\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 91, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/91\hfil Distribution-valued weak solutions]
{Distribution-valued weak solutions to a parabolic
problem arising in financial mathematics}

\author[M. Eydenberg, M. C. Mariani\hfil EJDE-2009/91\hfilneg]
{Michael Eydenberg, Maria Cristina Mariani}  % in alphabetical order

\address{Michael Eydenberg \newline
Department of Mathematical Sciences \\
New Mexico State University \\
Las Cruces, NM 88003-8001, USA}
\email{mseyden@nmsu.edu}

\address{Maria Christina Mariani \newline
Department of Mathematical Sciences\\
University of Texas, El Paso, Bell Hall 124\\
El Paso, Texas 79968-0514, USA}
\email{mcmariani@utep.edu}

\thanks{Submitted September 10, 2008. Published July 30, 2009.}
\subjclass[2000]{35K10, 35D30, 91B28}
\keywords{Weak solutions; parabolic differential equations; 
\hfill\break\indent Black-Scholes type equations}

\begin{abstract}
 We study distribution-valued solutions to a parabolic problem that
 arises from a model of the Black-Scholes equation in option pricing.
 We give a minor generalization of known existence and uniqueness
 results for solutions in bounded domains
 $\Omega \subset \mathbb{R}^{n+1}$ to give existence of solutions
 for certain classes of distributions $f\in \mathcal{D}'(\Omega)$.
 We also study growth conditions for smooth solutions of certain
 parabolic equations on $\mathbb{R}^n\times (0,T)$ that have
 initial values in the space of distributions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction and Motivation}

Recently, there has been an increased interest in the study of parabolic
differential equations that arise in financial mathematics.  A particular
instance of this is the Black-Scholes model of option pricing via a
reversed-time parabolic differential equation \cite{BS}. In 1973 Black
and Scholes developed a theory of market dynamic assumptions, now known as
the Black-Scholes model, to which the It\^o calculus can be applied.
Merton \cite{Me} further added to this theory completing a system  for
measuring, pricing and hedging basic options.
The pricing formula  for basic options is known as the
Black-Scholes formula, and is numerically found by
solving  a parabolic partial differential equation using It\^o's formula.
In this frame, general parabolic equations in multidimensional domains
arise in problems for barrier options for several assets \cite{W}.

Much of the current research in mathematical finance deals with removing
the simplifying assumptions of the Black-Scholes model. In this model, an
important quantity is the volatility that is a measure of the fluctuation
(i.e. risk) in the asset prices; it corresponds to the diffusion coefficient
in the Black-Scholes equation.  While in the standard Black-Scholes model
the volatility is assumed constant, recent variations of this model allow
for the volatility to take the form of a stochastic variable \cite{HES}.
In this approach the underlying security $S$ follows, as in the classical
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,
however, the variance $v(t)=(\sigma (t)) ^{2}$ also follows a
stochastic process given by
\[
dv_{t}=\kappa (\theta -v(t)) dt+\gamma \sqrt{v_{t}}dW_{t}
\]
where $W$ is another standard Brownian motion.  The correlation coefficient
between $W$ and $Z$ is denoted by $\rho $:
\[
E(dZ_{t},dW_{t}) =\rho dt.
\]
This leads to the generalized Black-Scholes equation
\begin{align*}
&\frac{1}{2}vS^{2}(D_{SS}U) +\rho \gamma vS(
D_{v}D_{s}U) +\frac{1}{2}v\gamma ^{2}(D_{vv}U) +rSD_{S}U \\
&+ [ \kappa (\theta -v) -\lambda v]
D_{v}U-rU+D_{t}u=0.
\end{align*}
Introducing the change of variables given by $y=\ln S$,
$x=\frac{v}{\gamma }$, $\tau =T-t$, we see that  $u(x,y)=U(S,v)$ satisfies
\[
D_{\tau }u=\frac{1}{2}\gamma x[ \Delta u+2\rho D_{xy}u] +\frac{1}{
\gamma }[ \kappa (\theta -\gamma x)-\lambda \gamma x]
D_{x}u+(r-\frac{\gamma x}{2}) D_{y}u-ru
\]
in the cylindrical domain $\Omega \times (0,T) $ with $\Omega
\subset \mathbb{R}^{2}$.  Using the Feynman-Kac relation, more general models with stochastic volatility have been
considered (see  \cite{BBF}) leading to systems such as
\begin{gather*}
D_{\tau }u =\frac{1}{2}\mathop{\rm trace}(M(x,\tau)
D^{2}u) +q(x,\tau) \cdot Du \\
u(x,0) = u_{0}(x)
\end{gather*}
for some diffusion matrix $M$ and payoff function $u_{0}$.

