\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 231, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/231\hfil
Solutions to a partial integro-differential equations]
{Solutions to a partial integro-differential parabolic system arising
in the pricing of financial options in regime-switching jump diffusion models}
\author[I. Florescu, R. Liu, M. C. Mariani \hfil EJDE-2012/231\hfilneg]
{Ionut Florescu, Ruihua Liu, Maria Cristina Mariani} % in alphabetical order
\address{Ionut Florescu \newline
Department of Mathematical Sciences,
Stevens Institute of Technology,
Castle Point on Hudson, Hoboken, NJ 07030, USA}
\email{Ionut.Florescu@stevens.edu}
\address{Ruihua Liu \newline
Department of Mathematics,
University of Dayton, 300 College Park, Dayton, OH 45469-2316, USA}
\email{rliu01@udayton.edu}
\address{Maria Cristina Mariani \newline
Department of Mathematical Sciences,
The University of Texas at El Paso,
Bell Hall 124, El Paso, TX 79968-0514, USA}
\email{mcmariani@utep.edu}
\thanks{Submitted December 15, 2011. Published December 21, 2012.}
\subjclass[2000]{35K99, 35F99, 45K05, 45K99}
\keywords{Partial integro-differential equations; option pricing;
\hfill\break\indent regime-switching jump diffusion;
upper and lower solutions}
\begin{abstract}
We study a complex system of partial integro-differential equations
(PIDE) of parabolic type modeling the option pricing problem in
a regime-switching jump diffusion model. Under suitable conditions,
we prove the existence of solutions of the PIDE system in a general
domain by using the method of upper and lower solutions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}
The problem of pricing derivatives in financial mathematics often leads
to studying partial differential and/or integral equations.
The typical differential equations obtained are of parabolic type.
In recent years, the complexity of the equations studied has increased,
due to the inclusion of stochastic volatility, stochastic interest rate,
and jumps in the mathematical models governing the dynamics of the
underlying asset prices. The integral terms in a partial differential
equation with integral terms (henceforth PIDE) come from modeling jumps
in the underlying asset prices.
Florescu and Mariani \cite{FlorescuM} considered a continuous time
asset price model containing both stochastic volatility and discontinuous jumps.
In this model, the volatility is driven by a second correlated
Brownian motion and the jump is modeled by a compound Poisson process.
Standard risk-neutral pricing principle is used to obtain a single second-order
partial integro-differential equation (PIDE) for the prices of European
options written on the asset. Motivated by this financial mathematics problem,
a general integro-differential parabolic problem is posed and studied in
the cited work \cite{FlorescuM}. The existence of solution is proved by
employing a method of upper and lower solutions and a diagonal argument.
Moreover, the proof can provide an approximation method for numerically
finding the solution of the general type PIDE which was later implemented
in \cite{FlorescuMS}. In the current work we are discussing a more general
model capable of producing realistic paths. The resulting option price
may be found as the solution of a system of PIDE's, which to our knowledge,
have never been studied before by the method employed in this work.
The main result of this paper is Theorem \ref{main-result} which provides
conditions on the integral terms in the PIDE system which guarantee the
existence of the solution to this system. The emphasis in this work is on
the applied mathematical methods rather than the stochastic process due
to the technical nature of this result.
\section{Motivating the PIDE System under Study}
In this section, we introduce and motivate the regime-switching jump
diffusion model, the option pricing problem, and the resulting system
of partial integro-differential equations we will study in the next section.
\subsection{About the suitability of the stochastic model postulated}
From the beginning of the $20$-th century starting with Louis
Jean-Baptiste Alphonse Bachelier (1870-1946) researchers have been
looking for mathematical models which are capable of capturing the
main features of an observed price path. The most famous attempt is
the Black-Scholes-Merton model \cite{black:1973,merton:1973} which
influenced so much of the literature on asset pricing. Of course,
the model is now known to be too simple for high frequency data
and many attempts have been made in the last 20 years to capture
the complexity exhibited by the evolution of asset prices.
In recent years, considerable attention has been drawn to regime-switching
models in financial mathematics aiming to include the influence of
macroeconomic factors on the individual asset price behavior.
See, for example \cite{{Guo}, {Liu}, {LiuZY}}. In this setting,
asset prices are dictated by a number of stochastic differential equations
coupled by a finite-state Markov chain,
which represents various randomly changing economical factors.
Mathematically, the regime-switching models
generalize the traditional models in such a way that various coefficients
in the models depend on the Markov chain. Consequently, a system
(not a single one) of coupled PDEs (or PIDEs) is obtained for
option prices.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{fig1}
\caption{Tick data for one trading day and a certain equity}
\label{fig:sampleStockPath}
\end{figure}
To further illustrate the motivation of this study,
in Figure \ref{fig:sampleStockPath} we present the one day evolution
of high frequency data (all trades) for a particular equity gathered
from a single exchange. This image or sample path is generally representative
for many traded assets in any markets during any given day.
