
Proposition~\ref{p3.2} is also regular. To study the regularity of
SC-semigroups, we need some preliminary results. The proofs in the
sequel rely heavily on estimates derived from the relations
(\ref{3.5})--(\ref{3.8}).


\bproposition\label{p3.3} Suppose that (\ref{3.3}) and (\ref{3.4})
define a stochastically continuous SC-semigroup
$(\gamma(t))_{t\ge0}$. Then
 \begin{eqnarray}\label{3.12}
A(t) := b_1(t) + b_2^2(t) + a(t) + \int_D [\chi(\xi_1) +
\chi^2(\xi_2)] m(t,d\xi)
 \end{eqnarray}
is a locally bounded function of $t\ge0$. Moreover, we have $A(t)
\to 0$ as $t\to0$. \eproposition

\proof The stochastic continuity of the SC-semigroup implies that
$\phi(t,u)$ is jointly continuous in $(t,u)$. Since $\phi(t,u) \to
0$ as $t\to0$, the results follow by slight modifications of the
arguments in the proof of Proposition~\ref{p2.3}.  \qed



Let $B(\cdot)$ and $c_0(\cdot)$ be given respectively by
Propositions~\ref{p2.3} and~\ref{p2.4}. In the next two lemmas, we
fix a constant $T\ge0$ and let $C(T) \ge 0$ be a constant such
that
 \begin{eqnarray}\label{3.13}
\max\{B(t), c_0(t), \beta_{22}^2(t)\} \le C(T), \qquad 0\le t\le
T.
 \end{eqnarray}
For $0\le r_1 < t_1 < r_2 < t_2 < \cdots$ we set $\sigma_n =
\sum_{j=1}^n (t_j-r_j)$.

\blemma\label{l3.1} Suppose that (\ref{3.3}) and (\ref{3.4})
define a stochastically continuous SC-semigroup
$(\gamma(t))_{t\ge0}$. Then for $0\le r_1 < t_1 < r_2 < t_2 <
\cdots \le T$ we have
 \begin{eqnarray}\label{3.14}
\sum_{j=1}^n [b_1(t_j) - b_1(r_j)] \le C(T)b_1(\sigma_n)
 \end{eqnarray}
and
 \begin{eqnarray}\label{3.15}
\sum_{j=1}^n [a(t_j) - a(r_j)] \le C(T)[b_1(\sigma_n) +
a(\sigma_n)].
 \end{eqnarray}
Consequently, $b_1(t)$ and $a(t)$ are absolutely continuous in
$t\ge0$. \elemma

\proof We shall only give the proof of (\ref{3.15}) since the
proof of (\ref{3.14}) is similar. By (\ref{3.7}) we find that $t
\mapsto a(t)$ is non-decreasing and
 \begin{eqnarray*}
a(t_1)-a(r_1)
 \ar=\ar
b_1(t_1-r_1)\alpha(r_1) + a(t_1-r_1)\beta_{22}^2(r_1)  \\
 \ar\le\ar
C(T)[b_1(t_1-r_1) + a(t_1-r_1)],
 \end{eqnarray*}
that is, (\ref{3.15}) holds for $n=1$. Now suppose that
(\ref{3.15}) is true for $n-1$. By (\ref{3.13}) and
Propositions~\ref{p2.2} and~\ref{p3.1},
 \begin{eqnarray*}
\sum_{j=1}^n [a(t_j)-a(r_j)]
 \ar\le\ar
[a(t_n) - a(r_n)] + C(T)[b_1(\sigma_{n-1}) +
a(\sigma_{n-1})]  \\
 \ar=\ar
b_1(t_n-r_n)\alpha(r_n) + a(t_n-r_n)\beta_{22}^2(r_n)  \\
 \ar \ar
+\, C(T)[b_1(\sigma_{n-1}) + a(\sigma_{n-1})]  \\
 \ar=\ar
b_1(t_n-r_n)[\beta_{11}(\sigma_{n-1})\alpha(r_n-\sigma_{n-1})  \\
 \ar \ar
+\, \alpha(\sigma_{n-1})\beta_{22}^2(r_n-\sigma_{n-1})]
+ a(t_n-r_n)\beta_{22}^2(r_n) \\
 \ar \ar
+\, C(T)[b_1(\sigma_{n-1}) + a(\sigma_{n-1})]  \\
 \ar\le\ar
C(T)b_1(t_n-r_n)\beta_{11}(\sigma_{n-1}) + C(T)
b_1(t_n-r_n)\alpha(\sigma_{n-1})  \\
 \ar \ar
+\, C(T)[a(t_n-r_n)\beta_{22}^2(\sigma_{n-1}) + a(\sigma_{n-1})]  \\
 \ar \ar
+\, C(T)b_1(\sigma_{n-1})  \\
 \ar=\ar
C(T)b_1(\sigma_n) + C(T)a(\sigma_n).
 \end{eqnarray*}
That proves (\ref{3.15}) by induction. The absolute continuity of
$b_1(t)$ and $a(t)$ follows by Proposition~\ref{p3.3}. \qed



\blemma\label{l3.2} Suppose that (\ref{3.3}) and (\ref{3.4})
define a stochastically continuous SC-semigroup
$(\gamma(t))_{t\ge0}$. Set
 \begin{eqnarray}\label{3.16}
f(t) = \int_D \chi(\xi_1) m(t,d\xi)
 \quad\mbox{and}\quad
g(t) = \int_D \chi^2(\xi_2) m(t,d\xi).
 \end{eqnarray}
Then for any $0\le r_1 < t_1 < \cdots < r_n < t_n \le T$ we have
 \begin{eqnarray}\label{3.17}
\sum_{j=1}^n [f(t_j)-f(r_j)] \le C(T)[b_1(\sigma_n) +
f(\sigma_n)],
 \end{eqnarray}
and
 \begin{eqnarray}\label{3.18}
\sum_{j=1}^n [g(t_j)-g(r_j)] \le C(T)[b_1(\sigma_n) + f(\sigma_n)
+ g(\sigma_n)].
 \end{eqnarray}
Consequently, $f(t)$ and $g(t)$ are absolutely continuous in
$t\ge0$. \elemma


\proof The proofs of (\ref{3.17}) and (\ref{3.18}) are based on
ideas similar to those in the proof of the last lemma. We here
give the proof of (\ref{3.18}) since it involves more careful
calculations. By (\ref{3.8}) we find that $t\mapsto g(t)$ is
non-decreasing and
 \begin{eqnarray*}
g(t_1)-g(r_1)
 \ar=\ar
\int_D m(t_1-r_1,d\xi)\int_D \chi^2(\eta_2) Q(r_1,\xi,d\eta)  \\
 \ar \ar
+\, b_1(t_1-r_1)\int_D \chi^2(\xi_2) \mu(r_1,d\xi)  \\
 \ar\le\ar
C(T)\int_D [\chi(\xi_1) + \chi^2(\xi_2)] m(t_1-r_1,d\xi) +
C(T) b_1(t_1-r_1)  \\
 \ar=\ar
C(T)[b_1(t_1-r_1) + f(t_1-r_1) + g(t_1-r_1)],
 \end{eqnarray*}
where we used Proposition~\ref{p2.4} for the inequality. Then
(\ref{3.18}) holds for $n=1$. Suppose the inequality is true for
$n-1$. By (\ref{3.13}) and the results of Propositions~\ref{p2.4}
and~\ref{p3.1} we have
 \begin{eqnarray*}
\sum_{j=1}^n [g(t_j)-g(r_j)]
 \ar\le\ar
[g(t_n) - g(r_n)] + C(T)[b_1(\sigma_{n-1}) + f(\sigma_{n-1})
+ g(\sigma_{n-1})]  \\
 \ar=\ar
\int_Dm(t_n-r_n,d\xi)\int_D \chi^2(\eta_2) Q(r_n,\xi,d\eta)   \\
  \ar \ar
+\, b_1(t_n-r_n)\int_D \chi^2(\xi_2) \mu(r_n,d\xi)   \\
  \ar \ar
+\, C(T)[b_1(\sigma_{n-1}) + f(\sigma_{n-1}) + g(\sigma_{n-1})]  \\
 \ar\le\ar
C(T)\int_D m(t_n-r_n,d\xi)\int_D [\chi(\eta_1) + \chi^2(\eta_2)]
Q(\sigma_{n-1},\xi,d\eta)   \\
  \ar \ar
+\, b_1(t_n-r_n)\int_D \mu(\sigma_{n-1},d\xi)\int_D \chi^2(\eta_2)
Q(r_n-\sigma_{n-1},\xi,d\eta)   \\
  \ar \ar
+\, b_1(t_n-r_n)\beta_{11}(\sigma_{n-1})\int_D \chi^2(\eta_2)
\mu(r_n-\sigma_{n-1},d\eta)   \\
  \ar \ar
+\, C(T)[b_1(\sigma_{n-1}) + f(\sigma_{n-1}) + g(\sigma_{n-1})]  \\
 \ar\le\ar
C(T)\int_D m(t_n-r_n,d\xi)\int_D [\chi(\eta_1) + \chi^2(\eta_2)]
Q(\sigma_{n-1},\xi,d\eta)   \\
  \ar \ar
+\, C(T)b_1(t_n-r_n)\int_D [\chi(\xi_1) + \chi^2(\xi_2)]
\mu(\sigma_{n-1},d\xi)   \\
  \ar \ar
+\, C(T)[b_1(\sigma_n) + f(\sigma_{n-1}) + g(\sigma_{n-1})]  \\
 \ar=\ar
C(T)[b_1(\sigma_n) + f(\sigma_n) + g(\sigma_n)].
 \end{eqnarray*}
Then (\ref{3.18}) follows by induction. The second assertion
follows by Proposition~\ref{p3.3}. \qed

\blemma\label{l3.3} Suppose that (\ref{3.3}) and (\ref{3.4})
determine a stochastically continuous SC-semigroup
$(\gamma(t))_{t\ge0}$. Then there is a $\sigma$-finite kernel
$m^\prime(s, d\xi)$ from $(0,\infty)$ to $D$ supported by $D
\setminus \{0\}$ so that
 \begin{eqnarray}\label{3.19}
m(t,d\xi) = \int_0^tm^\prime(s,d\xi)ds, \qquad t\ge0.
 \end{eqnarray}
\elemma

\proof {From} (\ref{3.8}) we see that $m(t,d\xi)$ is increasing in
$t\ge0$. By Lemma~\ref{l3.2}, the function
 \begin{eqnarray*}
t \mapsto \int_D [\chi(\xi_1) + \chi^2(\xi_2)] m(t,d\xi)
 \end{eqnarray*}
is absolutely continuous. Then the assertion follows as in the
proof of Dawson \textit{et al.}\ (2004, Theorem~2.2). \qed


\btheorem\label{t3.1} Let $(\gamma(t))_{t\ge0}$ be a
stochastically continuous SC-semigroup given by (\ref{3.3}) and
(\ref{3.4}). Then $(\gamma(t))_{t\ge0}$ is regular if and only if
the function $t\mapsto b_2(t)$ is absolutely continuous on
$[0,\infty)$. \etheorem

\proof Suppose that $t\mapsto b_2(t)$ is absolutely continuous. In
view of Lemma~\ref{l3.1}, we can find Borel measurable functions
$a^\prime(\cdot) \ge0$ and $b_j^\prime(\cdot)$ such that
 \begin{eqnarray*}
a(t) = \int_0^t a^\prime(s) ds \quad\mbox{and}\quad b_j(t) =
\int_0^t b_j^\prime(s) ds, \qquad j=1,2.
 \end{eqnarray*}
Let $m^\prime(s,d\xi)$ be given by Lemma~\ref{l3.3} and let
 \begin{eqnarray*}
\phi^\prime(s,u) = b_1^\prime(s)u_1 + b_2^\prime(s)u_2 +
a^\prime(s)u_2^2 + \int_D (e^{\<u,\xi\>} - 1 - \chi(\xi_2)u_2)
m^\prime(s,d\xi).
 \end{eqnarray*}
Then we have
 \begin{eqnarray}\label{3.20}
\phi(t,u)
 =
\int_0^t \phi^\prime(s,u) ds.
 \end{eqnarray}
By (\ref{3.9}) and (\ref{3.20}) it is easy to show that
 \begin{eqnarray}\label{3.21}
\int_0^r \phi^\prime(s+t,u) ds
 =
\int_0^r \phi^\prime(s,\psi(t,u)) ds.
 \end{eqnarray}
Let $\nu_s$ be the infinitely divisible probability measure on $D$
such that $\hat\nu_s(u) = \exp\{\phi^\prime(s,u)\}$. Based on
(\ref{3.21}), it is easy to modify the definitions of
$\phi^\prime(s,\cdot)$ and $\nu_s$ accordingly so that
$\nu_t=\nu_sQ_{t-s}$ for all $t>s>0$ while (\ref{3.20}) remains
true; see Li (1996a). In other words, $(\nu_s)_{s>0}$ is an
entrance law for $(Q(t))_{t\ge0}$. But $(Q(t))_{t\ge0}$ is a
Feller semigroup by Duffie \textit{et al.}\ (2003,
Proposition~8.2). Then the Ray-Knight compactification of $D$ with
respect to $(Q(t))_{t\ge0}$ coincides with its one point
compactification $\bar D := D \cup \{\partial\}$ and the
Ray-Knight extension of $(Q(t))_{t\ge0}$ satisfies
$Q(t,\partial,\cdot) = \delta_\partial$ and $Q(t,x,\{\partial\}) =
0$ for every $x\in D$. It follows that the entrance space for
$(Q(t))_{t\ge0}$ is just $D$. By Sharpe (1988, p.196), there is a
probability measure $\nu_0$ on $D$ such that $\nu_s = \nu_0 Q_s$
for all $s>0$. Then $\phi^\prime(s,u) = \log \hat\nu_0(\psi(s,u))$
and hence $(\gamma(t))_{t\ge0}$ is regular. Conversely, if
$(\gamma(t))_{t\ge0}$ is regular, the function $t\mapsto
\phi(t,u)$ is absolutely continuous on $[0,\infty)$ for every
$u\in U$. Then (\ref{3.4}) and Lemmas~\ref{l3.1} and \ref{l3.3}
imply that $t\mapsto b_2(t)$ is absolutely continuous. \qed