These considerations motivate the study of the general parabolic equation
\begin{equation} \label{L_gen}
\begin{gathered}
Lv =f(v,x,t)\quad\text{in }\Omega  \\
v(x,t) =v_{0}(x,t)\quad\text{on }\mathcal{P}\Omega
\end{gathered}
\end{equation}
where $\Omega \subset
\mathbb{R}
^{n+1}$ is a smooth domain, $f:\mathbb{R}^{n+2}\mapsto\mathbb{R}$
is continuous and continuously differentiable with respect to $v$,
$v_{0}\in C(\mathcal{P}\Omega) $, and $\mathcal{P}\Omega $ is
the parabolic boundary of $\Omega $.  Here, $L$ is a second order elliptic
operator of the form
\begin{equation} \label{L_form}
Lv=\sum_{i,j=1}^na_{ij}(x,t)D_{ij}v+
\sum_{i=1}^nb_{i}(x,t)D_{i}v+c(x,t)v-\eta D_{t}v
\end{equation}
where $\eta \in (0,1) $ and $a_{ij}$, $b_{i}$, $c$ satisfy the
following 4 conditions:
\begin{gather}
a_{ij}, b_{i},c  \in C(\overline{\Omega })  \label{cond_1} \\
 \lambda \|\xi \|^{2}\leq \sum_{ij}a_{ij}(x,t)\xi _{i}\xi _{j}
\leq \Lambda \|\xi \|^{2}, \quad (0<\lambda \leq \Lambda)  \label{cond_2} \\
\|a_{ij}\|_{\infty},\|b_{i}\|_{\infty },
\|c\|_{\infty }<\infty \label{cond_3} \\
 c\leq 0.  \label{cond_4}
\end{gather}
Existence and uniqueness results for (\ref{L_gen}) when $\Omega $ is a
bounded domain and the coefficients belong to the H\"{o}lder space
$C^{\delta ,\delta /2}(\overline{\Omega }) $ have been
well-established (c.f. \cite{L} and \cite{KRY}).  Extensions of these
results to domains of the form $\Omega \times (0,T) $ where
$\Omega \subset \mathbb{R}^n$ is in general an unbounded domain
are also given, as in \cite{AMM1} and \cite{AMM2}.

Our concern in this work, however, is in the interpretation and solution
of (\ref{L_gen}) in the sense of distributions. This is inspired
primarily by the study in \cite{L}, Chapter 3, which obtains weak
solutions $v$ of the divergence-form operator
\[
\sum_{i,j=1}^nD_{i}(a_{ij}D_{j}v) -\eta D_{t}v=f
\]
where the matrix $a_{ij}$ is constant and $f$ belongs to the Sobolev space
$W^{1,\infty }(\Omega) $, where $\Omega \subset \mathbb{R}^{n+1}$ is a
bounded domain. The solutions $v$ are weak in the sense that
the derivatives of $v$ can only be defined in the context of distributions,
as we discuss in more detail below. Our goal is to generalize these results
to the well-known classical space $\mathcal{D}(\Omega)$ of test
functions and its strong-dual space,
$\mathcal{D}'(\Omega) $. In particular, we let
$f\in \mathcal{D}'(\Omega) $ be of the form $f=D_\alpha g$ for some
$g\in C(\overline{\Omega })$, and ask what conditions are sufficient
on $f$ and the coefficients $a_{ij}$, $b_{i}$, and $c$ so that $Lv=f$
makes sense for some other $v\in \mathcal{D}'(\Omega) $.

Another facet of this question, however, is to consider characterizations of
classical solutions to parabolic differential equations that define
distributions at their boundary.  This problem has been extensively studied
in the case that $L$ is associated with an operator semigroup, beginning
with the work of \cite{[H]} and \cite{MAT1} to realize various spaces of
distributions as initial values to solutions of the heat equation.  The
problem is to consider the action of a solution $v(x,t)$ to the heat
equation on
$\mathbb{R}^n\times (0,T)$ on a test function $\phi $ in the following
sense:
\begin{equation}
(v,\phi) =\lim_{t\to 0^{+}}\int_{
\mathbb{R}^n}v(x,t)\phi (x)dx.  \label{lim}
\end{equation}
 The authors in \cite{MAT2} and \cite{K} characterize those solutions $v$
for which (\ref{lim}) defines a hyperfunction in terms of a suitable growth
condition on the solution $v(x,t)$, while \cite{CK} extends these results
to describe solutions with initial values in the spaces of Fourier
hyperfunctions and infra-exponentially tempered distributions.  \cite{D}
gives a characterization of the growth of smooth solutions to the Hermite
heat equation $L=\bigtriangleup -| x|^{2}-D_{t}$ with
initial values in the space of tempered distributions.  In all of these
cases, the ability to express a solution $v$ of the equation $Lv=0$ as
integration against an operator kernel (the heat kernel for the Heat
semigroup and the Mehler kernel \cite{T} for the Hermite heat semigroup)
plays an important role in establishing sufficient and necessary growth
conditions.  While this is not possible for a general parabolic operator of
the form (\ref{L_form}), in this paper we propose a sufficient growth
condition for a solution of $Lv=0$ on
$\mathbb{R}^n\times (0,T) $ to define a particular type of distribution,
and we show the necessity of this condition in a few special cases.

The terminology we use in this paper is standard.  We will denote
 $X=(x,t)$ as an element of
$\mathbb{R}^{n+1}$ where $x\in \mathbb{R}^n$.  Derivatives will be
denoted by $D_{i}$ with $1\leq i\leq n$ or
$D_{t}$ for single derivatives, and by $D_{\alpha }$ with
$\alpha \in\mathbb{N}^n$ for higher-order derivatives.
If $\alpha \in\mathbb{N}^n$ then $| \alpha |$ denotes the sum
\[
| \alpha |=\alpha _{1}+\dots +\alpha _{n}.
\]
Constants will generally be denoted by $C$, $K$, $M$, etc. with indices
representing their dependence on certain parameters of the equation.

