\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 62, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/62\hfil
Solutions to integro-differential parabolic problems]
{Solutions to integro-differential parabolic problems
arising in the pricing of financial options in a Levy market}
\author[I. Florescu, M. C. Mariani\hfil EJDE-2010/62\hfilneg]
{Ionu\c{t} Florescu, Maria Cristina Mariani} % in alphabetical order
\address{Ionu\c{t} Florescu \newline
Department of Mathematical Sciences,
Stevens Institute of Technology \\
Castle Point on Hudson, Hoboken, NJ 07030, USA}
\email{Ionut.Florescu@stevens.edu}
\address{Maria Christina 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 September 10, 2009. Published May 5, 2010.}
\subjclass[2000]{35K99, 45K05, 91680}
\keywords{Integro-differential parabolic equations; financial mathematics,
\hfill\break\indent Levy markets; jumps processes; stochastic volatility}
\begin{abstract}
We study an integro-differential parabolic problem modelling
a process with jumps and stochastic volatility in financial
mathematics. Under suitable conditions, we prove the existence
of solutions in a general domain using the method of upper
and lower solutions and a diagonal argument.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
In recent years there has been an increasing interest in solving PDE
problems arising in Financial Mathematics and in particular on
option pricing. The standard approach to this problem leads to the
study of equations of parabolic type.
In financial mathematics, usually the Black-Scholes model
(\cite{BS}, \cite{DU}, \cite{H},
\cite{I}, \cite{J}, \cite{ME}) is used for pricing derivatives,
by means of a backward parabolic
partial differential equation. A probability approach of the
fundamental theorem of asset pricing is given in \cite{ds}.
In this model, an important quantity is the volatility which
is a measure of the fluctuation (risk) in the asset prices,
and corresponds to the diffusion coefficient in the
Black-Scholes equation.
In the standard Black-Scholes model, a basic assumption is that
the volatility
is constant. Several models proposed in recent years, however,
allowed the volatility to be non constant or a stochastic variable.
For instance, in \cite{He} a model with stochastic volatility is proposed.
In this model the underlying security S follows,
as in the Black-Scholes model, a stochastic process
$$
dS_t = \mu S_t \,dt + \sigma_t S_t \,dZ_t,
$$
where $Z$ is a standard Brownian motion.
Unlike the classical model, the variance $v(t) = \sigma^2(t)$ also
follows a stochastic process given by
$$
d v_t = \kappa (\theta -v(t)) \,dt + \gamma \sqrt{v_t} \,dW_t,
$$
where $W$ is another standard Brownian motion. The correlation
coefficient between $W$ and $Z$ is denoted by $\rho$:
$$
\mathop{\rm Cov}(dZ_t,dW_t) = \rho \,dt
$$
This leads to a generalized Black-Scholes equation:
\begin{align*}
&\frac 12 vS^2\frac{\partial^2 U}{\partial S^2} + \rho\gamma vS
\frac{\partial^2 U}{\partial v\partial S} + \frac 12
v\gamma^2\frac{\partial^2 U}{\partial v^2} + rS \frac{\partial
U}{\partial S}\\
&+ [ \kappa(\theta -v) - \lambda v]\frac{\partial U}{\partial
v}-rU + \frac{\partial U}{\partial t} = 0.
\end{align*}
A similar model has been considered in \cite{AAM},
\cite{AZ}.
More general models with stochastic volatility have been
considered for example in \cite{BBF}, where the following problem is
derived using the Feynman-Kac lemma:
\begin{gather*}
u_t = \frac 12 Tr\left(M(x,\tau)D^2u\right) + q(x,\tau)\cdot
Du,\\
u(x,0) = u_0(x)
\end{gather*}
for some diffusion matrix $M$ and a payoff function $u_0$.
In the above $D$ denotes the gradient vector and $D^2$ the Hessian
matrix, when $x\in \mathbb R^n$.
The Black-Scholes models with jumps arise from the fact that
the driving Brownian motion is a continuous process, thus it
has difficulties fitting the financial data presenting large
fluctuations. The necessity of taking into account the large
market movements, and a great amount of information arriving
suddenly (i.e. a jump) has led to the study of partial
integro-differential equations (PIDE) in which the integral
term is modelling the jump.
In \cite{AA}, \cite{ME} the following PIDE in the variables $t$
and $S$ is obtained:
\begin{equation}
\label{problem1}
\frac 12 \sigma^2 S^2 F_{SS}+(r-\lambda k)SF_S+F_t
-rF+\lambda E\{F(SY,t)-F(S,t)\}=0.