\bcorollary\label{c3.1} Let $(\gamma(t))_{t\ge0}$ be a
stochastically continuous SC-semigroup defined by (\ref{3.1}).
Then $(\gamma(t))_{t\ge0}$ is regular if and only if $t \mapsto
\hat \gamma(t,u)$ is absolutely continuous for all $u\in U$.
\ecorollary

\proof By (\ref{3.4}) and Lemmas~\ref{l3.1} and~\ref{l3.3},
$t\mapsto b_2(t)$ is absolutely continuous if and only if
$t\mapsto \phi(t,u)$ is absolutely continuous on $[0,\infty)$ for
all $u\in U$. Then the result follows from Theorem~\ref{t3.1}.
\qed


%%%%%%%%%%%%%%%%%%%% (Section 4) %%%%%%%%%%%%%%%%%%%%%


\section{Regularities under moment conditions}

\setcounter{equation}{0}



In this section, we prove the regularities of HA-semigroups and
their associated SC-semigroups under some conditions on the first
moments. Suppose that $(Q(t))_{t\ge0}$ is a stochastically
continuous HA-semigroup defined by (\ref{2.1}), where $\psi(t,u) =
(\psi_1(t,u)$, $\beta_{22}(t)u_2)$ is given by
Corollary~\ref{c2.1} and Proposition~\ref{p2.2}. Let us consider
the following hypothesis.


\bhypothesis\label{h4.1} Suppose that
 \begin{eqnarray}\label{4.1}
\int_D [\xi_1 + l_{12}(\xi_2)] \mu(t,d\xi) < \infty
 \end{eqnarray}
for all $t\ge0$ or, equivalently,
 \begin{eqnarray}\label{4.2}
\int_D |\xi| Q(t,x,d\xi) < \infty
 \end{eqnarray}
for all $t\ge0$ and $x\in D$. \ehypothesis



If Hypothesis~\ref{h4.1} holds, we have a more convenient
representation for the function $\psi_1(t,u)$. Indeed, we may
differentiate both sides of (\ref{2.6}) to see that
 \begin{eqnarray}\label{4.3}
q_{11}(t) := \frac{\partial\psi_1}{\partial u_1}(t,u)\Big|_{u=0} =
\beta_{11}(t) + \int_D \xi_1 \mu(t,d\xi)
 \end{eqnarray}
and
 \begin{eqnarray}\label{4.4}
q_{12}(t) := \frac{\partial\psi_1}{\partial u_2}(t,u)\Big|_{u=0} =
\beta_{12}(t) + \int_D [\xi_2-\chi(\xi_2)] \mu(t,d\xi).
 \end{eqnarray}
On the other hand, differentiating both sides of (\ref{2.1}) we
find that
 \begin{eqnarray}\label{4.5}
\int_D \xi_1 Q(t,x,d\xi) = x_1q_{11}(t)
 \end{eqnarray}
and
 \begin{eqnarray}\label{4.6}
\int_D \xi_2 Q(t,x,d\xi) = x_1q_{12}(t) + x_2\beta_{22}(t).
 \end{eqnarray}



\bproposition\label{p4.1} If Hypothesis~\ref{h4.1} holds, we have
 \begin{eqnarray}\label{4.7}
\psi_1(t,u) \ar=\ar \beta_{11}(t)u_1 + q_{12}(t)u_2 \nnm \\
 \ar \ar
+\, \alpha(t)u_2^2 + \int_D (e^{\<u,\xi\>} - 1 - u_2\xi_2)
\mu(t,d\xi),
 \end{eqnarray}
where $\alpha(t)$, $\beta_{11}(t)$ and $\mu(t,d\xi)$ are as in
Proposition~\ref{p3.1} and $q_{12}(t)$ is given by (\ref{4.4}) and
satisfies
 \begin{eqnarray}\label{4.8}
q_{12}(r+t) = q_{11}(r)q_{12}(t) + q_{12}(r) \beta_{22}(t), \qquad
r,t\ge0.
 \end{eqnarray}
\eproposition

\proof The representation (\ref{4.7}) follows immediately from
(\ref{2.6}). By Proposition~\ref{p2.2},
 \begin{eqnarray*}
q_{12}(r+t)
 \ar=\ar
\beta_{12}(r+t) + \int_D[\xi_2-\chi(\xi_2)]\mu(r+t,d\xi)
\\
 \ar=\ar
\beta_{11}(r)\beta_{12}(t) + \beta_{12}(r) \beta_{22}(t)  \\
 \ar \ar
+ \int_D[Q(t)\chi_2(\xi) - \beta_{22}(t) \chi(\xi_2)]
\mu(r,d\xi)  \\
 \ar \ar
+ \int_D[\xi_1q_{12}(t) + \xi_2\beta_{22}(t) -
Q(t)\chi_2(\xi)] \mu(r,d\xi)  \\
 \ar \ar
+\, \beta_{11}(r)\int_D[\xi_2 - \chi(\xi_2)]
\mu(t,d\xi)  \\
 \ar=\ar
\beta_{11}(r)\beta_{12}(t) + \beta_{12}(r) \beta_{22}(t) +
\beta_{22}(t) \int_D[\xi_2 - \chi(\xi_2)]
\mu(r,d\xi)  \\
 \ar \ar
+\, q_{12}(t)\int_D\xi_1\mu(r,d\xi) + \beta_{11}(r)
\int_D[\xi_2 - \chi(\xi_2)] \mu(t,d\xi)  \\
 \ar=\ar
\beta_{11}(r)q_{12}(t) + q_{12}(r) \beta_{22}(t) + q_{12}(t)\int_D
\xi_1 \mu(r,d\xi).
 \end{eqnarray*}
Then we get (\ref{4.8}) from (\ref{4.3}). \qed


\bhypothesis\label{h4.2} Suppose that (\ref{4.2}) holds and for
every fixed $x \in D$ the mapping
 \begin{eqnarray}\label{4.9}
t \mapsto |\xi|Q(t,x,d\xi)
 \end{eqnarray}
is continuous by the weak convergence of finite measures.
\ehypothesis


The following theorem can be regarded as an extension of Watanabe
(1969, Theorem~5) to the state space of the positive half plane.


\btheorem\label{t4.1} If Hypothesis~\ref{h4.2} holds, the
HA-semigroup $(Q(t))_{t\ge0}$ is regular. \etheorem


\proof Under the hypothesis, we may differentiate both sides of
(\ref{2.1}) with respect to $u_1$ and $u_2$ to get
 \begin{eqnarray*}
\int_{D} \xi_j\exp\{\<u,\xi\>\} Q(t,(x_1,0),d\xi)
 =
x_1\psi^\prime_{1,u_j}(t,u)\exp\{x_1\psi_1(t,u)\}
 \end{eqnarray*}
for $j=1,2$. On the other hand, since (\ref{4.9}) depends on
$t\ge0$ continuously, we have
 \begin{eqnarray*}
\lim_{t\to0}\int_{D} \xi_1\exp\{\<u,\xi\>\} Q(t,(x_1,0),d\xi)
 =
x_1\exp\{u_1x_1\}
 \end{eqnarray*}
and
 \begin{eqnarray*}
\lim_{t\to0}\int_{D} \xi_2\exp\{\<u,\xi\>\} Q(t,(x_1,0),d\xi) = 0.
 \end{eqnarray*}
Comparing the above equalities we obtain
 \begin{eqnarray}\label{4.10}
\lim_{t\to0} \psi^\prime_{1,u_1}(t,u) = 1
 \quad\mbox{and}\quad
\lim_{t\to0} \psi^\prime_{1,u_2}(t,u) = 0.
 \end{eqnarray}
For $t>0$ let
 \begin{eqnarray*}
p(t,u)
 =
\frac{1}{t} \int_0^t \psi(s,u) ds.
 \end{eqnarray*}
Then (\ref{4.10}) implies that $\lim_{t\to0}p^\prime_{1,u_1}(t,u)
= 1$ and $\lim_{t\to0}p^\prime_{1,u_2}(t,u) = 0$. By
Proposition~\ref{p2.1} and Corollary~\ref{c2.1} we have
$p^\prime_{2,u_1} (t,u) = 0$ and $\lim_{t\to0} p^\prime_{2,u_2}
(t,u) = 1$. Let $U_1$ be any fixed bounded subset of $U$. It is
easy to see that the above limits hold with uniform convergence on
$U_1$. Then we can choose sufficiently small $r>0$ so that the
matrix
 \begin{eqnarray}\label{4.11}
\partial p(r,u) := \left(\begin{array}{cc}
p^\prime_{1,u_1}(r,u), \ar p^\prime_{2,u_1}(r,u)  \\
p^\prime_{1,u_2}(r,u), \ar p^\prime_{2,u_2}(r,u)
\end{array}\right)
 \end{eqnarray}
is invertible for all $u\in U_1$. Observe that
 \begin{eqnarray*}
p(r,\psi(t,u)) - p(r,u)
 \ar=\ar
\frac{1}{r}\bigg[\int_0^r\psi(s,\psi(t,u))ds - \int_0^r\psi(s,u)]ds\bigg] \\
 \ar=\ar
\frac{1}{r}\bigg[\int_t^{r+t}\psi(s,u)ds - \int_0^r\psi(s,u)]ds\bigg] \\
 \ar=\ar
\frac{1}{r}\int_0^t [\psi(r+s,u) - \psi(s,u)]ds.
 \end{eqnarray*}
It follows immediately that
 \begin{eqnarray}\label{4.12}
\lim_{t\to0} \frac{1}{t}[p(r,\psi(t,u)) - p(r,u)]
 =
\frac{1}{r}(\psi(r,u) - u).
 \end{eqnarray}
For $t>0$ and $u\in U_1$ let
 \begin{eqnarray*}
q(r,t,u)
 :=
[p(r,\psi(t,u)) - p(r,u)](\partial p(r,u))^{-1}.
 \end{eqnarray*}
Then we have
 \begin{eqnarray}\label{4.13}
\lim_{t\to0} \frac{1}{t}q(r,t,u)
 =
\frac{1}{r}(\psi(r,u) - u)(\partial p(r,u))^{-1}.
 \end{eqnarray}
By Taylor's expansion,
 \begin{eqnarray*}
p(r,\psi(t,u)) - p(r,u)
 =
(\psi(t,u) - u) (\partial p(r,u))
 +
o(|\psi(t,u) - u|)
 \end{eqnarray*}
and consequently
 \begin{eqnarray}\label{4.14}
q(r,t,u)
 =
(\psi(t,u) - u)
 +
o(|\psi(t,u) - u|)
 \end{eqnarray}
as $t\to0$. It follows that
 \begin{eqnarray*}
|q(r,t,u)|
 =
|\psi(t,u) - u|
 +
o(|\psi(t,u) - u|)
 \end{eqnarray*}
and hence
 \begin{eqnarray*}
\lim_{t\to0}|q(r,t,u)|\,|\psi(t,u) - u|^{-1} = 1,
 \end{eqnarray*}
which together with (\ref{4.13}) implies that
 \begin{eqnarray*}
\lim_{t\to0} \frac{1}{t}|\psi(t,u) - u|
 =
\lim_{t\to0} \frac{1}{t}|q(r,t,u)|
 =
\frac{1}{r}|(\psi(r,u) - u)(\partial p(r,u))^{-1}|.
 \end{eqnarray*}
Then we get from (\ref{4.14}) that
 \begin{eqnarray}\label{4.15}
q(r,t,u) = (\psi(t,u) - u) + o(t).
 \end{eqnarray}
{From} (\ref{4.13}) and (\ref{4.15}) it follows that
 \begin{eqnarray*}
\lim_{t\to0} \frac{1}{t}(\psi(t,u) - u)
 =
\frac{1}{r}(\psi(r,u) - u)(\partial p(r,u))^{-1}
 \end{eqnarray*}
for all $u\in U_1$. Clearly, the entries of $\partial p(r,u)$ and
hence those of $(\partial p(r,u))^{-1}$ are continuous in $u\in
U_1$. Then the derivative $\psi^\prime_t(0,u)$ exists and is
continuous in $u\in U_1$. Since $U_1$ can be arbitrary, we get the
desired regularity. \qed

Now let $(\gamma(t))_{t\ge0}$ be a stochastically continuous
SC-semigroup associated with $(Q(t))_{t\ge0}$ with characteristic
function determined by (\ref{3.3}) and (\ref{3.4}). We consider
the following

\bhypothesis\label{h4.3} Suppose that
 \begin{eqnarray}\label{4.16}
\int_D [\xi_1 + l_{12}(\xi_2)] m(t,d\xi) < \infty
 \end{eqnarray}
or, equivalently,
 \begin{eqnarray}\label{4.17}
\int_D |\xi| \gamma(t,d\xi) < \infty
 \end{eqnarray}
for all $t\ge0$. \ehypothesis


Under the above hypothesis, there is a more convenient
representation for the function $\phi(t,u)$. Indeed, from
(\ref{3.4}) it follows that
 \begin{eqnarray}\label{4.18}
h_1(t) := \frac{\partial\phi}{\partial u_1}(t,u)\Big|_{u=0} =
b_1(t) + \int_D \xi_1 m(t,d\xi)
 \end{eqnarray}
and
 \begin{eqnarray}\label{4.19}
h_2(t) := \frac{\partial\phi}{\partial u_2}(t,u)\Big|_{u=0} =
b_2(t) + \int_D [\xi_2-\chi(\xi_2)] m(t,d\xi).
 \end{eqnarray}
By differentiating both sides of (\ref{3.3}) we get
 \begin{eqnarray}\label{4.20}
\int_D \xi_1 \gamma(t,d\xi) = h_1(t)
 \quad\mbox{and}\quad
\int_D \xi_2 \gamma(t,d\xi) = h_2(t).
 \end{eqnarray}
The proof of the next proposition is similar to that of
Proposition~\ref{p4.1}.

