# CBI-time-changed Lévy processes

Claudio Fontana  Department of Mathematics “Tullio Levi Civita”, University of
Padova (Italy<EMAIL_ADDRESS>,  Alessandro Gnoatto  Department of
Economics, University of Verona (Italy<EMAIL_ADDRESS>and
Guillaume Szulda  CERMICS, Ecole des Ponts, Marne-la-Vallée (France).
<EMAIL_ADDRESS>
###### Abstract.

We introduce and study the class of CBI-time-changed Lévy processes (CBITCL),
obtained by time-changing a Lévy process with respect to an integrated
continuous-state branching process with immigration (CBI). We characterize
CBITCL processes as solutions to a certain stochastic integral equation and
relate them to affine stochastic volatility processes. We provide a complete
analysis of the time of explosion of exponential moments of CBITCL processes
and study their asymptotic behavior. In addition, we show that CBITCL
processes are stable with respect to a suitable class of equivalent changes of
measure. As illustrated by some examples, CBITCL processes are flexible and
tractable processes with a significant potential for applications in finance.

###### Key words and phrases:

Branching process; change of time; affine process; stochastic volatility;
moment explosion.

###### 2010 Mathematics Subject Classification:

60G44, 60G51, 60H20, 60J25, 60J80, 91G20, 91G30.

_Acknowledgements:_ The authors are thankful to two anonymous Reviewers for
their constructive remarks which helped to improve the paper. C.F. is grateful
to the Europlace Institute of Finance for financial support to this work. G.S.
acknowledges hospitality and financial support from the University of Verona,
where part of this work has been conducted. This work is part of the project
BIRD 2019 “Term structure dynamics in interest rate and energy markets:
modelling and numerics” and of the project STARS@UNIPD - PRISMA (Probabilistic
Methods for Information in Security Markets), both funded by the University of
Padova.

##  1\. Introduction

Since their introduction in [KW71], continuous-state branching processes with
immigration (CBI processes) have represented a major topic of research in the
theory of stochastic processes (we refer to [Li20] for a recent overview of
some of the main developments in the field). CBI processes belong to the class
of affine processes (see [DFS03]) and, due to their analytical tractability,
have found important applications in mathematical finance, especially in
interest rate modelling (see [Fil01]).
In recent years, CBI processes have attracted a renewed interest in financial
modelling, due to their capability of reproducing empirical features of
financial time series such as volatility clustering and self-exciting jumps.
In particular, self-exciting CBI processes have been exploited for the
construction of single-curve and multi-curve interest rate models in [JMS17,
FGS21], for the modelling of energy prices in [JMSS19, CMS22] and for
stochastic volatility modelling in [JMSZ21].

In this paper, we introduce and study the class of CBI-time-changed Lévy
processes (CBITCL), obtained by time-changing a Lévy process with respect to
the time integral of a CBI process. This construction combines the
distributional flexibility of Lévy processes with the self-exciting behavior
of CBI processes, while retaining full analytical tractability. CBITCL
processes have a significant potential for applications in finance, notably in
markets with stochastic volatility, as illustrated in the companion paper
[FGS22]. In this work, we lay the theoretical foundations of CBITCL processes
and derive some results that are especially motivated by financial
applications.

The main contributions of the present paper can be outlined as follows:

  * •

We characterize CBITCL processes as solutions to a stochastic integral
equation which generalizes the well-known Dawson-Li representation of a CBI
process (see [DL06]) and we provide an additional characterization in terms of
their semimartingale characteristics. Moreover, we study the relation with
affine stochastic volatility processes, adopting an extension of the original
definition of [KR11], and derive necessary and sufficient conditions for a
general affine stochastic volatility process to be a CBITCL process.

  * •

By combining techniques of affine processes and specific properties of CBITCL
processes, we analyse the time of explosion of exponential moments of CBITCL
processes, under a suitable assumption on their dependence structure. This has
important consequences in finance, where a CBITCL process can be used to
represent the log-price of an asset. We also study the asymptotic behavior of
CBITCL processes. While the existence of a stationary distribution of a CBI
process is well understood (see, e.g., [Li20, Section 10]), we provide an
asymptotic result analogous to [KR11, Theorem 3.4] for the distribution of a
CBITCL process, making use of our characterization of the lifetime of
exponential moments.

  * •

In view of financial applications, we derive a class of equivalent changes of
probability that leave invariant the class of CBITCL processes, up to a change
in their parameters. Moreover, we illustrate how our results can be applied to
some specifications of CBITCL processes that have been recently considered in
mathematical finance.

We emphasize that, while some of our results can be derived from the general
theory of affine processes, we obtain more refined statements under minimal
technical assumptions by exploiting the specific structure of CBITCL
processes.

Our work is naturally related to the use of stochastic changes of time in
finance. Starting from the seminal work [Cla73], time-changed processes have
been widely adopted as models for asset prices and we refer to [BNS15, Swi16]
for detailed accounts on the topic. In particular, our work builds on the
contributions of [CGMY03, CW04], where stochastic volatility models have been
constructed by relying on time-changed Lévy processes. An empirical analysis
of several specifications of such models has been conducted in [HW04]. While
their analysis only considers changes of time driven by square-root
diffusions, [HW04] point out that a promising direction of research is the
study of models where the activity rate of the time change process exhibits
high-frequency jumps. This has been confirmed by the empirical analysis
conducted in [FHM21]. Our work contributes to this line of research by
developing a general theoretical framework for the use of CBI processes as
changes of time for Lévy processes, also allowing for the presence of self-
exciting jumps in the activity rate of the time change and for dependence
between the Lévy process and the activity rate.

Structure of the paper. In Section 2, we state the definition of CBITCL
processes, characterize them as solutions to a stochastic integral equation
and study their relation to affine stochastic volatility processes. In Section
3, under a suitable assumption on the dependence structure of a CBITCL
process, we derive an explicit characterization of the lifetime of exponential
moments and of the asymptotic behavior of CBITCL processes. In Section 4, we
present a class of equivalent changes of probability that leave invariant the
class of CBITCL processes. Finally, in Section 5 we present two examples of
CBITCL processes that are relevant for financial applications and discuss the
applications of CBITCL processes to option pricing.

Notation. In the following, we denote by $\mathbb{R}_{+}$ ($\mathbb{R}_{-}$,
resp.) the set of non-negative (non-positive, resp.) real numbers. We denote
by $\mathbb{C}_{+}$ ($\mathbb{C}_{-}$, resp.) the set of complex numbers with
non-negative (non-positive, resp.) real part. The set $\mathsf{i}\mathbb{R}$
is the set of complex numbers with null real part.

##  2\. Definition and characterization

In this section, we state the definition of a CBITCL process and prove some
foundational results for this class of processes. We let
$(\Omega,\mathcal{F},\mathbb{P})$ be a probability space supporting the
following objects:

  * (i)

a one-dimensional Lévy process $K=(K_{t})_{t\geq 0}$ with triplet
$(\beta,0,\nu)$, where $\beta\geq 0$ and $\nu$ is a Lévy measure111We recall
that a measure $\gamma$ on $\mathbb{R}^{n}$ is a Lévy measure if
$\gamma(\\{0\\})=0$ and $\int_{\mathbb{R}^{n}}(|x|^{2}\wedge
1)\gamma(\mathrm{d}x)<+\infty$. on $\mathbb{R}_{+}$ such that
$\int_{0}^{1}{x\,\nu(\mathrm{d}x)}<+\infty$. The exponent
$\Psi:\mathbb{C}_{-}\to\mathbb{C}$ is given by

(2.1) $\Psi(u)=\beta
u+\int_{\mathbb{R}_{+}}{(e^{ux}-1)\nu(\mathrm{d}x)},\qquad\text{for all
}u\in\mathbb{C}_{-};$

  * (ii)

a two-dimensional Lévy process $(M,N)=((M_{t},N_{t}))_{t\geq 0}$, independent
of $K$, with triplet $(b,\sigma^{2},\Pi)$, where
$\Pi(\mathrm{d}x,\mathrm{d}z)$ is a Lévy measure on
$\mathbb{R}_{+}\times\mathbb{R}$ and

(2.2) $b=\begin{pmatrix}-b_{X}\\\
b_{Z}\end{pmatrix},\qquad\sigma^{2}=\begin{pmatrix}\sigma_{X}^{2}&\rho\,\sigma_{X}\sigma_{Z}\\\
\rho\,\sigma_{X}\sigma_{Z}&\sigma_{Z}^{2}\end{pmatrix},\qquad\int_{[1,+\infty)\times\mathbb{R}}{x\,\Pi(\mathrm{d}x,\mathrm{d}z)<+\infty},$

with $b_{X},b_{Z}\in\mathbb{R}$, $\sigma_{X},\sigma_{Z}\in\mathbb{R}_{+}$ and
$\rho\in[-1,1]$. The exponent
$\Lambda:\mathbb{C}_{-}\times\mathsf{i}\mathbb{R}\rightarrow\mathbb{C}$ is
given by

(2.3) $\displaystyle\Lambda(u_{1},u_{2})$
$\displaystyle=-b_{X}u_{1}+b_{Z}u_{2}+\frac{1}{2}\sigma_{X}^{2}u_{1}^{2}+\rho\,\sigma_{X}\sigma_{Z}\,u_{1}u_{2}+\frac{1}{2}\sigma_{Z}^{2}u_{2}^{2}$
$\displaystyle\quad+\int_{\mathbb{R}_{+}\times\mathbb{R}}{\bigl{(}e^{u_{1}x+u_{2}z}-1-u_{1}x-u_{2}z\bm{1}_{\\{|z|<1\\}}\bigr{)}\Pi(\mathrm{d}x,\mathrm{d}z)},$

for all $(u_{1},u_{2})\in\mathbb{C}_{-}\times\mathsf{i}\mathbb{R}$. The
process $M$ is therefore a spectrally positive Lévy process with finite first
moment and exponent $\Phi:\mathbb{C}_{-}\to\mathbb{C}$ given by

(2.4)
$\Phi(u)=-b_{X}u+\frac{1}{2}\sigma_{X}^{2}u^{2}+\int_{\mathbb{R}_{+}\times\mathbb{R}}{(e^{ux}-1-ux)\Pi(\mathrm{d}x,\mathrm{d}z)},\qquad\text{for
all }u\in\mathbb{C}_{-},$

whereas $N$ is a general Lévy process whose exponent
$\Xi:\mathsf{i}\mathbb{R}\to\mathbb{C}$ is given by

(2.5)
$\Xi(u)=b_{Z}u+\frac{1}{2}\sigma_{Z}^{2}u^{2}+\int_{\mathbb{R}_{+}\times\mathbb{R}}{(e^{uz}-1-uz\bm{1}_{\\{|z|<1\\}})\Pi(\mathrm{d}x,\mathrm{d}z)},\qquad\text{for
all }u\in\mathsf{i}\mathbb{R}.$

######  Definition 2.1.

A process $(X,Z)=((X_{t},Z_{t}))_{t\geq 0}$ taking values in the state space
$\mathbb{R}_{+}\times\mathbb{R}$ is a _CBI-time-changed Lévy process_ (CBITCL)
if it holds that

(2.6) $\begin{cases}X_{t}=X_{0}+M_{Y_{t}}+K_{t},\\\
Z_{t}=N_{Y_{t}},\end{cases}$

for all $t\geq 0$, with $X_{0}\geq 0$ and where $Y=(Y_{t})_{t\geq 0}$ denotes
the process $Y:=\int_{0}^{\cdot}{X_{s}\,\mathrm{d}s}$.

The above definition is well-posed due to the existence of a unique strong
solution $(X,Z)$ to the stochastic time change equation (2.6), which follows
as a special case of [CPGUB17, Theorem 1]. In addition, by [CPGUB13, Theorem
2], the first component $X$ of a CBITCL process $(X,Z)$ is a conservative
stochastically continuous continuous-state branching process with immigration
(CBI) in the sense of [KW71], with initial value $X_{0}$, immigration
mechanism $\Psi$ and branching mechanism $\Phi$. Note that, since $X$ has
càdlàg paths, the integral $Y:=\int_{0}^{\cdot}X_{s}\,\mathrm{d}s$ is always
well defined as a non-decreasing process and can therefore be used as a change
of time.

In the following, we write that a process $(X,Z)$ is
$\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$ as a shorthand notation to denote the
fact that $(X,Z)$ is a CBI-time-changed Lévy process in the sense of
Definition 2.1, with $X_{0}\geq 0$ and where the exponents $\Psi$ and
$\Lambda$ are given in (2.1) and (2.3), respectively.

######  Remark 2.2.

Definition 2.1 can be extended to processes taking values on the state space
$\mathbb{R}^{m}_{+}\times\mathbb{R}^{n}$, for $m,n\in\mathbb{N}$. As shown in
[CPGUB17], this would generate processes that belong to the affine class.
However, it seems difficult to derive explicit results such as a
characterization of the finiteness of exponential moments in the multi-
dimensional case. For this reason and for ease of exposition, in the present
work we restrict our attention to two-dimensional CBITCL processes.

###  2.1. CBITCL processes as solutions to stochastic integral equations

It is well known that a CBI process can be characterized in two equivalent
ways: as the solution to the stochastic integral equation of Dawson and Li
(see [DL06]) and as the solution to a stochastic time change equation of
Lamperti-type (see [CPGUB13]). It can be shown that these two representations
of a CBI process are equivalent in a weak sense (see [Szu21, Theorem 2.12]).
In the present context of CBITCL processes, Definition 2.1 builds upon the
Lamperti-type representation of a CBI process. In this section, we show that
the Dawson-Li representation can be extended to CBITCL processes. To this
effect, let us introduce the following objects, assumed to be mutually
independent:

  * •

a two-dimensional Brownian motion $B=(B_{t})_{t\geq 0}$ with independent
components;

  * •

a Poisson random measure $N_{0}(\mathrm{d}t,\mathrm{d}x,\mathrm{d}z)$ on
$(0,+\infty)\times\mathbb{R}_{+}\times\mathbb{R}$ of compensator
$\mathrm{d}t\,\nu(\mathrm{d}x)\,\delta_{0}(\mathrm{d}z)$;

  * •

a Poisson random measure
$N_{1}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}x,\mathrm{d}z)$ on
$(0,+\infty)^{2}\times\mathbb{R}_{+}\times\mathbb{R}$ of compensator
$\mathrm{d}t\,\mathrm{d}u\,\Pi(\mathrm{d}x,\mathrm{d}z)$.

In the following, we shall always use the tilde notation to denote compensated
random measures. Let $\sigma\in\mathbb{R}^{2\times 2}$ be a matrix with non-
negative diagonal elements such that $\sigma\sigma^{\top}=\sigma^{2}$ and, for
$X_{0}\geq 0$, consider the following two-dimensional stochastic integral
equation:

$\displaystyle\begin{pmatrix}X_{t}\\\ Z_{t}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}X_{0}\\\
0\end{pmatrix}+\int_{0}^{t}{\left(\begin{pmatrix}\beta\\\
0\end{pmatrix}+X_{s}\,b\right)\mathrm{d}s}+\sigma\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}B_{s}}+\int_{0}^{t}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}x\\\
0\end{pmatrix}N_{0}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)}$ (2.7)
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{X_{s^{-}}}\\!\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}0\\\
z\bm{1}_{\\{|z|\geq
1\\}}\end{pmatrix}N_{1}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x,\mathrm{d}z)}$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{X_{s^{-}}}\\!\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}x\\\
z\bm{1}_{\\{|z|<1\\}}\end{pmatrix}\widetilde{N}_{1}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x,\mathrm{d}z)}.$

If $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ is
a filtered probability space supporting $B$, $N_{0}$ and $N_{1}$ with the
above properties, then equation (2.7) admits a unique strong solution $(X,Z)$
defined on $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$. Indeed, by [FL10,
Corollary 5.2], there exists a unique strong solution $X$ to the first
component of the stochastic integral equation (2.7), which corresponds to the
Dawson-Li representation of the CBI process $X$, see [DL06]. In turn, noting
that the right-hand side of (2.7) depends only on the process $X$, this
implies the uniqueness of the process $Z$ defined by the second component of
(2.7).