\end{equation}
Here $r$ denotes the riskless rate, $\lambda$ the jump intensity,
and $k=E(Y-1)$, where
$E$ is the expectation operator and
the random variable $Y-1$ measures the percentage change
in the stock price if the jump
-- modelled by a Poisson process -- occurs
(for details see \cite{AA}, \cite{ME}).
The following problem is a generalization of \eqref{problem1} for
$N$ assets with prices $S_1,\dots, S_N$:
\begin{align*}
&\sum_{i=1}^N\frac 12 \sigma_i^2 S_i^2
\frac{\partial^2 F}{\partial S_i^2}+
\sum_{i\ne j}\frac 12 \rho_{ij}\sigma_i\sigma_j S_iS_j
\frac{\partial^2 F}{\partial S_i\partial S_j}+
\sum_{i=1}^N(r-\lambda k_i)S_i\frac{\partial F}{\partial S_i}+
\frac{\partial F}{\partial t}
-rF\\
&+\lambda\int[F(S_1Y_1,\dots ,S_dY_d,t)-F(S_1,\dots ,S_d,t)]g(Y_1,\dots ,
Y_d) dY_1\dots dY_d=0
\end{align*}
with
$$
\rho_{ij}dt=E\{dz_i,dz_j\}
$$
the correlation coefficients and $g$ the probability density function
of the random variable $(Y_1,\ldots,Y_n)$ modelling the jump sizes.
We recall that the case in which $F$ is increasing and all jumps are
negative corresponds to the evolution of a call option near a crash,
see \cite{ct} and the references therein.
In applications when modelling high frequency data by a Levy
--like stochastic process appears to be the best fit (see \cite{MF}
and its references). Jump-diffusion models are a particular case of
Levy processes and indeed stock evolution was soon modelled using
various classes of Levy processes
(see \cite{Barndoff},
\cite{Madan}).
When using these more general models, option prices are once again
found by solving the resulting partial integro-differential equations.
For example, integro-differential equations appear in exponential
Levy models, when the market price of an asset is represented as
the exponential of a Levy stochastic process. The exponential Levy
models have been discussed by several authors
(see for example \cite{ct,hg}).
When the volatility is stochastic we may consider the process
\begin{gather*}
dS = S\sigma dZ +S\mu dt,\\
d\sigma = \beta \sigma dW + \alpha \sigma dt
\end{gather*}
where $Z$ and $W$
are two standard Brownian motion with correlation coefficient
$\rho$.
If $F ( S, \sigma, t )$
is the price of an option depending on
the price of the asset $S$, then by
Ito's lemma \cite{I}, it holds:
$$
dF(S,\sigma,t) = F_S dS + F_\sigma d\sigma +\mathcal{L}F dt
$$
where $\mathcal{L}$ is given by
$$
\mathcal{L}
= \partial_t + \frac 12 \sigma^2 S^2
\frac {\partial^2}{\partial S^2} +
\frac 12 \beta^2 \sigma^2
\frac {\partial^2}{\partial \sigma^2}
+ \rho \sigma^2 S\beta
\frac {\partial^2}{\partial S\partial \sigma}
$$
Under an appropriate choice of the portfolio the
stochastic term of the equation vanishes (for details, see \cite{AZ}).
A generalized tree process has been developed in
\cite{flo1,flo2} that \emph{approximates any
Stochastic Volatility model}.
Unlike the non-random volatility case the tree construction is
stochastic every time it is created, since that is the only way we
are able to deal with the huge complexity involved.
If in this model we add a jump component modelled by a compound
Poisson process to the process $S$, and we follow Merton \cite{ME}
we obtain the following PIDE, which is a generalization of the
previous equation for stochastic volatility:
\begin{equation}
\label{problem2}
\begin{aligned}
&\frac{\partial F}{\partial t} +
\frac 12 \sigma^2 S^2\frac {\partial^2 F}{\partial S^2}
+\frac 12 \sigma^2 \beta^2\frac {\partial^2 F}{\partial \sigma^2}
+\rho \sigma^2 \beta S\frac {\partial^2 F}{\partial S \partial \sigma}
+(r-\lambda k)S\frac{\partial F}{\partial S}\\
& - \frac 12 \rho \sigma^2\beta\frac {\partial F}{\partial \sigma}
+ \lambda\int_{\mathbb R}[F(SY,\sigma ,t)-F(S,\sigma,t)]g(Y)
dY-r F=0
\end{aligned}
\end{equation}
Here, $r$ denotes the riskless rate, $\lambda $ the jump intensity, and
$k=E (Y-1) $, where $E$ is the
expectation operator, $g$ is the density of the $Y$ random variable
and the quantity $Y-1$ measures the percentage change in the stock
price if the jump occurs. The jump times are modelled by a Poisson
process, for details see \cite[9.2]{ME} .