Looking at the image we recognize several characteristics which can
be captured by using a regime-switching jump diffusion model.
The price path seems to jump in several places during the day
(either up or down) and in between these jumps it seems to follow processes
with perhaps different parameters. For example, the variability at the
beginning of the day seems to be larger than the variability in the
middle of the day. As described next, in a regime-switching jump diffusion
model the process jumps at random times by a random amount and, in
between jumps, the process could follow diffusions with distinct coefficients.
We believe such a model is appropriate for describing the observed features
of the asset price during the day.
\subsection{Regime-switching jump diffusion model}
We assume that all the stochastic processes in this paper are defined on
some underlying complete probability space $(\mathcal S, \mathcal{F}, \mathcal{P})$.
Let $B_t$ be a one-dimensional standard Brownian motion.
Let $\alpha_t$ be a continuous-time Markov chain with state space
$\mathcal{M} : =\{1,\dots,m\}$. Let $Q=(q_{ij})_{m\times m} $ be the
intensity matrix (or the generator) of $\alpha_t$. In this context
the generator $q_{ij}$, $i,j=1,\dots, m$ satisfy:
\begin{itemize}
\item[(I)] $q_{ij}\geq 0$ if $i\neq j$;
\item[(II)] $q_{ii} = -\sum_{j\neq i}q_{ij}$ for each $i=1, \dots, m$.
\end{itemize}
We assume that the Brownian motion $B_t$ and the Markov chain
$\alpha_t$ are independent.
Let $N_t$ be a Cox process (a specialized non-homogeneous Poisson process)
with regime-dependent intensity $\lambda_{\alpha_t}$. Thus, when the
current state is $\alpha_t=i$, the time until the next jump is given
by an exponential random variable with mean $1/\lambda_i$. $N_t$ models
the number of the jumps in the asset price up to time $t$.
Let the jump sizes be given by a sequence of iid random variables
$Y_i, i=1,2,\dots ,$ with probability density $g(y)$.
Assume that the jump sizes $Y_i$, $i=1,2,\dots ,$ are independent
of $B_t$ and $\alpha_t$.
We model the time evolution of the asset price $S_t$ by using the
regime-switching jump diffusion:
\begin{equation} \label{asset-price-model}
\frac{d S_t}{ S_t}= \mu_{\alpha_t} dt +\sigma_{\alpha_t} d B_t + dJ_t,\quad t
\geq 0,
\end{equation}
where $ \mu_{\alpha_t}$ and $\sigma_{\alpha_t}$ are the appreciation
rate and the volatility rate of the asset $S_t$, respectively. $J_t$
is the jump component given by
\begin{equation} \label{jump-component}
J_t=\sum_{k=1}^{N_t} (Y_k-1).
\end{equation}
The $Y_i-1$ values represent the percentage of the asset price by which
the process jumps. Note that, in between switching times the process
follows a regular jump diffusion with constant coefficients.
However, the coefficients are switching as governed by the corresponding
state of the Markov chain.
In the model setting \eqref{asset-price-model} the volatility is modeled
as a finite-state stochastic Markov chain $\sigma_{\alpha_t}$.
As further reference for the model usefulness, \eqref{asset-price-model}
may be considered as a discrete approximation of a continuous-time
diffusion model for the stochastic volatility (e.g. the Heston's model).
See Liu \cite{Liu} and references therein for more details.
\subsection{The option pricing problem}
Given that the asset price process follows the hypothesized
model \eqref{asset-price-model} we look into the problem of derivative
pricing written on the corresponding asset.
To this end denote $r_{\alpha_t}$ the risk-free interest rate corresponding
to the state $\alpha_t$ of the Markov chain.
We consider an European type option written on the asset $S_t$ with
maturity $T<\infty$. Let $V_i(S, t)$ denote the option value functions
at time to maturity $t$, when the asset price $S_t=S$ and the regime
$\alpha_t=i$ (assuming that the regime $\alpha_t$ is observable).
Under these assumptions the value functions $V_i(S, t)$, $i=1, \dots, m$,
satisfy the system of PIDEs
\begin{equation} \label{PIDE-regime-switching}
\begin{aligned}
&\frac{1}{2} \sigma^2_i S^2
\frac{\partial^2 V_i }{\partial S^2}
+ (r_i-\lambda_i \kappa) S \frac{\partial V_i }{\partial S}- r_i V_i
-\frac{\partial V_i}{\partial t} \\
&+ \lambda_i E[ V_i(SY, t)-V_i(S, t) ]
+ \sum_{j\neq i} q_{i j}[V_j-V_i] =0,
\end{aligned}
\end{equation}
where we use the notation $\kappa = E[Y-1]=\int (y-1)g(y) dy$.