The next theorem asserts that defining CBITCL processes as solutions to the
stochastic integral equation (2.7) is equivalent to Definition 2.1. In the
following, we say that a process $(X,Z)$ defined on the probability space
$(\Omega,\mathcal{F},\mathbb{P})$ is a weak solution to (2.7) if there exists
an enlargement of the probability space supporting a two-dimensional Brownian
motion $B$ and Poisson random measures $N_{0}$ and $N_{1}$ as above such that
$(X,Z)$ satisfies (2.7). In a similar manner, we say that a process $(X,Z)$
defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ is equivalent
to a $\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$ if there exists an enlargement of
the probability space supporting independent Lévy processes $K$ and $(M,N)$
with exponents $\Psi$ and $\Lambda$, respectively, such that $(X,Z)$ satisfies
(2.6).

######  Theorem 2.3.

If the process $(X,Z)$ is a $\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$, then it is
a weak solution to (2.7). Conversely, if the process $(X,Z)$ is solution to
(2.7) on some filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, then it is equivalent to a
$\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$, with $\Psi$ and $\Lambda$ given by
(2.1) and (2.3), respectively.

###### Proof.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$ and denote by
$\mathbb{F}^{(M,N)}$ the natural filtration of $(M,N)$. Making use of the
Lévy-Itô decomposition of $(M,N)$, we have that

$\begin{pmatrix}M_{Y_{t}}\\\
N_{Y_{t}}\end{pmatrix}=b\,Y_{t}+\sigma\,W_{Y_{t}}+\int_{0}^{Y_{t}}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}0\\\
z\bm{1}_{\\{|z|\geq
1\\}}\end{pmatrix}\mu(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)}+\int_{0}^{Y_{t}}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}x\\\
z\bm{1}_{\\{|z|<1\\}}\end{pmatrix}\tilde{\mu}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)},$

for all $t\geq 0$, where $(W_{t})_{t\geq 0}$ is two-dimensional Brownian
motion in $\mathbb{F}^{(M,N)}$ and $\mu(\mathrm{d}t,\mathrm{d}x,\mathrm{d}z)$
is a Poisson random measure with compensator
$\mathrm{d}t\,\Pi(\mathrm{d}x,\mathrm{d}z)$ in $\mathbb{F}^{(M,N)}$,
independent of $W$. Let the filtration $\mathbb{G}^{(M,N),K}$ be defined as
the initial enlargement of $\mathbb{F}^{(M,N)}$ with respect to
$\sigma(K_{s};s\geq 0)$, i.e.,
$\mathcal{G}^{(M,N),K}_{t}:=\mathcal{F}^{(M,N)}_{t}\vee\sigma(K_{s};s\geq 0)$,
for all $t\geq 0$. By part (2) of [CPGUB13, Lemma 5], it holds that $Y_{t}$ is
a $\mathbb{G}^{(M,N),K}$-stopping time, for each $t\geq 0$. Therefore, the
process $(Y_{t})_{t\geq 0}$ is a change of time in $\mathbb{G}^{(M,N),K}$, in
the sense of [Jac79, Definition 10.1]. By continuity of $Y$, the random
measure $\mu(\mathrm{d}t,\mathrm{d}x,\mathrm{d}z)$ is adapted to $Y$ (see
[Jac79, Definition 10.25]) and, due to the independence of $(M,N)$ and $K$, it
has the same compensator $\mathrm{d}t\,\Pi(\mathrm{d}x,\mathrm{d}z)$ in both
filtrations $\mathbb{G}^{(M,N),K}$ and $\mathbb{F}^{(M,N)}$. Applying [Jac79,
Theorems 10.27 and 10.28], we can perform a change of time and obtain

$\displaystyle\int_{0}^{Y_{t}}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}0\\\
z\bm{1}_{\\{|z|\geq
1\\}}\end{pmatrix}\mu(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}0\\\
z\bm{1}_{\\{|z|\geq
1\\}}\end{pmatrix}\mu(X_{s}\,\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)},$
$\displaystyle\int_{0}^{Y_{t}}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}x\\\
z\bm{1}_{\\{|z|<1\\}}\end{pmatrix}\tilde{\mu}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}x\\\
z\bm{1}_{\\{|z|<1\\}}\end{pmatrix}\tilde{\mu}(X_{s}\,\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)},$

for all $t\geq 0$. By [IW89, Theorem II.7.4], on a suitable extension of the
probability space there exists a Poisson random measure
$N_{1}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}x,\mathrm{d}z)$ with compensator
$\mathrm{d}t\,\mathrm{d}u\,\Pi(\mathrm{d}x,\mathrm{d}z)$ such that

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}0\\\
z\bm{1}_{\\{|z|\geq
1\\}}\end{pmatrix}\mu(X_{s}\,\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)}$
$\displaystyle=\int_{0}^{t}\int_{0}^{X_{s^{-}}}\\!\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}0\\\
z\bm{1}_{\\{|z|\geq
1\\}}\end{pmatrix}N_{1}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x,\mathrm{d}z)},$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}x\\\
z\bm{1}_{\\{|z|<1\\}}\end{pmatrix}\tilde{\mu}(X_{s}\,\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)}$
$\displaystyle=\int_{0}^{t}\int_{0}^{X_{s^{-}}}\\!\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}x\\\
z\bm{1}_{\\{|z|<1\\}}\end{pmatrix}\widetilde{N}_{1}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x,\mathrm{d}z)},$

for all $t\geq 0$. Proceeding in a similar way and making use of [IW89,
Theorem 7.1’], on a suitable extension of the probability space there exists a
two-dimensional Brownian motion $B=(B_{t})_{t\geq 0}$ such that
$W_{Y_{t}}=\int_{0}^{t}\sqrt{X_{s}}\mathrm{d}B_{s}$, for all $t\geq 0$. We
have therefore obtained that

(2.8) $\displaystyle\begin{pmatrix}M_{Y_{t}}\\\ N_{Y_{t}}\end{pmatrix}$
$\displaystyle=\int_{0}^{t}{X_{s}\,b\,\mathrm{d}s}+\sigma\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}B_{s}}+\int_{0}^{t}\int_{0}^{X_{s^{-}}}\\!\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}0\\\
z\bm{1}_{\\{|z|\geq
1\\}}\end{pmatrix}N_{1}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x,\mathrm{d}z)}$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{X_{s^{-}}}\\!\int_{\mathbb{R}_{+}\times\mathbb{R}}{\begin{pmatrix}x\\\
z\bm{1}_{\\{|z|<1\\}}\end{pmatrix}\widetilde{N}_{1}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x,\mathrm{d}z)}.$

The Lévy-Itô decomposition of the process $K$ can be written as $K_{t}=\beta
t+\int_{0}^{t}\int_{\mathbb{R}_{+}\times\mathbb{R}}{x\,N_{0}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)}$,
for all $t\geq 0$, where $N_{0}(\mathrm{d}t,\mathrm{d}x,\mathrm{d}z)$ is a
Poisson random measure with compensator
$\mathrm{d}t\,\nu(\mathrm{d}x)\,\delta_{0}(\mathrm{d}z)$. Adding the Lévy-Itô
decomposition of $K$ to (2.8) and comparing with (2.6), we have proved that
$(X,Z)$ solves equation (2.7), on a suitable extension of the original
probability space.

Conversely, suppose that $(X,Z)$ solves (2.7) on some filtered probability
space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$. We first observe that the
process $K=(K_{t})_{t\geq 0}$ defined by $K_{t}:=\beta
t+\int_{0}^{t}\int_{\mathbb{R}_{+}\times\mathbb{R}}{x\,N_{0}(\mathrm{d}s,\mathrm{d}x,\mathrm{d}z)}$,
for all $t\geq 0$, is a Lévy process with triplet $(\beta,0,\nu)$ and exponent
$\Psi$ given by (2.1). Consider then the two-dimensional process $V:=(X-K,Z)$.
In order to show that $V$ is a time-changed Lévy process, we follow the proof
of [Kal06, Theorem 3.2]. We first note that the process $V$ is constant on
each interval $[r,s]\subseteq\mathbb{R}_{+}$ such that $Y_{r}=Y_{s}$ a.s.
Indeed, for any such interval, it necessarily holds that $X=0$ a.s. on $[r,s]$
(Lebesgue-a.e.), and, therefore, (2.7) implies that $V$ is a.s. constant on
$[r,s]$. Let $Y_{\infty}:=\lim_{t\to+\infty}Y_{t}$, which is well-defined
since the process $Y$ is non-decreasing, and define the inverse time change
$\tau(t):=\inf\\{s\geq 0:Y_{s}>t\\}$, for all $t\geq 0$. Define the process
$L^{1}=(L^{1}_{t})_{t<Y_{\infty}}$ by222We point out that it may happen that
$Y_{\infty}<+\infty$, since zero is an absorbing state for $X$ when
$\Psi\equiv 0$.

$L^{1}_{t}:=V_{\tau(t)},\qquad\text{ for all }t<Y_{\infty}.$

By [Jac79, Lemma 10.14], $V$ is adapted to $(\tau(t))_{t\geq 0}$ and it holds
that $V_{t}=L^{1}_{Y_{t}}$, for all $t\geq 0$. Since $(X,Z)$ solves (2.7), the
semimartingale $V=(X-K,Z)$ has characteristics
$(Yb,Y\sigma^{2},\mathrm{d}Y_{t}\,\Pi(\mathrm{d}x,\mathrm{d}z))$ in the
filtration $\mathbb{F}$ with respect to the truncation function $h$ given by
$h(x):=(x_{1},x_{2}\bm{1}_{\\{|x_{2}|<1\\}})$, for
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$. By [Jac79, Theorem 10.16], $L^{1}$ is a
semimartingale in the time-changed filtration $(\mathcal{F}_{\tau(t)})_{t\geq
0}$ on the stochastic interval $\llbracket 0,Y_{\infty}\llbracket$. Similarly
as in the proof of [Kal06, Theorem 3.2], [Jac79, Theorems 10.17 and 10.27]
together with the fact that $Y_{\tau(t)}=t$, for all $t<Y_{\infty}$, imply
that the semimartingale characteristics $(B,C,F)$ of $L^{1}$ in
$(\mathcal{F}_{\tau(t)})_{t\geq 0}$ are given by

(2.9) $B_{t}=Y_{\tau(t)}\,b=t\,b,\quad
C_{t}=Y_{\tau(t)}\,\sigma^{2}=t\,\sigma^{2},\quad
F(\mathrm{d}t,\mathrm{d}x,\mathrm{d}z)=\mathrm{d}Y_{\tau(t)}\,\Pi(\mathrm{d}x,\mathrm{d}z)=\mathrm{d}t\,\Pi(\mathrm{d}x,\mathrm{d}z),$

for all $t<Y_{\infty}$. Let $L^{2}=(L_{t}^{2})_{t\geq 0}$ be a two-dimensional
Lévy process with triplet $(b,\sigma^{2},\Pi)$ and exponent (2.3), independent
of $V$ and $K$, possibly defined on a suitable enlargement of the probability
space. Define the process $L=(L_{t})_{t\geq 0}$ by

$L_{t}:=L^{1}_{t}\bm{1}_{\\{t<Y_{\infty}\\}}+(L^{1}_{Y_{\infty}}+L^{2}_{t-Y_{\infty}})\bm{1}_{\\{t\geq
Y_{\infty}\\}},\qquad\text{ for all }t\geq 0,$

where $L^{1}_{Y_{\infty}}:=\lim_{t\to+\infty}V_{t}=:V_{\infty}$ on
$\\{Y_{\infty}<+\infty\\}$, which is well-defined since $V$ is a
semimartingale up to infinity on $\\{Y_{\infty}<+\infty\\}$, see [CS05]. Note
that $Y_{\infty}$ is a stopping time in the time-changed filtration
$(\mathcal{F}_{\tau(t)})_{t\geq 0}$ and, therefore, $L$ is a semimartingale in
$(\mathcal{F}_{\tau(t)})_{t\geq 0}$. Moreover, by the results of [Jac79,
Section 10.2b], it can be shown that the semimartingale characteristics of $L$
are given by (2.9), which by [JS03, Corollary II.4.19] implies that $L$ is a
Lévy process with triplet $(b,\sigma^{2},\Pi)$. Since $V_{t}=L_{Y_{t}}$, for
all $t\geq 0$, to complete the proof it remains to show that $L$ is
independent of $K$. To this effect, in view of [HWY92, Theorem 11.43], it
suffices to prove that $|\Delta L_{t}\,\Delta K_{t}|=0$ a.s. for all $t\geq
0$. Using the definition of $L$ and [Jac79, Theorem 10.27], we can compute

(2.10) $\displaystyle\mathbb{E}\bigl{[}|\Delta L_{t}\,\Delta K_{t}|\bigr{]}$
$\displaystyle=\mathbb{E}\bigl{[}\bm{1}_{\\{t<Y_{\infty}\\}}|\Delta
L^{1}_{t}\,\Delta K_{t}|\bigr{]}$
$\displaystyle=\mathbb{E}\bigl{[}\bm{1}_{\\{\tau(t)<+\infty\\}}|\Delta
V_{\tau(t-)}\,\Delta K_{t}|\bigr{]}$
$\displaystyle\leq\mathbb{E}\bigl{[}\bm{1}_{\\{\tau(t-)<t\\}}|\Delta
V_{\tau(t-)}\,\Delta
K_{t}|\bigr{]}+\mathbb{E}\bigl{[}\bm{1}_{\\{t<\tau(t-)<+\infty\\}}|\Delta
V_{\tau(t-)}\,\Delta K_{t}|\bigr{]},$

where we have used the fact that $\mathbb{E}[\bm{1}_{\\{\tau(t-)=t\\}}|\Delta
V_{\tau(t-)}\,\Delta K_{t}|]=0$ since the random measures $N_{0}$ and $N_{1}$
in (2.7) are assumed to be independent. Moreover, using the fact that
$\tau(t-)$ is a predictable time in the filtration $\mathbb{F}$ together with
[JS03, Proposition II.1.17], it holds that

$\mathbb{E}\bigl{[}\bm{1}_{\\{t<\tau(t-)<+\infty\\}}|\Delta
V_{\tau(t-)}\,\Delta
K_{t}|\bigr{]}=\mathbb{E}\bigl{[}\bm{1}_{\\{t<\tau(t-)<+\infty\\}}|\Delta
K_{t}|\mathbb{E}[|\Delta V_{\tau(t-)}||\mathcal{F}_{\tau(t-)-}]\bigr{]}=0.$

In an analogous way, since $K$ is quasi-left-continuous, we have that

$\mathbb{E}\bigl{[}\bm{1}_{\\{\tau(t-)<t\\}}|\Delta V_{\tau(t-)}\,\Delta
K_{t}|\bigr{]}=\mathbb{E}\bigl{[}\bm{1}_{\\{\tau(t-)<t\\}}|\Delta
V_{\tau(t-)}|\mathbb{E}[|\Delta K_{t}||\mathcal{F}_{t-}]\bigr{]}=0.$

In view of (2.10), this implies that $|\Delta L_{t}\,\Delta K_{t}|=0$ a.s. for
all $t\geq 0$, thus completing the proof. ∎

From now on, if a CBITCL process $(X,Z)$ is directly defined as the unique
strong solution to (2.7) on some filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, we will say that $(X,Z)$ is
given by its _extended Dawson-Li representation_. Representation (2.7) can be
especially useful for the numerical simulation of CBITCL processes (compare
with [FGS21, Appendix B] in the case CBI processes).

######  Remark 2.4.

One of the characteristic features of CBI processes is their self-exciting
behavior. As can be deduced from equation (2.7), this behavior is inherited by
the class of CBITCL processes. In particular, we can observe the following:

  * •

The local martingale terms of the process $X$ depend on the current level of
the process itself. This generates self-excitation since large values of the
process increase its volatility. In particular, a large jump of $X$ increases
the likelihood of further jumps of $X$.

  * •

The volatility of $Z$ is determined by $X$. Therefore, large values of $X$ are
associated to an increased volatility of both processes, thus generating
volatility clustering effects in $(X,Y)$.

As mentioned in the introduction, self-exciting and volatility clustering
phenomena are often present in financial time series. Together with the
remarkable analytical tractability, this makes CBITCL processes especially
appropriate for financial modeling, as illustrated in [FGS22] in the context
of foreign currency markets with stochastic volatility (see also the examples
in Section 5).

The next proposition characterizes CBITCL processes in terms of their
semimartingale differential characteristics (see [Kal06] for additional
information on this notion of characteristics).