The previous discussion motivates to consider more general
integro-differential parabolic problems.
This work is devoted to the study of the solutions to the
following general partial integro-differential equation
in an unbounded smooth domain:
\begin{equation}
\label{problem}
\begin{gathered}
Lu - u_t = \mathcal{G}(t,u) \quad\text{in } \Omega\times (0,T)\\
u(x,0) = u_0(x) \quad\text{on } \Omega\times\{ 0\}\\
u(x,t) = h(x,t) \quad\text{on } \partial\Omega\times (0,T)\\
\end{gathered}
\end{equation}
We shall assume that $\Omega\subset\mathbb{R}^d$ is an unbounded smooth domain,
$L$ is a second order elliptic operator in non divergence form, namely
$$
Lu := \sum_{i,j = 1}^d a^{ij}(x,t) u_{x_ix_j} +
\sum_{i= 1}^d b^{i}(x,t) u_{x_i} + c(x,t)u,
$$
where the coefficients of $L$ belong to the H\"older Space
$C^{\delta,\delta/2}(\overline\Omega\times [0,T])$
and satisfy the following two conditions
\begin{gather*}
\Lambda |v|^2 \ge \sum_{i,j = 1}^d a^{ij}(x,t) v_iv_j \ge \lambda
|v|^2\quad (\Lambda \ge \lambda >0),\\
|b^i(x,t)| \le C, \quad c(x,t)\le 0.
\end{gather*}
The operator $\mathcal{G}$ is a completely continuous integral
operator as the ones defined in \eqref{problem1} and
\eqref{problem2}, modelling the jump. More precisely, we shall
assume that $\mathcal{G}(t,u) =\int_\Omega g(x,t,u)dx$, where $g$
is a continuous function. In this general model, the case in which
$g$ is increasing with respect to $u$ and all jumps are positive
corresponds to the evolution of a call option near a crash.
Furthermore, we shall assume that $u_0\in C^{2+\delta}(\overline\Omega)$,
$h\in C^{2+\delta,1+\delta/2}(\overline\Omega\times [0,T])$ and satisfy
the compatibility condition
\begin{equation}
\label{comp}
h(x,0) = u_0(x)\quad \forall\; x\in \partial\Omega
\end{equation}
We shall prove the existence of solutions of \eqref{problem},
using the method of upper and lower solutions.
We recall that a smooth function
$u$ is called an upper (lower) solution of problem
\eqref{problem} if
\begin{gather*}
Lu - u_t \le \;(\ge) \mathcal{G}(t,u) \quad\text{in } \Omega\times (0,T)\\
u(x,0) \ge \;(\le) u_0(x) \quad\text{on } \Omega\times\{ 0\}\\
u(x,t) \ge \;(\le) h(x,t) \quad\text{on } \partial\Omega\times (0,T)
\end{gather*}
Our main result reads as follows:
\begin{theorem} \label{main}
Let $L$ and $\mathcal{G}$ be the operators defined above.
Assume that either:
\begin{itemize}
\item $\mathcal{G}$ is nonincreasing with respect to $u$, or
\item there exist some continuous, and increasing one dimensional
function $f$ such that $\mathcal{G}(t,u) - f(u)$ is nonincreasing
with respect to $u$
\end{itemize}
Furthermore, assume there exist $\alpha$ and $\beta$ a lower and an upper
solution of the problem with $\alpha\le \beta$ in $\Omega\times (0,T)$.
Then \eqref{problem} admits a solution $u$ such that
$\alpha\le u \le \beta$ in $\Omega\times (0,T)$.
\end{theorem}
\begin{remark} \rm
The existence result above is applicable for any sub-linear
$\mathcal{G}$, or more generally, for any $\mathcal{G}$ that
is bounded by a polynomial.
\end{remark}
\section{The method of upper and lower solutions}
\label{upper-lower}
In this section we give a proof of Theorem \ref{main}.