We give also a brief introduction to the theory of weak solutions and
distributions as they pertain to our results. For $n\geq 1$, take
$\Omega\subset \mathbb{R}^{n+1}$ to be open. Let
$u,v \in L^1_{\mathrm{loc}}(\Omega)$ and
$\alpha \in\mathbb{N}^n$. We say that $v$ is the weak partial
derivative of $u$ of order
$|\alpha|$, denoted simply by $D_\alpha u=v$, provided that
\[
\int_\Omega u (D_\alpha\phi) dx = (-1)^{|\alpha|}
\int_\Omega v\phi dx
\]
for all test functions $\phi \in C^{\infty}_0(\Omega)$. Observe
that $v$ is unique only up to a set of zero measure. This leads to the
following definition of the Sobolev space $W^{k,p}(\Omega)$:

Let $p \in [1,\infty)$, $k\in \mathbb{N}$, and
$\Omega \subset\mathbb{R}^{n+1}$ be open. We define the Sobolev space
$W^{k,p}(\Omega)$ as those $u \in L_{\rm loc}^{1}(\Omega)$ for which
the weak derivatives $D_\alpha u$ are defined and belong to $L^p(\Omega)$
for each $0 \leq |\alpha| \leq k$.
Observe that $W^{k,p}(\Omega)$ is a Banach space with the norm
\[
\|u \|_{k,p}= \sum_{0\leq
|\alpha|\leq k} \|D_\alpha u\|_{L^p(\Omega)}.
\]
Furthermore, we denote by $W^{k,p}_0(\Omega)$ the closure of the
test-function space $C^\infty_0(\Omega)$ under the Sobolev norm
$\| \cdot \|_{k,p}$.

The classical space $\mathcal{D}(\Omega)$ of test functions with
support in the domain $\Omega \subset \mathbb{R}^{n+1}$ originates
from the constructions of \cite{SCH}. To begin, let
$K \subset \Omega$ be a regular, compact set. We denote by
$\mathcal{D}_k(K)$ the space of functions $\phi \in C^\infty_0(K)$
for which
\[
\|\phi \|_{k,K}= \|(1+|x|)^k\hat{\phi}(x) \|_\infty<\infty.
\]
In fact, the norm $\|\cdot \|_{k,K}$
makes $\mathcal{D}_k(K)$ into a Banach space of smooth functions
with support contained in $K$. Observe that the sequence
$\mathcal{D}_k(K)$ for $k\in \mathbb{N}$
 is a sequence of Banach spaces with the property that
\[
\mathcal{D}_{k+1}(K) \subset \mathcal{D}_k(K)
\]
for each $k$, where the inclusion is continuous. It follows that we may take
the projective limit of these spaces to define the space
\[
\mathcal{D}(K)=\mathop{\rm proj}_{k\to \infty} \mathcal{D}_k(K)
\]
of test functions $\phi$ which satisfy $\|\phi\|_{k,K}<\infty$
for every $k\in\mathbb{N}$.

Now, let $K_{i}$ be an increasing sequence of compact subsets of $\Omega $
whose union is all of $\Omega $. We refer to such a sequence as a compact
exhaustion of $\Omega $. Then we have the continuous inclusions
\[
\mathcal{D}(K_{i}) \subset \mathcal{D}(K_{i+1})
\]
for each $i$. Thus, we may take an inductive limit to define
\[
\mathcal{D}(\Omega) =\mathop{\rm ind}_{i\to \infty }\mathcal{D}
(K_{i}) .
\]
This is a space of continuous functions $\phi $ for which there exists a
compact set $K\subset \Omega $ with
$\|\phi ||_{k,K}<\infty $ for all $k\in\mathbb{N}$.
The topology on this space can equivalently be described as follows: a
sequence $\phi _{i}$ in $\mathcal{D}(\Omega) $ converges to 0
if and only if there is a compact set $K\subset \Omega $ such that
$\{ \phi_{i}\} _{i=1}^{\infty }\subset \mathcal{D}(K) $ and
$\|\phi _{i}\|_{k,K}\to 0$
for each $k$.

We consolidate these statements in the following definition:

\begin{definition} \label{def2} \rm
Let $\Omega \subset \mathbb{R}^{n+1}$ be an open set with a countable,
compact exhaustion $K_i$. We define
$\mathcal{D}(\Omega)$ as the locally convex topological vector
space
\[
\mathcal{D}(\Omega) = \mathop{\rm ind}_{i \to \infty} \mathop{\rm proj}_{k
\to \infty} \mathcal{D}_k(K_i).
\]
\end{definition}

The space $\mathcal{D}(\Omega)$ is separable, complete, and
bornologic. We recall that a locally convex topological vector space
$X$ is bornologic if and only if the continuous linear operators from
$X$ to another locally convex topological vector space $Y$ are exactly
the bounded linear operators from $X$ to $Y$. We denote by
$\mathcal{D}'(\Omega)$ the topological dual of this space with the
strong-operator topology, also referred to as a space of distributions. The
space $\mathcal{D}'(\Omega)$ includes such objects as
$u=\sum_\alpha D_\alpha g$, where $g\in C(\Omega)$. In particular,
the action of $u$ on a test function $\phi$ is interpreted in the weak
sense:
\[
u(\phi)=\sum_\alpha (-1)^{|\alpha|} \int_\Omega g D_\alpha(\phi)
dx.
\]

The layout of this paper is as follows: In Section 2 we give existence
and uniqueness results to certain divergence-form parabolic differential
equations in sufficiently small domains
$\Omega \subset\mathbb{R} ^{n+1}$.  In Section 3 we extend these
results to general bounded domains
in the constant-coefficient case.  We employ the Perron process
\cite{L,DO} to obtain solutions to (\ref{L_gen}) when
$f\in W^{1,\infty }(\Omega) $ and $v_{0}=0$, and then show how these
can be used to obtain solutions for certain types of distributions.
Section 4 discusses growth conditions on solutions to (\ref{L_gen})
when $\Omega =\mathbb{R}^n\times (0,T)$ that define distributions
in the sense of (\ref{lim}).
We make use of a technique of \cite{CK} to write the integral
appearing in (\ref{lim}) as the difference of two other functionals,
both of which have a limit as $t\to 0^{+}$.  Using this, we obtain
a sufficient growth criterion and explore its necessity in a
few settings.