\bproposition\label{p4.2} If Hypotheses~\ref{h4.1} and~\ref{h4.3}
hold, the function $\phi(t,u)$ has representation
 \begin{eqnarray}\label{4.21}
\phi(t,u) = b_1(t)u_1 + h_2(t)u_2 + a(t)u_2^2 + \int_D
(e^{\<u,\xi\>} - 1 - u_2\xi_2) m(t,d\xi),
 \end{eqnarray}
where $a(t)$, $b_1(t)$ and $m(t,d\xi)$ are as in
Proposition~\ref{p3.1} and $h_2(t)$ is defined by (\ref{4.19}) and
satisfies
 \begin{eqnarray}\label{4.22}
h_2(r+t) = h_1(r)q_{12}(t) + h_2(r) \beta_{22}(t) + h_2(t), \qquad
r,t\ge0.
 \end{eqnarray}
\eproposition


\bproposition\label{p4.3} If Hypotheses~\ref{h4.2} and~\ref{h4.3}
hold, the function $t \mapsto h_1(t)$ is continuously
differentiable on $[0,\infty)$. \eproposition

\proof By (\ref{3.11}), for any $u_1\in \mbb{C}_-$ we have
 \begin{eqnarray}\label{4.23}
\int_{D} \exp\{u_1\xi_1\} P(t,x,d\xi)
 =
\exp\{x_1\psi_1(t,(u_1,0)) + \phi(t,(u_1,0))\}, \qquad u_1\in
\mbb{C}_-.
 \end{eqnarray}
Then the projection of $P(t,x,\cdot)$ to $\mbb{R}_+$ is
independent of $x_2\in \mbb{R}$. Let $P_1(t,x_1,\cdot)$ denote
this projection. In view of (\ref{4.23}), we see that
$(P_1(t))_{t\ge0}$ is the transition semigroup of a CBI-process in
the sense of Kawazu and Watanabe (1971). By Kawazu and Watanabe
(1971, Theorem~1.1), we have the representation
 \begin{eqnarray}\label{4.24}
\phi(t,(u_1,0)) \ar=\ar \int_0^t\bigg[b_1\psi_1(s,(u_1,0)) \nnm  \\
 \ar \ar\qquad
+ \int_0^\infty (\exp\{\psi_1(s,(u_1,0))\xi_1\} - 1)
m_1(d\xi_1)\bigg]ds,
 \end{eqnarray}
where $b_1 \ge 0$ is a constant and $m_1(d\xi_1)$ is a
$\sigma$-finite measure on $(0,\infty)$ such that
 \begin{eqnarray*}
\int_0^\infty \chi(\xi_1) m_1(d\xi_1) < \infty.
 \end{eqnarray*}
By differentiating both sides of (\ref{4.24}) with respect to
$u_1$ at zero and appealing to (\ref{4.3}) and (\ref{4.18}) it is
easy to show that
 \begin{eqnarray}\label{4.25}
h_1(t)
 =
\bigg(b_1 + \int_D \xi_1 m_1(d\xi_1)\bigg) \int_0^t q_{11}(s) ds.
 \end{eqnarray}
Then we must have
 \begin{eqnarray*}
\int_0^\infty \xi_1 m_1(d\xi_1) < \infty.
 \end{eqnarray*}
In view of (\ref{4.5}), Hypothesis~\ref{h4.2} implies that
$q_{11}(s)$ is continuous in $s\ge0$ with $q_{11}(0)=1$. By
(\ref{4.25}) we find that $h_1(t)$ is differentiable in $t\ge0$.
\qed


\btheorem\label{t4.2} Suppose that Hypotheses~\ref{h4.1}
and~\ref{h4.3} hold. Then $(\gamma(t))_{t\ge0}$ is regular if and
only if $t \mapsto h_2(t)$ is absolutely continuous on
$[0,\infty)$. \etheorem

\proof Based on representation (\ref{4.21}), this follows as in
the proof of Theorem~\ref{t3.1}. \qed


\bcorollary\label{c4.1} Suppose that Hypotheses~\ref{h4.2}
and~\ref{h4.3} hold. Then $(\gamma(t))_{t\ge0}$ is regular if and
only if the function $h_2(\cdot)$ is differentiable at some and
hence all $t\ge0$. \ecorollary


\proof By Proposition~\ref{p4.3}, the function $t\mapsto h_1(t)$
is continuously differentiable. {From} (\ref{4.22}) we know that
the differentiability of $h_2(\cdot)$ at any $t_0\ge 0$ implies
its differentiability at $0$. Using relation (\ref{4.22}) once
again we see that $h_2(\cdot)$ has right derivative at every $t\ge
0$ with
 \begin{eqnarray*}
h_2^\prime(t+) = h_1^\prime(0+)q_{12}(t) + h_2^\prime(0+)
\beta_{22}(t).
 \end{eqnarray*}
This function is continuous in $t\ge0$, so $h_2(\cdot)$ is
absolutely continuous. Then the desired result follows from
Theorem~\ref{t4.2}. \qed







%%%%%%%%%%%%%%%%%%%% (Section 5) %%%%%%%%%%%%%%%%%%%%%


\section{One-dimensional stochastic equations}

\setcounter{equation}{0}

In this section, we prove the existence and pathwise uniqueness of
solution of a one-dimensional stochastic equation with
non-Lipschitz coefficients and jumps of Poisson type. To simplify
the calculations, we only consider a special case for the
coefficients which is sufficient for the applications in the next
section. The result may be regarded as an extension of the
well-known result of Yamada and Watanabe (1971); see also Fang and
Zhang (2005) and the references therein for various
generalizations of their result in the setting of diffusion
processes. For the general background and notation of stochastic
equations we refer to Ikeda and Watanabe (1989).

Let $\theta_0\ge0$ and $\theta_1\ge 0$ be constants and let
$m(d\xi)$ and $\mu(d\xi)$ be $\sigma$-finite measures on
$(0,\infty)$ satisfying $m(l_1) + \mu(l_{12}) < \infty$. Suppose
that $(\Omega, {\mcr F}, {\mcr F}_t, \mbf{P})$ is a filtered
probability space satisfying the usual hypotheses on which the
following are defined:
\begin{itemize}

\item[(a)]
an $r$-dimensional Brownian motion $B(\cdot) = (B_1(\cdot),
\cdots, B_r(\cdot))$;

\item[(b)]
a Poisson random measure $N_0(ds,d\xi)$ on $(0,\infty)^2$ with
intensity $dsm(d\xi)$;

\item[(c)]
a Poisson random measure $N_1(ds,du,d\xi)$ on $(0,\infty)^3$ with
intensity $dsdu\mu(d\xi)$;

\item[(d)]
an $r$-dimensional progressive process $\sigma(\cdot) =
(\sigma_1(\cdot), \cdots, \sigma_r(\cdot))$ such that $|\sigma(t)|
\le \bar\sigma(t)$ for all $t\ge0$ and a non-negative
deterministic increasing function $\bar\sigma(\cdot)$ on
$[0,\infty)$;

\item[(e)]
a non-negative progressive process $b(\cdot)$ such that $b(t) \le
\bar b(t)$ for all $t\ge0$ and a non-negative deterministic
increasing function $\bar b(\cdot)$ on $[0,\infty)$;

\item[(f)]
a progressive process $\beta(\cdot)$ such that $|\beta(t)| \le
\bar \beta(t)$ for all $t\ge0$ and a non-negative deterministic
increasing function $\bar \beta(\cdot)$ on $[0,\infty)$;

\item[(g)]
a non-negative progressive process $l(\cdot)$ such that $l(t) \le
\bar l(t)$ for all $t\ge0$ and a non-negative deterministic
increasing function $\bar l(\cdot)$ on $[0,\infty)$.

\end{itemize}
We assume that $B(\cdot)$, $N_0$ and $N_1$ are independent of each
other and $({\mcr F}_t)_{t\ge0}$ is the augmented natural
filtration generated by them. Consider the stochastic integral
equation
 \begin{eqnarray}\label{5.1}
x(t) \ar=\ar x(0) + \int_0^t(b(s) + \beta(s)x(s))ds  \nnm \\
 \ar \ar
+ \sum_{j=1}^r\int_0^t\sigma_j(s)\sqrt{2x(s)} dB_j(s)
+ \int_0^t\int_0^\infty \theta_0\xi N_0(ds,d\xi)  \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{l(s)x(s-)}\int_0^\infty \theta_1\xi \tilde
N_1(ds,du,d\xi),
 \end{eqnarray}
where $\tilde N_1(ds,du,d\xi) = N_1(ds,du,d\xi) - dsdu\mu(d\xi)$.
Clearly, the diffusion coefficients of (\ref{5.1}) do not meet the
requirements of Ikeda and Watanabe (1989, p.265). Observe also
that integration in the last term on the right-hand side is taken
over a random set. By a \textit{solution} of (\ref{5.1}) we mean a
non-negative c\`adl\`ag progressive process $x(\cdot)$ satisfying
the equation a.s.\ for each $t\ge0$. We say \textit{pathwise
uniqueness} of solution holds for (\ref{5.1}) if any two solutions
of the equation with the same initial state are indistinguishable.


\bproposition\label{p5.1} Let $x(\cdot)$ be a solution of
(\ref{5.1}) satisfying $\mbf{E}[x(0)] < \infty$. Then we have
 \begin{eqnarray}\label{5.2}
\mbf{E}[x(t)]
 \le
\{\mbf{E}[x(0)] + t\bar b(t) + \theta_0m(l_1)t\}
\exp\{t\bar\beta(t)\}
 \end{eqnarray}
for all $t\ge0$. \eproposition


\proof Let $\tau_n = \inf\{t>0: x(t)\ge n\}$ and $x_n(t) =
x(t\land\tau_n)$. By Ethier and Kurtz (1986, p.131), we have
$\mbf{E}[x_n(s)] = \mbf{E}[x_n(s-)] \le \mbf{E}[n\vee x(0)] <
\infty$ for almost all $s\ge0$. Since $\bar b(t)$ and $\bar\beta(t)$
are both increasing in $t\ge0$, it follows from (\ref{5.1}) that
 \begin{eqnarray*}
\mbf{E}[x_n(t)]
 \le
\mbf{E}[x(0)] + t\bar b(t) + \theta_0m(l_1)t + \bar\beta(t)
\int_0^t \mbf{E}[x_n(s)]ds.
 \end{eqnarray*}
By Gronwall's inequality we get
 \begin{eqnarray*}
\mbf{E}[x_n(t)]
 \le
\{\mbf{E}[x(0)] + t\bar b(t) + \theta_0m(l_1)t\} \exp\{\bar
\beta(t)t\}.
 \end{eqnarray*}
Then (\ref{5.2}) follows by Fatou's lemma. \qed


\bproposition\label{p5.2} Let $x(\cdot)$ be a solution of
(\ref{5.1}) satisfying $\mbf{E}[x(0)] < \infty$. Then we have
 \begin{eqnarray}\label{5.3}
\mbf{E}\Big[\sup_{0\le s\le t}x(s)\Big]
 \ar\le\ar
\mbf{E}[x(0)] + [\bar b(t) + \theta_0m(l_1)]t  \nnm \\
 \ar \ar
+ \left[\bar\beta(t) + \theta_1\bar l(t)\mu(l_{12})\right]\int_0^t
\mbf{E}[x(s)]ds \nnm \\
 \ar \ar
+\, 4\Big[2r\bar\sigma(t) + \theta_1\sqrt{\bar l(t)}
\mu(l_{12})\Big]\bigg(1 + \int_0^t\mbf{E}[x(s)]ds\bigg)
 \end{eqnarray}
for all $t\ge0$. \eproposition