######  Proposition 2.5.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$. Then, $(X,Z)$ is a
semimartingale with differential characteristics $(B,C,F)$ relative to the
truncation function $h(x)=x\bm{1}_{\\{|x|<1\\}}$ given by

(2.11)
$\begin{gathered}B_{t}=\begin{pmatrix}\beta+\int_{0}^{1}{x\,\nu(\mathrm{d}x)}\\\
0\end{pmatrix}+X_{t^{-}}\Biggl{(}b-\begin{pmatrix}\int_{[1,+\infty)\times\mathbb{R}}{x\,\Pi(\mathrm{d}x,\mathrm{d}z)}\\\
0\end{pmatrix}\Biggr{)},\qquad C_{t}=X_{t^{-}}\,\sigma^{2},\\\
F_{t}(\mathrm{d}x,\mathrm{d}z)=\nu(\mathrm{d}x)\delta_{0}(\mathrm{d}z)+X_{t^{-}}\Pi(\mathrm{d}x,\mathrm{d}z),\qquad\text{
for all }t\geq 0.\end{gathered}$

Conversely, if $(X,Z)$ is a semimartingale with differential characteristics
given as in (2.11), with $\beta\geq 0$, $b$ and $\sigma$ as in (2.2), $\nu$
and $\Pi$ Lévy measures on $\mathbb{R}_{+}$ and
$\mathbb{R}_{+}\times\mathbb{R}$, respectively, such that
$\int_{0}^{1}x\,\nu(\mathrm{d}x)<+\infty$ and
$\int_{[1,+\infty)\times\mathbb{R}}{x\,\Pi(\mathrm{d}x,\mathrm{d}z)<+\infty}$,
then it is equivalent to a $\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$, where $\Psi$
and $\Lambda$ are determined by $(\beta,b,\sigma^{2},\nu,\Pi)$ as in (2.1) and
(2.3), respectively.

###### Proof.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$ on
$(\Omega,\mathcal{F},\mathbb{P})$. By Theorem 2.3, the process $(X,Z)$ is
solution to equation (2.7) on an extended filtered probability space
$(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{F}^{\prime},\mathbb{P}^{\prime})$
(see [IW89, Definition II.7.1]). This implies that $(X,Z)$ is a quasi-left-
continuous semimartingale on
$(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{F}^{\prime},\mathbb{P}^{\prime})$
and its differential characteristics as stated in (2.11) can be directly
deduced from (2.7). In view of [Jac79, Remark 10.40], $(X,Z)$ is also a
semimartingale on $(\Omega,\mathcal{F},\mathbb{F}^{(X,Z)},\mathbb{P})$, where
$\mathbb{F}^{(X,Z)}=(\mathcal{F}^{(X,Z)}_{t})_{t\geq 0}$ denotes the natural
filtration of $(X,Z)$, with the same differential characteristics given in
(2.11).

Conversely, suppose that $(X,Z)$ is a semimartingale with differential
characteristics given as in (2.11). Writing the canonical representation of
$(X,Z)$ (which is determined by the differential characteristics, see [JS03,
Theorem II.2.34]) and arguing as in the first part of the proof of Theorem
2.3, it can be easily shown that $(X,Z)$ satisfies (2.7) on a suitable
extension of the probability space. Therefore, it follows from Theorem 2.3
that $(X,Z)$ is equivalent to a $\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$. ∎

###  2.2. CBITCL processes as affine processes

In this section, we regard CBITCL processes as affine processes in the sense
of [DFS03]. More specifically, we connect CBITCL processes to affine
stochastic volatility processes as considered in [KR11]. We first state the
following proposition, which provides the Fourier–Laplace transform of the
joint process $(X,Y,Z)$, where $Y:=\int_{0}^{\cdot}{X_{s}\,\mathrm{d}s}$. When
restricted to the pair $(X,Z)$, this result follows as a consequence of
[CPGUB17, Theorem 1]. Without loss of generality (see Theorem 2.3), we suppose
that the CBITCL process $(X,Z)$ is given by its extended Dawson-Li
representation on a filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ supporting the processes
introduced at the beginning of Section 2.1.

######  Proposition 2.6.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Lambda)$ and consider the
process $(X,Y,Z)$, where $Y:=\int_{0}^{\cdot}{X_{s}\,\mathrm{d}s}$. Then,
$(X,Z)$ and $(X,Y,Z)$ are affine procesess on the state spaces
$\mathbb{R}_{+}\times\mathbb{R}$ and $\mathbb{R}_{+}^{2}\times\mathbb{R}$,
respectively, and it holds that

(2.12)
$\mathbb{E}\bigl{[}e^{u_{1}X_{T}+u_{2}Y_{T}+u_{3}Z_{T}}\big{|}\mathcal{F}_{t}\bigr{]}=\exp\bigl{(}\mathcal{U}(T-t,u_{1},u_{2},u_{3})+\mathcal{V}(T-t,u_{1},u_{2},u_{3})X_{t}+u_{2}Y_{t}+u_{3}Z_{t}\bigr{)},$

for all $(u_{1},u_{2},u_{3})\in\mathbb{C}_{-}^{2}\times\mathsf{i}\mathbb{R}$
and $0\leq t\leq T<+\infty$, where the functions
$\mathcal{U}(\cdot,u_{1},u_{2},u_{3}):\mathbb{R}_{+}\rightarrow\mathbb{C}$ and
$\mathcal{V}(\cdot,u_{1},u_{2},u_{3}):\mathbb{R}_{+}\rightarrow\mathbb{C}_{-}$
are solutions to

(2.13) $\displaystyle\mathcal{U}(t,u_{1},u_{2},u_{3})$
$\displaystyle=\int_{0}^{t}{\Psi\bigl{(}\mathcal{V}(s,u_{1},u_{2},u_{3})\bigr{)}\,\mathrm{d}s},$
(2.14) $\displaystyle\frac{\partial\mathcal{V}}{\partial
t}(t,u_{1},u_{2},u_{3})$
$\displaystyle=\Lambda\bigl{(}\mathcal{V}(t,u_{1},u_{2},u_{3}),u_{3}\bigr{)}+u_{2},\qquad\mathcal{V}(0,u_{1},u_{2},u_{3})=u_{1},$

where $\Psi:\mathbb{C}_{-}\rightarrow\mathbb{C}$ and
$\Lambda:\mathbb{C}_{-}\times\mathsf{i}\mathbb{R}\rightarrow\mathbb{C}$ denote
the functions given by (2.1) and (2.3), respectively.

###### Proof.

Since $(X,Z)$ is the unique strong solution to (2.7) on
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, it is a strong Markov process
and a semimartingale with characteristics given by (2.11) (see Proposition
2.5). By [DFS03, Theorem 2.12], it follows that $(X,Z)$ is an affine process
on $\mathbb{R}_{+}\times\mathbb{R}$ with functional characteristics $(F,R)$
given by

$F(v_{1},v_{2})=\Psi(v_{1}),\qquad R(v_{1},v_{2})=\Lambda(v_{1},v_{2}),$

for all $(v_{1},v_{2})\in\mathbb{C}_{-}\times\mathsf{i}\mathbb{R}$. By
applying [KR09, Theorem 4.10], we obtain that the process $(X,Y,Z)$ is an
affine process on $\mathbb{R}_{+}^{2}\times\mathbb{R}$ with functional
characteristics $(\widetilde{F},\widetilde{R})$ given by

$\widetilde{F}(u_{1},u_{2},u_{3})=F(u_{1},u_{3})=\Psi(u_{1}),\qquad\widetilde{R}(u_{1},u_{2},u_{3})=R(u_{1},u_{3})+u_{2}=\Lambda(u_{1},u_{3})+u_{2},$

for all $(u_{1},u_{2},u_{3})\in\mathbb{C}_{-}^{2}\times\mathsf{i}\mathbb{R}$.
The affine transform formula (2.12) and the Riccati equations (2.13)-(2.14)
are then a consequence of [DFS03, Theorem 2.7]. ∎

The availability of the explicit characterization of the conditional Fourier-
Laplace transform stated in Proposition 2.6 is of great usefulness for the
application of CBITCL processes in finance (see also Section 5.3). More
specifically, many pricing applications require the computation of conditional
expectations of the form (2.12), with $Y$ typically playing the role of a
discount factor.

In line with [KR11], we say that a process $(X,Z)$ taking values in
$\mathbb{R}_{+}\times\mathbb{R}$ is an affine stochastic volatility process if
it is an affine process and its Fourier-Laplace transform has the structure
(2.12) (with $u_{2}=0$), for some functions $\mathcal{U}$ and $\mathcal{V}$.
This terminology is explained by the fact that, in financial applications, the
process $Z$ usually plays the role of the log-price process of a risky asset,
while $X$ represents its instantaneous variance.333We point out that the
notion of affine stochastic volatility process that we adopt in this paper is
more general than [KR11, Definition 2.8]. Indeed, we do not require that
$\exp(Z)$ is a martingale nor impose a non-degeneracy condition on $\Xi$
(corresponding to conditions (A3)-(A4) in [KR11]). An explicit
characterization of the martingale property of $\exp(Z)$ will be given below
in Corollary 4.3. Proposition 2.6 directly implies that CBITCL processes
belong to the class of affine stochastic volatility processes. In the next
proposition, we provide conditions for the validity of the converse
implication. We recall that, if $(X,Z)$ is an affine process on
$\mathbb{R}_{+}\times\mathbb{R}$, then the compensator of its jump measure is
of the form
$\nu^{(X,Z)}(\mathrm{d}t,\mathrm{d}x,\mathrm{d}z)=(m_{0}(\mathrm{d}x,\mathrm{d}z)+X_{t-}m_{1}(\mathrm{d}x,\mathrm{d}z))\mathrm{d}t$,
where $m_{0}$ and $m_{1}$ are Lévy measures on
$\mathbb{R}_{+}\times\mathbb{R}$ satisfying
$\int_{\mathbb{R}_{+}\times\mathbb{R}}(1\wedge(x+z^{2}))m_{0}(\mathrm{d}x,\mathrm{d}z)<+\infty$
(see, e.g., [KR11, Section 2.1]).

######  Proposition 2.7.

Let $(X,Z)$ be an affine stochastic volatility process. Then, $(X,Z)$ is
equivalent to a CBITCL process if the following three conditions hold:

  1. (i)

$\int_{[1,+\infty)\times\mathbb{R}}x\,m_{1}(\mathrm{d}x,\mathrm{d}z)<+\infty$;

  2. (ii)

$m_{0}(\mathrm{d}x,\mathrm{d}z)=m_{0}^{X}(\mathrm{d}x)\,\delta_{0}(\mathrm{d}z)+\delta_{0}(\mathrm{d}x)\,m_{0}^{Z}(\mathrm{d}z)$;

  3. (iii)

$Z$ is a.s. constant on every interval $[r,s]\subseteq\mathbb{R}_{+}$ such
that $\int_{r}^{s}X_{u-}\mathrm{d}u=0$ a.s.

###### Proof.

If $(X,Z)$ is an affine stochastic volatility process, then by [DFS03, Theorem
2.12] it is a semimartingale with differential characteristics $(B,C,F)$ given
by

$B_{t}=\begin{pmatrix}\beta_{1}\\\
\beta_{2}\end{pmatrix}+X_{t-}\begin{pmatrix}b_{1}\\\ b_{2}\end{pmatrix},\qquad
C_{t}=\begin{pmatrix}0&0\\\
0&\alpha_{2}\end{pmatrix}+X_{t-}\begin{pmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{pmatrix},$
$F_{t}(\mathrm{d}x,\mathrm{d}z)=m_{0}(\mathrm{d}x,\mathrm{d}z)+X_{t-}m_{1}(\mathrm{d}x,\mathrm{d}z),$

where $(\beta_{1},\beta_{2})\in\mathbb{R}_{+}\times\mathbb{R}$,
$(b_{1},b_{2})\in\mathbb{R}^{2}$, $\alpha_{2}\in\mathbb{R}_{+}$ and
$\bigl{(}\begin{smallmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{smallmatrix}\bigr{)}$ is a symmetric positive semi-definite
matrix. By condition (iii), the process $Z$ is constant on all intervals
$[r,s]\subseteq\mathbb{R}_{+}$ such that $X_{u-}=0$ a.s. for a.e. $u\in[r,s]$,
which implies that $\beta_{2}=0$ and $\alpha_{2}=0$. Moreover, conditions (ii)
and (iii) together imply that $m_{0}^{Z}=0$. Using condition (i) and recalling
that $\int_{0}^{1}x\,m^{X}_{0}(\mathrm{d}x)<+\infty$, since $(X,Z)$ is assumed
to be an affine stochastic volatility process, we have thus shown that
$(B,C,F)$ can be written in the form (2.11), where the Lévy measures
$\nu(\mathrm{d}x)$ and $\Pi(\mathrm{d}x,\mathrm{d}z)$ correspond to
$m_{0}^{X}(\mathrm{d}x)$ and $m_{1}(\mathrm{d}x,\mathrm{d}z)$, respectively.
The claim then follows by Proposition 2.5. ∎

In the remaining sections of the paper, we shall restrict our attention to
CBITCL processes satisfying the following assumption on the dependence
structure of the Lévy process $(M,N)$.

######  Assumption 2.8.

The Lévy measure $\Pi$ is of the form

(2.15)
$\Pi(\mathrm{d}x,\mathrm{d}z)=\pi(\mathrm{d}x)\,\delta_{0}(\mathrm{d}z)+\delta_{0}(\mathrm{d}x)\,\gamma(\mathrm{d}z),$

where $\pi$ is a Lévy measure on $\mathbb{R}_{+}$ such that
$\int_{1}^{+\infty}{x\,\pi(\mathrm{d}x)}<+\infty$ and $\gamma$ is a Lévy
measure on $\mathbb{R}$.

Assumption 2.8 amounts to excluding common jumps in the processes $X$ and $Z$,
while allowing for non-zero correlation in their continuous local martingale
parts. This assumption enables us to derive explicit results in the following
sections, in particular an explicit characterization of the lifetime of
exponential moments. Such explicit results would be impossible to obtain under
a general Lévy measure, similarly to the case of general affine stochastic
volatility processes (see [KR11]).

Under Assumption 2.8, the Lévy exponent $\Lambda$ given in (2.3) takes the
following simpler form:

$\Lambda(u_{1},u_{2})=\Phi(u_{1})+\rho\,\sigma_{X}\sigma_{Z}\,u_{1}u_{2}+\Xi(u_{2}),\qquad\text{for
all }(u_{1},u_{2})\in\mathbb{C}_{-}\times\mathsf{i}\mathbb{R},$

where $\Phi$ and $\Xi$ are given by

(2.16) $\displaystyle\Phi(u_{1})$
$\displaystyle=-b_{X}u_{1}+\frac{1}{2}\sigma_{X}^{2}u_{1}^{2}+\int_{\mathbb{R}_{+}}{(e^{u_{1}x}-1-u_{1}x)\pi(\mathrm{d}x)},\qquad\text{
for all }u_{1}\in\mathbb{C}_{-},$ (2.17) $\displaystyle\Xi(u_{2})$
$\displaystyle=b_{Z}u_{2}+\frac{1}{2}\sigma_{Z}^{2}u_{2}^{2}+\int_{\mathbb{R}}{(e^{u_{2}z}-1-u_{2}z\bm{1}_{\\{|z|<1\\}})\gamma(\mathrm{d}z)},\qquad\text{
for all }u_{2}\in\mathsf{i}\mathbb{R}.$

In the following, we shall write that a CBITCL process $(X,Z)$ is a
$\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ if Assumption 2.8 holds, with
$\Psi$, $\Phi$, $\Xi$ respectively given by (2.1), (2.16), (2.17) and
$\rho\in[-1,1]$.

For later use, we note that under Assumption 2.8 the stochastic integral
equation (2.7) takes the following form:

$\displaystyle X_{t}$ $\displaystyle=X_{0}+\int_{0}^{t}{(\beta-
b_{X}X_{s})\,\mathrm{d}s}+\sigma_{X}\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}B_{s}^{X}}+\int_{0}^{t}\int_{0}^{+\infty}{x\,N_{0}(\mathrm{d}s,\mathrm{d}x)}$
(2.18)
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{X_{s-}}\\!\int_{0}^{+\infty}{x\,\widetilde{N}_{1}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x)},$
$\displaystyle Z_{t}$
$\displaystyle=b_{Z}\int_{0}^{t}{X_{s}\,\mathrm{d}s}+\sigma_{Z}\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}B_{s}^{Z}}+\int_{0}^{t}\int_{0}^{X_{s-}}\\!\int_{|z|\geq
1}{z\,N_{2}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)}$ (2.19)
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{X_{s-}}\\!\int_{|z|<1}{z\,\widetilde{N}_{2}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)},$

where $B^{X}=(B_{t}^{X})_{t\geq 0}$ and $B^{Z}=(B_{t}^{Z})_{t\geq 0}$ are one-
dimensional Brownian motions with correlation $\rho$,
$N_{0}(\mathrm{d}t,\mathrm{d}x)$ is a Poisson random measure on
$(0,+\infty)\times\mathbb{R}_{+}$ with compensator
$\mathrm{d}t\,\nu(\mathrm{d}x)$, $N_{1}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}x)$
is a Poisson random measure on $(0,+\infty)\times\mathbb{R}_{+}^{2}$ with
compensator $\mathrm{d}t\,\mathrm{d}u\,\pi(\mathrm{d}x)$ and
$N_{2}(\mathrm{d}t,\mathrm{d}x,\mathrm{d}z)$ is a Poisson random measure on
$(0,+\infty)\times\mathbb{R}_{+}\times\mathbb{R}$ with compensator
$\mathrm{d}t\,\mathrm{d}u\,\gamma(\mathrm{d}z)$. Moreover, the random measures
$N_{0}$, $N_{1}$ and $N_{2}$ are mutually independent and also independent of
the Brownian motions $B^{X}$ and $B^{Z}$.