\section{Weak $W^{1,2}$-solutions in small balls}

 We begin with establishing some basic existence and uniqueness
results for solutions to divergence-form operators that are weak in a
particular sense.  Our methodology is based on that of
\cite[Chapter 3.3]{L}, , with minor generalizations to the hypotheses.
This approach has the
advantage in that it allows us to work with the relatively simple Sobolev
spaces as opposed to the H\"{o}lder spaces, and also that it gives existence
results in small balls $B$ that can be generalized to arbitrary bounded
domains $\Omega $.  To begin, we must describe the the type of weak
solutions we are looking for:  let $\Omega \subset \mathbb{R}^{n+1}$
be a bounded domain, and define the diameter
$2R=\mathop{\rm diam}(\Omega) $ by
\[
2R=\sup_{(x,t),(y,s)\in \Omega }|x-y|.
\]
For $1\leq i,j\leq n$, let $a_{ij}$, $b_{i}$, and $c$ be elements of
$C(\overline{\Omega }) $ that satisfy \eqref{cond_1}-\eqref{cond_4},
and assume in addition that the matrix $a_{ij}$ is symmetric.
Then, for any fixed $\varepsilon ,\eta \in (0,1] $, we define
divergence-form operator $L_{\varepsilon ,\eta }$ as
\[
L_{\varepsilon ,\eta }v=\sum D_{i}(a_{ij}D_{j}v) +\sum
b_{i}D_{i}v+cv+D_{t}(\varepsilon D_{t}v) -\eta D_{t}v.
\]
Now consider the Sobolev space $W^{1,2}(\Omega) $, and let
$W_{0}^{1,2}(\Omega) $ be the closure of
$C_{0}^{\infty }(\Omega) $ under the Sobolev norm
$\|\cdot\|_{1,2}$.  Choose any $f\in L^{2}(\Omega) $ and
$v_{0}\in W^{1,2}(\Omega) $.  Using the
terminology of \cite{L}, we say that $v$ is a weak
$W^{1,2}$-solution of the problem
\begin{equation}  \label{L_w}
\begin{gathered}
L_{\varepsilon ,\eta }v = f\quad \text{in }\Omega  \\
v  = v_{0}\quad \text{on }\partial \Omega
\end{gathered}
\end{equation}
if $v-v_{0}\in W_{0}^{1,2}(\Omega) $ and, for all $\phi \in
\mathcal{C}_{0}^{2}(\Omega) $,
\begin{align*}
&\int_{\Omega }-\sum_{ij}a_{ij}(D_{j}v) (D_{i}\phi)
+\sum_{i}b_{i}(D_{i}v) (\phi) +cv\phi
-\varepsilon (D_{t}v) (D_{t}\phi) -\eta (
D_{t}v) \phi \,dx\,dt\\
&=\int_{\Omega }f\phi \,dx\,dt.
\end{align*}
We begin with the following proposition concerning the existence and
uniqueness of $W^{1,2}$-solutions to \eqref{L_w} in bounded domains; see
also  \cite[Theorem 8.3]{GT} for an alternative proof that employs the
Fredholm alternative for the operator $L_{\varepsilon ,\eta }$:

\begin{proposition} \label{weak_sol}
Let $\Omega \subset \mathbb{R}^{n+1}$ be a bounded domain and set
$2R=\mathop{\rm diam}(\Omega) $.  Assume $a_{ij}$, $b_{i}$, and
$c$ are in $C(\overline{\Omega }) $ and satisfy
\eqref{cond_1}-\eqref{cond_4} with $a_{ij}$ symmetric. Then there
exists a constant $K_{n,a,b}$ such that if $R<K$, then for any
$f\in L^{2}(\Omega) $ and $v_{0}\in W^{1,2}(\Omega) $
there is a unique $W^{1,2}$-solution of \eqref{L_w}.
\end{proposition}

\begin{proof}
We first prove the proposition for $v_{0}=0$.  Assume, at first, that the
$b_{i}$, $c$, and $\eta $ are all $0$.  As a consequence of (\ref{cond_2}),
we may define an inner product on $W_{0}^{1,2}(\Omega) $ by
\[
\langle \phi ,\psi \rangle =\int_{\Omega }\sum_{ij}a_{ij}(
D_{j}\phi) (D_{i}\psi) +\varepsilon (D_{t}\phi
) (D_{t}\psi) \,dx\,dt
\]
and observe that $W_{0}^{1,2}$ is complete with respect to this inner
product.  Now, $f\in L^{2}(\Omega) $ defines a linear
functional on $W_{0}^{1,2}(\Omega) $ via the integral
\[
F(\phi) =-\int_{\Omega }f\phi \,dx\,dt.
\]
The Riesz Representation Theorem gives a unique function
$v\in W_{0}^{1,2}(\Omega) $ such that
$\langle v,\phi \rangle =F(\phi) $, and this is the unique solution of
(\ref{L_w}) for this case.