\proof Applying Doob's inequality to the martingale terms in
(\ref{5.1}),
 \begin{eqnarray*}
\mbf{E}\Big[\sup_{0\le s\le t}x(s)\Big]
 \ar\le\ar
\mbf{E}[x(0)] + \int_0^t \mbf{E}[\bar b(s) + \bar\beta(s)x(s)]ds +
\theta_0t \int_0^\infty \xi m(d\xi)  \nnm \\
 \ar \ar
+\, 4\sum_{j=1}^r\mbf{E}^{\frac{1}{2}}\bigg[\bigg(\int_0^t
\sigma_j(s)\sqrt{2x(s)}dB_j(s)\bigg)^2\bigg]  \nnm \\
 \ar \ar
+\, 4\mbf{E}^{\frac{1}{2}}\bigg[\bigg(\int_0^t \int_0^{l(s)x(s-)}
\int_0^1 \theta_1\xi \tilde N_1(ds,du,d\xi)\bigg)^2
\bigg] \nnm \\
 \ar \ar
+\, \mbf{E}\bigg[\int_0^t \int_0^{l(s)x(s-)} \int_1^\infty
\theta_1\xi N_1(ds,du,d\xi)\bigg] \nnm \\
 \ar\le\ar
\mbf{E}[x(0)] + \int_0^t \bar b(s) ds + \int_0^t \bar\beta(s)
\mbf{E}[x(s)]ds + \theta_0 tm(l_1)  \nnm \\
 \ar \ar
+\, 8\sum_{j=1}^r\bigg(\int_0^t\mbf{E}[\sigma_j^2(s)x(s)]ds\bigg)
^{\frac{1}{2}}   \\
 \ar \ar
+\, 4\theta_1\mu(l_{12})\bigg(\int_0^t \mbf{E}[l(s)x(s)]
ds\bigg)^{\frac{1}{2}} \nnm \\
 \ar \ar
+\, \theta_1\mu(l_{12})\int_0^t \mbf{E}[l(s) x(s)]ds.
 \end{eqnarray*}
Then we obtain (\ref{5.3}) by combining the terms.  \qed



\btheorem\label{t5.1} Suppose that $x_1(\cdot)$ and $x_2(\cdot)$
are two solutions of (\ref{5.1}) satisfying $\mbf{E}[x_2(0) +
x_1(0)] < \infty$. Then we have
 \begin{eqnarray}\label{5.4}
\mbf{E}[|x_2(t) - x_1(t)|]
 \le
\mbf{E}[|x_2(0) - x_1(0)|] \exp\{t\bar\beta(t)\}.
 \end{eqnarray}
Consequently, the pathwise uniqueness of solution holds for
(\ref{5.1}). \etheorem

\proof By Proposition~\ref{p5.1} it is easy to find that
$\mbf{E}[x_2(t) + x_1(t)]$ is locally bounded in $t\ge0$. Let
$z(t) = x_2(t) - x_1(t)$. Then we have
 \begin{eqnarray}\label{5.5}
z(t) \ar=\ar z(0) + \int_0^t \beta(s)z(s)ds \nnm \\
 \ar \ar
+ \sum_{j=1}^r \int_0^t \sigma_j(s) \Big(\sqrt{2x_2(s)} -
\sqrt{2x_1(s)}\Big)
dB_j(s) \nnm \\
 \ar \ar
+ \int_0^t \int_{l(s) x_1(s-)}^{l(s)x_2(s-)} \int_0^\infty
\theta_1\xi 1_{\{x_1(s-) \le x_2(s-)\}}\tilde
N_1(ds,du,d\xi) \nnm \\
 \ar \ar
- \int_0^t \int_{l(s) x_2(s-)}^{l(s)x_1(s-)} \int_0^\infty
\theta_1\xi 1_{\{x_1(s-) > x_2(s-)\}}\tilde N_1(ds,du,d\xi).
 \end{eqnarray}
Let $\{a_k\}$ be the sequence defined inductively by $a_0=1$ and
$a_k = a_{k-1} e^{-k}$ for $k\ge1$. It is easy to check that
$\int_{a_k} ^{a_{k-1}} (ku)^{-1}du = 1$. For $k\ge1$ let $g_k$ be
a non-negative continuous function on $[0,\infty)$ which has
support contained in $[a_k, a_{k-1}]$ and satisfies $0\le g_k(u)
\le 2(ku)^{-1}$ and $\int_{a_k} ^{a_{k-1}} g_k(u) du = 1$. Then
 \begin{eqnarray*}
h_k(x) = \int_0^{|x|}dy \int_0^yg_k(u)du, \qquad x\in \mbb{R},
 \end{eqnarray*}
defines a twice continuously differentiable function $h_k$ such
that $h_k(x) \to |x|$ increasingly as $k\to \infty$. Set
$H_k(x,\xi) = \Delta_\xi h_k(x) - h_k^\prime (x)\xi$. By
(\ref{5.5}) and It\^o's formula,
 \begin{eqnarray}\label{5.6}
h_k(z(t))
 \ar=\ar
h_k(z(0)) + \int_0^t h_k^\prime(z(s))\beta(s)z(s)ds
+ \mbox{mart.} \nnm \\
 \ar \ar
+ \sum_{j=1}^r\int_0^t h_k^{\prime\prime}(z(s)) \sigma_j^2(s)
\Big(\sqrt{x_2(s)} - \sqrt{x_1(s)}\Big)^2 ds \nnm \\
 \ar \ar
+ \int_0^t ds \int_0^\infty H_k(z(s),\theta_1\xi)
l(s)z(s)1_{\{z(s)\ge0\}} \mu(d\xi) \nnm \\
 \ar \ar
- \int_0^t ds \int_0^\infty H_k(z(s),-\theta_1\xi)
l(s)z(s)1_{\{z(s)<0\}} \mu(d\xi);
 \end{eqnarray}
see, e.g., Dellacherie and Meyer (1982, p.334-335). Note that
$|h_k^\prime(x)| \le 1$ and $0\le h_k^{\prime\prime} (x) =
g_k(|x|) \le 2|kx|^{-1}$. It follows that
 \begin{eqnarray*}
h_k^{\prime\prime}(z(s))\Big(\sqrt{x_2(s)} - \sqrt{x_1(s)}\Big)^2
 \le
h_k^{\prime\prime}(z(s))|x_2(s) - x_1(s)|
 \le
2/k.
 \end{eqnarray*}
By the mean-value theorem and Taylor's expansion it is easy to
show that $|H_k(x,\xi)x| \le |2\xi x| \land |k^{-1}\xi^2|$
whenever $x \xi \ge0$. Then we may take the expectations in
(\ref{5.6}) to find
 \begin{eqnarray}\label{5.7}
\mbf{E}[h_k(z(t))]
 \ar\le\ar
\mbf{E}[h_k(z(0))] + \bar\beta(t) \int_0^t \mbf{E}[|z(s)|]ds +
\frac{2}{k}\sum_{j=1}^r\int_0^t \mbf{E}[\sigma_j^2(s)]
ds \nnm \\
 \ar \ar
+\, \theta_1\bar l(t)\int_0^t ds \int_0^\infty\mbf{E}[2\xi|z(s)|]
\land (k^{-1}\theta_1\xi^2) \mu(d\xi).
 \end{eqnarray}
Letting $k\to\infty$ in (\ref{5.7}) we obtain
 \begin{eqnarray*}
\mbf{E}[|z(t)|]
 \le
\mbf{E}[|z(0)|] + \bar\beta(t) \int_0^t \mbf{E}[|z(s)|] ds.
 \end{eqnarray*}
Then (\ref{5.4}) follows by Gronwall's inequality. \qed

Now we turn to the existence of the solution of (\ref{5.1}). The
Picard iteration method fails for this equation because the
diffusion coefficients are not Lipschitz. Since the coefficients
are random, we cannot follow the standard argument of martingale
problem. In the approach given below, we first approximate the
random coefficients by some simple processes and consider a
sequence of equations without small and large jumps. The original
coefficients and the small jumps are retrieved by a limit argument
based on the second moment analysis. Finally, we obtain the
solution of (\ref{5.1}) by adding the large jumps.


A stochastic process $q(\cdot)$ defined on $(\Omega, {\mcr F},
{\mcr F}_t, \mbf{P})$ is called a \textit{simple process} if there
is a sequence $0=r_0 < r_1 < r_2 < \cdots$ increasing to infinity
and a sequence of random variables $\{\eta_k\}$ such that $\eta_k$
is ${\mcr F}_{r_k}$-measurable and
 \begin{eqnarray}\label{5.8}
q(t) = \eta_0 1_{\{0\}}(t) + \sum_{k=0}^\infty \eta_k 1_{(r_k,
r_{k+1}]}(t), \qquad t\ge0.
 \end{eqnarray}
We approximate the coefficients of (\ref{5.1}) in the following
way:
\begin{itemize}

\item[(a)]
Let $\{\sigma_n\}$ be a sequence of $r$-dimensional simple
processes such that $|\sigma_n(t)| \le \bar \sigma(t)$ for all
$t\ge0$ and $\sigma_n(\cdot) \to \sigma(\cdot)$ a.s.\ in
$L^2([0,J], \lambda)$ for all integers $J\ge1$.

\item[(b)]
Let $\{b_n\}$ be a sequence of non-negative simple processes such
that $b_n(t) \le \bar b(t)$ for all $t\ge0$ and $b_n(\cdot) \to
b(\cdot)$ a.s.\ in $L^2([0,J], \lambda)$ for all integers $J\ge1$.

\item[(c)]
Let $\{\beta_n\}$ be a sequence of simple processes such that
$|\beta_n(t)| \le \bar \beta(t)$ for all $t\ge0$ and
$\beta_n(\cdot) \to \beta(\cdot)$ a.s.\ in $L^2([0,J], \lambda)$
for all integers $J\ge1$.

\item[(d)]
Let $\{l_n\}$ be a sequence of simple processes such that $l_n(t)
\le \bar l(t)$ for all $t\ge0$ and $l_n(\cdot) \to l(\cdot)$ a.s.\
in $L^2([0,J], \lambda)$ for all integers $J\ge1$.

\end{itemize}
Let $L\ge1$ be an integer and let $\{\varepsilon_n\}$ be a
decreasing sequence such that $\mu(\{\varepsilon_n: n\ge 1\}) =
0$, $0< \varepsilon_n \le 1$ and $\varepsilon_n \to 0$ as
$n\to\infty$. Suppose that $x(0)$ is a non-negative ${\mcr
F}_0$-measurable random variable satisfying $\mbf{E}[x(0)] <
\infty$. Let $x_n(\cdot)$ denote the non-negative solution of the
stochastic equation
 \begin{eqnarray}\label{5.9}
x_n(t) \ar=\ar x(0) + \int_0^t(b_n(s) + \beta_n(s)x_n(s))ds  \nnm \\
 \ar \ar
+
\sum_{j=1}^r\int_0^t\sigma_{n,j}(s)\sqrt{2x_n(s)} dB_j(s)
+ \int_0^t\int_{\varepsilon_n}^L \theta_0\xi N_0(ds,d\xi)  \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{l_n(s)x_n(s-)}\int_{\varepsilon_n}^L
\theta_1\xi \tilde N_1(ds,du,d\xi).
 \end{eqnarray}
Based on Proposition~\ref{p5.2} and the results in Ikeda and
Watanabe (1989, pp.235-237), the existence of the strong solution
the above equation follows by arguments similar to those of Ikeda
and Watanabe (1989, pp.245-246). Let
 \begin{eqnarray}\label{5.10}
y_{n,j}(t) := \int_0^t\sigma_{n,j}(s)\sqrt{2x_n(s)} dB_j(s)
 \end{eqnarray}
and
 \begin{eqnarray}\label{5.11}
z_n(t) := \int_0^t \int_0 ^{l_n(s)x_n(s-)}\int_{\varepsilon_n}^L
\xi \tilde N_1(ds,du,d\xi).
 \end{eqnarray}


\blemma\label{l5.1} For $1\le j\le r$ the sequence
$y_{n,j}(\cdot)$ is tight in $C([0,\infty), \mbb{R})$, and the
sequences $x_n(\cdot)$ and $z_n(\cdot)$ are tight in
$D([0,\infty), \mbb{R})$. \elemma