First we solve an analogous problem in a bounded domain;
with this aim, we extend the boundary data to
the interior of $\Omega\times (0,T)$:
\begin{lemma}\label{phi}
Let $U$ be a smooth and bounded subset of $\Omega$. Then
there exists a unique function
$\varphi_U\in C^{2+\delta,1+\delta/2}(\overline U \times [0,T])$
such that
\begin{gather*}
L\varphi_U - (\varphi_U)_t = 0, \\
\varphi_U(x,0) = u_0(x) \quad x\in U \\
\varphi_U(x,t) = h(x,t) \quad (x,t) \in \partial U\times [0,T]
\end{gather*}
Moreover, if $\alpha$ and $\beta$
are a lower and an upper solution of this reduced problem with
$\alpha\le \beta$ in $\Omega\times (0,T)$, then
$$
\alpha(x,t) \le \varphi_U(x,t) \le \beta(x,t)
$$
for $(x,t)\in \overline U\times [0,T]$.
\end{lemma}
\begin{proof}
Existence and uniqueness follow immediately from \cite[Thm. 10.4.1]{K}
and the compatibility condition \eqref{comp}.
By the maximum principle, it is clear that if
$\alpha\le \beta$ are a lower and an upper solution, then
$$
\alpha(x,t) \le \varphi_U(x,t) \le \beta(x,t)
$$
\end{proof}
\begin{lemma}\label{lem:lemma2}
Let $U\subset \mathbb{R}^d$ a bounded smooth domain,
let $\tilde T \beta(x_0,t_0)$
for some $(x_0,t_0) \in \overline U\times [0,\tilde T]$.
As $u^1|_{\partial\overline U\times [0,\tilde T]} \le
\beta|_{\partial\overline U\times [0,\tilde T]} $,
we deduce that $(x_0,t_0) \in U\times (0,\tilde T)$,
and we may assume that
$(x_0,t_0)$ is a maximum of $u^1-\beta$.
Therefore, $\nabla (u^1-\beta)(x_0,t_0)=0$
(then $(u^1-\beta)_t(x_0,t_0)=0$) and $\Delta (u^1-\beta)(x_0,t_0)<0$,
and as $L$ is strictly elliptic,
$L (u^1-\beta)(x_0,t_0)<0$.
We have that
$L (u^1-\beta)- (u^1-\beta)_t=\mathcal{G}(t,\alpha)- (L(\beta)-(\beta)_t)\ge
\mathcal{G}(t,\alpha)- \mathcal{G}(t,\beta)\ge 0$.
Then
$L (u^1-\beta)(x_0,t_0)\ge (u^1-\beta)_t(x_0,t_0)=0$,
which is a contradiction.
Next, we assume as inductive hypothesis that
$u^n\ge u^{n+1}$. As before, if
$u^n(x_0,t_0)- u^{n+1}(x_0,t_0) > 0$ is a maximum,
we will obtain the same contradiction with $L (u^n- u^{n+1})(x_0,t_0)\ge
(u^n- u^{n+1})_t(x_0,t_0)=0$.
In the same way as before,
it follows that $u^{n+1}\le\beta$.
We now define
$$
u(x,t) = \lim_{n\to \infty} u^n(x,t).
$$
In \cite[Chapter 7]{L} the $L^p$-estimates for this type
of differential equations are given, specifically that
the $W^{2,1}_p$-norm of $u^n-u^m$ can be controlled
by its $L^p$-norm and
the $L^p$-norm of its image by the operator
$L-\partial_t$, what results in the following:
\begin{align*}
&\|D^2 (u^n-u^m)\|_{L^p(V)}+ \|(u^n-u^m)_t\|_{L^p(V)} \\
&\le c \left(\|L(u^n-u^m)-(u^n-u^m)_t\|_{L^p(V)}+
\|u^n-u^m\|_{L^p(V)} \right).
\end{align*}
By construction,
$$
L(u^n-u^m)-(u^n-u^m)_t =
\mathcal{G}(\cdot,u^{n-1}) - \mathcal{G}(\cdot,u^{m-1}).
$$
As $\mathcal{G}$ is a completely
continuous operator, using the fact
that $\alpha\le u^n\le \beta$ and Lebesgue's dominated convergence
theorem it follows that
$\{u^n\}$ is a Cauchy sequence in $W^{2,1}_p(V)$. Hence $u^n\to u$ in the
$W^{2,1}_p$-norm, and then
$u$ is a strong solution of the problem.