To extend this to nonzero $b_{i}$, $c$, and $\eta $, we use the method of
continuity \cite{KRY,L}.  For $h\in [ 0,1] $,
define the operator $\mathcal{L}_{h}:W_{0}^{1,2}(\Omega)
\mapsto W_{0}^{1,2}(\Omega) $ as follows: given
$v\in W_{0}^{1,2}(\Omega) $ let $\mathcal{L}_{h}v(\phi) $
be the linear functional defined on $W_{0}^{1,2}(\Omega) $ by
\begin{align*}
\mathcal{L}_{h}v(\phi)
&= \int_{\Omega }-\sum_{ij}a_{ij}(
D_{j}v) (D_{i}\phi) +h\sum_{i}b_{i}(D_{i}v)
(\phi) \\
&\quad +hcv\phi -\varepsilon (D_{t}v) (D_{t}\phi) -h\eta
(D_{t}v) \phi \,dx\,dt.
\end{align*}
Then set $\mathcal{L}_{h}(v) =g$ where
$g\in W_{0}^{1,2}(\Omega) $ is the unique element for which
$\langle g,\phi \rangle =\mathcal{L}_{h}v(\phi) $ under the Riesz Representation Theorem.
Observe that $\mathcal{L}_{h}$ is linear and bounded for every $h$ and, by
what we have just proved, $\mathcal{L}_{0}$ is invertible.  Now, assume
$\mathcal{L}_{h}(v) =g$.  Then
\[
\langle v,v\rangle =-\langle g,v\rangle +\int_{\Omega
}h\sum_{i}b_{i}(D_{i}v) (v) +hcv^{2}-h\eta (
D_{t}v) v\,dx\,dt.
\]
Since $c\leq 0$ and
$\int_{\Omega }(D_{t}v) v\,dx\,dt
=\frac{1}{2}\int_{\Omega }D_{t}(v^{2}) \,dx\,dt=0$, this implies
\begin{equation}
\begin{aligned}
\langle v,v\rangle &\leq -\langle g,v\rangle
+h\int_{\Omega }\sum_{i}b_{i}(D_{i}v) (v) \,dx\,dt\\
&\leq \theta \langle v,v\rangle +\frac{1}{\theta }\langle
g,g\rangle +| \int_{\Omega }\sum_{i}b_{i}(D_{i}v)
(v) \,dx\,dt|
\end{aligned}\label{meth_1}
\end{equation}
for any $\theta >0$.

Consider now the term $| \int_{\Omega }\sum_{i}b_{i}(
D_{i}v) (v) \,dx\,dt|$.  Let $a=\inf_{(x,t)\in
\Omega }x_{1}$ and $b=\sup_{(x,t)\in \Omega }x_{1}$, so that $b-a\leq 2R$
and $(x,t)\in \Omega $ implies $x_{1}\in (a,b)$.  Then, for
$v\in C_{0}^{\infty }(\Omega) $, we have
\begin{align*}
\big| \int_{\Omega }\sum_{i}b_{i}(D_{i}v) (v)\,dx\,dt\big|
&\leq \int_{\Omega }\sum_{i}| b_{i}\|D_{i}v| |v|\,dx\,dt \\
&=\int_{\Omega }\sum_{i}| b_{i}||D_{i}v|\big|
\int_{a}^{x_{1}}D_{1}v(s,x',t)ds\big|\,dx\,dt
\end{align*}
where we write $x'$ for the $n-1$-tuple $(x_{2},\dots x_{n})$.
Using the Cauchy-Schwartz inequality for the $ds$ integral, this becomes
\begin{align*}
&\int_{\Omega }\sum_{i}| b_{i}\|
D_{i}v|\int_{a}^{b}| D_{1}v(s,x',t)ds|\,dx\,dt \\
&\leq (2R) ^{1/2}\int_{\Omega }\sum_{i}|
b_{i}|| D_{i}v|\Big(\int_{a}^{b}[
D_{1}v(s,x',t)] ^{2}ds\Big) ^{1/2}\,dx\,dt.
\end{align*}
We can then separate the terms in the sum to obtain
\[
 (2R) ^{1/2}\Big[\theta '\int_{\Omega
}\sum_{i}| b_{i}|^{2}| D_{i}v|
^{2}\,dx\,dt+\frac{n}{\theta '}\int_{\Omega
}\int_{a}^{b}[ D_{1}v(s,x',t)] ^{2}\,ds\,dx'dt\Big].
\]
for any $\theta '>0$.  Setting $\theta '=1$ and using the
Fubini-Tonelli theorem for the second integral, we get the estimate
\begin{align*}
& (2R) ^{1/2}\Big[C_{b}\int_{\Omega }\sum_{i}|
D_{i}v|^{2}\,dx\,dt+nR\int_{\Omega }[
D_{1}v(s,x',t)] ^{2}\,ds\,dx'dt\Big] \\
&\leq (2R) ^{1/2}\Big[C_{b}\lambda \int_{\Omega }\frac{1}{\lambda }
\sum_{i}| D_{i}v|^{2}\,dx\,dt
+nR\lambda \int_{\Omega }\frac{1}{\lambda }\sum_{i}[ D_{i}v(x,t)] ^{2}\,dx\,dt\Big] \\
&\leq C_{n,a,b}\big(R^{1/2}+R^{3/2}\big) \langle v,v\rangle
\end{align*}
where the constant $C_{n,a,b}$ depends only on $n$, $a$
(through $\lambda $), and $b$.  Hence,
\[
\big| \int_{\Omega }\sum_{i}b_{i}(D_{i}v) (v)
\,dx\,dt\big|\leq C_{n,a,b}(R^{1/2}+R^{3/2}) \langle
v,v\rangle
\]
for all $v\in C_{0}^{\infty }(\Omega) $, a result which extends
to all $v\in W_{0}^{1,2}(\Omega) $ by density.  Thus, we see
that there is a $K_{n,a,b}$ such that $R<K$ implies
\[
\big| \int_{\Omega }\sum_{i}b_{i}(D_{i}v) (v)
\,dx\,dt\big|\leq \frac{1}{2}\langle v,v\rangle .
\]
 Placing this into (\ref{meth_1}), it follows that with such $R$ we may
choose $\theta >0$ so that $\langle v,v\rangle \leq \beta
\langle g,g\rangle $ for some positive $\beta $ that is
independent of $h$.  The method of continuity then implies that
$\mathcal{L}_{h}$ is invertible for all $h\in [ 0,1] $, and in
particular for $h=1$.  Hence, given $f\in L^{2}(\Omega) $ we may use
the Riesz Representation Theorem
to find a $g\in W_{0}^{1,2}(\Omega) $ for which
$\langle g,\phi \rangle =\int_{\Omega }f\phi \,dx\,dt$, and then use the
invertibility of $\mathcal{L}_{h}$ to obtain the weak $W^{1,2}$-solution
to (\ref{L_w}) with $v_{0}=0$.