\proof By Proposition~\ref{p5.1} it is easy to show that $C(t) :=
\sup_{n\ge1} \mbf{E} [x_n(t)]$ is a locally bounded function of
$t\ge0$. By (\ref{5.10}) we have
 \begin{eqnarray}\label{5.12}
\mbf{E} [|y_{n,j}(t)|^2]
 =
2\int_0^t\mbf{E}[\sigma_{n,j}^2(s)x_n(s)] ds
 \le
2\int_0^t\bar\sigma^2(s)C(s) ds
 \end{eqnarray}
and
 \begin{eqnarray}\label{5.13}
\mbf{E} [|z_n(t)|^2]
 =
\int_0^t\mbf{E}[l_n(s)x_n(s)] ds \int_{\varepsilon_n}^L \xi^2
\mu(d\xi)
 \le
\int_0^t\bar l(s)C(s)ds \int_0^L \xi^2 \mu(d\xi).
 \end{eqnarray}
Then $y_{n,j}(t)$ and $z_n(t)$ are tight sequences of random
variables for every fixed $t\ge0$. Now let $\{\tau_n\}$ be a
sequence of stopping times bounded above by $T\ge0$. By the
properties of independent increments of the Brownian motion and
the Poisson process we obtain as in the calculations in
(\ref{5.12}) and (\ref{5.13}) that
 \begin{eqnarray*}
\mbf{E}[|y_{n,j}(\tau_n+t) - y_{n,j}(\tau_n)|^2]
 \le
2\int_0^t\bar\sigma^2(T+s)C(T+s) ds
 \end{eqnarray*}
and
 \begin{eqnarray*}
\mbf{E} [|z_n(\tau_n+t) - z_n(\tau_n)|^2]
 \le
\int_0^t\bar l(T+s)C(T+s)ds \int_0^L \xi^2 \mu(d\xi).
 \end{eqnarray*}
Then $y_{n,j}(\cdot)$ and $z_n(\cdot)$ are tight in $D([0,\infty),
\mbb{R})$ by the criterion of Aldous (1978). Since $C([0,\infty),
\mbb{R})$ is a closed subset of $D([0,\infty), \mbb{R})$, we infer
that $y_{n,j}(\cdot)$ is also tight in $C([0,\infty), \mbb{R})$.
By similar calculations for other terms on the right-hand side of
(\ref{5.9}) we find that $x_n(\cdot)$ is tight in $D([0,\infty),
\mbb{R}_+)$. \qed



By Lemma~\ref{l5.1} we may construct a new filtered probability
space $(\Omega, {\mcr F}, {\mcr F}_t, \mbf{P})$ satisfying the
usual hypotheses on which the following stochastic equations are
realized:
 \begin{eqnarray}\label{5.14}
x_n(t) \ar=\ar x_n(0) + \int_0^t(b_n(s) + \beta_n(s)x_n(s))ds \nnm \\
 \ar \ar
+ \sum_{j=1}^r\int_0^t \sigma_{n,j}(s) \sqrt{2x_n(s)} dB_{n,j}(s)
+ \int_0^t\int_{\varepsilon_n}^L \theta_0\xi N_{n,0} (ds,d\xi) \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{l_n(s)x_n(s-)} \int_{\varepsilon_n}^L
\theta_1\xi \tilde N_{n,1} (ds,du,d\xi),
 \end{eqnarray}
where the processes $\{x_n, B_n, \sigma_n, b_n, \beta_n, l_n\}$
and the random measures $\{N_{n,0}, N_{n,1}\}$ are distributed as
$\{x_n, B, \sigma_n, b_n, \beta_n, l_n\}$ and $\{N_0, N_1\}$ in
(\ref{5.9}). Moreover, as $n\to \infty$ we have
\begin{itemize}

\item[(a)]
$x_n(\cdot) \to$ a process $x(\cdot)$ a.s.\ by the topology of
$D([0,\infty), \mbb{R}_+)$;

\item[(b)]
$B_n(\cdot) \to$ an $r$-dimensional Brownian motion $B(\cdot)$
a.s.\ by the topology of $C([0,\infty), \mbb{R}^r)$;

\item[(c)]
$\xi N_{n,0}(ds,d\xi) \to \xi N_0(ds,d\xi)$ a.s.\ by the weak
convergence of finite measures on $(0,J] \times (0,L]$ for all
integers $J\ge1$, where $N_0(ds,dy)$ is a Poisson random measure
on $(0,\infty)^2$ with intensity $dsm(dy)$;

\item[(d)]
$\xi^2 N_{n,1}(ds,du,d\xi) \to \xi^2 N_1(ds,du,d\xi)$ a.s.\ by the
weak convergence of finite measures on $(0,J]^2 \times (0,L]$ for
all integers $J\ge1$, where $N_1(ds,du,dy)$ is a Poisson random
measure on $(0,\infty)^3$ with intensity $dsdu\mu(dy)$;

\item[(e)]
$\sigma_n(\cdot)$, $b_n(\cdot)$, $\beta_n(\cdot)$ and $l_n(\cdot)$
converge a.s.\ to processes $\sigma(\cdot)$, $b(\cdot)$,
$\beta(\cdot)$ and $l(\cdot)$, respectively, by the topology of
$L^2([0,J], \lambda)$ for each integer $J\ge1$;

\item[(f)]
for each $1\le j\le r$ it holds that
 \begin{eqnarray}\label{5.15}
y_{n,j}(t) := \int_0^t\sigma_{n,j}(s) \sqrt{2x_n (s)} dB_{n,j}(s)
 \to~\mbox{a process $y_j(t)$ a.s.}
 \end{eqnarray}
by the topology of $C([0,\infty), \mbb{R})$;

\item[(g)]
it holds that
 \begin{eqnarray}\label{5.16}
z_n(t) := \int_0^t \int_0 ^{l_n(s)x_n(s-)} \int_{\varepsilon_n}^L
\xi \tilde N_{n,1}(ds,du,d\xi)
 \to~\mbox{a process $z(t)$ a.s.}
 \end{eqnarray}
by the topology of $D([0,\infty), \mbb{R})$.

\end{itemize}
The existence of such a probability space follows by the Skorokhod
representation; see, e.g., Ethier and Kurtz (1986, p.102). Indeed,
we can and do assume that the probability space is constructed so
that the above assertions hold simultaneously for all integers
$L\ge1$. Of course, the processes $\{x_n(\cdot), y_{n,j}(\cdot),
z_n(\cdot), x(\cdot), y_j(\cdot), z(\cdot)\}$ all depend on
$L\ge1$. We suppress this dependence for simplicity of the
notation. Note also that the processes $\{B(\cdot), \sigma(\cdot),
b(\cdot), \beta(\cdot), l(\cdot)\}$ and the random measures
$\{N_0(ds,d\xi), N_1(ds,du,d\xi)\}$ are distributed as those in
(\ref{5.1}).


\blemma\label{l5.2} For each $t\ge0$ we have a.s.\
 \begin{eqnarray}\label{5.17}
y_j(t) = \int_0^t\sigma_j(s) \sqrt{2x (s)} dB_j(s)
 \end{eqnarray}
and
 \begin{eqnarray}\label{5.18}
z(t) = \int_0^t \int_0 ^{l(s)x(s-)}\int_0^L \xi \tilde
N_{1,n}(ds,du,d\xi).
 \end{eqnarray}
\elemma

\proof For $m\ge1$ let $\tau_m = \inf\{t>0: x(t)\ge m$ or $x_n(t)\ge
m$ for some $n\ge 1\}$. Let $\{q_{k,m} (\cdot)\}$ be a sequence of
non-negative simple processes bounded above by $m$ such that
$q_{k,m}(s) \to x(s)1_{\{s\le \tau_m\}}$ a.s.\ by the topology of
$L^2([0,T], \lambda)$ for each $T\ge0$. By (\ref{5.15}) we have
 \begin{eqnarray}\label{5.19}
y_{n,j}(t\land \tau_m) = \int_0^t \sigma_{k,j}(s)
\sqrt{2q_{k,m}(s)}dB_{n,j}(s) + \eta_{n,k,m,j}(t),
 \end{eqnarray}
where
 \begin{eqnarray*}
\eta_{n,k,m,j}(t) \ar=\ar \int_0^t \sigma_{n,j}(s) \Big(\sqrt{2x_n
(s)} - \sqrt{2x (s)}\Big) 1_{\{s\le \tau_m\}} dB_{n,j}(s)   \\
 \ar \ar
+ \int_0^t \sigma_{n,j}(s) \Big(\sqrt{2x (s)} -
\sqrt{2q_{k,m}(s)}\Big) 1_{\{s\le \tau_m\}} dB_{n,j}(s) \\
 \ar \ar
+ \int_0^t (\sigma_{n,j}(s) - \sigma_{k,j}(s)) \sqrt{2q_{k,m}(s)}
dB_{n,j}(s).
 \end{eqnarray*}
It is simple to see that
 \begin{eqnarray}\label{5.20}
\mbf{E}[\eta_{n,k,m,j}^2(t)]
 \ar\le\ar
6\bar\sigma(t)^2 \int_0^t \mbf{E}\Big[\Big(\sqrt{x_n (s)} -
\sqrt{x (s)}\Big)^21_{\{s\le \tau_m\}}\Big] ds \nnm \\
 \ar \ar
+\, 6\bar\sigma(t)^2 \int_0^t \mbf{E}\Big[\Big(\sqrt{x (s)} -
\sqrt{q_{k,m}(s)}\Big)^21_{\{s\le \tau_m\}}\Big] ds \nnm  \\
 \ar \ar
+\, 6m\int_0^t \mbf{E}\left[(\sigma_{n,j}(s) - \sigma_{k,j}
(s))^2\right] ds.
 \end{eqnarray}
In view of (\ref{5.19}), the limit $\eta_{k,m,j}(t) =
\lim_{n\to\infty} \eta_{n,k,m,j}(t)$ exists and
 \begin{eqnarray}\label{5.21}
y_j(t\land \tau_m) = \int_0^t \sigma_{k,j}(s)
\sqrt{2q_{k,m}(s)}dB_j(s) + \eta_{k,m,j}(t).
 \end{eqnarray}
By (\ref{5.20}) and Fatou's lemma,
 \begin{eqnarray*}
\mbf{E}[\eta_{k,m,j}^2(t)]
 \ar\le\ar
6\bar\sigma(t)^2 \int_0^t \mbf{E}\Big[\Big(\sqrt{x (s)} -
\sqrt{q_{k,m}(s)}\Big)^2 1_{\{s\le \tau_m\}}\Big] ds \nnm  \\
 \ar \ar
+\, 6m\int_0^t \mbf{E}\left[(\sigma_j(s) - \sigma_{k,j}
(s))^2\right] ds,
 \end{eqnarray*}
which goes to zero as $k\to\infty$. Now we can take the limit in
(\ref{5.21}) to obtain
 \begin{eqnarray*}
y_j(t\land \tau_m) = \int_0^t \sigma_j(s) \sqrt{2x(s)}1_{\{s\le
\tau_m\}}dB_j(s).
 \end{eqnarray*}
Then we have (\ref{5.17}) since $\tau_m \to \infty$ as $m \to
\infty$. The equality (\ref{5.18}) can be proved using similar
ideas. \qed



\blemma\label{l5.3} For each $t\ge0$ we have a.s.\
 \begin{eqnarray}\label{5.22}
x(t) \ar=\ar x(0) + \int_0^t(b(s) + \beta (s)x(s))ds \nnm \\
 \ar \ar
+ \sum_{j=1}^r\int_0^t \sigma_j(s) \sqrt{2x(s)}dB_j(s)
+ \int_0^t\int_0^L \theta_0\xi N_0 (ds,d\xi) \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{l(s)x(s-)} \int_0^L \theta_1\xi \tilde N_1
(ds,du,d\xi).
 \end{eqnarray}
\elemma

\proof By dominated convergence we have a.s.\
 \begin{eqnarray*}
\lim_{n\to \infty}\int_0^t(b_n(s) + \beta_n(s) x_n(s))ds
 =
\int_0^t(b(s) + \beta (s)x(s))ds.
 \end{eqnarray*}
On the other hand, it is easy to show that a.s.\
 \begin{eqnarray*}
\lim_{n\to \infty} \int_0^t\int_{\varepsilon_n}^L \xi N_{n,0}
(ds,d\xi)
 =
\int_0^t\int_0^L \xi N_0 (ds,d\xi).
 \end{eqnarray*}
Then (\ref{5.22}) follows from (\ref{5.14}) and Lemma~\ref{l5.2}.
\qed

\btheorem\label{t5.2} There is a solution $x(\cdot)$ of
(\ref{5.1}). \etheorem


\proof By Lemma~\ref{l5.3} there is a sequence of processes
$\{x_k(\cdot)\}$ satisfying the equations
 \begin{eqnarray*}
x_k(t) \ar=\ar x(0) + \int_0^t(b(s) + \beta(s)x_k(s))ds +
\sum_{j=1}^r
\int_0^t \sigma_j(s)\sqrt{2x_k(s)}dB_j(s) \nnm \\
 \ar \ar
+ \int_0^t\int_0^{k} \theta_0\xi N_0 (ds,d\xi) + \int_0^t \int_0
^{l(s)x_k(s-)} \int_0^1 \theta_1\xi \tilde N_1 (ds,du,d\xi) \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{l(s)x_k(s-)} \int_1^{k} \theta_1\xi N_1
(ds,du,d\xi) \nnm \\
 \ar \ar
- \int_1^\infty \theta_1\xi \mu(d\xi)\int_0^t l(s)x_k(s)ds.
 \end{eqnarray*}
The pathwise uniqueness of solutions holds for those equations by
Theorem~\ref{t5.1}. Then it is easy to show that $x_k(\cdot)$ is
increasing in $k\ge1$. Let $x(\cdot) := \lim_{k\to \infty}
x_k(\cdot)$. By Propositions~\ref{p5.1} and~\ref{p5.2} and Fatou's
lemma we conclude that $\mbf{E} [\sup_{0\le s\le T}x(s)] < \infty$
for each $T\ge0$. Then we infer that $x(\cdot)$ satisfies
(\ref{5.1}). \qed


In particular, if $\{\sigma, b, \beta,l\}$ are all deterministic
constants, Theorems~\ref{t5.1} and~\ref{t5.2} imply that
(\ref{5.1}) has a unique \textit{strong solution} $x(\cdot)$ and
the solution is a strong Markov process; see, e.g., Ikeda and
Watanabe (1989, pp.163-166 and p.215). By It\^o's formula, we find
that $x(\cdot)$ has generator $L$ determined by
 \begin{eqnarray}\label{5.23}
L f(x)
 \ar=\ar
\alpha xf^{\prime\prime}(x) + (b + \beta x) f^\prime(x)
+ \int_0^\infty \Delta_{\theta_0\xi} f(x) m(d\xi) \nnm \\
 \ar \ar
+\, \int_0^\infty (\Delta_{\theta_1\xi} f(x) - f^\prime(x)
\theta_1\xi)lx\mu(d\xi),
 \end{eqnarray}
where $\alpha = \sum_{j=1}^r\sigma_j^2$. Then $x(\cdot)$ is a
CBI-process; see Kawazu and Watanabe (1971) and Shiga and Watanabe
(1973). The stochastic equation (\ref{5.1}) gives explicit
representations of the two types of jumps of the process in terms
of the Poisson random measures $N_0(ds, d\xi)$ and $N_1(ds, du,
d\xi)$. As far as we know, this characterization of the
CBI-process has not appeared in the literature. In the general
case, the solution of (\ref{5.1}) can be regarded as a
\textit{generalized CBI-process} with random parameters.