Suppose now that the only condition on $\mathcal{G}(t,u)$ is
that there exist some increasing, continuous function $f$ such
that $\mathcal{G}(t,u) - f(u)$ is nonincreasing with respect to $u$.
We define
$u^{n+1} \in W^{2,1}_p(V)$
as the unique solution of the problem
\begin{equation} \label{e2.4}
\begin{gathered}
Lu^{n+1} - u^{n+1}_t -f( u^{n+1}) = \mathcal{G}(t,u^{n})-f( u^{n})
\quad\text{in } U\times (0,\tilde T)\\
u^{n+1}(x,0) = u_0(x) \quad\text{in } U\times\{ 0\}\\
u^{n+1}(x,t) = \varphi_U(x,t) \quad\text{in } \partial U\times (0,\tilde T)\\
\end{gathered}
\end{equation}
As before, we claim that
$$
0\le u^n(x,t)\le u^{n+1}(x,t) \le \beta(x,t) \quad
\forall (x,t)\in \overline U\times [0,\tilde T],\;
\forall n \in\mathbb{N}_0
$$
Indeed, by the maximum principle it follows that $u^1\ge 0$;
moreover,
$$
Lu^{1} - u^{1}_t -f( u^{1}) =
\mathcal{G}(t,0)-f(0)\ge \mathcal{G}(t,\beta) - f (\beta) \ge
L\beta - \beta_t -f( \beta)
$$
and hence $u^1 \le\beta$.
Inductively,
\begin{align*}
Lu^{n+1} - u^{n+1}_t -f( u^{n+1})
&= \mathcal{G}(t,u^{n})-f( u^{n}) \\
& \le \mathcal{G}(t,u^{n-1})-f( u^{n-1})=Lu^{n} - u^{n}_t -f( u^{n})
\end{align*}
Thus $u^{n+1} \ge u^n$. In the same way as before
it follows that $u^{n+1}\le\beta$. Proving any of these claims is similar
with what we did earlier. Indeed, assume that for some
$(x_0,t_0)\in \bar{U}\times[0,\tilde{T}]$ we have
$u^{n+1}(x_0,t_0) < u^n(x_0,t_0)$. By continuity we may assume that
$(x_0,t_0)$ is a maximum of $u^n-u^{n+1}$.
As before, $L (u^n-u^{n+1})(x_0,t_0)<0$, but from the induction
hypothesis above:
$$
L (u^n-u^{n+1})(x_0,t_0)
\ge (u^{n}_t-u^{n+1}_t)(x_0,t_0)+f( u^{n}(x_0,t_0))-f( u^{n+1}(x_0,t_0))
\geq 0,
$$
contradiction. The expression is non-negative since $(x_0,t_0)$
is a maximum point and $f$ is increasing.
The rest of the proof follows as in the other case.
\end{proof}
\subsection*{Proof of Theorem \ref{main}}
We approximate the 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 $\Omega_N \cap \Omega$.
Then, using Lemma \ref{lem:lemma2} define $u^N$ as a solution of
the problem
\begin{equation} \label{problem-N}
\begin{gathered}
Lu - u_t = \mathcal{G}(t,u) \quad\text{in }
\Omega_N\times (0,T-\frac 1N)\\
u(x,0) = u_0(x) \quad\text{in } \Omega_N\times\{ 0\}\\
u(x,t) = h(x,t) \quad\text{in } \partial\Omega_N\times (0,T-\frac 1N)
\end{gathered}
\end{equation}
such that $\alpha=0\le u^N\le \beta$ in $\Omega_N\times (0,T-\frac 1N)$.
Define $V_N = \Omega_N\times (0,T-\frac 1N)$ and choose
$p>d$. For $M > N$, we have that
\begin{align*}
&\| D^2(u^M)\|_{L^p(V_N)} + \| (u^M)_t\|_{L^p(V_N)}\\
&\le c \left(\|Lu^M -(u^M)_t\|_{L^p(V_N)}+
\|u^M\|_{L^p(V_N)} \right)\\
&\le c \left(\|\mathcal{G}(t,u^M) \|_{L^p(V_N)}+
\|\beta \|_{L^p(V_N)} \right)
\le C
\end{align*}
for some constant $C$ depending only on $N$. By Morrey imbedding,
there exists a subsequence that converges uniformly on $\overline V_N$.