Finally, let $v_{0}\in W^{1,2}(\Omega) $ be nonzero.  Observe
that $L_{\varepsilon ,\eta }v_{0}(\phi) $ also defines a
linear, continuous functional on $W_{0}^{1,2}(\Omega) $, and
thus $L_{\varepsilon ,\eta }(v_{0}) $ defines an element of
$W_{0}^{1,2}(\Omega) $ by the Riesz Representation Theorem,
and in particular an element of $L^{2}(\Omega) $.  Let $w$ be the
unique weak $W^{1,2}$-solution to
\begin{gather*}
L_{\varepsilon ,\eta }
w =g\quad \text{in }\Omega \\
w =0\quad \text{on }\partial \Omega
\end{gather*}
where $g=f-L_{\varepsilon ,\eta }(v_{0}) $.  Then $v=w+v_{0}$
is the solution to \eqref{L_w}.

It is possible to extend this existence result to $\varepsilon =0$ if the
coefficients $a_{ij}$ and $b_{i}$ are constant in addition to satisfying the
hypotheses of Proposition \ref{weak_sol}.  The basic strategy is to obtain
a uniform estimate on the derivatives of solutions $v_{\varepsilon }$ to (
\ref{L_w}) with $\eta $ fixed and $\varepsilon \in (0,1] $.
This will require us to also strengthen our hypotheses on the $v_{0}$, $f$,
and $\Omega $.  The first result we need is a maximal property that holds
when $v_{0}$ has a continuous extension to the boundary of $\Omega $.
\end{proof}

\begin{lemma}\label{est}
Let $\Omega $ be a bounded domain, and assume
$v_{0}\in W^{1,2}(\Omega) \cap C(\overline{\Omega }) $
satisfies the inequality $v_{0}\leq M$ on $\partial \Omega $ for some
constant $M\geq 0$.  Assume further that $v\in W^{1,2}(\Omega)$
is such that $v-v_{0}\in W_{0}^{1,2}(\Omega) $.
\begin{itemize}
\item[(a)]  If $u=(v-M) ^{+}$, then $u\in W_{0}^{1,2}(\Omega)$

\item[(b)]  If $R=\mathop{\rm diam}(\Omega) <K$ and
$L_{\varepsilon,\eta }v(\phi) \geq 0$ for all nonnegative
$\phi \in W_{0}^{1,2}(\Omega)$, then $v\leq M$ in $\Omega $.
\end{itemize}
\end{lemma}