%%%%%%%%%%%%%%%%%%%% (Section 6) %%%%%%%%%%%%%%%%%%%%%


\section{Constructions of the two-dimensional processes}

\setcounter{equation}{0}

Based on the results in the last section, we here construct two
classes of Markov processes as strong solutions of stochastic
integral equations. The first class is the regular affine process
and the second is the catalytic CBI-process. The characterizations
of those processes in terms of stochastic equations play the key
role in the study of the limit theorems in the next section. To
simplify the discussions, we impose some conditions on the jumps
so that the processes possess finite first moments.


\bdefinition\label{d6.1} A set of parameters $(a, (\alpha_{ij}),
(b_1,b_2), (\beta_{ij}), m, \mu)$ is called \textit{admissible} if
\begin{itemize}

\item[(i)]
$a\in \mbb{R}_+$ is a constant;

\item[(ii)]
$(\alpha_{ij})$ is a symmetric non-negative definite $(2\times
2)$-matrix;

\item[(iii)]
$(b_1,b_2)\in D$ is a vector;

\item[(iv)]
$(\beta_{ij})$ is a $(2\times 2)$-matrix with $\beta_{12}=0$;

\item[(v)]
$m(d\xi)$ is a $\sigma$-finite measure on $D$ supported by $D
\setminus \{0\}$ such that
 \begin{eqnarray*}
\int_D [l_1(\xi_1) + l_{12}(\xi_2)] m(d\xi) < \infty;
 \end{eqnarray*}

\item[(vi)]
$\mu(d\xi)$ is a $\sigma$-finite measure on $D$ supported by $D
\setminus \{0\}$ such that
 \begin{eqnarray*}
\int_D [l_{12}(\xi_1) + l_{12}(\xi_2)] \mu(d\xi) < \infty.
 \end{eqnarray*}

\end{itemize}
\edefinition


\btheorem\label{t6.1} {\rm (Duffie \textit{et al.}\, 2003)}
Suppose that $(a, (\alpha_{ij}), (b_1,b_2), (\beta_{ij}), m, \mu)$
is a set of admissible parameters. For $u = (u_1,u_2)\in U$ set
 \begin{eqnarray}\label{6.1}
F(u)
 \ar=\ar
b_1u_1 + b_2u_2 + au_2^2 + \int_D (e^{\<u,\xi\>} - 1 - \xi_2u_2)
m(d\xi)
 \end{eqnarray}
and
 \begin{eqnarray}\label{6.2}
R(u)
 \ar=\ar
\beta_{11}u_1 + \beta_{21}u_2 + \alpha_{11}u_1^2 +
2\alpha_{12}u_1u_2  \nnm \\
 \ar \ar\qquad\quad
+\, \alpha_{22}u_2^2 + \int_D (e^{\<u,\xi\>} - 1 - \xi_1u_1 -
\xi_2u_2) \mu(d\xi).
 \end{eqnarray}
Then there is a unique regular affine semigroup $(P(t))_{t\ge0}$
determined by (\ref{3.11}) where $\psi_2(t,u) = e^{\beta_{22}t}
u_2$, $\psi_1(t,u)$ solves the generalized Riccati equation
 \begin{eqnarray}\label{6.3}
\frac{\partial}{\partial t}\psi_1(t,u) =
R(\psi_1(t,u),e^{\beta_{22}t}u_2), \qquad \psi_1(0,u) = u_1,
 \end{eqnarray}
and
 \begin{eqnarray}\label{6.4}
\phi(t,u) = \int_0^t F(\psi_1(s,u),e^{\beta_{22}s}u_2)ds.
 \end{eqnarray}
\etheorem


Let $(a, (\alpha_{ij}), (\beta_{ij}), (b_j), m, \mu)$ be a set of
admissible parameters and let $A$ be the generator of the regular
affine semigroup $(P(t))_{t\ge0}$ characterized by
Theorem~\ref{t6.1}. It is not hard to show that
 \begin{eqnarray}\label{6.5}
Af(x)
 \ar=\ar
\alpha_{11}x_1f^{\prime\prime}_{11}(x) +
2\alpha_{12}x_1f^{\prime\prime}_{12}(x) +
\alpha_{22}x_1f^{\prime\prime}_{22}(x) +
af^{\prime\prime}_{22}(x) \nnm \\
 \ar \ar
+\, (b_1+\beta_{11}x_1)f^\prime_1(x) + (b_2 +
\beta_{21}x_1 + \beta_{22}x_2)f^\prime_2(x) \nnm \\
 \ar \ar
+ \int_D (\Delta_\xi f(x) - f^\prime_2(x)
\xi_2)m(d\xi) \nnm \\
 \ar \ar
+ \int_D (\Delta_\xi f(x)- \<\nabla f(x),\xi\>)x_1\mu(d\xi)
 \end{eqnarray}
for $f\in C^2(D)$, where $\nabla f(x) = (f^\prime_1(x),
f^\prime_2(x))$.

Let $\sigma_0 = \sqrt{a}$ and let $(\sigma_{ij})$ be a $(2\times
2)$-matrix satisfying $(\alpha_{ij}) = (\sigma_{ij})
(\sigma_{ij})^\tau$. Let $(\Omega, {\mcr F}, {\mcr F}_t, \mbf{P})$
be a filtered probability space satisfying the usual hypotheses.
Suppose that on this probability space the following objects are
defined:
\begin{itemize}

\item[(a)]
a $3$-dimensional Brownian motion $B(\cdot) = (B_0(\cdot),
B_1(\cdot), B_2(\cdot))$;

\item[(b)]
a Poisson random measure $N_0(ds,d\xi)$ on $(0,\infty) \times D$
with intensity $dsm(d\xi)$;

\item[(c)]
a Poisson random measure $N_1(ds,du,d\xi)$ on $(0,\infty)^2 \times
D$ with intensity $dsdu\mu(d\xi)$.

\end{itemize}
We assume that $B(\cdot)$, $N_0$ and $N_1$ are independent of each
other and $({\mcr F}_t)_{t\ge0}$ is the augmented natural
filtration generated by them. Let $x(0)$ be a non-negative ${\mcr
F}_0$-measurable random variable defined on $(\Omega, {\mcr F},
{\mcr F}_t, \mbf{P})$. By Theorems~\ref{t5.1} and~\ref{t5.2} there
is a unique strong solution $x(\cdot)$ of
 \begin{eqnarray}\label{6.6}
x(t) \ar=\ar x(0) + \int_0^t(b_1 + \beta_{11} x(s))ds +
\int_0^t\sigma_{11}\sqrt{2x(s)} dB_1(s) \nnm \\
 \ar \ar
+ \int_0^t\sigma_{12}\sqrt{2x(s)} dB_2(s) + \int_0^t\int_D
\xi_1 N_0(ds,d\xi) \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{x(s-)}\int_D \xi_1 \tilde N_1(ds,du,d\xi).
 \end{eqnarray}
As explained at the end of the last section, $x(\cdot)$ is a
CBI-process. In addition, let $z(0)$ be an ${\mcr F}_0$-measurable
random variable defined on $(\Omega, {\mcr F}, {\mcr F}_t,
\mbf{P})$. We consider the equation
 \begin{eqnarray}\label{6.7}
z(t) \ar=\ar z(0) + \int_0^t(b_2 + \beta_{21} x(s) + \beta_{22}
z(s))ds + \int_0^t\sqrt{2}\sigma_0dB_0(s) \nnm \\
 \ar \ar
+ \int_0^t\sigma_{21}\sqrt{2x(s)} dB_1(s)
+ \int_0^t\sigma_{22}\sqrt{2x(s)} dB_2(s)  \nnm \\
 \ar \ar
+ \int_0^t\int_D \xi_2  \tilde N_0(ds,d\xi) + \int_0^t \int_0
^{x(s-)}\int_D \xi_2 \tilde N_1(ds,du,d\xi).
 \end{eqnarray}


\btheorem\label{t6.2} The equation system (\ref{6.6}) and
(\ref{6.7}) has a unique strong solution $(x(\cdot), z(\cdot))$.
Moreover, $(x(\cdot), z(\cdot))$ is an affine Markov process with
generator $A$ given by (\ref{6.5}). \etheorem

\proof By It\^o's formula it is not hard to show that
 \begin{eqnarray}\label{6.8}
z(t) \ar=\ar e^{\beta_{22}t}z(0) + e^{\beta_{22}t} \int_0^t
e^{-\beta_{22}s}(b_2 + \beta_{21} x(s))ds  \nnm \\
 \ar \ar
+\, e^{\beta_{22}t} \int_0^t\sqrt{2}\sigma_0 e^{-\beta_{22}s}
dB_0(s) + e^{\beta_{22}t}\int_0^t\sigma_{21}e^{-\beta_{22}s}
\sqrt{2x(s)}dB_1(s)  \nnm \\
 \ar \ar
+\, e^{\beta_{22}t}\int_0^t\sigma_{22} e^{-\beta_{22}s}
\sqrt{2x(s)} dB_2(s) + e^{\beta_{22}t}\int_0^t \int_D
e^{-\beta_{22}s}\xi_2  \tilde N_0(ds,d\xi)  \nnm \\
 \ar \ar
+\, e^{\beta_{22}t}\int_0^t \int_0 ^{x(s-)}\int_D
e^{-\beta_{22}s}\xi_2 \tilde N_1(ds,du,d\xi)
 \end{eqnarray}
defines a solution of (\ref{6.7}) and conversely any solution of
(\ref{6.7}) must be given by (\ref{6.8}). The uniqueness implies
the strong Markov property of $(x(\cdot), z(\cdot))$. By It\^o's
formula, we find that the Markov process $(x(\cdot), z(\cdot))$
has generator $A$. \qed


Now suppose that $b_2\ge 0$ and $m_2(l_1)< \infty$, where $m_2$
denotes the projection of $m$ to $\mbb{R}$. Let $D_+ = \mbb{R}_+^2
\subset D$. Given another non-negative ${\mcr F}_0$-measurable
random variable $y(0)$ defined on $(\Omega, {\mcr F}, {\mcr F}_t,
\mbf{P})$, we consider the equation
 \begin{eqnarray}\label{6.9}
y(t) \ar=\ar y(0) + \int_0^t(b_2 + \beta_{21} x(s)y(s) +
\beta_{22}
y(s))ds + \int_0^t\sigma_0\sqrt{2y(s)} dB_0(s) \nnm \\
 \ar \ar
+ \int_0^t\sigma_{21}\sqrt{2x(s)y(s)} dB_1(s)
+ \int_0^t\sigma_{22}\sqrt{2x(s)y(s)} dB_2(s)  \nnm \\
 \ar \ar
+ \int_0^t\int_{D_+} \xi_2 N_0(ds,d\xi) + \int_0^t \int_0
^{lx(s-)y(s-)}\int_{D_+} \xi_2 \tilde N_1(ds,du,d\xi).
 \end{eqnarray}
A solution $y(\cdot)$ of (\ref{6.9}) can be regarded as a
generalized CBI-process with random parameters governed by the
process $x(\cdot)$. Following Dawson and Fleischmann (1997), we
shall call the pair $(x(\cdot), y(\cdot))$ a \textit{catalytic
CBI-process}, where $x(\cdot)$ is the \textit{catalyst process}
and $y(\cdot)$ is the \textit{reactant process}.