Using a standard diagonal argument, we may extract a subsequence
(still denoted $\{u^M\}$) such that
$u^M$ converges uniformly to some function $u$ over compact subsets of
$\Omega\times (0,T)$.
For $V = U \times (0,\tilde T)$, $U\subset \Omega$, $\tilde
T < T$, taking $M,N$ large enough we have that
\begin{align*}
&\|D^2 (u^N -u^M)\|_{L^p(V)}+
\|(u^N-u^M)_t\|_{L^p(V)} \\
&\le c \left(\|L(u^N-u^M)-(u^N-u^M)_t\|_{L^p(V)}+
\|u^N-u^M\|_{L^p(V)} \right)
\end{align*}
By construction,
$$
L(u^N-u^M)-(u^N-u^M)_t =
\mathcal{G}(t,u^{N-1}) - \mathcal{G}(t,u^{M-1})
$$
As before, using that $\mathcal{G}$ is
continuous, and that $\alpha \le u^N\le \beta$, by dominated
convergence it follows that
$\{u^N\}$ is a Cauchy sequence in $W^{2,1}_p(V)$.
Hence $u^N\to u$ in the
$W^{2,1}_p$-norm, and then
$u$ is a classical solution in $V$. It follows that $u$ satisfies
the equation on $\Omega\times (0,T)$. Furthermore, it is clear
that
$u(x,0) = u_0(x)$.
For $M>N$ we have that $u_M(x,t)
= u_N (x,t)$ for $x\in \partial\Omega\cap \partial\Omega_N$,
$t \in (0,T-\frac 1N)$.
Thus, it follows
that
$u$ satisfies the boundary condition $u(x,t) = h(x,t)$ on
$\partial\Omega\times [0,T)$.
\qed
\begin{example} Consider the problem
\begin{equation} \label{example1}
\begin{gathered}
Lu - u_t = \int_{\mathbb R} \left(u(x,t)-u(xy,t)\right)\nu(y)dy
\quad\text{in } \Omega\times (0,T)\\
u(x,0) = u_0(x) \quad\text{on } \Omega\times\{ 0\}\\
u(x,t) = h(x,t) \quad\text{on } \partial\Omega\times (0,T)
\end{gathered}
\end{equation}
where $\nu(y)=\frac{M}{\sqrt{\pi}} e^{-M^2 y^2}$, a Gaussian kernel,
and $\Omega\subseteq \mathbb R ^d$.
\end{example}
We shall verify that we can apply Theorem \ref{main}.
To do so we need to show that the operator $\mathcal{G}$ has
the properties of the theorem and that the problem admits an upper
and a lower solution.
Note that we can write $\mathcal{G}$ as:
$$
\mathcal{G}(t,x,u)= u(x,t) - E[u(xY,t)],
$$
where the expectation is calculated with respect to some variable $Y$
with density $\nu(y)$. Now take $f(u)=u$ which is obviously a continuous
and increasing functional and observe that:
$$
\mathcal{G}(t,x,u)-f(u)= - E[ u(xY,t)],
$$
which is clearly non-increasing with respect to $u$ (the expectation
is an increasing operator).
Thus, the operator $\mathcal{G}$ satisfies the hypothesis of the
theorem and we just need to find upper and lower solutions for
the problem. To do so, notice that $\mathcal{G}(t,x,0)=0$,
thus $\alpha\equiv0$ is a lower solution of the problem
(with suitable boundary conditions). Let us define
$\beta(x,t) = k (T-t)^{-\frac d2}e^{\frac \theta{T-t} |x|^2}$,
with $k>0$, $\theta>0$. We will show that $\beta$ is an upper
solution to the problem.
A straightforward computation shows that $\beta$ satisfies
\begin{align*}
L\beta - \beta_t
&= \beta\Big\{ \big(\frac {2\theta}{T-t}\big)^2
\sum_{i,j = 1}^d a^{ij} x_ix_j +
\frac {2\theta}{T-t}\sum_{i = 1}^d a^{ii} \\
&\quad +
\frac {2\theta}{T-t} \sum_{i = 1}^d b^{i}x_i- \big[\frac {d}{2(T-t)} +
\frac {\theta}{(T-t)^2}|x|^2\big]
\Big\}
\end{align*}
Using the fact that
$\sum_{i = 1}^d a^{ii} \le \Lambda$, and that
$2\sum_{i = 1}^d b^{i}x_i \le \varepsilon |x|^2 + \frac 1\varepsilon
\|b\|_\infty^2$ for some small $\varepsilon$, we deduce that
$$
\frac 1\beta (L\beta - \beta_t) \le
\left(4\theta\Lambda - 1 +\varepsilon (T-t)\right)
\frac{\theta |x|^2}{(T-t)^2} +
\frac{1}{T-t} [2\theta \Lambda -\frac d2 + \frac 1\varepsilon
\theta \|b\|_\infty^2 ].