##  3\. Finiteness of exponential moments and asymptotic behavior

In this section, assuming the validity of Assumption 2.8, we study the
existence of (discounted) exponential moments of CBITCL processes and we
characterize their asymptotic behavior. These properties are intimately
connected to the maximal lifetime of the solutions to the Riccati equations,
explicitly characterized in Theorem 3.4 below. We let $(X,Z)$ be a
$\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ given by its extended Dawson-Li
representation on a given filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$.

###  3.1. Finiteness of exponential moments

In financial applications, the finiteness of (discounted) exponential moments
often represents an indispensable requirement (see, e.g., [FGS21, FGS22]). In
order to make use of some results of [KRM15] for general affine processes, let
us define

(3.1)
$\mathcal{D}_{X}:=\biggl{\\{}u\in\mathbb{R}:\int_{1}^{+\infty}{e^{ux}\,(\nu+\pi)(\mathrm{d}x)}<+\infty\biggr{\\}}.$

The convex set $\mathcal{D}_{X}$ is non-empty and represents the effective
domain of the functions $\Psi$ and $\Phi$, which can be extended to finite-
valued convex functions on $\mathcal{D}_{X}$. Similarly, let us define

(3.2) $\mathcal{D}_{Z}:=\biggl{\\{}u\in\mathbb{R}:\int_{|z|\geq
1}{e^{uz}\,\gamma(\mathrm{d}z)}<+\infty\biggr{\\}},$

which represents the effective domain of the Lévy exponent $\Xi$ when
restricted to real arguments. By standard results on exponential moments of
Lévy measures (see, e.g., [Sat99, Theorem 25.17]), the Lévy exponent $\Xi$ can
be extended as a finite-valued convex function on $\mathcal{D}_{Z}$.

Adapting [KRM15, Definition 2.10] to the present setup, we introduce the
following definition.

######  Definition 3.1.

For every
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$,
we say that
$(\mathcal{U}(\cdot,u_{1},u_{2},u_{3}),\mathcal{V}(\cdot,u_{1},u_{2},u_{3}))$
is a solution to the _extended Riccati system_ if it solves the following
system:

(3.3) $\displaystyle\mathcal{U}(t,u_{1},u_{2},u_{3})$
$\displaystyle=\int_{0}^{t}\Psi\bigl{(}\mathcal{V}(s,u_{1},u_{2},u_{3})\bigr{)}\mathrm{d}s,$
(3.4) $\displaystyle\frac{\partial\mathcal{V}}{\partial
t}(t,u_{1},u_{2},u_{3})$
$\displaystyle=\Phi\bigl{(}\mathcal{V}(t,u_{1},u_{2},u_{3})\bigr{)}+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,\mathcal{V}(t,u_{1},u_{2},u_{3})+\Xi(u_{3}),$
$\displaystyle\mathcal{V}(0,u_{1},u_{2},u_{3})$ $\displaystyle=u_{1},$

up to a time $\mathrm{T}^{(u_{1},u_{2},u_{3})}\in[0,+\infty]$, with
$\mathrm{T}^{(u_{1},u_{2},u_{3})}$ denoting the joint lifetime of the
functions
$\mathcal{U}(\cdot,u_{1},u_{2},u_{3}):[0,\mathrm{T}^{(u_{1},u_{2},u_{3})})\rightarrow\mathbb{R}$
and
$\mathcal{V}(\cdot,u_{1},u_{2},u_{3}):[0,\mathrm{T}^{(u_{1},u_{2},u_{3})})\rightarrow\mathcal{D}_{X}$.

Definition 3.1 extends the Riccati system (2.13)-(2.14) by allowing for the
possibility of explosion in finite time. In some situations, which will be
precisely characterized in Theorem 3.4 below, the lifetime
$\mathrm{T}^{(u_{1},u_{2},u_{3})}$ turns out to be infinite, in which case
(3.3)-(3.4) admit a global solution.

It is well known that the branching mechanism $\Phi$ is a locally Lipschitz
continuous function on the interior $\mathcal{D}_{X}^{\circ}$ of the set
$\mathcal{D}_{X}$, but it may fail to be so at the boundary
$\partial\mathcal{D}_{X}$. Therefore, a solution
$\mathcal{V}(\cdot,u_{1},u_{2},u_{3})$ to the ODE (3.4) may not be unique when
it starts at $\partial\mathcal{D}_{X}$ or reaches it at a later time. This
observation motivates the introduction of the concept of minimal solution in
[KRM15]. In our setup, for the sake of tractability, we prefer to impose an
additional mild technical assumption which guarantees uniqueness of the
(local) solution to the ODE (3.4) for every
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$.
To this effect, we define

(3.5) $\psi:=\sup\bigl{\\{}x\geq
0:\Psi(x)<+\infty\bigr{\\}}\qquad\text{and}\qquad\phi:=\sup\bigl{\\{}x\geq
0:\Phi(x)<+\infty\bigr{\\}}.$

Since $\mathcal{D}_{X}$ is a convex set containing $\mathbb{R}_{-}$, it can be
written as $\mathcal{D}_{X}=(-\infty,\psi\wedge\phi)$, or
$(-\infty,\psi\wedge\phi]$ when
$\Psi(\psi\wedge\phi)\vee\Phi(\psi\wedge\phi)<+\infty$ (which is equivalent to
$\int_{1}^{+\infty}{e^{(\psi\wedge\phi)x}(\nu+\pi)(\mathrm{d}x)}<+\infty$).
The function $\Phi$ is a finite-valued convex function on $\mathcal{D}_{X}$
and, hence, differentiable a.e. on $\mathcal{D}_{X}^{\circ}$, with

(3.6)
$\Phi^{\prime}(u)=-b_{X}+\sigma_{X}^{2}u+\int_{0}^{+\infty}{x(e^{ux}-1)\pi(\mathrm{d}x)},\qquad\text{
for all }u\in\mathcal{D}_{X}^{\circ}.$

If $\psi\wedge\phi=+\infty$, then $\mathcal{D}_{X}=\mathbb{R}$, in which case
$\Phi\in\mathcal{C}^{1}(\mathbb{R},\mathbb{R})$ obviously holds. If
$\psi\wedge\phi<+\infty$, the following assumption ensures that
$\Phi^{\prime}(\psi\wedge\phi)<+\infty$, which in turn implies that
$\Phi\in\mathcal{C}^{1}(\mathcal{D}_{X},\mathbb{R})$.

######  Assumption 3.2.

If $\psi\wedge\phi<+\infty$, then
$\int_{1}^{+\infty}{x\,e^{(\psi\wedge\phi)x}\pi(\mathrm{d}x)}<+\infty$.

The validity of Assumption 3.2 can be easily verified for specific types of
CBITCL process. Moreover, as illustrated in Section 5, it is satisfied by a
large class of models. Under Assumption 3.2, there exists a unique solution
$(\mathcal{U}(\cdot,u_{1},u_{2},u_{3}),\mathcal{V}(\cdot,u_{1},u_{2},u_{3}))$
to the extended Riccati system (3.3)-(3.4) up to time
$\mathrm{T}^{(u_{1},u_{2},u_{3})}$, for all
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$.
This enables us to state the following result, which extends Proposition 2.6
and follows directly from [KRM15, Theorem 2.14].

######  Lemma 3.3.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ and
$Y:=\int_{0}^{\cdot}{X_{s}\,\mathrm{d}s}$. Suppose that Assumption 3.2 holds.
Then, for all
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$
and $0\leq t\leq T<\mathrm{T}^{(u_{1},u_{2},u_{3})}$, it holds that

(3.7)
$\mathbb{E}\bigl{[}e^{u_{1}X_{T}+u_{2}Y_{T}+u_{3}Z_{T}}\bigr{|}\mathcal{F}_{t}\bigr{]}=\exp\bigl{(}\mathcal{U}(T-t,u_{1},u_{2},u_{3})+\mathcal{V}(T-t,u_{1},u_{2},u_{3})X_{t}+u_{2}Y_{t}+u_{3}Z_{t}\bigr{)},$

where
$(\mathcal{U}(\cdot,u_{1},u_{2},u_{3}),\mathcal{V}(\cdot,u_{1},u_{2},u_{3}))$
is the unique solution to the extended Riccati system (3.3)-(3.4) defined up
to time $\mathrm{T}^{(u_{1},u_{2},u_{3})}$.

The lifetime $\mathrm{T}^{(u_{1},u_{2},u_{3})}$ is closely connected to the
finiteness of exponential moments of the process $(X,Y,Z)$. Indeed, in view of
[KRM15, Proposition 3.3], it holds that

(3.8) $\mathrm{T}^{(u_{1},u_{2},u_{3})}=\sup\bigl{\\{}t\geq
0:\mathbb{E}[e^{u_{1}X_{t}+u_{2}Y_{t}+u_{3}Z_{t}}]<+\infty\bigr{\\}}.$

The next theorem is the main result of this section and provides an explicit
formula for $\mathrm{T}^{(u_{1},u_{2},u_{3})}$, for every
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$.
The proof is based on techniques similar to [KR11, Theorem 4.1], which covers
the case of affine stochastic volatility processes. However, our theorem is
specific to CBITCL processes and avoids the additional assumptions of [KR11,
Theorem 4.1]. In particular, it allows for CBI processes $X$ with an arbitrary
(not necessarily strictly subcritical, i.e. $b_{X}>0$) branching mechanism
$\Phi$. For $(u_{2},u_{3})\in\mathbb{R}\times\mathcal{D}_{Z}$, we introduce
the following notation:

$\displaystyle\mathcal{S}^{(u_{2},u_{3})}$
$\displaystyle:=\bigl{\\{}x\in\mathcal{D}_{X}:\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})\leq
0\bigr{\\}},$ $\displaystyle\chi^{(u_{2},u_{3})}$
$\displaystyle:=\sup\mathcal{S}^{(u_{2},u_{3})}\in[-\infty,\psi\wedge\phi],$

with the convention $\chi^{(u_{2},u_{3})}=-\infty$ if the set
$\mathcal{S}^{(u_{2},u_{3})}$ is empty.

######  Theorem 3.4.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ and suppose that
Assumption 3.2 holds. Then, for every
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$,
the lifetime $\mathrm{T}^{(u_{1},u_{2},u_{3})}$ is given as follows:

  * (i)

if $u_{1}\leq\chi^{(u_{2},u_{3})}$, then
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=+\infty$;

  * (ii)

if $u_{1}>\chi^{(u_{2},u_{3})}$, then

(3.9)
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=\int_{u_{1}}^{\psi\wedge\phi}{\frac{\mathrm{d}x}{\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})}}.$

###### Proof.

For simplicity of notation, for fixed
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$,
we denote by $\mathrm{T}^{\mathcal{U}}$ and $\mathrm{T}^{\mathcal{V}}$ the
lifetimes of the functions $\mathcal{U}(\cdot,u_{1},u_{2},u_{3})$ and
$\mathcal{V}(\cdot,u_{1},u_{2},u_{3})$ solutions to (3.3)-(3.4), respectively.
Making use of this notation, the lifetime $\mathrm{T}^{(u_{1},u_{2},u_{3})}$
can be decomposed as
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=\mathrm{T}^{\mathcal{U}}\wedge\mathrm{T}^{\mathcal{V}}$.
Always for simplicity of notation, we omit to write the superscript
$(u_{2},u_{3})$ in $\chi^{(u_{2},u_{3})}$ and $\mathcal{S}^{(u_{2},u_{3})}$.
Let us first consider the case $u_{1}\leq\chi$. If
$\Phi(u_{1})+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{1}\,u_{3}+\Xi(u_{3})=0$,
then the constant function $\mathcal{V}(\cdot,u_{1},u_{2},u_{3})\equiv u_{1}$
is the unique solution to (3.4), so that $\mathrm{T}^{\mathcal{V}}=+\infty$.
Since $u_{1}\in\mathcal{D}_{X}$, we also have
$\mathrm{T}^{\mathcal{U}}=+\infty$, which implies that
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=+\infty$. Suppose now that
$\Phi(u_{1})+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{1}\,u_{3}+\Xi(u_{3})<0$.
Let us define $\xi:=\inf\mathcal{S}$, with $\xi=+\infty$ if the set
$\mathcal{S}$ is empty. Note that, in the present case, $\xi<u_{1}$ and, if
$\xi>-\infty$, then
$\Phi(\xi)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,\xi+\Xi(u_{3})=0$. By
convexity of $\Phi$, it holds that
$\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})<0$, for all
$x\in(\xi,u_{1}]$. Therefore, equation (3.3) implies that the function
$\mathcal{V}(\cdot,u_{1},u_{2},u_{3})$ is strictly decreasing and we can write

$t=-\int_{\mathcal{V}(t,u_{1},u_{2},u_{3})}^{u_{1}}\frac{\mathrm{d}x}{\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})},\qquad\text{
for all }t\geq 0.$

Letting $t\to+\infty$ on both sides of this identity, we obtain that
$\mathcal{V}(t,u_{1},u_{2},u_{3})\to\xi$ as $t\to+\infty$, while
$\xi<\mathcal{V}(t,u_{1},u_{2},u_{3})\leq u_{1}$ for all $t\geq 0$. This shows
that $\mathrm{T}^{\mathcal{V}}=+\infty$. Moreover, making use of the structure
of $\Psi$, we obtain $-\infty<\mathcal{U}(t,u_{1},u_{2},u_{3})\leq
t\Psi(u_{1})$ for all $t\geq 0$, implying that
$\mathrm{T}^{\mathcal{U}}=+\infty$. We have thus shown that
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=+\infty$. If $u_{1}\leq\chi$ and
$\Phi(u_{1})+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{1}\,u_{3}+\Xi(u_{3})>0$,
then we necessarily have $u_{1}<\xi\in\mathcal{D}_{X}$ and
$\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})>0$, for all
$x\in[u_{1},\xi)$. Arguing similarly as above, this implies that
$u_{1}\leq\mathcal{V}(t,u_{1},u_{2},u_{3})<\xi$ for all $t\geq 0$, which in
turn leads to $\mathrm{T}^{(u_{1},u_{2},u_{3})}=+\infty$.

Let us now consider the case $u_{1}>\chi$ (which includes the case
$\chi=-\infty$). By using the convexity of $\Phi$, we have
$\Phi(u_{1})+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{1}\,u_{3}+\Xi(u_{3})>0$,
implying that the function $\mathcal{V}(\cdot,u_{1},u_{2},u_{3})$ is strictly
increasing with values in $[u_{1},\phi]$. The function
$\mathcal{V}(\cdot,u_{1},u_{2},u_{3})$ can be extended to a maximal interval
of existence $[0,\mathrm{T}^{*})$ such that one of the following two cases
occurs:

  1. (i)

$\mathrm{T}^{*}=+\infty$;

  2. (ii)

$\mathrm{T}^{*}<+\infty$ and
$\lim_{t\to\mathrm{T}^{*}}\mathcal{V}(t,u_{1},u_{2},u_{3})=\phi$.