Recalling that $q_{ii} = -\sum_{j\neq i}q_{ij}$ and using the density
$g(y)$, we can rewrite \eqref{PIDE-regime-switching} as
\begin{equation}\label{PIDE-regime-switching-01}
\begin{aligned}
&\frac{1}{2} \sigma^2_i S^2
\frac{\partial^2 V_i }{\partial S^2} + (r_i-\lambda_i \kappa) S \frac{\partial V_i }{\partial S}- (r_i+\lambda_i-q_{ii}) V_i
-\frac{\partial V_i}{\partial t} \\
& = -\lambda_i \int V_i(Sy, t)g(y) dy
-\sum_{j\neq i} q_{ij} V_j.
\end{aligned}
\end{equation}
Standard risk-neutral pricing principle is used for the derivation
of equation \eqref{PIDE-regime-switching} from the
dynamics \eqref{asset-price-model} (not presented here),
we refer for instance to \cite{Guo, karatzasS}.
Such types of systems are complicated and hard to approach.
In \cite{FlorescuM} we analyze a single PIDE which appears when
the process exhibit jumps and has stochastic volatility.
The approach was further implemented and an algorithm to calculate
the solution was provided in \cite{FlorescuMS}.
The current problem is more complex by involving a system of PIDE's. However,
note that the system is coupled only through the final term in
the equation \eqref{PIDE-regime-switching-01}, the rest of the terms
in each equation $i$ are in the respective $V_i(\cdot,\cdot)$.
This fact provides hope that an existence proof
(and a potential solving algorithm) may be provided in the current
situation as well.
As a historical note William Feller (1906-1970) and his students
developed the semigroup theory for Markov Processes and there is
a well known direct link through them with the resulting PDE's
for option pricing (see e.g., \cite{Ethier:2005} or \cite{Strook:1979}
for excellent reviews of this connection). However, they worked with
diffusion processes (and later jump diffusion processes)
characterizing Markov processes and these models lead to simple PIDE's.
In the case presented here, while the regime switching is governed
by a con\-tinuous-time Markov chain and while each process being switched
is indeed a continuous-time Markov process (jump diffusion),
the overall structure may not be described by a simple Markov
process with a diffusion + density type infinitesimal generator.
Instead, the resulting overall Markov process is complex and produces
the type of coupled systems of PIDE's studied in this paper.
The work we present proves an existence of solution theorem for such systems.
This system is very different from the work published in
Pitt's dissertation in 1967 \cite{Pitt} and naturally the analysis
follows different techniques, thus our proof (about a different problem)
is different than the analysis done by Pitt, that was later extended
to time dependent coefficients on a simpler Markov process.
\section{The General PIDE System}
To obtain a solution to the system \eqref{PIDE-regime-switching-01}
we formulate the problem using more general terms.
This will provide a universal approach to the kind of PIDE systems
arising when solving complex option pricing problems.
We first recall that the Black-Scholes equation
$$
{\partial V\over\partial t} +{1\over 2}\sigma^2S^2{\partial^2V\over\partial S^2}
+rS{\partial V\over\partial S}-rV=0
$$
becomes a heat type equation after performing the classical
(Euler type) change of variable: $S=Ee^x$ and $t=T-{2\tau\over \sigma^2}$,
where $E,T,\sigma$ are constants, see for example \cite{whd}.
From now on, we assume that this classical change of variable
for Black-Scholes type equations was performed.
To this end, let $\Omega\subset \mathbb{R}^d$ be an unbounded smooth domain,
and we consider a collection of $m$ functions $u_i(x,t)$, $i=1,\dots , m$,
where $x=(x_1,x_2,\dots ,x_d)$ ($u_i:\mathbb R^d \times [0, T] \to \mathbb R$).
Let the operator $\mathcal{L}_i$ be defined by:
\begin{equation} \label{operator-L-i}
\mathcal{L}_i u_i=\sum_{j=1}^{d}\sum_{k=1}^{d} a_{jk}^i (x,t)
\frac{\partial u_i}{\partial x_j\partial x_k} + \sum_{j=1}^{d} b_{j}^i (x,t)
\frac{\partial u_i}{\partial x_j } +c^i(x,t)u_i, \; i=1,\dots , m,
\end{equation}
where the coefficients $a_{jk}^i$, $b_{j}^i$ and $c^i$,
$i\in\{1,\dots , m\}$; $j, k\in\{1,\dots , d\}$ belong to the H\"older
space $C^{\delta, \delta/2}(\overline{\Omega} \times [0, T])$ and
satisfy the following conditions:
\begin{itemize}
\item There exist two constants $\Lambda_1$, $\Lambda_2$ with
$0<\Lambda_1\leq \Lambda_2 <\infty$ such that
\begin{equation} \label{operator-L-i-con-1}
\Lambda_1 |v|^2 \leq \sum_{j=1}^{d}\sum_{k=1}^{d} a_{jk}^i (x,t) v_jv_k
\leq \Lambda_2 |v|^2 \; \text{ for } v=(v_1,\dots, v_d)^T \in \mathbb{R}^d.
\end{equation}
\item There exists a constant $C>0$ such that
\begin{equation} \label{operator-L-i-con-2}
|b_{j}^i (x,t)| \leq C.