\btheorem\label{t6.3} The equation system (\ref{6.6}) and
(\ref{6.9}) has a unique strong solution $(x(\cdot), y(\cdot))$.
\etheorem

\proof It suffices to consider the case where the initial states
$x(0)$ and $y(0)$ are deterministic. For $n\ge x(0)$ let $\tau_n =
\inf\{s>0: x(s)\ge n\}$ and $x_n(t) = x(t \land \tau_n)$. By
Theorems~\ref{t5.1} and~\ref{t5.2}, there is a unique strong
solution $(x(\cdot), y_n(\cdot))$ of the equation system formed by
(\ref{6.6}) and
 \begin{eqnarray}\label{6.10}
y_n(t) \ar=\ar y(0) + \int_0^t(b_2 + \beta_{21} x_n(s)y_n(s)
+ \beta_{22}y_n(s))ds   \nnm \\
 \ar \ar
+ \int_0^t\sigma_0\sqrt{2y_n(s)} dB_0(s) + \int_0^t\sigma_{21}
\sqrt{2x_n(s)y_n(s)} dB_1(s)   \nnm \\
 \ar \ar
+ \int_0^t\sigma_{22}\sqrt{2x_n(s)y_n(s)} dB_2(s) + \int_0^t
\int_{D_+} \xi_2 N_0(ds,d\xi)   \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{l x_n(s-)y_n(s-)}\int_{D_+} \xi_2 \tilde
N_1(ds,du,d\xi).
 \end{eqnarray}
By the uniqueness, for any $n\ge m\ge x(0)$ the two processes
$y_n(t \land \tau_m)$ and $y_m(t\land\tau_m)$ are
indistinguishable. Since $\tau_n \to \infty$ as $n\to \infty$, it
is easy to see that $y(t) := \lim_{n\to \infty} y_n(t)$ is the
unique solution of (\ref{6.9}). \qed


By Theorem~\ref{t6.3}, the catalytic CBI-process $(x(\cdot),
y(\cdot))$ is a strong Markov process with state space $D_+$. Let
$D_- = \mbb{R}_+ \times \mbb{R}_-$. By It\^o's formula we find
that $(x(\cdot),y(\cdot))$ has generator $L$ determined by
 \begin{eqnarray}\label{6.11}
L f(x)
 \ar=\ar
\alpha_{11}x_1f^{\prime\prime}_{11}(x) + 2\alpha_{12} x_1
\sqrt{x_2}f^{\prime\prime}_{12}(x) + \alpha_{22}x_1x_2
f^{\prime\prime}_{22}(x) + ax_2f^{\prime\prime}_{22}(x) \nnm \\
 \ar \ar
+\, (b_1+\beta_{11}x_1)f^\prime_1(x) + (b_2 + \beta_{21}x_1x_2 +
\beta_{22}x_2)f^\prime_2(x) \nnm \\
 \ar \ar
+ \int_{D_+} \Delta_\xi f(x) m(d\xi)
+ \int_{D_-} \Delta_{(\xi_1,0)} f(x) m(d\xi) \nnm \\
 \ar \ar
+ \int_{D_+} [\Delta_\xi f(x) - \<\nabla
f(x),\xi\>](x_1\land lx_1x_2)\mu(d\xi) \nnm \\
 \ar \ar
+ \int_{D_+}[\Delta_{(\xi_1,0)} f(x) - f^\prime_1(x)
\xi_1][x_1 - (x_1\land lx_1x_2)]\mu(d\xi) \nnm \\
 \ar \ar
+ \int_{D_+}[\Delta_{(0,\xi_2)} f(x) - f^\prime_2(x)
\xi_2][lx_1x_2 - (x_1\land lx_1x_2)]\mu(d\xi) \nnm \\
 \ar \ar
+ \int_{D_-} [\Delta_{(\xi_1,0)} f(x) - f^\prime_1(x)
\xi_1]x_1\mu(d\xi).
 \end{eqnarray}






%%%%%%%%%%%%%%%%%%%% (Section 7) %%%%%%%%%%%%%%%%%%%%%


\section{Fluctuation limit theorems}

\setcounter{equation}{0}

In this section, we show that an affine process arises naturally
from a limit theorem based on catalytic CBI-processes. By virtue of
the characterizations given in the last section, we can establish
the limit theorem in the sense of convergence in probability. Let
$(\Omega, {\mcr F}, {\mcr F}_t, \mbf{P})$ be a filtered probability
space satisfying the usual hypotheses and let $B(\cdot)$, $N_0$ and
$N_1$ be given as in the last section. Let $(a, (\alpha_{ij}),
(\beta_{ij}), (b_j), m, \mu)$ be admissible parameters with
$\beta_{22}< 0$ and $m_2(l_1)< \infty$. Let $\sigma_0 = \sqrt{a}$
and let $(\sigma_{ij})$ be a $(2\times 2)$-matrix satisfying
$(\alpha_{ij}) = (\sigma_{ij}) (\sigma_{ij})^\tau$.

Let $\{\theta_k\}$ be a sequence such that $1\le \theta_k \to
\infty$ as $k\to\infty$. For each $k \ge 1$ let $y_k(0)$ be an
${\mcr F}_0$-measurable random variable and let $y_k(\cdot)$ be
the solution of
 \begin{eqnarray}\label{7.1}
y_k(t) \ar=\ar y_k(0) + \int_0^t (-\theta_k\beta_{22} + \beta_{21}
x(s)\tilde y_k(s) + b_2\tilde y_k(s) + \beta_{22}y_k(s))ds
\nnm \\
 \ar \ar
+ \int_0^t\sigma_0\sqrt{2\tilde y_k(s)} dB_0(s)+
\int_0^t\sigma_{21}\sqrt{2x(s)\tilde y_k(s)} dB_1(s)
\nnm \\
 \ar \ar
+ \int_0^t\sigma_{22}\sqrt{2x(s)\tilde y_k(s)} dB_2(s) +
\int_0^t\int_{D_+} \xi_2 \tilde N_0(ds,d\xi)  \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{x(s-)\tilde y_k(s-)} \int_{D_+} \xi_2 \tilde
N_1(ds,du,d\xi),
 \end{eqnarray}
where $\tilde y_k(t) = y_k(t)/\theta_k$ and $x(\cdot)$ is defined by
(\ref{6.6}). When $\theta_k$ is sufficiently large, (\ref{7.1}) is
essentially a special form of (\ref{6.9}). Then the pair $(x(\cdot),
y_k(\cdot))$ is a catalytic CBI-process. Set $z_k(t) = y_k(t) -
\theta_k$.

\btheorem\label{t7.1} Suppose that $z(0)$ is an ${\mcr
F}_0$-measurable random variable such that $\mbf{E}[|z(0)|] <
\infty$ and
 \begin{eqnarray}\label{7.2}
\lim_{k\to \infty}\mbf{E}[|z_k(0) - z(0)|] = 0.
 \end{eqnarray}
Then $z_k(\cdot)$ converges in probability by the topology of
$D([0,\infty), \mbb{R})$ to the solution $z(\cdot)$ of
 \begin{eqnarray}\label{7.3}
z(t) \ar=\ar z(0) + \int_0^t (b_2 + \beta_{21}x(s) +
\beta_{22}z(s))ds
+ \int_0^t\sqrt{2}\sigma_0 dB_0(s) \nnm \\
 \ar \ar
+ \int_0^t \sigma_{21}\sqrt{2x(s)} dB_1(s) + \int_0^t \sigma_{22}
\sqrt{2x(s)} dB_2(s)  \nnm \\
 \ar \ar
+ \int_0^t \int_{D_+} \xi_2 \tilde N_0(ds,d\xi) + \int_0^t \int_0
^{x(s-)} \int_{D_+} \xi_2 \tilde N_1(ds,du,d\xi).
 \end{eqnarray}
\etheorem


\proof {From} (\ref{7.1}) we get
 \begin{eqnarray}\label{7.4}
\tilde y_k(t) \ar=\ar \tilde y_k(0) + \int_0^t (-\beta_{22} +
\theta_k^{-1} \beta_{21}x(s)\tilde y_k(s) + \theta_k^{-1} b_2
\tilde y_k(s) + \beta_{22}\tilde y_k(s))ds \nnm \\
 \ar \ar
+ \int_0^t\theta_k^{-1}\sigma_0\sqrt{2\tilde y_k(s)} dB_0(s)+
\int_0^t\theta_k^{-1}\sigma_{21}\sqrt{2x(s)\tilde y_k(s)} dB_1(s)
 \nnm \\
 \ar \ar
+ \int_0^t\theta_k^{-1}\sigma_{22}\sqrt{2x(s)\tilde y_k(s)}
dB_2(s) + \int_0^t\int_{D_+} \theta_k^{-1}\xi_2 \tilde
N_0(ds,d\xi)  \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{x(s-)\tilde y_k(s-)} \int_{D_+}
\theta_k^{-1}\xi_2 \tilde N_1(ds,du,d\xi).
 \end{eqnarray}
For $n\ge 1$ let $\tau_n = \inf\{s>0: x(s)\ge n\}$. Then $\tau_n \to
\infty$ as $n\to \infty$. Under condition (\ref{7.2}) we clearly
have $\sup_{k\ge1}\mbf{E}[\tilde y_k(0)] < \infty$. By
Proposition~\ref{p5.2},
 \begin{eqnarray}\label{7.5}
\sup_{k\ge1}\mbf{E}\Big[\sup_{0\le s\le T} \tilde y_k(s \land
\tau_n)\Big] < \infty.
 \end{eqnarray}
Let $\eta_{n,k}(t) = \tilde y_k(t\land\tau_n) - 1$. By (\ref{7.4})
and Doob's martingale inequality we get
 \begin{eqnarray}\label{7.6}
\mbf{E}[|\eta_{n,k}(t)|]
 \ar\le\ar
\mbf{E}[|\eta_{n,k}(0)|] + |\beta_{22}|\int_0^t
\mbf{E}[|\eta_{n,k}(s)|]ds \nnm \\
 \ar \ar
+\, \theta_k^{-1}\int_0^t (b_2 + n|\beta_{21}|)\mbf{E}[\tilde
y_k(s\land\tau_n)]ds   \nnm \\
 \ar \ar
+\, \theta_k^{-1}(\sqrt{2}\sigma_0 + \sqrt{2n}\sigma_{21} +
\sqrt{2n}\sigma_{22}) \bigg(\int_0^t \mbf{E}[\tilde y_k
(s\land\tau_n)]ds\bigg)^{\frac{1}{2}} \nnm \\
 \ar \ar
+\, \theta_k^{-1}\sqrt{tm_2(l_{12})} + 2\theta_k^{-1}t
m_2(l_{12}) \nnm \\
 \ar \ar
+\, \theta_k^{-1}\sqrt{n\mu_2(l_{12})}\bigg(\int_0^t\mbf{E}[\tilde
y_k(s\land\tau_n)]ds\bigg)^{\frac{1}{2}} \nnm \\
 \ar \ar
+\, 2n\theta_k^{-1}\mu_2(l_{12})\int_0^t\mbf{E}[\tilde
y_k(s\land\tau_n)]ds,
 \end{eqnarray}
where $m_2$ and $\mu_2$ denote respectively the projections of $m$
and $\mu$ to $\mbb{R}$. An application of Gronwall's inequality
shows that
 \begin{eqnarray}\label{7.7}
\mbf{E}[|\eta_{n,k}(t)|] = \mbf{E}[|\tilde y_k(t\land\tau_n)-1|]
\to 0
 \end{eqnarray}
as $k\to \infty$. {From} (\ref{7.1}) we see that $z_k(\cdot)$
satisfies
 \begin{eqnarray}\label{7.8}
z_k(t) \ar=\ar z_k(0) + \int_0^t (b_2\tilde y_k(s) +
\beta_{21}x(s)
\tilde y_k(s) + \beta_{22}z_k(s))ds \nnm \\
 \ar \ar
+ \int_0^t\sigma_0\sqrt{2\tilde y_k(s)} dB_0(s)+ \int_0^t
\sigma_{21}\sqrt{2x(s)\tilde y_k(s)} dB_1(s) \nnm \\
 \ar \ar
+ \int_0^t\sigma_{22}\sqrt{2x(s)\tilde y_k(s)} dB_2(s) +
\int_0^t\int_{D_+} \xi_2 \tilde N_0(ds,d\xi)  \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{x(s-)\tilde y_k(s-)} \int_{D_+} \xi_2 \tilde
N_1(ds,du,d\xi).
 \end{eqnarray}
Let $\zeta_{n,k}(t) = z_k(t\land\tau_n) - z(t\land\tau_n)$. Then
we have
 \begin{eqnarray*}
\zeta_{n,k}(t) \ar=\ar \zeta_{n,k}(0) + \int_0^{t\land\tau_n} (b_2
+
\beta_{21} x(s)) \eta_{n,k}(s)ds \nnm \\
 \ar \ar
+\, \beta_{22}\int_0^{t\land\tau_n} \zeta_{n,k}(s)ds +
\int_0^{t\land\tau_n} \sqrt{2}\sigma_0(\sqrt{\tilde y_k(s)}-1)
dB_0(s) \nnm \\
 \ar \ar
+ \int_0^{t\land\tau_n}\sigma_{21}\sqrt{2x(s)}(\sqrt{\tilde
y_k(s)}-1) dB_1(s) \nnm \\
 \ar \ar
+ \int_0^{t\land\tau_n}\sigma_{22}\sqrt{2x(s)}(\sqrt{\tilde
y_k(s)}-1) dB_2(s) \nnm \\
 \ar \ar
+ \int_0^{t\land\tau_n} \int_{x(s-)} ^{x(s-)\tilde y_k(s-)}
\int_{D_+} \xi_2 \tilde N_1(ds,du,d\xi).
 \end{eqnarray*}
By It\^o's formula,
 \begin{eqnarray*}
e^{-\beta_{22}t}\zeta_{n,k}(t) \ar=\ar \zeta_{n,k}(0) +
\int_0^{t\land \tau_n} e^{-\beta_{22}s}(b_2 + \beta_{21} x(s))
\eta_{n,k}(s)ds
\nnm \\
 \ar \ar
+\, \int_0^{t\land\tau_n} \sqrt{2}\sigma_0e^{-\beta_{22}s}
(\sqrt{\tilde y_k(s)}-1) dB_0(s) \nnm \\
 \ar \ar
+ \int_0^{t\land\tau_n}\sigma_{21}e^{-\beta_{22}s}\sqrt{2x(s)}
(\sqrt{\tilde y_k(s)}-1) dB_1(s) \nnm \\
 \ar \ar
+ \int_0^{t\land\tau_n}\sigma_{22}e^{-\beta_{22}s}\sqrt{2x(s)}
(\sqrt{\tilde y_k(s)}-1) dB_2(s) \nnm \\
 \ar \ar
+ \int_0^{t\land\tau_n} \int_{x(s-)} ^{x(s-)\tilde y_k(s-)}
\int_{D_+} e^{-\beta_{22}s}\xi_2 1_{\{|\xi_2|\le
1\}} \tilde N_1(ds,du,d\xi) \nnm \\
 \ar \ar
+ \int_0^{t\land\tau_n} \int_{x(s-)} ^{x(s-)\tilde y_k(s-)}
\int_{D_+} e^{-\beta_{22}s}\xi_2 1_{\{|\xi_2|>1\}}
N_1(ds,du,d\xi) \nnm \\
 \ar \ar
- \int_0^{t\land\tau_n}e^{-\beta_{22}s}x(s)[\tilde y_k(s)-1]ds
\int_{D_+} \xi_2 1_{\{|\xi_2|>1\}} \mu(d\xi).
 \end{eqnarray*}
By Doob's inequality we get
 \begin{eqnarray*}
\ar \ar\mbf{E}\Big[\sup_{0\le s\le t}|e^{-\beta_{22}s}
\zeta_{n,k}(s)|\Big]  \nnm \\
 \ar \ar\qquad
\le~\mbf{E}[\zeta_{n,k}(0)] + \int_0^t e^{-\beta_{22}s}(b_2 +
n|\beta_{21}|)\mbf{E}[|\eta_{n,k}(s)|]ds \nnm \\
 \ar \ar\qquad\qquad
+\, 4\sqrt{2}\sigma_0\bigg(\int_0^t e^{-2\beta_{22}s}
\mbf{E}[(\sqrt{\tilde y_k(s\land\tau_n)}-1)^2]
ds\bigg)^{\frac{1}{2}} \nnm \\
 \ar \ar\qquad\qquad
+\, 4\sqrt{2n}(\sigma_{22} + \sigma_{21}) \bigg(\int_0^t
e^{-2\beta_{22}s} \mbf{E}[(\sqrt{\tilde y_k(s\land\tau_n)}-1)^2]
ds\bigg)^{\frac{1}{2}} \nnm \\
 \ar \ar\qquad\qquad
+\, 4\sqrt{n\mu_2(l_{12})}\bigg(\int_0^t e^{-2\beta_{22}s}
\mbf{E}[|\eta_{n,k}(s)|] ds\bigg)^{\frac{1}{2}} \nnm \\
 \ar \ar\qquad\qquad
+\, 2n\mu_2(l_{12})\int_0^t e^{-\beta_{22}s}
\mbf{E}[|\eta_{n,k}(s)|] ds,
 \end{eqnarray*}
where
 \begin{eqnarray*}
\mbf{E}[(\sqrt{\tilde y_k(s\land\tau_n)}-1)^2] \le \mbf{E}[|\tilde
y_k(s\land\tau_n)-1|] = \mbf{E}[|\eta_{n,k}(s)|].
 \end{eqnarray*}
Then (\ref{7.2}) and (\ref{7.7}) imply that
 \begin{eqnarray}\label{7.9}
\mbf{E} \Big[\sup_{0\le s\le t}|e^{-\beta_{22}s}
\zeta_{n,k}(s)|\Big] \to 0
 \end{eqnarray}
as $k\to \infty$. For any $\varepsilon >0$, $\eta>0$ and $T\ge0$
we first choose $n$ so that $\mbf{P}\{\tau_n\le T\} \le
\varepsilon/2$. In view of (\ref{7.9}), there is some $k_0$ so
that
 \begin{eqnarray*}
\mbf{P}\Big\{\sup_{0\le s\le T}|\zeta_{n,k}(s)| \ge \eta\Big\} \le
\eta^{-1} \mbf{E}\Big[\sup_{0\le s\le T}|\zeta_{n,k}(s)|\Big] \le
\varepsilon/2
 \end{eqnarray*}
for every $k\ge k_0$. It then follows that
 \begin{eqnarray*}
\mbf{P}\Big\{\sup_{0\le s\le T}|z_k(s) - z(s)| \ge \eta\Big\} \le
\mbf{P}\{\tau_n\le T\} + \mbf{P}\Big\{\sup_{0\le s\le T}|
\zeta_{n,k} (s)| \ge \eta\Big\} \le \varepsilon.
 \end{eqnarray*}
Then $z_k(\cdot)$ converges to $z(\cdot)$ in probability by the
topology of $D([0,\infty), \mbb{R})$. \qed