\begin{proof}
(a)  From  of \cite[Lemma 3.7]{L}, we have that if
$f\in W^{1,2}(\Omega) $, then $f^{+}\in W^{1,2}(\Omega) $ with
\[
D_{\alpha }f^{+}=\chi _{A}D_{\alpha }f,
\]
where $| \alpha |=1$ and $A=\{ x:f(x)>0\} $.
 Let $v_{k}\in C_{0}^{\infty }(\Omega) $ be such that
$v_{k}\to v-v_{0}$ in $W^{1,2}(\Omega) $, and define
$w=v_{0}-M\in C(\overline{\Omega }) \cap W^{1,2}(\Omega) $.
Then for every integer $k>0$, the function
$(v_{k}+w-\frac{1}{k}) ^{+}\in W^{1,2}(\Omega) $ is compactly
supported in $\Omega $, and so belongs to $W_{0}^{1,2}(\Omega) $
by convolution.  Now
$(v_{k}+w-\frac{1}{k}) ^{+}\to (v-M) ^{+}\in L^{2}(\Omega) $ as
$k\to \infty $.  Furthermore, for $| \alpha |=1$ we have
\[
\|D_{\alpha }(v_{k}+w-\frac{1}{k})
^{+}-D_{\alpha }(v-M) ^{+}\|_{2}
=\|\chi _{E_{k}}D_{\alpha }(v_{k}+v_{0})
-\chi _{E}D_{\alpha }v\|_{2}
\]
where
\[
E_{k}=\{ x:v_{k}(x)+w(x)-\frac{1}{k}>0\} , \quad
E=\{x:v(x)-M>0\} .
\]
From this we obtain the estimate
\begin{align*}
&\|\chi _{E_{k}}D_{\alpha }(v_{k}+v_{0})
-\chi _{E}D_{\alpha }v\|_{2}\\
&\leq \|\chi _{E_{k}}[ D_{\alpha }(
v_{k}+v_{0}) -D_{\alpha }v] \|
_{2}+\|(\chi _{E_{k}}-\chi _{E}) D_{\alpha
}v\|_{2} \\
&\leq \|\chi _{E_{k}}[ D_{\alpha }(
 v_{k}+v_{0}) -D_{\alpha }v] \|
 _{2}+\|\chi _{B}(\chi _{E_{k}}-\chi _{E})
  D_{\alpha }v\|_{2}\\
&\quad +\|\chi _{\Omega \backslash B}(\chi _{E_{k}}-\chi _{E}) D_{\alpha }v\|_{2},
\end{align*}
where $B=\{ x:v(x)=M\} $.  Now, since
$v_{k}+w+\frac{1}{k}\to v-M$ in $L^{2}(\Omega) $ it follows that
$v_{k}+w+\frac{1}{k}\to v-M$ in measure, and so there is a subsequence
$v_{k_{n}}+w+\frac{1}{k_{n}}$ that converges to $v-M$ pointwise a.e..
 Since $\chi _{E_{k_{n}}}\to \chi _{E}$ a.e. on
$\chi _{\Omega \backslash B}$
while $D_{\alpha }v=0$ a.e. on $\chi _{B}$ (c.f. \cite[Lemma 3.7]{L}
again), we conclude that
\[
\|\chi _{E_{k_{n}}}D_{\alpha }(
v_{k_{n}}+v_{0}) -\chi _{E}D_{\alpha }v\|
_{2}\to 0
\]
as $n\to \infty $, and thus $(v-M) ^{+}\in W_{0}^{1,2}(\Omega) $.

(b)  Let $u=(v-M) ^{+}\in W_{0}^{1,2}(\Omega)$.
Then $L_{\varepsilon ,\eta }v(u) \geq 0$, that is
\[
\int_{\Omega }\sum a_{ij}(D_{j}v) (D_{i}u) -\sum
b_{i}(D_{i}v) (u) -cvu
+\varepsilon (
D_{t}v) (D_{t}u) +\eta (D_{t}v) u\,dx\,dt\leq 0.
\]
Observe, however, that the left hand side of this expression is equal to
\[
\int_{\Omega }\sum_{ij}a_{ij}(D_{j}u) (D_{i}u)
-\sum_{i}b_{i}(D_{i}u) (u) -cv(v-M)^{+}
+\varepsilon (D_{t}u) (D_{t}u) +\eta (
D_{t}u) u\,dx\,dt.
\]
We have that $cv(v-M)^{+}\leq 0$ and and
$\int_{\Omega }\eta (D_{t}u) u\,dx\,dt=0$; so this implies
\[
\int_{\Omega }\sum_{ij}a_{ij}(D_{j}u) (D_{i}u)
-\sum_{i}b_{i}(D_{i}u) (u) +\varepsilon (
D_{t}u) (D_{t}u) \,dx\,dt\leq 0.
\]
However, since $R<K$, the proof of Lemma \ref{weak_sol} gives
\[
\int_{\Omega }\sum_{ij}a_{ij}(D_{j}u) (D_{i}u)
-\sum_{i}b_{i}(D_{i}u) (u) +\varepsilon (
D_{t}u) (D_{t}u) \,dx\,dt\geq \frac{1}{2}\langle
u,u\rangle .
\]
Thus, $\langle u,u\rangle \leq 0$; i.e., $u=0$.
\end{proof}

Using Lemma \ref{est}, we can obtain the desired equicontinuity in the
case that the domain $\Omega $ has the form of a small ball; i.e.,
$$
\Omega =B(R) =\{ (x,t) :| x|^{2}+t^{2}<R^{2}\}.
$$
We will also require that the coefficients
$a_{ij}$, $b_{i}$ of $L_{\varepsilon ,\eta }$ be constant while
$c\in C^{1}(\overline{\Omega }) $.  Furthermore, let
$v_{0}\in C^{2}(\overline{\Omega }) $ and
$f\in W^{1,\infty }(\Omega) \subset W^{1,2}(\Omega) $, so that there
are constants $V$, $F$ for which
\begin{equation}
\begin{gathered}
| v_{0}|+ \sum_{i}| D_{i}v_{0}|+ \sum_{ij}| D_{ij}v_{0}|
+| D_{t}v_{0}|+ | D_{tt}v_{0}|<V,  \\
| f|+\sum_{i}| D_{i}f|+|f_{t}|<F.
\end{gathered} \label{eff}
\end{equation}

\begin{lemma}\label{equi}
Let $\Omega =B(R) $ and assume that $L_{\varepsilon,\eta }$ satisfies
the hypotheses of Proposition \ref{weak_sol} in addition
to the following:  the coefficients $a_{ij}$, $b_{i}$ are constant and
$c\in C^{1}(\overline{\Omega }) $.  Assume also that
$v_{0}\in C^{2}(\overline{\Omega }) $,
$f\in W^{1,\infty }(\Omega) $, and let $V$, $F$ be as in (\ref{eff}).
Then there are constants $K_{n,a,b,c,\eta }'$ and $C_{n,a,b,c,\eta ,V,F}$
such that if $R<K_{n,a,b,c}'$, then for any weak $W^{1,2}$-solution $v$
of \eqref{L_w}, we have
\begin{equation}
\sum | D_{i}v|+| D_{t}v|\leq C
\label{equi_est}
\end{equation}
where, in particular, $C$ is independent of $\varepsilon \in (0,1] $.
\end{lemma}