\end{equation}
\item The functions
\begin{equation} \label{operator-L-i-con-3}
c^i(x,t) \leq 0.
\end{equation}
\end{itemize}
This general formulation encompasses all models presented including
as degenerate cases the diffusion model of Black Scholes and
the jump diffusion of Merton. The conditions are needed to ensure
the existence of solution for a system of the type
\eqref{PIDE-regime-switching-01}. Generally, these conditions
are satisfied by most option pricing equations arising in finance.
The generalized problem corresponding to the system of PIDE's in
equation \eqref{PIDE-regime-switching-01} on an unbounded smooth
domain $\Omega$ is:
\begin{equation} \label{general-system-PIDE}
\begin{gathered}
\mathcal{L}_i u_i -\frac{\partial u_i}{\partial t}
= \mathcal{G}_i(t,u_i) - \sum_{j\neq i} q_{ij} u_j \quad \text{in }\Omega \times (0,T)
\\
u_i(x,0)= u_{i,0}(x) \quad \text{on }\Omega \times \{0\}
\\
u_i(x,t)=h_i(x,t) \quad\text{on }\partial \Omega \times (0,T)
\end{gathered}
\end{equation}
for $i=1,\dots , m$,
where $\mathcal{G}_i$, are continuous integral operators.
We assume that the boundary conditions
$u_{i,0} \in C^{2+\delta}(\overline{\Omega})$, and
$h_i \in C^{2+\delta, 1+\delta/2}(\overline{\Omega} \times [0, T])$
satisfy the compatibility condition
\begin{equation} \label{compatibility-condition}
h_i(x,0)=u_{i,0}(x), \quad
\text{for any }x\in \partial \Omega, \; i=1,\dots , m.
\end{equation}
We note that as applied to problem \eqref{PIDE-regime-switching-01}
the operators $\mathcal{L}_i$ and $\mathcal{G}_i$ differ in the parameter values only,
not in functional form. However, the general problem formulation
as described above contains the case when the option is written on a
basket of assets (not only a single stock) which are all modeled by
different jump-diffusion type processes and they are all dependent
on the same regime-switching Markov process $\alpha_t$.
The goal is to establish the existence of a solution to the
system \eqref{general-system-PIDE} using the method of upper and
lower solutions.
\begin{definition} \label{defn-upper-lower-solutions} \rm
A collection of $m$ smooth functions $u=\{u_i, 1\leq i \leq m\}$
is called an upper (lower) solution of problem \eqref{general-system-PIDE}
if:
\begin{equation} \label{upper-lower-PIDE}
\begin{gathered}
\mathcal{L}_i u_i -\frac{\partial u_i}{\partial t}\leq (\geq)
\mathcal{G}_i(t,u_i) - \sum_{j\neq i} q_{i
j} u_j \quad \text{in }\Omega \times (0,T) \\
u_i(x,0) \geq (\leq ) u_{i,0}(x) \quad \text{on }\Omega \times \{0\}\\
u_i(x,t) \geq (\leq ) h_i(x,t) \quad\text{on }\partial \Omega \times (0,T)
\end{gathered}
\end{equation}
for $ i=1,\dots , m$.
\end{definition}
Our main result is stated in the following theorem.
\begin{theorem} \label{main-result}
Let the operators $\mathcal{L}_i$ and $\mathcal{G}_i$, $1\leq i \leq m$ be as defined above.
Assume that either:
\begin{itemize}
\item for each $1\leq i \leq m$, $\mathcal{G}_i$ is non-increasing with
respect to $u_i$, or
\item for each $1\leq i \leq m$, there exists a continuous and
increasing one-dimen\-sional function $f_i$ such that $\mathcal{G}_i(t,u_i)-f_i(u_i)$
is non-increasing with respect to $u_i$.
\end{itemize}
Furthermore, assume there exist a lower solution
$\alpha=\{\alpha_i, 1\leq i \leq m\}$ and an upper solution
$\beta=\{\beta_i, 1\leq i \leq m\}$ of problem \eqref{general-system-PIDE}
satisfying $\alpha \leq \beta$ componentwise (i.e., $\alpha_i \leq \beta_i$,
$1\leq i \leq m$) in $\Omega \times (0,T)$. Then \eqref{general-system-PIDE}
admits a solution $u$ such that $\alpha\leq u\leq \beta$ in
$\Omega \times (0,T)$.
\end{theorem}
\subsection{The method of upper and lower solutions}
In this section we present a proof of our main result,
Theorem \ref{main-result}. To this end, we first solve an
analogous problem in a bounded domain and then extend the solution
to the unbounded domain $\Omega \times (0,T)$.
We note that we need this extension since in general option problems
are solved on $(S_1,\dots,S_d,t)\in (0,\infty)^d\times [0,T]$.