In case (i), since $\mathcal{V}(\cdot,u_{1},u_{2},u_{3})$ is strictly
increasing, the limit $l:=\lim_{t\to+\infty}\mathcal{V}(t,u_{1},u_{2},u_{3})$
is well-defined with values in $(u_{1},\phi]\cup\\{+\infty\\}$. Suppose that
$l<+\infty$, i.e., the line $y=l$ is a horizontal asymptote for
$\mathcal{V}(\cdot,u_{1},u_{2},u_{3})$ as $t\to+\infty$. This implies that
$\frac{\partial\mathcal{V}}{\partial t}(t,u_{1},u_{2},u_{3})\to 0$ as
$t\to+\infty$. Letting $t\to+\infty$ on both sides of (3.3), this yields
$\Phi(l)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,l+\Xi(u_{3})=0$,
contradicting the fact that
$\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})>0$ for all
$x>\chi$. Therefore, the limit $l$ must necessarily be infinite, which can
only happen if $\phi=+\infty$ and, in this case,
$\lim_{t\to+\infty}\mathcal{V}(t,u_{1},u_{2},u_{3})=\phi$, analogously to case
(ii). In case (ii), let $(\mathrm{T}_{n})_{n\in\mathbb{N}}$ be an increasing
sequence such that $\mathrm{T}_{n}\to\mathrm{T}^{*}$ as $n\to+\infty$.
Similarly as above, making use of equation (3.3), we can write

(3.10)
$\mathrm{T}_{n}=\int_{u_{1}}^{\mathcal{V}(\mathrm{T}_{n},u_{1},u_{2},u_{3})}{\frac{\mathrm{d}x}{\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})}},\qquad\text{
for all }n\in\mathbb{N}.$

Letting $n\to+\infty$ on both sides of (3.10) yields

$\mathrm{T}^{*}=\int_{u_{1}}^{\phi}{\frac{\mathrm{d}x}{\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})}},$

which represents the lifetime $\mathrm{T}^{\mathcal{V}}$ of the function
$\mathcal{V}(\cdot,u_{1},u_{2},u_{3})$. To complete the proof, it suffices to
observe that, if $\phi\leq\psi$, then
$\int_{0}^{t}\Psi(\mathcal{V}(s,u_{1},u_{2},u_{3}))\mathrm{d}s$ is always
finite whenever $\mathcal{V}(t,u_{1},u_{2},u_{3})$ is finite, so that
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=\mathrm{T}^{\mathcal{V}}$. If $\phi>\psi$,
then
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=\inf\\{t\in\mathbb{R}_{+}:\mathcal{V}(t,u_{1},u_{2},u_{3})=\psi\\}$.
Without loss of generality, we can assume that there exists $n\in\mathbb{N}$
such that $\mathrm{T}_{n}=\mathrm{T}^{(u_{1},u_{2},u_{3})}$ and
$\mathcal{V}(\mathrm{T}_{n},u_{1},u_{2},u_{3})=\psi$. Inserting this into
equation (3.10) and combining the two cases $\phi\leq\psi$ and $\psi<\phi$ we
obtain formula (3.9). ∎

######  Remark 3.5.

(1) The result of Theorem 3.4 is useful for financial applications (see
Section 5.3). Indeed, many derivatives can be efficiently priced by resorting
to Fourier representations of their payoffs and exploiting the knowledge of
the conditional characteristic function of $(X,Y,Z)$ (see, e.g., [Fil09,
Section 10.3]). This requires an extension of (3.7) to the complex domain. As
shown in [KRM15], the feasibility of this extension crucially depends on the
fact that the lifetime $\mathrm{T}^{(u_{1},u_{2},u_{3})}$ is greater than the
maturity of the payoff to be priced, for suitable
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$.

(2) In asset pricing models, the finiteness of the lifetime
$\mathrm{T}^{(u_{1},u_{2},u_{3})}$, for
$(u_{1},u_{2},u_{3})\in\mathcal{D}_{X}\times\mathbb{R}\times\mathcal{D}_{Z}$,
is intimately related to the shape of the implied volatility smile at extreme
strikes, see [Lee04b] and [KR11, Section 5.1]. Therefore, the availability of
an explicit description of $\mathrm{T}^{(u_{1},u_{2},u_{3})}$ in Theorem 3.4
permits to characterize the tail behavior of the implied volatility smile in
financial models driven by CBITCL processes, as will be illustrated in the
examples considered in Section 5.

(3) In the case of classical CBI processes, a characterization of the lifetime
of exponential moments has been obtained in [FGS21, Theorem 2.7], which can be
recovered as a special case of Theorem 3.4 by taking $u_{2}\in\mathbb{R}_{-}$
and $u_{3}=0$.

The following corollary provides a necessary and sufficient condition for the
finiteness of exponential moments for all $u_{1}\in\mathcal{D}_{X}$, for fixed
but arbitrary $(u_{2},u_{3})\in\mathbb{R}\times\mathcal{D}_{Z}$. Whenever
$\psi\wedge\phi=+\infty$, we denote
$\Phi(\psi\wedge\phi):=\lim_{u\to+\infty}\Phi(u)$, which is well-defined with
values in $\\{-\infty,+\infty\\}$.

######  Corollary 3.6.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ and suppose that
Assumption 3.2 holds. Assume that, if $\psi<+\infty$ and $\psi\leq\phi$, then
$\int_{1}^{+\infty}e^{\psi x}\nu(\mathrm{d}x)<+\infty$. Let
$(u_{2},u_{3})\in\mathbb{R}\times\mathcal{D}_{Z}$. Then,
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=+\infty$ holds for all
$u_{1}\in\mathcal{D}_{X}$ if and only if
$\Phi(\psi\wedge\phi)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}(\psi\wedge\phi)+\Xi(u_{3})\leq
0$.

###### Proof.

Note first that, under the present assumptions,
$\mathcal{D}_{X}=(-\infty,\psi\wedge\phi]$ whenever $\psi\wedge\phi<+\infty$.
Suppose first that
$\Phi(\psi\wedge\phi)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}(\psi\wedge\phi)+\Xi(u_{3})\leq
0$. In this case, $\chi^{(u_{2},u_{3})}=\psi\wedge\phi$ (in both cases
$\psi\wedge\phi<+\infty$ and $\psi\wedge\phi=+\infty$). By Theorem 3.4, it
follows that $\mathrm{T}^{(u_{1},u_{2},u_{3})}=+\infty$ for all
$u_{1}\in\mathcal{D}_{X}$. Conversely, suppose that
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=+\infty$ for all $u_{1}\in\mathcal{D}_{X}$.
If $\psi\wedge\phi<+\infty$, then $\psi\wedge\phi\in\mathcal{D}_{X}$ and,
therefore, $\mathrm{T}^{(\psi\wedge\phi,u_{2},u_{3})}=+\infty$. Arguing by
contradiction, suppose that
$\Phi(\psi\wedge\phi)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}(\psi\wedge\phi)+\Xi(u_{3})>0$.
In that case, by the properties of the function $\Phi$, we would have
$\chi^{(u_{2},u_{3})}<\psi\wedge\phi$. But then formula (3.9) would imply that
$\mathrm{T}^{(\psi\wedge\phi,u_{2},u_{3})}=0$, thus leading to a
contradiction. On the other hand, if $\psi\wedge\phi=+\infty$, then
$\mathrm{T}^{(u_{1},u_{2},u_{3})}=+\infty$ for all
$u_{1}\in\mathcal{D}_{X}=\mathbb{R}$. Arguing again by contradiction, suppose
that
$\Phi(\psi\wedge\phi)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}(\psi\wedge\phi)+\Xi(u_{3})>0$.
In this case, there exists $M>0$ such that
$\Phi(x)+u_{2}+\rho\,\sigma_{X}\,\sigma_{Z}\,u_{3}\,x+\Xi(u_{3})>0$ for all
$x\geq M$. In turn, this yields $\chi^{(u_{2},u_{3})}<M$, which by Theorem 3.4
would imply that $\mathrm{T}^{(M,u_{2},u_{3})}<+\infty$, thus leading to a
contradiction. ∎

###  3.2. Asymptotic behavior of CBITCL processes

In this section, we study the long-term behavior of a CBITCL process $(X,Z)$
satisfying Assumption 2.8. Let us first recall that, if the CBI process $X$ is
strictly subcritical (i.e., $b_{X}>0$ in the branching mechanism (2.16)), then
it converges in law to a unique stationary distribution $\eta$ as
$t\to+\infty$, with Laplace transform

$L_{\eta}(\lambda)=\exp\left(\int_{\lambda}^{0}\frac{\Psi(x)}{\Phi(x)}\mathrm{d}x\right),\qquad\text{
for all }\lambda\leq 0,$

see [Li20, Theorem 10.4]. In general, the time-changed Lévy process $Z$ does
not admit an ergodic distribution. However, similarly as in [KR11, Section
3.2] but under weaker technical requirements, we can prove that the rescaled
cumulant generating function $\frac{1}{t}\log\mathbb{E}[e^{uZ_{t}}]$ converges
as $t\to+\infty$ to a limit corresponding to the cumulant generating function
of an infinitely divisible random variable.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ and suppose that
Assumption 3.2 is satisfied. We recall from Lemma 3.3 that

$\mathbb{E}\bigl{[}e^{uZ_{t}}\bigr{]}=\exp\bigl{(}\mathcal{U}(t,0,0,u)+\mathcal{V}(t,0,0,u)X_{0}\bigr{)},\qquad\text{
for all }u\in\mathcal{D}_{Z},$

where $(\mathcal{U}(\cdot,0,0,u),\mathcal{V}(\cdot,0,0,u))$ is the unique
solution to the following extended Riccati system:

(3.11) $\displaystyle\mathcal{U}(t,0,0,u)$
$\displaystyle=\int_{0}^{t}{\Psi\bigl{(}\mathcal{V}(s,0,0,u)\bigr{)}\mathrm{d}s},$
(3.12) $\displaystyle\frac{\partial\mathcal{V}}{\partial t}(t,0,0,u)$
$\displaystyle=\Phi\bigl{(}\mathcal{V}(t,0,0,u)\bigr{)}+\rho\,\sigma_{X}\,\sigma_{Z}\,u\,\mathcal{V}(t,0,0,u)+\Xi(u),\qquad\mathcal{V}(0,0,0,u)=0,$

for all $0\leq t<\mathrm{T}(u)$, where $\mathrm{T}(u):=\mathrm{T}^{(0,0,u)}$
denotes the maximal lifetime of
$(\mathcal{U}(\cdot,0,0,u),\mathcal{V}(\cdot,0,0,u))$. The next corollary
follows directly from Theorem 3.4 and provides an explicit description of
$\mathrm{T}(u)$. For simplicity of notation, we denote
$\chi(u):=\chi^{(0,u)}$.

######  Corollary 3.7.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ and suppose that
Assumption 3.2 holds. Then, for every $u\in\mathcal{D}_{Z}$, the lifetime
$\mathrm{T}(u)$ is given as follows:

  1. (i)

if $\chi(u)\geq 0$, then $\mathrm{T}(u)=+\infty$;

  2. (ii)

if $\chi(u)<0$, then

(3.13)
$\mathrm{T}(u)=\int_{0}^{\psi\wedge\phi}{\frac{\mathrm{d}x}{\Phi(x)+\rho\,\sigma_{Z}\,\sigma_{Z}\,u\,x+\Xi(u)}}.$

By Corollary 3.7, $\mathrm{T}(u)=+\infty$ for all $u\in\mathcal{D}_{Z}$ such
that $\chi(u)\geq 0$, meaning that the functions $\mathcal{U}(t,0,0,u)$ and
$\mathcal{V}(t,0,0,u)$ are finite for all $t\geq 0$. The study of the
asymptotic behavior of $\mathbb{E}[e^{uZ_{t}}]$ therefore requires analysing
the asymptotic properties of the functions $\mathcal{U}(\cdot,0,0,u)$ and
$\mathcal{V}(\cdot,0,0,u)$ for all
$u\in\mathcal{X}:=\\{u\in\mathcal{D}_{Z}:\chi(u)\geq 0\\}$. This is the
content of the next proposition, which specializes [KR11, Theorem 3.4] to the
case of CBITCL processes. More precisely, by relying on Corollary 3.7 and
exploiting the specific structure of a CBITCL process, we obtain an asymptotic
result which only requires the CBI process $X$ to be strictly subcritical,
besides the technical requirement of Assumption 3.2, thereby weakening some of
the assumptions of [KR11].

######  Proposition 3.8.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ with $b_{X}>0$
and suppose that Assumption 3.2 holds. For every $u\in\mathcal{X}$, define
$\xi(u):=\inf\bigl{\\{}x\in\mathcal{D}_{X}:\Phi(x)+\rho\,\sigma_{Z}\,\sigma_{Z}\,u\,x+\Xi(u)\leq
0\bigr{\\}}$ if $\Xi(u)\neq 0$ and $\xi(u):=0$ otherwise. Then, for every
$u\in\mathcal{X}$, it holds that

$\underset{t\to+\infty}{\lim}\,\mathcal{V}(t,0,0,u)=\xi(u)\qquad\text{and}\qquad\underset{t\to+\infty}{\lim}\,\frac{1}{t}\,\mathcal{U}(t,0,0,u)=\Psi\bigl{(}\xi(u)\bigr{)}.$

###### Proof.

The branching mechanism $\Phi$ satisfies $\Phi(0)=0$ and is continuous and
convex. Moreover, if $b_{X}>0$, then
$\lim_{x\to-\infty}(\Phi(x)+\rho\,\sigma_{X}\,\sigma_{Z}\,u\,x)=+\infty$, for
every $u\in\mathcal{X}$. Making use of these properties, the fact that
$\\{x\in\mathcal{D}_{X}:\Phi(x)+\rho\,\sigma_{X}\,\sigma_{Z}\,u\,x+\Xi(u)\leq
0\\}\neq\emptyset$, for every $u\in\mathcal{X}$, implies that the quantity
$\xi(u)$ is always finite and belongs to $\mathcal{D}_{X}$. In addition, by
continuity of $\Phi$, it holds that

(3.14)
$\Phi\bigl{(}\xi(u)\bigr{)}+\rho\,\sigma_{X}\,\sigma_{Z}\,u\,\xi(u)+\Xi(u)=0,\qquad\text{for
all }u\in\mathcal{X}.$

Let us now consider separately the three cases $\xi(u)=0$, $\xi(u)<0$ and
$\xi(u)>0$. If $\xi(u)=0$, then $\Xi(u)=0$ by (3.14) or by definition of
$\xi(u)$ and the function $\mathcal{V}(\cdot,0,0,u)\equiv 0$ is the unique
solution to (3.12), so that $\mathcal{V}(t,0,0,u)\to\xi(u)$ as $t\to+\infty$
trivially holds. If $\xi(u)<0$, then the fact that $\chi(u)\geq 0$ together
with the convexity of $\Phi$ implies that $\Xi(u)<0$ necessarily holds.
Therefore, by convexity of $\Phi$, it holds that
$\Phi(x)+\rho\,\sigma_{X}\,\sigma_{Z}\,u\,x+\Xi(u)<0$ for all
$x\in(\xi(u),0]$. Equation (3.12) then implies that $\mathcal{V}(\cdot,0,0,u)$
is strictly decreasing and satisfies $\xi(u)<\mathcal{V}(t,0,0,u)\leq 0$ and

$t=\int_{\mathcal{V}(t,0,0,u)}^{0}{\frac{-\mathrm{d}x}{\Phi(x)+\rho\,\sigma_{X}\,\sigma_{Z}\,u\,x+\Xi(u)}},\qquad\text{
for all }t\geq 0.$

Letting $t\to+\infty$ on both sides of this identity and recalling (3.14), we
obtain $\mathcal{V}(t,0,0,u)\to\xi(u)$ as $t\to+\infty$. If $\xi(u)>0$, then
we necessarily have $\Phi(x)+\rho\,\sigma_{X}\,\sigma_{Z}\,u\,x+\Xi(u)>0$, for
all $x\in[0,\xi(u))$, implying that $\mathcal{V}(\cdot,0,0,u)$ is strictly
increasing and satisfies $0\leq\mathcal{V}(t,0,0,u)<\xi(u)$ and

$t=\int_{\mathcal{V}(t,0,0,u)}^{0}{\frac{-\mathrm{d}x}{\Phi(x)+\rho\,\sigma_{X}\,\sigma_{Z}\,u\,x+\Xi(u)}},\qquad\text{
for all }t\geq 0.$

Analogously to the preceding case, letting $t\to+\infty$ on both sides of the
latter identity and making use of (3.14), we obtain that
$\mathcal{V}(t,0,0,u)\to\xi(u)$ as $t\to+\infty$. Finally, for all
$u\in\mathcal{X}$, the convergence of the function
$t\mapsto(1/t)\,\mathcal{U}(t,0,0,u)$ directly follows from equation (3.11):

$\frac{1}{t}\,\mathcal{U}(t,0,0,u)=\frac{1}{t}\,\int_{0}^{t}{\Psi\bigl{(}\mathcal{V}(s,0,0,u)\bigr{)}\,\mathrm{d}s}\longrightarrow\Psi\bigl{(}\xi(u)\bigr{)}\qquad\text{
as }t\to+\infty.$

∎

Proposition 3.8 yields the following long-term behavior of the time-changed
Lévy process $Z$:

$\frac{1}{t}\log\mathbb{E}\bigl{[}e^{uZ_{t}}\bigr{]}=\frac{1}{t}\,\mathcal{U}(t,0,0,u)+\frac{1}{t}\,\mathcal{V}(t,0,0,u)X_{0}\underset{t\to+\infty}{\longrightarrow}\Psi\bigl{(}\xi(u)\bigr{)},\qquad\text{
for all $u\in\mathcal{X}$.}$

Similarly as in [KR11, Theorem 3.4], it can be shown that $\xi(\cdot)$ and
$\Psi(\xi(\cdot))$ are cumulant generating functions of infinitely divisible
random variables. We can therefore conclude that the marginal distributions of
$Z$ are asymptotically equivalent to those of a Lévy process with
characteristic exponent $\Psi(\xi(\cdot))$. Notice that $\Psi(\xi(\cdot))$
corresponds to the exponent obtained by subordinating a Lévy process with
exponent $\xi$ by an independent Lévy process with exponent $\Psi$, see
[Sat99, Theorem 30.1]. In particular, this subordinator is equivalent to the
Lévy process $K=(K_{t})_{t\geq 0}$ appearing in the Lamperti-type
representation of the CBI process $X$ (recall the stochastic time change
equation (2.6), see also [CPGUB13] and [Szu21, Section 2.5]).