$$
Taking $\varepsilon< 1/T$, and
$$
\theta \le \min \big\{\frac {1-T\varepsilon}{4\Lambda},
\frac {d\varepsilon}{2\|b\|_\infty^2+4\Lambda}\big\},
$$
it follows that with this particular choice of $\theta$, the $\beta$
defined above has the property that
$$
L\beta - \beta_t \le 0.
$$
Finally, if we show that $\mathcal{G}(t,x,\beta)\ge 0$ and if the
boundary conditions are chosen so that $0\le u_0(x)\le \beta(x,0)$
and $0\le h(x,t)\le \beta(x,t)$ for $x\in \partial\Omega$,
then we have that $\alpha\equiv 0$ and
$\beta$ are respectively a lower and an upper solution of
the problem.
Let us show that $\mathcal{G}(t,x,\beta)\ge 0$ for our $\beta$.
To simplify notation write $\beta=ce^{\alpha x^2}$, with
$c=k (T-t)^{-\frac d2}$, $\alpha=\frac \theta{T-t}$ and assume
that $ x^2<\frac{M^2}{ 2\alpha}=\frac{M^2}{2\theta^2 (T-t)^2}$,
where $M$ is the positive constant from the definition of $\nu$.
For convenience, we work in a one dimensional domain and we require
that $M>\sqrt[4]{2}$. We note that $\theta$ in the definition of
$\beta$ may be chosen as small as we want so the restriction on
the domain of $x$ is irrelevant. Then
$$
\mathcal{G}(t,x,\beta)=\int_{\mathbb{R}}\big( ce^{\alpha x^2}- ce^{\alpha x^2y^2}\big)
\nu(y)dy
= ce^{\alpha x^2}-c \frac{M}{\sqrt{M^2-x^2\alpha}}=f(x)
$$
and
$$
\frac{df}{dx}(x)= c\alpha x\Big(2e^{\alpha x^2}
-\frac{M}{\sqrt{(M^2-x^2\alpha)^3}}\Big).
$$
We note that $c,\alpha>0$, and the term within parentheses is always
positive for any $x$ within the domain
$\{x\in R: x^2<\frac{M^2}{2\alpha}\}$ if $M>\sqrt[4]{2}$. Indeed,
since $\alpha x^2>0$ we always have
$$
2e^{\alpha x^2}-\frac{M}{\sqrt{(M^2-x^2\alpha)^3}}\ge 2
-\frac{M}{\sqrt{(M^2-x^2\alpha)^3}}\ge 0
$$
The last inequality holds for any $x$ with property
$$
x^2\le \frac{1}{\alpha} \Big(M^2- \frac{M^{2/3}}{2^{2/3}}\Big).
$$
However, recall that the domain of $x$ is $x^2<\frac{M^2}{2\alpha}$
and note that if $M>\sqrt[4]{2}$,
$$
\frac{M^2}{2\alpha}< \frac{1}{\alpha} \Big(M^2- \frac{M^{2/3}}{2^{2/3}}\Big),
$$
thus the term within parentheses in $\frac{df}{dx}$ is positive for
all $x$ within the domain.
Therefore, the sign of the derivative is the sign of $x$, which means
that the function $f(x)$ attains its minimum at $x=0$. As $f(0)=0$,
we conclude that $f(x)\ge 0$ for all $x$ in the domain and therefore
$\beta$ is an upper solution of the problem.
Finally, we note that we can use the procedure outlined within
the proof of the Lemma \ref{lem:lemma2} to construct
an approximate solution of the original problem.
As this construction is straightforward we do not insist on details.
\subsection*{Acknowledgments}
The authors are especially grateful to the reviewers for their
careful reading of the manuscript and their fruitful remarks.
\begin{thebibliography}{10}
\bibitem{AAM}
P.~Amster, C.~Averbuj, P.~{De Napoli}, and M.C. Mariani.
\newblock A parabolic problem arising on financial mathematics.
\newblock {\em Nonlinear Analysis: Real World Applications}, 2009.
\newblock (in press).