Please also note that the theory may be used just as well for perpetual
options (when $T=\infty$).
\begin{lemma} \label{homogeneous-decoupled-problem}
Let $U$ be a smooth and bounded subset of $\Omega$. Then, there exists a
unique collection of functions
$\varphi_U=\{\varphi_{U,i}, 1\leq i \leq m\}$ with
$\varphi_{U,i}\in C^{2+\delta, 1+\delta/2}(\overline{U} \times [0, T])$ such that
\begin{equation} \label{homogeneous-decoupled-problem-soln}
\begin{gathered}
\mathcal{L}_i \varphi_{U,i} -\frac{\partial \varphi_{U,i}}{\partial t}= 0, \quad
(x,t) \in U \times (0,T), \\
\varphi_{U,i}(x,0)=u_{i,0}(x), \quad x \in U,\\
\varphi_{U,i}(x,t)=h_i(x,t), \quad (x,t)\in \partial U\times [0,T],
\end{gathered}
\end{equation}
for $i=1,\dots , m$.
Moreover, if $\alpha$ and $\beta$ are respectively a lower and an upper
solution of this reduced problem \eqref{homogeneous-decoupled-problem-soln}
with $\alpha \leq \beta$ in $U\times (0, T)$, then
\begin{equation} \label{homogeneous-decoupled-problem-soln-upper-lower}
\alpha(x,t)\leq \varphi_U(x,t) \leq \beta(x,t), \quad (x,t) \in \overline{U}
\times [0,T]. \end{equation}
\end{lemma}
\begin{proof}
Note that the homogeneous system \eqref{homogeneous-decoupled-problem-soln}
is decoupled. Thus, solving the system means solving the individual PDE's.
Applying Florescu and Mariani \cite[Lemma 2.1]{FlorescuM} to each of
the $m$ component equations, we obtain the expected result.
\end{proof}
The next result is crucial and it is the lemma that makes the transition
from simple PDE's to a complex system of PIDE's on a bounded domain.
\begin{lemma} \label{problem-bounded-domain}
Let $U\in\mathbb{R}^d$ be a smooth and bounded domain.
Let $0<\tilde{T} \beta_{i_0}(x_0,t_0)$.
Since $u^1_{i_0}|_{\partial \overline{U}
\times [0, \tilde{T}]} \leq \beta_{i_0}|_{\partial \overline{U}
\times [0, \tilde{T}]}$, we deduce that
$(x_0, t_0) \in U \times (0,\tilde{T})$ and furthermore we may
assume that $(x_0, t_0)$ is a maximum point of $u^1_{i_0}-\beta_{i_0}$.
It follows that $\nabla (u^1_{i_0}-\beta_{i_0})(x_0, t_0) =0$,
$\triangle (u^1_{i_0}-\beta_{i_0})(x_0, t_0) <0$ and
$\frac{\partial (u^1_{i_0}-\beta_{i_0})}{\partial t} (x_0, t_0)=0$.
Since $\mathcal{L}_{i_0}$ is strictly elliptic, we have
\begin{equation} \label{induction-u1-beta}
\mathcal{L}_{i_0}(u^1_{i_0}-\beta_{i_0})(x_0, t_0) <0.
\end{equation}
On the other hand, \eqref{u1-and-beta} implies that at the maximum
point $(x_0, t_0)$, $\mathcal{L}_{i_0}(u^1_{i_0}-\beta_{i_0})(x_0, t_0) \geq 0$,
a contradiction with \eqref{induction-u1-beta}.
In the general induction step, given
$\alpha \leq u^{n-1}\leq u^{n} \leq \beta$, we can use a similar
argument to show that $\alpha \leq u^{n }\leq u^{n+1} \leq \beta$.
First, we claim that $u^{n }\leq u^{n+1}$. If this is not true,
there exists an index $1\leq i_0 \leq m$ and a point
$(x_0, t_0) \in U \times (0,\tilde{T})$ such that
\begin{equation} \label{induction-un}
\mathcal{L}_{i_0}(u^{n}_{i_0}-u^{n+1}_{i_0})(x_0, t_0) <0.
\end{equation}
On the other hand, from the way the sequence is defined in
\eqref{PIDE-bounded-domain-known-RHS-un} and the fact that,
$\mathcal{G}_i$ is non-increasing with respect to $u_i$ for each
$1\leq i \leq m$ and $q_{ij}\geq 0$ for $i\neq j$, we have
\begin{equation} \label{un-and-un1}
\mathcal{L}_i u_i^{n+1} -\frac{\partial u^{n+1}_{i}}{\partial t}
= \mathcal{G}_{i}(t,u^{n}_{i}) - \sum_{j\neq i } q_{i j} u^{n}_{j}
\leq \mathcal{G}_{i}(t,u^{n-1}_{i}) - \sum_{j\neq i } q_{i j} u^{n-1}_{j}
= \mathcal{L}_i u_i^{n} -\frac{\partial u^{n}_{i}}{\partial t}.