##  4\. CBITCL-preserving changes of probability

In this section, we describe a class of equivalent changes of probability that
leave invariant the class of CBITCL processes. More precisely, we consider
Esscher-type changes of measure under which a CBITCL process remains a CBITCL
process, with modified branching and immigration mechanisms and Lévy exponent.
The results of this section are motivated by financial applications, where one
typically wants to ensure that a model preserves its structural
characteristics under both the statistical and the risk-neutral probability,
as well as under risk-neutral probabilities associated to different numéraires
(see [FGS22] for an application to a multi-currency market).

Similarly as in Section 3, we suppose that Assumption 2.8 holds. Let us fix
two constants $\zeta\in\mathbb{R}$ and $\lambda\in\mathbb{R}$ and consider the
process $\mathcal{W}=(\mathcal{W}_{t})_{t\geq 0}$ defined by

(4.1) $\mathcal{W}_{t}:=\zeta(X_{t}-X_{0})+\lambda Z_{t},\qquad\text{ for all
}t\geq 0.$

In view of [JS03, Proposition II.8.26], the process $\mathcal{W}$ is an
exponentially special semimartingale if and only if $\zeta\in\mathcal{D}_{X}$
and $\lambda\in\mathcal{D}_{Z}$. In such a case, $\mathcal{W}$ admits a unique
exponential compensator, i.e., a predictable finite variation process
$\mathcal{K}=(\mathcal{K}_{t})_{t\geq 0}$ such that
$\exp(\mathcal{W}-\mathcal{K})$ is a local martingale. The following lemma
provides the explicit representation of the exponential compensator
$\mathcal{K}$.

######  Lemma 4.1.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$. Consider the
process $\mathcal{W}$ defined by (4.1), with $\zeta\in\mathcal{D}_{X}$ and
$\lambda\in\mathcal{D}_{Z}$. Then, the exponential compensator $\mathcal{K}$
of $\mathcal{W}$ is given by

(4.2)
$\mathcal{K}_{t}=t\,\Psi(\zeta)+Y_{t}\,\bigl{(}\Phi(\zeta)+\zeta\,\lambda\,\rho\,\sigma_{X}\,\sigma_{Z}+\Xi(\lambda)\bigr{)},\qquad\text{
for all }t\geq 0.$

###### Proof.

In view of [KS02, Theorems 2.18 and 2.19], taking into account that CBITCL
processes are quasi-left-continuous, the exponential compensator $\mathcal{K}$
coincides with the modified Laplace cumulant process of $(X,Z)$ computed at
$\theta=(\zeta,\lambda)$. The latter can be explicitly expressed as follows in
terms of the semimartingale differential characteristics $(B,C,F)$ of $(X,Z)$:

$\mathcal{K}_{t}=\int_{0}^{t}\biggl{(}\theta^{\top}B_{s}+\frac{1}{2}\theta^{\top}C_{s}\,\theta+\int_{\mathbb{R}^{2}}{\bigl{(}e^{\theta^{\top}x}-1-\theta^{\top}x\bm{1}_{\\{|x|<1\\}}\bigr{)}\,F_{s}(\mathrm{d}x)}\biggr{)}\mathrm{d}s.$

Making use of the explicit representation (2.11) of the characteristics
$(B,C,F)$, we then obtain

$\displaystyle\mathcal{K}_{t}$
$\displaystyle=t\biggl{(}\beta\zeta+\int_{0}^{+\infty}{(e^{\zeta
x}-1)\nu(\mathrm{d}x)}\biggr{)}+Y_{t}\biggl{(}-b_{X}\zeta+\frac{1}{2}\sigma^{2}_{X}\zeta^{2}+\int_{0}^{+\infty}{(e^{\zeta
x}-1-\zeta x)\pi(\mathrm{d}x)}\biggr{)}$
$\displaystyle\quad+Y_{t}\,\zeta\,\lambda\,\rho\,\sigma_{X}\,\sigma_{Z}+Y_{t}\biggl{(}b_{Z}\lambda+\frac{1}{2}\sigma_{Z}^{2}\lambda^{2}+\int_{\mathbb{R}}{(e^{\lambda
z}-1-\lambda z\bm{1}_{\\{|z|<1\\}})\gamma(\mathrm{d}z)}\biggr{)}$
$\displaystyle=t\,\Psi(\zeta)+Y_{t}\,\bigl{(}\Phi(\zeta)+\zeta\,\lambda\,\rho\,\sigma_{X}\,\sigma_{Z}+\Xi(\lambda)\bigr{)}.$

∎

The process $\mathcal{W}$ introduced in (4.1) can be used to define an
equivalent change of probability that leaves invariant the class of CBITCL
processes. To this effect, we consider a time horizon $\mathcal{T}<+\infty$
and assume that $(X,Z)$ is directly given by its extended Dawson-Li
representation (2.7) on a filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$. In view of Theorem 2.3, this
entails no loss of generality.

######  Theorem 4.2.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ and suppose that
Assumption 3.2 holds. Consider the process $\mathcal{W}$ defined in (4.1),
with $\zeta\in\mathcal{D}_{X}$ and $\lambda\in\mathcal{D}_{Z}$, and its
exponential compensator $\mathcal{K}$ given by (4.2). Then, the process
$(\exp(\mathcal{W}_{t}-\mathcal{K}_{t}))_{t\in[0,\mathcal{T}]}$ is a
martingale. Moreover, under the probability measure
$\mathbb{P}^{\prime}\sim\mathbb{P}$ defined on $(\Omega,\mathcal{F})$ by

(4.3)
$\frac{\mathrm{d}\mathbb{P}^{\prime}}{\mathrm{d}\mathbb{P}}:=e^{\mathcal{W}_{\mathcal{T}}-\mathcal{K}_{\mathcal{T}}},$

the process $(X,Z)$ is equivalent to a CBITCL process, with parameters
$\beta^{\prime}$, $\nu^{\prime}$, $b_{X}^{\prime}$, $\sigma_{X}^{\prime}$,
$\pi^{\prime}$, $\rho^{\prime}$, $b_{Z}^{\prime}$, $\sigma_{Z}^{\prime}$, and
$\gamma^{\prime}$ reported in Table 1, and Assumption 2.8 remains satisfied
under $\mathbb{P}^{\prime}$.

CBITCL parameters under $\mathbb{P}^{\prime}$
---
$\beta^{\prime}:=\beta$
$\nu^{\prime}(\mathrm{d}x):=e^{\zeta x}\nu(\mathrm{d}x)$
$b_{X}^{\prime}:=b_{X}-\zeta\,\sigma_{X}^{2}-\lambda\,\rho\,\sigma_{X}\sigma_{Z}-\int_{0}^{+\infty}{x(e^{\zeta
x}-1)\pi(\mathrm{d}x)}$
$\sigma_{X}^{\prime}:=\sigma_{X}$
$\pi^{\prime}(\mathrm{d}x):=e^{\zeta x}\pi(\mathrm{d}x)$
$\rho^{\prime}:=\rho$
$b_{Z}^{\prime}:=b_{Z}+\zeta\,\rho\,\sigma_{X}\sigma_{Z}+\lambda\,\sigma^{2}_{Z}+\int_{|z|<1}{z\,(e^{\lambda
z}-1)\gamma(\mathrm{d}z)}$
$\sigma_{Z}^{\prime}:=\sigma_{Z}$
$\gamma^{\prime}(\mathrm{d}z):=e^{\lambda z}\,\gamma(\mathrm{d}z)$
Table 1. Parameter transformations from $\mathbb{P}$ to $\mathbb{P}^{\prime}$
for the CBITCL process $(X,Z)$.

###### Proof.

By Lemma 4.1, the process $\exp(\mathcal{W}-\mathcal{K})$ is a local
martingale and, by Fatou’s lemma, also a supermartingale. Therefore, to prove
the martingale property of
$(\exp(\mathcal{W}_{t}-\mathcal{K}_{t}))_{t\in[0,\mathcal{T}]}$, it suffices
to show that
$\mathbb{E}[\exp(\mathcal{W}_{\mathcal{T}}-\mathcal{K}_{\mathcal{T}})]=1$.
More specifically, making use of equations (4.1) and (4.2), we will prove that

(4.4) $e^{-\zeta X_{0}-\mathcal{T}\Psi(\zeta)}\,\mathbb{E}\bigl{[}e^{\zeta
X_{\mathcal{T}}-(\Phi(\zeta)+\zeta\,\lambda\,\rho\,\sigma_{X}\sigma_{Z}+\Xi(\lambda))Y_{\mathcal{T}}+\lambda
Z_{\mathcal{T}}}\bigr{]}=1.$

Recalling the notation introduced in Section 3.1, we observe that
$\zeta\leq\chi^{(-\Phi(\zeta)-\zeta\,\lambda\,\rho\,\sigma_{X}\,\sigma_{Z}-\Xi(\lambda),\lambda)}$.
Theorem 3.4 therefore implies that
$\mathrm{T}^{(\zeta,-\Phi(\zeta)-\zeta\,\lambda\,\rho\,\sigma_{X}\,\sigma_{Z}-\Xi(\lambda),\lambda)}=+\infty$,
thereby showing that the expectation in (4.4) is finite. Moreover, under
Assumption 3.2, there exists a unique solution to the extended Riccati system
(3.3)-(3.4) with
$(u_{1},u_{2},u_{3})=(\zeta,-\Phi(\zeta)-\zeta\,\lambda\,\rho\,\sigma_{X}\,\sigma_{Z}-\Xi(\lambda),\lambda)$.
The solution to (3.4) is given by the constant function
$\mathcal{V}(\cdot,\zeta,-\Phi(\zeta)-\zeta\,\lambda\,\rho\,\sigma_{X}\,\sigma_{Z}-\Xi(\lambda),\lambda)=\zeta$,
which in turn implies that
$\mathcal{U}(t,\zeta,-\Phi(\zeta)-\zeta\,\lambda\,\rho\,\sigma_{X}\,\sigma_{Z}-\Xi(\lambda),\lambda)=t\Psi(\zeta)$,
for all $t\geq 0$. The validity of (4.4) then follows directly from Lemma 3.3.
We have thus shown that (4.3) defines a probability measure
$\mathbb{P}^{\prime}\sim\mathbb{P}$ with density process
$(\exp(\mathcal{W}_{t}-\mathcal{K}_{t}))_{t\in[0,\mathcal{T}]}$.

In order to show that $(X,Z)$ is a CBITCL process under $\mathbb{P}^{\prime}$,
we first express
$(\exp(\mathcal{W}_{t}-\mathcal{K}_{t}))_{t\in[0,\mathcal{T}]}$ as a
stochastic exponential, making use of [JS03, Theorem II.8.10] together with
(2.18)-(2.19):

$\displaystyle e^{\mathcal{W}-\mathcal{K}}$
$\displaystyle=\mathcal{E}\biggl{(}\zeta\,\sigma_{X}\int_{0}^{\cdot}{\sqrt{X_{s}}\,\mathrm{d}B_{s}^{X}}+\lambda\,\sigma_{Z}\int_{0}^{\cdot}{\sqrt{X_{s}}\,\mathrm{d}B_{s}^{Z}}+\int_{0}^{\cdot}\int_{0}^{+\infty}{(e^{\zeta
x}-1)\widetilde{N}_{0}(\mathrm{d}s,\mathrm{d}x)}\biggr{)}$
$\displaystyle\quad\times\mathcal{E}\biggl{(}\int_{0}^{\cdot}\int_{0}^{X_{s-}}\int_{0}^{+\infty}{(e^{\zeta
x}-1)\widetilde{N}_{1}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x)}+\int_{0}^{\cdot}\int_{0}^{X_{s-}}\int_{\mathbb{R}}{(e^{\lambda
z}-1)\widetilde{N}_{2}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)}\biggr{)}.$

By Girsanov’s theorem, the processes
$B^{\prime,X}=(B_{t}^{\prime,X})_{t\in[0,\mathcal{T}]}$ and
$B^{\prime,Z}=(B_{t}^{\prime,Z})_{t\in[0,\mathcal{T}]}$ defined by

$B_{t}^{\prime,X}:=B_{t}^{X}-(\zeta\,\sigma_{X}+\lambda\,\rho\,\sigma_{Z})\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}s}\quad\text{
and }\quad
B_{t}^{\prime,Z}:=B_{t}^{Z}-(\zeta\,\rho\,\sigma_{X}+\lambda\,\sigma_{Z})\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}s},$

for all $t\in[0,\mathcal{T}]$, are Brownian motions under the probability
measure $\mathbb{P}^{\prime}$, with correlation $\rho$. Again by Girsanov’s
theorem, under $\mathbb{P}^{\prime}$ the compensated Poisson random measures
associated to $N_{0}(\mathrm{d}t,\mathrm{d}x)$,
$N_{1}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}x)$, and
$N_{2}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}z)$ are respectively given by

$\displaystyle\widetilde{N}_{0}^{\prime}(\mathrm{d}t,\mathrm{d}x)$
$\displaystyle:=N_{0}(\mathrm{d}t,\mathrm{d}x)-e^{\zeta
x}\nu(\mathrm{d}x)\,\mathrm{d}t,$
$\displaystyle\widetilde{N}_{1}^{\prime}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}x)$
$\displaystyle:=N_{1}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}x)-e^{\zeta
x}\pi(\mathrm{d}x)\,\mathrm{d}u\,\mathrm{d}t,$
$\displaystyle\widetilde{N}_{2}^{\prime}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}z)$
$\displaystyle:=N_{2}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}z)-e^{\lambda
z}\gamma(\mathrm{d}z)\,\mathrm{d}u\,\mathrm{d}t.$

Therefore, under the probability measure $\mathbb{P}^{\prime}$, the extended
Dawson-Li representation (2.18)-(2.19) of $(X,Z)$ can be rewritten as follows:

$\displaystyle X_{t}$
$\displaystyle=X_{0}+\int_{0}^{t}{(\beta^{\prime}-b_{X}^{\prime}X_{s})\mathrm{d}s}+\sigma_{X}\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}B_{s}^{\prime,X}}$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{+\infty}{x\,N_{0}(\mathrm{d}s,\mathrm{d}x)}+\int_{0}^{t}\int_{0}^{X_{s-}}\int_{0}^{+\infty}{x\widetilde{N}_{1}^{\prime}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}x)},$
$\displaystyle Z_{t}$
$\displaystyle=b_{Z}^{\prime}\int_{0}^{t}{X_{s}\,\mathrm{d}s}+\sigma_{Z}\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}B_{s}^{\prime,Z}}+\int_{0}^{t}\int_{0}^{X_{s-}}\int_{|z|\geq
1}{z\,N_{2}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)}$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{X_{s-}}\int_{|z|<1}{z\,\widetilde{N}_{2}^{\prime}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)},$

where the parameters $\beta^{\prime}$, $b_{X}^{\prime}$, $b^{\prime}_{Z}$,
$\sigma_{X}^{\prime}$, $\sigma_{Z}^{\prime}$, and $\rho^{\prime}$ are given as
in Table 1. By Theorem 2.3, under the probability measure
$\mathbb{P}^{\prime}$ the process $(X,Z)$ is therefore equivalent to a CBITCL
process. Moreover, Assumption 2.8 remains satisfied under
$\mathbb{P}^{\prime}$, as a consequence of the fact that
$\mathbb{P}^{\prime}\sim\mathbb{P}$. ∎

As pointed out at the end of Section 2.2, in financial applications the
component $Z$ of a CBITCL process $(X,Z)$ is typically related to the log-
price process of an asset. In order to ensure absence of arbitrage, it is
useful to have conditions characterizing the martingale property of $\exp(Z)$.
To this end, by exploiting the previous results, we can state the following
corollary.