\bibitem{AA}
L.~Andersen and J.~Andreasen.
\newblock Jump-diffusion processes: Volatility smile fitting and numerical
methods for option pricing.
\newblock {\em Review of Derivatives Research}, 4:231--262, 2000.
\bibitem{AZ}
M.~Avellaneda and Y.~Zhu.
\newblock Risk neutral stochastic volatily model.
\newblock {\em International Journal of Theory and Applied Finance},
1(2):289--310, 1998.
\bibitem{Barndoff}
O.~E. Barndorff-Nielsen, J.~L. Jensen, and M.~S\/{o}rensen.
\newblock Some stationary processes in discrete and continuous time.
\newblock {\em Advances in Applied Probability}, 30(4):989--1007, Dec. 1998.
\bibitem{BBF}
H.~Berestycki, J.~Busca, and I.~Florent.
\newblock Computing the implied volatility in stochastic volatility models.
\newblock {\em Communications on Pure and Applied Mathematics},
57(10):1352--1373, 2004.
\bibitem{BS}
F.~Black and M.~Scholes.
\newblock The valuation of options and corporate liability.
\newblock {\em Journal of Political Economy}, 81:637--654, 1973.
\bibitem{ct}
R.~Cont and P.~Tankov.
\newblock {\em Financial modeling with jumps processes}.
\newblock CRC Financial mathematics series. Chapman \& Hall, 2003.
\bibitem{ds}
F.~Delbaen and W.~Schachermayer.
\newblock A general version of the fundamental theorem of asset pricing.
\newblock {\em Mathematische Annalen}, (300):463--520, 1994.
\bibitem{DU}
D.~Duffie.
\newblock {\em Dynamic Asset Pricing Theory}.
\newblock Princeton University Press, 3rd edition, 2001.
\bibitem{flo1}
I.~Florescu.
\newblock {\em Stochastic Volatility Stock Price: Approximation and Valuation
Using a Recombining Tree. Sharp estimation of the Almost Sure Lyapunov
Exponent Estimation for the Anderson Model in Continuous Space}.
\newblock PhD thesis, Purdue University, West Lafayette, IN, December 2004.
\bibitem{flo2}
I.~Florescu and F.~Viens.
\newblock Stochastic volatility: option pricing using a multinomial recombining
tree.
\newblock {\em Applied Mathematical Finance}, 15(2):151--181, April 2008.
\bibitem{hg}
H.~Geman.
\newblock Pure jump {Levy} processes for asset price modeling.
\newblock {\em Journal of Banking and Finance}, 26:1297--1316, 2002.
\bibitem{He}
S.~L. Heston.
\newblock A closed-form solution for options with stochastic volatility with
applications to bond and currency options.
\newblock {\em Review of Financial Studies}, 6(2):327--343, 1993.
\bibitem{H}
J.~C. Hull.
\newblock {\em Options, Futures, and other Derivatives}.
\newblock Prentice Hall, 7th edition, 2008.
\bibitem{I}
N.~Ikeda.
\newblock {\em Stochastic Differential Equations and Diffusion Processes}.
\newblock North-Holland, 2nd revised edition, 1989.
\bibitem{J}
R.~A. Jarrow.
\newblock {\em Modelling Fixed Income Securities and Interest Rate Options}.
\newblock Stanford Economics and Finance, 2nd edition, 2002.
\bibitem{K}
N.V. Krylov.
\newblock {\em Lectures on elliptic and parabolic equations in H\"older
Spaces}, volume~12 of {\em Graduate Studies in Mathematics}.
\newblock American Mathematical Society, 1996.
\bibitem{L}
G.~M. Lieberman.
\newblock {\em Second order parabolic differential equations}.
\newblock World Scientific Publ., 1996.
\bibitem{Madan}
D.~B. Madan, P.~P. Carr, and E.~C. Chang.
\newblock The variance gamma process and option pricing.
\newblock {\em European Finance Review}, 2(1):79--105, 1998.
\bibitem{MF}
M.C. Mariani, I.~Florescu, M.P.~Beccar Varela, and E.~Ncheuguim.
\newblock Long correlations and levy models applied to the study of memory
effects in high frequency (tick) data.
\newblock {\em Physica A}, 388(8):1659--1664, April 2009.
\bibitem{ME}
R.C. Merton.
\newblock {\em Continuous-Time Finance}.
\newblock Wiley-Blackwell, 1992.
\end{thebibliography}
\end{document}