\end{equation}
It follows that at the maximum point $(x_0, t_0)$,
$ \mathcal{L}_{i_0}(u^{n}_{i_0}-u^{n+1}_{i_0})(x_0, t_0)\geq 0$,
a contradiction with \eqref{induction-un}. In a similar way,
we can show that $u^{n+1} \leq \beta$.
We now define:
\begin{equation} \label{limit-un} u(x,t) = \lim_{n\to \infty} u^n(x,t),
\end{equation}
or componentwise,
\begin{equation} \label{limit-un-1} u_i(x,t)
= \lim_{n\to \infty} u^n_i(x,t), \quad \forall
(x,t)\in \overline{U} \times [0,\tilde{T}], \; i=1,\dots m.
\end{equation}
Since $u^n \leq \beta$ and $\beta \in L^p(V)$, by the Lebesgue's dominated
convergence theorem, we obtain that $\{u^n_i\}_{n=1}^{\infty}$
is a convergent sequence, therefore a Cauchy sequence in the complete
space $L^p(V)$ for each $i=1,\dots m$.
Using the results in \cite[Chapter 7]{Lieberman}, the $W_p^{2,1}$-norm
of the difference $u_i^n-u_i^m$ can be controlled by its $L^p$-norm
and the $L^p$-norm of its image under the operator
$\mathcal{L}_i -\frac{\partial }{\partial t}$. Using these results,
there exists a constant $C>0$ such that
\begin{equation} \label{proof-Cauchy}
\begin{aligned}
&\|u_i^n-u_i^m\|_{W_p^{2,1}(V)}\\
&= \|D^2(u_i^n-u_i^m)\|_{L^p(V)} + \|(u_i^n-u_i^m)_t\|_{L^p(V)} \\
& \leq C\Big(\|\mathcal{L}_i(u_i^n-u_i^m)
-\frac{\partial (u_i^n-u_i^m) }{\partial t}\|_{L^p(V)}
+ \|u_i^n-u_i^m\|_{L^p(V)} \Big).
\end{aligned}
\end{equation}
By construction,
\begin{equation} \label{proof-Cauchy-1}
\mathcal{L}_i(u_i^n-u_i^m)-\frac{\partial (u_i^n-u_i^m) }{\partial t}
= \mathcal{G}_i(\cdot, u_i^{n-1})-\mathcal{G}_i(\cdot, u_i^{m-1})
- \sum_{j\neq i} q_{ij} (u_j^{n-1}-u_j^{m-1}).
\end{equation}
Since $\mathcal{G}_i$ is a completely continuous operator, there is a constant
$C_1>0$ such that,
\begin{equation} \label{proof-Cauchy-2}
\begin{split}
&\| \mathcal{G}_i(\cdot, u_i^{n-1})-\mathcal{G}_i(\cdot, u_i^{m-1})
- \sum_{j\neq i} q_{ij} (u_j^{n-1}-u_j^{m-1}) \|_{L^p(V)}\\
&\leq C_1 \sum_{j=1}^m \| u_j^{n-1}-u_j^{m-1} \|_{L^p(V)}.
\end{split}
\end{equation}
Combining \eqref{proof-Cauchy}, \eqref{proof-Cauchy-1}, \eqref{proof-Cauchy-2},
it follows that $\{u^n_i\}_{n=1}^{\infty}$ is a Cauchy sequence in
$W_p^{2,1}(V)$ for each $i=1,\dots m.$ Hence $u^n_i\to u_i$
in the $W_p^{2,1}$-norm, and thus $u=\{u_i,1\leq i\leq m\}$
is a strong solution of the problem \eqref{general-system-PIDE-bounded-domain}.
Now suppose the condition on $\mathcal{G}_i(t, u_i)$ is that for each
$1\leq i \leq m$, there exists a continuous and increasing
function $f_i$ such that $\mathcal{G}_i(t,u_i)-f_i(u_i)$ is non-increasing
with respect to $u_i$. Starting with $\tilde{u}^0=0$, we define
recursively a sequence $\{\tilde{u}^n, n=0,1,\dots \}$ such that
$\tilde{u}^{n+1}=\{\tilde{u}^{n+1}_{i}\in W_p^{2,1}(V), 1\leq i\leq m\}$
is the unique solution of the problem
\begin{equation} \label{PIDE-bounded-domain-known-RHS-un-2}
\begin{gathered}
\mathcal{L}_i \tilde{u}^{n+1}_i -\frac{\partial \tilde{u}^{n+1}_i}{\partial t}
-f_i(\tilde{u}^{n+1}_i)
= \mathcal{G}_i(t,\tilde{u}^{n}_i) - f_i(\tilde{u}^{n}_i) - \sum_{j\neq i} q_{i
j} \tilde{u}^{n}_j \quad \text{in }U \times (0,\tilde{T}) \\
\tilde{u}^{n+1}_i(x,0)=u_{i,0}(x) \quad \text{on }U \times \{0\}
\\
\tilde{u}^{n+1}_i(x,t)=\varphi_{U,i}(x,t) \quad \text{on }
\partial U \times (0,\tilde{T})
\end{gathered}
\end{equation}
for $i=1,\dots , m$.