Clearly, the pair $(x(\cdot), z(\cdot))$ defined by (\ref{6.6})
and (\ref{7.3}) is an affine process with non-negative jumps. In
other words, Theorem~\ref{t7.1} gives an interpretation of a
particular class of affine processes in terms of catalytic
CBI-processes. To consider general affine processes, we assume the
following decompositions of the parameters:
 \begin{eqnarray}\label{7.10}
\sigma_0 = \sigma_0^+ - \sigma_0^-,~ \sigma_{2j} = \sigma_{2j}^+ -
\sigma_{2j}^-,~ b_2 = b_2^+ - b_2^-,~ \beta_{21} = \beta_{21}^+ -
\beta_{21}^-.
 \end{eqnarray}
Let $x(\cdot)$ be defined by (\ref{6.6}) and let
$y_k^{\pm}(\cdot)$ be the solutions of the equations
 \begin{eqnarray}
y_k^+(t) \ar=\ar y_k^+(0) + \int_0^t (-\theta_k\beta_{22} +
\beta_{21}^+ x(s)\tilde y_k^+(s) + b_2^+\tilde y_k^+(s) +
\beta_{22}y_k^+(s))ds \nnm \\
 \ar \ar
+ \int_0^t\sigma_0^+\sqrt{2\tilde y_k^+(s)} dB_0(s) +
\int_0^t\sigma_{21}^+\sqrt{2x(s)\tilde y_k^+(s)} dB_1(s)
\nnm \\
 \ar \ar
+ \int_0^t\sigma_{22}^+\sqrt{2x(s)\tilde y_k^+(s)} dB_2(s) +
\int_0^t\int_{D_+} \xi_2 \tilde N_0(ds,d\xi)  \nnm \\
 \ar \ar
+ \int_0^t \int_0 ^{x(s-)\tilde y_k^+(s-)} \int_{D_+}
\xi_2 \tilde N_1(ds,du,d\xi), \label{7.11} \\
y_k^-(t) \ar=\ar y_k^-(0) + \int_0^t (-\theta_k\beta_{22} +
\beta_{21}^- x(s)\tilde y_k^-(s) + b_2^-\tilde y_k^-(s) +
\beta_{22}y_k^-(s))ds \nnm \\
 \ar \ar
+ \int_0^t\sigma_0^-\sqrt{2\tilde y_k^-(s)} dB_0(s) +
\int_0^t\sigma_{21}^-\sqrt{2x(s)\tilde y_k^-(s)} dB_1(s)
 \nnm \\
 \ar \ar
+ \int_0^t\sigma_{22}^-\sqrt{2x(s)\tilde y_k^-(s)} dB_2(s) -
\int_0^t\int_{D_-} \xi_2 \tilde N_0(ds,d\xi)  \nnm \\
 \ar \ar
- \int_0^t \int_0 ^{x(s-)\tilde y_k^-(s-)} \int_{D_-} \xi_2 \tilde
N_1(ds,du,d\xi),  \label{7.12}
 \end{eqnarray}
where $\tilde y_k^{\pm}(t) = y_k^{\pm}(t)/\theta_k$. We may regard
$(x(\cdot), y_k^+(\cdot), y_k^-(\cdot))$ as a \textit{catalytic
CBI-process with a pair of reactant processes}. Set $z_k^{\pm}(t)
= y_k^{\pm}(t) - \theta_k$ and $z_k(t) = z_k^+(t) - z_k^-(t) =
y_k^+(t) - y_k^-(t)$.


\btheorem\label{t7.2} Suppose that $z^+(0)$ and $z^-(0)$ are
${\mcr F}_0$-measurable random variables such that
$\mbf{E}[|z^+(0)| + |z^-(0)|] < \infty$ and
 \begin{eqnarray}\label{7.13}
\lim_{k\to \infty}\mbf{E}\Big[|z_k^+(0) - z^+(0)| + |z_k^-(0) -
z^-(0)|\Big] = 0.
 \end{eqnarray}
Then $z_k(\cdot)$ converges in probability by the topology of
$D([0,\infty), \mbb{R})$ to the solution $z(\cdot)$ of
 \begin{eqnarray}\label{7.14}
z(t) \ar=\ar z(0) + \int_0^t (b_2 + \beta_{21}x(s) +
\beta_{22}z(s))ds
+ \int_0^t\sqrt{2}\sigma_0 dB_0(s) \nnm \\
 \ar \ar
+ \int_0^t \sigma_{21}\sqrt{2x(s)} dB_1(s) + \int_0^t \sigma_{22}
\sqrt{2x(s)} dB_2(s)  \nnm \\
 \ar \ar
+ \int_0^t \int_D \xi_2 \tilde N_0(ds,d\xi) + \int_0^t \int_0
^{x(s-)} \int_D \xi_2 \tilde N_1(ds,du,d\xi),
 \end{eqnarray}
where $z(0) = z^+(0) - z^-(0)$. \etheorem


\proof By Theorem~\ref{t7.1}, the sequence $(z_k^+(\cdot),
z_k^-(\cdot))$ converges in probability by the topology of
$D([0,\infty), \mbb{R}^2)$ to the solution $(z^+(\cdot),
z^-(\cdot))$ of
 \begin{eqnarray*}
z^+(t) \ar=\ar z^+(0) + \int_0^t (b_2^+ + \beta_{21}^+ x(s) +
\beta_{22}z^+(s))ds + \int_0^t \sqrt{2}\sigma_0^+ dB_0(s)
\nnm \\
 \ar \ar
+ \int_0^t\sigma_{21}^+ \sqrt{2x(s)} dB_1(s) +
\int_0^t\sigma_{22}^+\sqrt{2x(s)} dB_2(s)  \nnm \\
 \ar \ar
+ \int_0^t \int_{D_+}\xi_2 \tilde N_0(ds,d\xi) + \int_0^t \int_0
^{x(s-)} \int_{D_+} \xi_2 \tilde N_1(ds,du,d\xi), \\
z^-(t) \ar=\ar z^-(0) + \int_0^t (b_2^- + \beta_{21}^- x(s) +
\beta_{22}z^-(s))ds + \int_0^t \sqrt{2}\sigma_0^- dB_0(s)
\nnm \\
 \ar \ar
+ \int_0^t \sigma_{21}^- \sqrt{2x(s)} dB_1(s) +
\int_0^t\sigma_{22}^-\sqrt{2x(s)} dB_2(s)  \nnm \\
 \ar \ar
- \int_0^t\int_{D_-} \xi_2 \tilde N_0(ds,d\xi) - \int_0^t \int_0
^{x(s-)} \int_{D_-} \xi_2 \tilde N_1(ds,du,d\xi).
 \end{eqnarray*}
It is simple to check that $z(\cdot) = z^+(\cdot) - z^-(\cdot)$
solves (\ref{6.7}). That proves the theorem. \qed


The pair $(x(\cdot), z(\cdot))$ defined by (\ref{6.6}) and
(\ref{7.14}) is an affine process with admissible parameters
$(\sigma_0^2, (\alpha_{ij}), (\beta_{ij}), (b_j), m, \mu)$. Then
the above theorem establishes a connection between catalytic
CBI-processes and affine processes. This result is of interest
since the studies of catalytic branching processes and affine
processes have been undergoing rapid developments in recent years
with rather different motivations; see, e.g., Dawson and
Fleischmann (2000) and Duffie \textit{et al.}\ (2003).

\bigskip

\textbf{Acknowledgments.} We would like to thank Professors
T.\,Shiga and S.\,Watanabe for helpful discussions and Professor
Z.Q.\,Chen for useful comments on the literature. We are grateful
to the two referees for pointing out a number of typos.






%%%%%%%%%%%%%%%%%%%% (Referenes) %%%%%%%%%%%%%%%%%%%%%


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\bigskip

\noindent School of Mathematics and Statistics \\
Carleton University \\
1125 Colonel By Drive \\
Ottawa, Canada K1S 5B6 \\
E-mail: {\tt ddawson@math.carleton.ca} \\
URL: {\tt http://lrsp.carleton.ca/directors/dawson/}

\bigskip

\noindent School of Mathematical Sciences \\
Beijing Normal University \\
Beijing 100875, P.\,R.\ China \\
E-mail: {\tt lizh@email.bnu.edu.cn} \\
URL: {\tt http://math.bnu.edu.cn/\~{}lizh/}


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