######  Corollary 4.3.

Let $(X,Z)$ be a $\mathrm{CBITCL}(X_{0},\Psi,\Phi,\Xi,\rho)$ and suppose that
Assumption 3.2 holds. Then, the process $(e^{Z_{t}})_{t\in[0,\mathcal{T}]}$ is
a martingale if and only if $1\in\mathcal{D}_{Z}$ and $\Xi(1)=0$.

###### Proof.

If $1\in\mathcal{D}_{Z}$ and $\Xi(1)=0$, by making use of (4.1) with
$(\zeta,\lambda)=(0,1)$ together with Lemma 4.1 and Theorem 4.2, we directly
obtain that $(e^{Z_{t}})_{t\in[0,\mathcal{T}]}$ is a martingale. Conversely,
if $(e^{Z_{t}})_{t\in[0,\mathcal{T}]}$ is a martingale, then
$\mathbb{E}[e^{Z_{\mathcal{T}}}]=1$. By [KRM15, Theorem 2.14-(a)], it follows
that $1\in\mathcal{D}_{Z}$. Moreover, in view of (3.8), we have that
$\mathrm{T}^{(0,0,1)}>\mathcal{T}$. By Lemma 3.3, the martingale property of
$(e^{Z_{t}})_{t\in[0,\mathcal{T}]}$ necessarily implies that
$\mathcal{V}(t,0,0,1)=0$, for all $t\in[0,\mathcal{T}]$. Since the ODE (3.4)
admits a unique solution under Assumption 3.2, it follows that $\Xi(1)=0$. ∎

##  5\. Examples and option pricing applications

In this section, we present some examples of CBITCL processes that possess a
self-exciting behavior and are particularly appropriate for financial
applications. In Section 5.1, we analyze from the viewpoint of CBITCL
processes the alpha-CIR process recently studied in [JMS17, JMSZ21]. In
Section 5.2, we investigate the CBITCL process adopted in [FGS22] for the
modelling of multi-currency markets with stochastic volatility. Finally, in
Section 5.3, we briefly discuss some general aspects of the use of CBITCL
processes for option pricing applications.

###  5.1. Alpha-CIR process and geometric Brownian motion

We say that a process $(X,Z)=((X_{t},Z_{t}))_{t\geq 0}$ is an _$\alpha$ -CIR-
time-changed geometric Brownian motion_ if it is given by

(5.1) $\left\\{\begin{aligned}
X_{t}&=X_{0}+\int_{0}^{t}{(\beta-b\,X_{s})\,\mathrm{d}s}+\sigma\,B^{X}_{Y_{t}}+\eta\,L_{Y_{t}},\\\
Z_{t}&=B^{Z}_{Y_{t}}-\frac{1}{2}Y_{t},\end{aligned}\right.$

with $Y_{t}=\int_{0}^{t}X_{s}\,\mathrm{d}s$, for all $t\geq 0$, and where
$X_{0}\geq 0$, $\beta\geq 0$, $b\geq 0$, $\sigma\geq 0$, $\eta\geq 0$, $B^{X}$
and $B^{Z}$ are one-dimensional Brownian motions with correlation $\rho$, with
$\rho\in[-1,1]$, and $L$ is a spectrally positive compensated $\alpha$-stable
Lévy process independent of $(B^{X},B^{Z})$, with stability parameter
$\alpha\in(1,2)$ and Lévy measure
$C_{\alpha}\,z^{-1-\alpha}\bm{1}_{\\{z>0\\}}\mathrm{d}z$ where $C_{\alpha}$ is
a suitable normalization constant.

This specification has been recently adopted for stochastic volatility
modelling in [JMSZ21], under the name of alpha-Heston stochastic volatility
model. The process defined in (5.1) belongs to the class of CBITCL processes
introduced in Definition 2.1, with $K_{t}:=\beta t$, $M_{t}:=-bt+\sigma
B_{t}^{X}+\eta L_{t}$, and $N_{t}:=B_{t}^{Z}-t/2$, for all $t\geq 0$. The
component $X$ is a CBI process with $\nu=0$ and
$\pi(\mathrm{d}z)=\eta^{\alpha}C_{\alpha}z^{-1-\alpha}\bm{1}_{\\{z>0\\}}\mathrm{d}z$,
with immigration mechanism $\Phi(x)=\beta x$ and branching mechanism

(5.2)
$\Phi(x)=-b\,x+\frac{1}{2}(\sigma\,x)^{2}+C_{\alpha}\,\Gamma(-\alpha)(-\eta\,x)^{\alpha},$

where $\Gamma$ denotes the Gamma function extended to
$\mathbb{R}\setminus\mathbb{Z}_{-}$ (see [Leb72]). Under this specification,
we have $\phi=0$ and $\psi=+\infty$, implying that
$\mathcal{D}_{X}=(-\infty,0]$. Since
$\int_{1}^{+\infty}z\pi(\mathrm{d}z)<+\infty$, Assumption 3.2 is satisfied.
Moreover, Assumption 2.8 is trivially satisfied. As a consequence of Theorem
3.4, the process $X$ does not admit exponential moments of any order, i.e.,
$\mathbb{E}[e^{uX_{t}}]=+\infty$ for all $u>0$ and $t>0$. This fact will
motivate the study of tempered $\alpha$-stable processes in Section 5.2.

######  Remark 5.1.

The process $X$ as introduced in (5.1) corresponds to the time change
representation of a stable Cox-Ingersoll-Ross process (also named _$\alpha$
-CIR process_, see [LM15, JMS17, JMSZ21] and [Szu21, Section 2.6.2]). An
$\alpha$-CIR process can be also defined by the following SDE:

$X_{t}=X_{0}+\int_{0}^{t}{(\beta-b\,X_{s})\mathrm{d}s}+\sigma\int_{0}^{t}{\sqrt{X_{s}}\,\mathrm{d}B^{X}_{s}}+\eta\int_{0}^{t}{\sqrt[\alpha]{X_{s-}}\,\mathrm{d}L_{s}}.$

By [FL10, Corollary 6.3], this SDE admits a unique strong solution that is a
CBI process.

In (5.1), the component $Z$ is defined as a time-changed Brownian motion with
drift. This specification ensures that $\exp(Z)=\mathcal{E}(B_{Y})$ is a
martingale (see Corollary 4.3). In view of financial applications, $Z$ can
therefore represent the discounted log-price process of a risky asset under a
risk-neutral probability measure. The associated Lévy exponent is given by
$\Xi(u)=u(u-1)/2$. Using the notation introduced in Section 3.2, we have that

$\begin{cases}\chi(u)=0,&\text{ for }u\in[0,1],\\\ \chi(u)<0,&\text{
otherwise}.\end{cases}$

Corollary 3.7 therefore implies that $\mathrm{T}(u)=+\infty$, for every
$u\in[0,1]$, while $\mathrm{T}(u)=0$, for every $u\notin[0,1]$. In other
words, for an $\alpha$-CIR-time-changed geometric Brownian motion it holds
that $\mathbb{E}[e^{uZ_{t}}]<+\infty$ for all $u\in[0,1]$ and $t>0$, while
$\mathbb{E}[e^{uZ_{t}}]=+\infty$ for all $u\notin[0,1]$ and $t>0$. These
results are consistent with [JMSZ21, Proposition 4.1 and Corollary 4.2], where
additional restrictions on the model parameters are required.

######  Remark 5.2.

As explained in part (2) of Remark 3.5, these results on the finiteness of
exponential moments of $Z$ can be used to study the behavior of the implied
volatility smile. Let us denote by $\sigma(T,k)$ the implied volatility of a
European Call option written on an asset with price process $\exp(Z)$, with
maturity $T$ and strike $\exp(k)$. By applying [Lee04b, Theorems 3.2 and 3.4],
we can deduce that the asymptotic behavior of $\sigma(T,k)$ at extreme strikes
is explicitly described as follows:

$\underset{k\to\pm\infty}{\limsup}\,\frac{\sigma^{2}(T,k)}{|k|}=\frac{2}{T},\qquad\text{
for all }T>0.$

Concerning the long-term behavior of the process $Z$, if we assume that $b>0$,
then by applying Proposition 3.8 we obtain

$\frac{1}{t}\log\mathbb{E}\bigl{[}e^{uZ_{t}}\bigr{]}\underset{t\to+\infty}{\longrightarrow}\beta\,\xi(u),\qquad\text{for
$u\in[0,1]$.}$

If we assume in addition that $\rho\leq 0$ (which is consistent with the so-
called leverage effect in stochastic volatility models, see for instance
[CW04]), then the quantity $\xi(u)$ can be explicitly computed as follows,
making use of equation (3.14):

$\xi(u)=\Phi_{u}^{-1}\biggl{(}\frac{u(1-u)}{2}\biggr{)},$

where $\Phi_{u}^{-1}$ denotes the inverse of the function
$x\mapsto\Phi_{u}(x):=\Phi(x)+\rho\,\sigma\,u\,x$, with $\Phi$ given by (5.2),
which can be easily seen to be a bijection from $\mathbb{R}_{-}$ to
$\mathbb{R}_{+}$, for every $u\in[0,1]$.

###  5.2. Tempered $\alpha$-stable CBI process and CGMY process

The CBITCL process considered in the previous subsection has the drawback that
its CBI component does not possess exponential moments. In finance
applications, the existence of exponential moments often represents an
essential requirement. For this reason, we now present a CBITCL process that
enjoys good integrability properties. The example considered in this
subsection relies on tempered $\alpha$-stable CBI processes, as introduced in
[FGS21] in interest rate modelling (see also [Szu21, Section 2.7]).

We recall from [FGS21] that a CBI process $X=(X_{t})_{t\geq 0}$ is said to be
tempered $\alpha$-stable if the Lévy measures appearing in (2.1) and (2.4) are
respectively given by

$\nu=0\qquad\text{ and
}\qquad\pi(\mathrm{d}z)=C_{\alpha}\,z^{-1-\alpha}\,e^{-\theta
z}\bm{1}_{\\{z>0\\}}\mathrm{d}z,$

where $\theta>0$, $\alpha\in(1,2)$ and $C_{\alpha}>0$ is a normalization
constant. Under this specification, the immigration mechanism reduces to
$\Psi(u)=\beta u$, while the branching mechanism can be explicitly computed as

(5.3) $\Phi(u)=-bu+\frac{1}{2}(\sigma
u)^{2}+C_{\alpha}\,\Gamma(-\alpha)\bigl{(}(\theta-u)^{\alpha}-\theta^{\alpha}+\alpha\theta^{\alpha-1}u\bigr{)},\qquad\text{
for all }u\leq\theta.$

It can be easily verified that $\mathcal{D}_{X}=(-\infty,\theta]$ and
Assumption 3.2 is satisfied (see also [Szu21, Lemma 2.19]). The existence of
exponential moments of $X$ can be characterized by relying on Corollary 3.6
(noting that, in the present case, $\phi=\theta$ and $\psi=+\infty$). Indeed,
taking $u_{2}=u_{3}=0$ in Corollary 3.6, it follows that
$\mathbb{E}[e^{uX_{T}}]<+\infty$ holds for all $u\leq\theta$ and $T>0$ if and
only if $\Phi(\theta)\leq 0$, namely, if and only if

(5.4)
$b\geq\frac{\sigma^{2}}{2}\theta+C_{\alpha}\,\Gamma(-\alpha)\,\theta^{\alpha-1}(\alpha-1).$

We have therefore shown that, in the case of tempered $\alpha$-stable CBI
processes, the existence of exponential moments amounts to a simple condition
on the parameters characterizing the process.

######  Remark 5.3.

As an example of application of Theorem 4.2, we show that tempered
$\alpha$-stable CBI processes can be easily constructed from non-tempered
$\alpha$-stable CBI processes by means of an equivalent change of probability
of the type (4.3). Let $X$ be an $\alpha$-CIR process, as in Section 5.1, and
define the process $\mathcal{W}$ as in (4.1) with $\zeta=-\theta$ and
$\lambda=0$, so that $\mathcal{W}=\theta(X_{0}-X)$, with $\theta>0$. By
Theorem 4.2, one can construct a probability measure
$\mathbb{P}^{\prime}\sim\mathbb{P}$ with density as in (4.3) such that under
$\mathbb{P}^{\prime}$ the Lévy measures of $X$ are given by $\nu^{\prime}=0$
and $\pi^{\prime}(\mathrm{d}z)=e^{-\theta
z}\eta^{\alpha}C_{\alpha}z^{-1-\alpha}\bm{1}_{\\{z>0\\}}\mathrm{d}z$. We have
therefore obtained that the process $X$ is a tempered $\alpha$-stable CBI
process under $\mathbb{P}^{\prime}$. An analogous change of measure technique
has been also employed in [JMS17, Proposition 4.1].

A CBITCL process $(X,Z)$ based on a tempered $\alpha$-stable CBI process can
be constructed as follows (see also [Szu21, Definition 2.18]):

(5.5) $\left\\{\begin{aligned}
X_{t}&=X_{0}+\int_{0}^{t}{(\beta-b\,X_{s})\mathrm{d}s}+\sigma\,B_{Y_{t}}+L_{Y_{t}}^{X},\\\
Z_{t}&=L_{Y_{t}}^{Z},\end{aligned}\right.$

with $Y_{t}=\int_{0}^{t}X_{s}\mathrm{d}s$, for all $t\geq 0$, where $X_{0}\geq
0$, $\beta\geq 0$, $b\in\mathbb{R}$, $\sigma\geq 0$, $B$ is a standard
Brownian motion independent of $L^{X}$, which is a spectrally positive
tempered $\alpha$-stable compensated Lévy process with Lévy measure
$C_{\alpha}\,z^{-1-\alpha}\,e^{-\theta z}\bm{1}_{\\{z>0\\}}\mathrm{d}z$, where
$\alpha\in(1,2)$, $\theta>0$,and $C_{\alpha}>0$ is a suitable normalization
constant, and $L^{Z}$ is a CGMY process (see [CGMY03]) independent of both $B$
and $L^{X}$. We recall that $L^{Z}$ is a CGMY process if its Lévy measure
$\gamma$ is of the form

(5.6)
$\gamma(\mathrm{d}z)=C_{Y}\bigl{(}z^{-1-Y}e^{-Mz}\bm{1}_{\\{z>0\\}}+|z|^{-1-Y}e^{-G|z|}\bm{1}_{\\{z<0\\}}\bigr{)}\mathrm{d}z,$

where the normalization constant can be chosen as $C_{Y}=1/\Gamma(-Y)$. The
parameters $G>0$ and $M>0$ temper the downward and the upward jumps,
respectively, while the parameter $Y\in(1,2)$ determines the local behavior of
$L^{Z}$, similarly to $\alpha$ above. The Lévy exponent $\Xi$ associated to a
CGMY process $L^{Z}$ with Lévy triplet $(0,0,\gamma\bigr{)}$ is given by

$\Xi(u)=\int_{\mathbb{R}}(e^{zu}-1-zu)\gamma_{Z}(\mathrm{d}z),\qquad\text{ for
all }u\in\mathsf{i}\mathbb{R}.$

It can be easily checked that $\mathcal{D}_{Z}=[-G,M]$ and the Lévy exponent
$\Xi$ takes the explicit form

$\Xi(u)=(M-u)^{Y}-M^{Y}+(G+u)^{Y}-G^{Y}+u\,Y\,(M^{Y-1}-G^{Y-1}),\qquad\text{
for all }u\in[-G,M].$

For simplicity of presentation, let us assume that $G=M$. In this case,
$\Xi:[-M,M]\to\mathbb{R}_{+}$ is a convex function with minimum $\Xi(0)=0$ and
maximum $\Xi(-M)=\Xi(M)=2M^{Y}(2^{Y-1}-1)$. If
$2M^{Y}(1-2^{Y-1})\geq\Phi(\theta)$, which can be rewritten in the form

(5.7)
$M\leq\biggl{(}\frac{\Phi(\theta)}{2\bigl{(}1-2^{Y-1}\bigr{)}}\biggr{)}^{1/Y},$

we have that $\chi(u)=\theta\geq 0$ for every $u\in[-M,M]$, using the notation
introduced in Section 3.2. Corollary 3.7 then implies that
$\mathrm{T}(u)=+\infty$ for every $u\in[-M,M]$, while $\mathrm{T}(u)=0$ for
every $u\notin[-M,M]$. Similarly as in Section 5.1, these results on the
finiteness of exponential moments of $Z$ can be used to characterize the tail
behavior of the implied volatility smile. Indeed, under condition (5.7) and
assuming $M>1$, an application of [Lee04b, Theorems 3.2 and 3.4] yields that

$\begin{aligned}
\underset{k\to-\infty}{\limsup}\,\frac{\sigma^{2}\bigl{(}T,k\bigr{)}}{|k|}&=\frac{2}{T}\bigl{(}1-2(\sqrt{M^{2}+M}-M)\bigr{)},\\\
\underset{k\to+\infty}{\limsup}\,\,\frac{\sigma^{2}\bigl{(}T,k\bigr{)}}{k}&=-\frac{2}{T}\bigl{(}1+2(\sqrt{M^{2}-M}-M)\bigr{)},\end{aligned}\qquad\qquad\text{
for all }T>0.$

The long-term behavior of the process $Z$ can be determined by relying on
Proposition 3.8. Under conditions (5.4) and (5.7), we have that

$\frac{1}{t}\log\mathbb{E}\bigl{[}e^{uZ_{t}}\bigr{]}\underset{t\to+\infty}{\longrightarrow}\beta\,\xi(u),\qquad\text{for
all }u\in[-M,M],$

where $\xi(u)$ is defined in the statement of Proposition 3.8.