The same arguments as before may be repeated almost verbatim to show that
\begin{equation} \label{increasing-sequence-un-2}
0 \leq \tilde{u}^n\leq \tilde{u}^{n+1} \leq \beta
\quad \text{in }\overline{U} \times [0,\tilde{T}],\;
\forall n\in \mathbb{N}.
\end{equation}
This will imply that,
$\{ \tilde{u}^n_i\}_{n=1}^\infty$ is a Cauchy sequence in $W_p^{2,1}(V)$
for each $i=1,\dots m$. If we denote with
$\tilde{u}_i=\lim_{n\to \infty} \tilde{u}^n_i$.
Then $\tilde{u}=\{\tilde{u}_i, 1\leq i\leq m\}$ is a strong solution
of problem \eqref{general-system-PIDE-bounded-domain}.
Note that the function $f$ is continuous and thus the solution of
the modified problem \eqref{PIDE-bounded-domain-known-RHS-un-2}
also solves the original system.
\end{proof}
Finally, all that remains is to extend the solution to the original
unbounded domain.
\begin{proof}[Proof of Theorem \ref{main-result}]
We first approximate the unbounded domain
$\Omega$ by a non-decreasing sequence $(\Omega_N)_{N\in \mathbb{N}}$
of bounded smooth sub-domains of $\Omega$, which can be chosen in
such a way that $\partial \Omega $
is also the union of the non-decreasing sequence
$ \partial \Omega_N \cap \partial \Omega$.
In view of Lemma \ref{problem-bounded-domain}, we define
$u^N=\{u_i^N, 1\leq i \leq m\}$ as a solution of the problem
\begin{equation} \label{general-system-PIDE-2}
\begin{gathered}
\mathcal{L}_i u_i -\frac{\partial u_i}{\partial t}
= \mathcal{G}_i(t,u_i) - \sum_{j\neq i} q_{i
j} u_j \quad \text{in } \Omega_N \times (0,T-\frac{1}{N}) \\
u_i(x,0)= u_{i,0}(x) \quad \text{on } \Omega_N \times \{0\}\\
u_i(x,t)= h_i(x,t) \quad \text{on } \partial \Omega_N \times (0,T-\frac{1}{N})
\end{gathered}
\end{equation}
for $i=1,\dots , m$, such that $0=\alpha \leq u^N \leq \beta$ in
$\Omega_N \times (0,T-\frac{1}{N})$.
Define $V_N=\Omega_N \times (0,T-\frac{1}{N})$ and choose $p>d$.
For $M>N$, we have:
\begin{equation} \label{proof-main-1}
\begin{aligned}
&\|D^2(u_i^M)\|_{L^p(V_N)} + \|(u_i^M)_t \|_{L^p(V_N)} \\
&\leq C_1\Big(\|\mathcal{L}_i u_i^M-\frac{\partial u_i^M}{\partial t}\|_{L^p(V_N)}
+ \|u_i^M\|_{L^p(V_N)} \Big) \\
& \leq C_1\Big(\| \mathcal{G}_i(t, u_i^M)-\sum_{j\neq i} q_{ij} u^M_j \|_{L^p(V_N)}
+ \|\beta\|_{L^p(V_N)} \Big)
\leq C ,
\end{aligned}
\end{equation}
for some constant $C $ depending only on $N$.
By Morrey embedding theorem, $W^{2,1}_p(V_N)\hookrightarrow C(\overline V_N)$
(see e. g. \cite{ad}), there exists a subsequence that
converges uniformly on $\overline V_N$.
Now, we apply the well known Cantor diagonal argument: for $N=1$,
we extract a subsequence of $u_i^M|_{\overline\Omega_1\times [0,T-1]}$
(still denoted $\{ u_i^M\}$ for notational simplicity)
that converges uniformly to some function $u_{i1}$ over
$\overline\Omega_1\times [0,T-1]$.
Next, we extract a subsequence of
$u_i^M|_{\overline\Omega_2\times [0,T-\frac 12]}$ for $M\ge 2$
(still denoted $\{ u_i^M\}$)
that converges uniformly to some function $u_{i2}$ over
$\overline\Omega_2\times [0,T-\frac 12]$, and so on.
As the families $\{\Omega_N\}$ and
$\{\partial\Omega_N \cap \partial\Omega\}$ are non-decreasing,
it is clear that $u_{iN}(x,0) = u_{iN}(x)$ for $x\in \Omega_N$,
and that $u_{iN}(x,t)= h(x,t)$ for
$x\in \partial \Omega\cap \partial \Omega_N$ and $t\in (0,T-\frac 1N)$.
Moreover, as $u_{i(N+1)}$ is constructed as the limit of a subsequence of
$u_i^M|_{\overline\Omega_{N+1}\times [0,T-\frac 1{N+1}]}$, which converges
uniformly to some function $u_{iN}$ over
$\overline\Omega_N\times [0,T-\frac 1N]$,
it follows that
$u_{i(N+1)}|_{\overline\Omega_N\times [0,T-\frac 1N]}=u_{iN}$ for every $N$.
Thus, the diagonal subsequence
(still denoted $\{ u_i^M\}$) converges uniformly
over compact subsets of
$\Omega\times (0,T)$ to the function $u_i$ defined as
$u_i = u_{iN}$ over $\overline\Omega_N\times [0,T- \frac 1N]$.
For $V = U \times (0,\tilde T)$, $U\subset \Omega$ and $\tilde
T < T$, taking $M,N \ge N_V$ for some $N_V$ large enough we have that
\begin{equation} \label{proof-main-2}
\begin{aligned}
&\|u_i^M-u_i^N\|_{W_p^{2,1}(V)} \\
&= \|D^2(u_i^M-u_i^N)\|_{L^p(V)} + \|(u_i^M-u_i^N)_t\|_{L^p(V)} \\
& \leq C\Big(\|\mathcal{L}_i(u_i^M-u_i^N)
-\frac{\partial (u_i^M-u_i^N) }{\partial t}\|_{L^p(V)}
+ \|u_i^M-u_i^N\|_{L^p(V)} \Big).
\end{aligned}
\end{equation}
By construction,
\begin{equation} \label{proof-main-3}
\mathcal{L}_i(u_i^M-u_i^N)-\frac{\partial (u_i^M-u_i^N) }{\partial t}
= \mathcal{G}_i(\cdot, u_i^{M-1})-\mathcal{G}_i(\cdot, u_i^{N-1})
- \sum_{j\neq i} q_{ij} (u_j^{M-1}-u_j^{N-1}).
\end{equation}
As in the proof of Lemma \ref{problem-bounded-domain}, since $\mathcal{G}_i$ is continuous,
$\alpha \leq u^N \leq \beta$, using the Lebesgue's dominated convergence theorem,
it follows that $\{u^N_i\} $ is a Cauchy sequence in $W_p^{2,1}(V)$ for each
$i=1,\dots m$. Hence $u_i^N \to u_i$ in the $W_p^{2,1}(V)$-norm, and then
$u=\{u_i, 1\leq i \leq m\}$ is a strong solution in $V$.
It follows that $u$ satisfies the
equation on $\Omega \times (0,T)$. Furthermore, it is clear that
$u_i(x, 0) = u_{i,0}(x) $. For $M>N$ we have that
$u^M_i(x,t)=u^N_i(x,t)= h_i(x,t)$ for
$x\in \partial \Omega_N \cap \partial \Omega$, $t\in (0, T-\frac{1}{N})$.
It then follows that $u$ satisfies the boundary conditions
$u_i(x,t)= h_i(x,t)$, $1\leq i \leq m$ on $\partial \Omega \times [0, T)$.
This completes the proof.
\end{proof}
\section{Conclusion}
In this paper we provided an existence proof of the solution of a
system of PIDE's coupled in a very specific way. This coupling
type arises in regime-switching models when the assets are all
changing their stochastic dynamics according to the same
continuous-time Markov chain $\alpha_t$ with intensity
matrix $Q=(q_{ij})_{m\times m}$. The proof of our main result,
Theorem \ref{main-result}, uses a construction that may be used
in a numerical scheme implementing a PDE solver.
Theorem \ref{main-result} is directly applicable to our motivating
system \eqref{PIDE-regime-switching-01}, noticing that in this case
$\mathcal{G}_i(t,u)=-\lambda_i \int u_i(Sy,t) g(y) dy$ is a non-increasing
continuous operator in $u_i$ and that
$\alpha=\{\alpha_i(S,t)=0, \; 1\leq i\leq m \}$
is a lower solution of the option problem since the boundary conditions
$u_{i,0}$ and $h_i$ are nonnegative functions
(represent the monetary value of the option on the boundaries).
The upper solution also exists in these cases but its specific
form depends on the jump distribution $g(y)$ and needs to be derived
in each case. Note that the construction in Theorem \ref{main-result}
does not use the upper solution at all but its existence guarantees
the convergence of the final solution. For specific examples of upper
solutions as depending on the distribution $g(y)$ we refer to \cite{FlorescuM}
and \cite{FlorescuMS}.
We want to add a remark about the general nature of Theorem \ref{main-result}.
The result is applicable whenever the jump-diffusion process and the regime
switching may be thought of as Markovian. In particular, a simple generalization
is to make the distribution of jumps dependent on the state of the regime
as in $g_{\alpha_t}(\cdot)$. This is directly solvable with the theory presented.
As mentioned in the paper, options written on a basket of stocks which all
follow different jump-diffusions but they are all dependent on the same
regime switching process $\alpha_t$ also solve a system of PIDE's of the
type analyzed in Theorem \ref{main-result}. Finally, the case when
the assets are characterized using different switching regimes (correlated)
is an example of a more complex case worthy of further investigation.
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\end{document}