The quantity $\xi(u)$ can be explicitly determined under the following
additional condition:

(5.8)
$b\geq\sigma^{2}\theta+C_{\alpha}\Gamma(-\alpha)\,\theta^{\alpha-1}\alpha.$

Condition (5.8) is stronger than condition (5.4) and, together with the
convexity and continuity of $\Phi$, it implies that $\Phi$ is decreasing on
$(-\infty,\theta]$. In this case, making use of (3.14), we have that

$\xi(u)=\Phi^{-1}\bigl{(}2M^{Y}-(M+u)^{Y}-(M-u)^{Y}\bigr{)},$

where $\Phi^{-1}$ denotes the inverse function of the branching mechanism
$\Phi$ given by (5.3), which under condition (5.8) is a bijection from
$(-\infty,\theta]$ to $[\Phi(\theta),+\infty)$.

######  Remark 5.4.

(1) The present specification does not necessarily guarantee the martingale
property of the process $\exp(Z)$ (compare with Corollary 4.3). Therefore, in
view of financial applications and similarly as in [FGS22], the CBITCL process
$(X,Z)$ may be used to model the discounted price process $S=(S_{t})_{t\geq
0}$ of a risky asset as follows:

(5.9) $\displaystyle\log S_{t}:=\lambda
Z_{t}+\zeta(X_{t}-X_{0})-\mathcal{K}_{t},\qquad\text{ for all }t\geq 0,$

where $\mathcal{K}$ denotes the exponential compensator (see Lemma 4.1), with
$\zeta\leq\theta$ and $\lambda\in[-G,M]$. Under this specification, the CBI
process $X$ plays the role of stochastic volatility, while the parameter
$\zeta$ determines the correlation between the log-price process and its
volatility. A direct application of Theorem 4.2 yields that $S$ is a
martingale, thereby ensuring absence of arbitrage. Moreover, since Assumption
3.2 is satisfied, the existence of moments $\mathbb{E}[S^{u}_{t}]$ can be
characterized by Theorem 3.4.

(2) As a direct consequence of Theorem 4.2, the class of CBITCL processes
considered in this subsection is stable with respect to equivalent changes of
probability of the form considered in Section 4. This property has been used
to construct risk-neutral measures that preserve the structure of the model in
[FGS22], where CBITCL processes have been applied to the modelling of multi-
currency markets in the presence of stochastic volatility and self-exciting
jumps.

###  5.3. Applications to option pricing

For simplicity of presentation, we assume that the discounted price process of
a risky asset is given by (5.9), with respect to a CBITCL process $(X,Z)$ and
with $(\zeta,\lambda)\in\mathcal{D}_{X}\times\mathcal{D}_{Z}$. In view of
Theorem 4.2, this implies the martingale property of the price process $S$,
thereby ensuring absence of arbitrage. In this specification, the process $X$
represents stochastic volatility, possibly correlated and with common jumps
with the asset returns.

For illustrative purposes, let us consider the problem of pricing a European
call option, with payoff $(S_{T}-K)^{+}$ delivered at maturity $T$. To solve
this pricing problem, we will rely on Fourier techniques, which are viable in
the present setting due to the analytical tractability of CBITCL processes.
Indeed, the characteristic function
$\varphi_{t}(u):=\mathbb{E}[e^{\mathsf{i}u\log S_{t}}]$, for $u\in\mathbb{R}$
and $t\geq 0$, is available in semi-closed form, up to the solution of the
system of Riccati ODEs. The following result is directly obtained as a special
case of Lemma 3.3 (see [FGS22, Lemma 3.9] in the specific case of a tempered
$\alpha$-stable CBI process, as considered in Section 5.2).

######  Corollary 5.5.

For every $u\in\mathbb{R}$ and $t\geq 0$, the characteristic function
$\varphi_{t}(u)$ is given by

$\varphi_{t}(u)=\exp\bigl{(}-\mathsf{i}u\zeta
X_{0}-\mathsf{i}ut\Psi(\zeta)+\mathcal{U}(t,u_{1},u_{2},u_{3})+\mathcal{V}(t,u_{1},u_{2},u_{3})X_{t}\bigr{)},$

where the functions $\mathcal{U}$ and $\mathcal{V}$ solve (2.13)-(2.14) with

$u_{1}:=\mathsf{i}u\zeta,\qquad
u_{2}:=-\mathsf{i}u\bigl{(}\Phi(\zeta)+\zeta\,\lambda\,\rho\,\sigma_{X}\sigma_{Z}+\Xi(\lambda)\bigr{)},\qquad
u_{3}:=\mathsf{i}u\lambda.$

Starting from the above corollary, the pricing problem can be directly solved
by relying on the Fourier-based method of [Lee04a]. This approach requires an
extension of the domain of $\varphi_{T}$ to a suitable subset of $\mathbb{C}$.
While precise results in this direction can be obtained from Theorem 3.4, for
the specific case of a call option one can resort to a simpler argument.
Indeed, $\varphi_{T}$ is obviously well posed in $u=0$ and, due to the
martingale property of $S$, also in $u=-\mathsf{i}$. In turn, this implies
that $\varphi_{T}$ can be analytically extended to the set
$\\{u\in\mathbb{C}:-1<\mathrm{Im}(u)<0\\}$. We can then apply [Lee04a, Theorem
5.1] and deduce the following pricing formula, which is valid for every
$-1<\alpha<0$:

(5.10)
$\mathbb{E}[(S_{T}-K)^{+}]=\varphi_{T}(-\mathsf{i})+\frac{1}{\pi}\int_{0-\alpha\mathsf{i}}^{\infty-\alpha\mathsf{i}}\mathrm{Re}\left(e^{-\mathsf{i}z\log
K}\frac{\varphi_{T}(z-\mathsf{i})}{-z(z-\mathsf{i})}\right)\mathrm{d}z.$

######  Remark 5.6.

In the context of CBITCL processes, the scope of applicability of this
Fourier-based pricing technique is not limited to European call options.
Indeed, whenever the Fourier transform of the payoff function is known, one
can derive a pricing formula along the lines of (5.10). Several examples of
equity payoff functions for which the Fourier transform is known are provided
in [EGP10]. In addition, several payoff functions related to the
volatility/variance of the asset price process can be priced by applying the
techniques developed in [KMKV11], [ZK14] and [DGG15], which are all viable in
the present context of CBITCL processes. We also mention [HZ10] and [CFGG16],
where payoffs depending on several assets are considered, most notably basket
options.

## References

  * [BNS15] O.E. Barndorff-Nielsen and A. Shiryaev.  Change of Time and Change of Measure.  World Scientific, Singapore, second edition, 2015.
  * [CFGG16] R. Caldana, G. Fusai, A. Gnoatto, and M. Grasselli.  General closed-form basket option pricing bounds.  Quantitative Finance, 16(4):535–554, 2016.
  * [CGMY03] P. Carr, H. Geman, D.B. Madan, and M. Yor.  Stochastic volatility for Lévy processes.  Mathematical Finance, 13(3):345–382, 2003.
  * [Cla73] P.K. Clark.  A subordinated stochastic process model with finite variance for speculative prices.  Econometrica, 41(1):135–155, 1973.
  * [CMS22] G. Callegaro, A. Mazzoran, and C. Sgarra.  A self-exciting modeling framework for forward prices in power markets.  Applied Stochastic Models in Business and Industry, 38(1):27–48, 2022.
  * [CPGUB13] M.E. Caballero, J.L. Pérez Garmendia, and G. Uribe Bravo.  A Lamperti-type representation of continuous-state branching processes with immigration.  Annals of Probability, 41(3):1585–1627, 2013.
  * [CPGUB17] M.E. Caballero, J.L. Pérez Garmendia, and G. Uribe Bravo.  Affine processes on $\mathbb{R}^{m}_{+}\times\mathbb{R}^{n}$ and multiparameter time changes.  Annales de l’Institut Henri Poincaré - Probabilités et Statistiques, 53(3):1280–1304, 2017.
  * [CS05] A. Cherny and A. Shiryaev.  On stochastic integrals up to infinity and predictable criteria for integrability.  In M. Émery, M. Ledoux, and M. Yor, editors, Séminaire de Probabilités XXXVIII, pages 165–185. Springer, Berlin-Heidelberg, 2005.
  * [CW04] P. Carr and L. Wu.  Time-changed Lévy processes and option pricing.  Journal of Financial Economics, 71:113–141, 2004.
  * [DFS03] D. Duffie, D. Filipović, and W. Schachermayer.  Affine processes and applications in finance.  Annals of Applied Probability, 13(3):984–1053, 2003.
  * [DGG15] J. Da Fonseca, A. Gnoatto, and M. Grasselli.  Analytic pricing of volatility-equity options within Wishart-based stochastic volatility models.  Operations Research Letters, 43(6):601–607, 2015.
  * [DL06] D.A. Dawson and Z. Li.  Skew convolution semigroups and affine Markov processes.  Annals of Probability, 34(3):1103–1142, 2006.
  * [EGP10] E. Eberlein, K. Glau, and A. Papapantoleon.  Analysis of fourier transform valuation formulas and applications.  Applied Mathematical Finance, 17(3):211–240, 2010.
  * [FGS21] C. Fontana, A. Gnoatto, and G. Szulda.  Multiple yield curve modeling with CBI processes.  Mathematics and Financial Economics, 15(2):579–610, 2021.
  * [FGS22] C. Fontana, A. Gnoatto, and G. Szulda.  CBI-time-changed Lévy processes for multi-currency modeling.  Annals of Operations Research, forthcoming, 2022.
  * [FHM21] H. Fallahgoul, J. Hugonnier, and L. Mancini.  Risk premia and Lévy jumps: theory and evidence.  Journal of Financial Econometrics, forthcoming, 2021.
  * [Fil01] D. Filipović.  A general characterization of one factor affine term structure models.  Finance and Stochastics, 5(3):389–412, 2001.
  * [Fil09] D. Filipović.  Term-Structure Models: A Graduate Course.  Springer, Berlin-Heidelberg, 2009.
  * [FL10] Z. Fu and Z. Li.  Stochastic equations of non-negative processes with jumps.  Stochastic Processes and their Applications, 120(3):306–330, 2010\.
  * [FO09] F. Fang and C. W. Oosterlee.  Pricing early-exercise and discrete barrier options by fourier-cosine series expansions.  Numerische Mathematik, 114(1):27, 2009.
  * [HW04] J.Z. Huang and L. Wu.  Specification analysis of option pricing models based on time-changed Lévy processes.  The Journal of Finance, 59(3):1405–1439, 2004.
  * [HWY92] S.-W. He, J.-G. Wang, and J.-A. Yan.  Semimartingale Theory and Stochastic Calculus.  Science Press - CRC Press, Beijing, 1992.
  * [HZ10] T.R. Hurd and Z. Zhou.  A fourier transform method for spread option pricing.  SIAM Journal on Financial Mathematics, 1(1):142–157, 2010.
  * [IW89] N. Ikeda and S. Watanabe.  Stochastic differential equations and diffusion processes.  North-Holland mathematical library. North-Holland, Amsterdam-Oxford-New York, second edition, 1989.
  * [Jac79] J. Jacod.  Calcul Stochastique et Problèmes de Martingales.  Springer, Berlin-Heidelberg-New York, 1979.
  * [JMS17] Y. Jiao, C. Ma, and S. Scotti.  Alpha-CIR model with branching processes in sovereign interest rate modeling.  Finance and Stochastics, 21(3):789–813, 2017.
  * [JMSS19] Y. Jiao, C. Ma, S. Scotti, and C. Sgarra.  A branching process approach to power markets.  Energy Economics, 79:144–156, 2019.
  * [JMSZ21] Y. Jiao, C. Ma, S. Scotti, and C. Zhou.  The Alpha-Heston stochastic volatility model.  Mathematical Finance, 31(3):943–978, 2021.
  * [JS03] J. Jacod and A. Shiryaev.  Limit Theorems for Stochastic Processes.  Springer, Berlin-Heidelberg-New York, second edition, 2003.
  * [Kal06] J. Kallsen.  A didactic note on affine stochastic volatility models.  In Y. Kabanov, R. Lipster, and J. Stoyanov, editors, From Stochastic Calculus to Mathematical Finance - The Shiryaev Festschrift, pages 343–368. Springer, Berlin, 2006.
  * [KMKV11] J. Kallsen, J. Muhle-Karbe, and M. Voß.  Pricing options on variance in affine stochastic volatility models.  Mathematical Finance, 21(4):627–641, 2011.
  * [KR09] M. Keller-Ressel.  Affine Processes - Theory and Applications in Finance.  PhD thesis, Vienna University of Technology, 2009.
  * [KR11] M. Keller-Ressel.  Moment explosions and long-term behavior of affine stochastic volatility models.  Mathematical Finance, 21(1):73–98, 2011.
  * [KRM15] M. Keller-Ressel and E. Mayerhofer.  Exponential moments of affine processes.  Annals of Applied Probability, 25(2):714–752, 2015.
  * [KS02] J. Kallsen and A. Shiryaev.  The cumulant process and Esscher’s change of measure.  Finance and Stochastics, 6:397–428, 2002.
  * [KW71] K. Kawazu and S. Watanabe.  Branching processes with immigration and related limit theorems.  Theory of Probability and its Applications, 16(1):36–54, 1971.
  * [Leb72] N.N. Lebedev.  Special Functions and their Applications.  Prentice-Hall, Englewood Cliffs (N.J.), 1972.
  * [Lee04a] R. Lee.  Option pricing by transform methods: Extensions, unification and error control.  Journal of Computational Finance, 7(3):51–86, 2004.
  * [Lee04b] R.W. Lee.  The moment formula for implied volatility at extreme strikes.  Mathematical Finance, 14(3):469–480, 2004.
  * [Li20] Z. Li.  Continuous-state branching processes with immigration.  In Y. Jiao, editor, From Probability to Finance - Lecture Notes of BICMR Summer School on Financial Mathematics, pages 1–70. Springer, Singapore, 2020.
  * [LM15] Z. Li and C. Ma.  Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model.  Stochastic Processes and their Applications, 125(8):3196–3233, 2015\.
  * [Sat99] K. Sato.  Lévy Processes and Infinitely Divisible Distributions.  Cambridge University Press, Cambridge, 1999.
  * [Swi16] A. Swishchuk.  Change of Time Methods in Quantitative Finance.  Springer, 2016.
  * [Szu21] G. Szulda.  Branching Processes and Multiple Term Structure Modeling.  PhD thesis, Université de Paris, 2021.
  * [ZK14] W. Zheng and Y.K. Kwok.  Closed form pricing formulas for discretely sampled generalized variance swaps.  Mathematical Finance, 24(4):855–881, 2014.