\begin{proof}
Let $w=R^{2}-| x|^{2}-t^{2}\in W_{0}^{1,2}(\Omega) $.
Then for any $\phi \in W_{0}^{1,2}(\Omega) $, we
have
\begin{align*}
L_{\varepsilon ,\eta }w(\phi)
&=\int_{\Omega}\sum_{ij}a_{ij}(2x_{j}) (D_{i}\phi)
+\sum_{i}b_{i}(-2x_{i}) \phi +c(R^{2}-|x|^{2}-t^{2}) \phi \\
&\quad +\varepsilon (2t) (D_{t}\phi) -\eta (
-2t) \phi \,dx\,dt \\
&= \int_{\Omega }\sum_{i}-2a_{ii}\phi -\sum_{i}b_{i}(2x_{i})
\phi +c(R^{2}-| x|^{2}-t^{2}) \phi \\
&\quad -2\varepsilon \phi +2\eta t\phi \,dx\,dt.
\end{align*}
Thus, we may write $L_{\varepsilon ,\eta }w=g\in L^{2}(\Omega)$,
where
\begin{align*}
g &= -2\mathop{\rm trace}(a_{ij}) -\sum_{i}2b_{i}x_{i}+c(
R^{2}-| x|^{2}-t^{2}) -2\varepsilon +2\eta t \\
&\leq -2n\lambda +nR\sup_{i}| b_{i}|+\|c\|_{\infty }R^{2}+2| \eta |R.
\end{align*}
Thus, it follows that we may choose $K_{n,a,b,c,\eta }'<K$ so that
$R<K_{n,a,b,\eta }'$ implies $g\leq -n\lambda $.  Similarly, if
$L_{\varepsilon ,\eta }v_{0}=h\in L^{2}(\Omega) $, then a
straightforward calculation shows that for $R<K'$ we have
$| h(x,t)|\leq C_{n,a,b,c,\eta ,V}$ for some constant $C$
independent of $\varepsilon $.  In particular, since
$L_{\varepsilon ,\eta}v=f$, there is a constant
$C_{n,a,b,c,\eta ,V,F}'$ for which
$| L_{\varepsilon ,\eta }(v-v_{0}) |=| f-h|\leq C'$.  Now, for such
$R$, if we define
\[
u_{\pm }=\pm \frac{n\lambda }{C'}(v-v_{0}) -w\in
W_{0}^{1,2}(\Omega) ,
\]
then $L_{\varepsilon ,\eta }u_{\pm }\geq 0$ in the sense of Lemma \ref{est}
and $u_{\pm }\leq 0$ on $\partial \Omega $, so by Lemma \ref{est}
it follows that $u_{\pm }\leq 0$ on $\Omega $; that is,
$| v-v_{0}|\leq \frac{C^{\prime \prime }}{n\lambda }w$.

Now, let $X=(x,t)\in \Omega $ and $Y=(y,s)\in \partial \Omega $, so that
\begin{align*}
| v(X)-v(Y)|=| v(X)-v_{0}(Y)|
&\leq | v(X)-v_{0}(X)|+| v_{0}(X)-v_{0}(Y)|\\
&\leq \frac{C'}{n\lambda }w(X)+2RV| X-Y|
\end{align*}
where the latter estimate follows from the Mean Value Theorem.  The Mean
Value Theorem also implies
\[
w(X)=w(X)-w(Y)\leq (\sup_{\Omega }| \nabla w|) | X-Y|\leq C_{n,R}''| X-Y|,
\]
and so, assuming $R<K'$, there is a constant $M_{n,a,b,c,\eta ,V,F}$
for which
$$
| v(X)-v(Y)|\leq M| X-Y|.
$$
In particular, for any $Y\in \partial \Omega $ and any
$\tau \in \mathbb{R}^{n+1}$ such that $Y+\tau \in \Omega $, we have
\[
| v(Y+\tau) -v(Y)|\leq M| \tau |.
\]
 Our goal is to extend this Lipschitz condition to all $X\in \Omega $.
Choose $\tau $ so that
$\Omega _{\tau }=\{ X\in \Omega :X+\tau \in \Omega \} $ is nonempty,
 and let $N$ be a constant to be determined later.  We define
$\rho _{\pm }\in W^{1,2}(\Omega _{\tau }) $
by
\[
\rho _{\pm }(X) =\pm [ v(X+\tau) -v(X)]
-M| \tau |-N| \tau |w(X).
\]
By a direct calculation, we find that for any $\phi \geq 0$ in
$W_{0}^{1,2}(\Omega _{\tau }) $ that
\begin{align*}
L_{\rho _{\pm }}(\phi)
&=\int_{\Omega _{\tau }}[ \pm
(f(X+\tau )-f(X)) -cM| \tau |-N|\tau |g(X)] \phi (X)dX \\
&\quad-\int_{\Omega _{\tau }}[ c(X+\tau )-c(X)] v(X+\tau )\phi (X)dX.
\end{align*}
 Recall that $c\in C^{1}(\Omega) $.  Observe also that given
any $X\in \Omega $ and $Y\in \partial \Omega $, we have
\begin{align*}
| v(X)|&\leq | v(X)-v(Y)| +| v(Y)|\\
&=| v(X)-v_{0}(Y)|+| v_{0}(Y)|\\
&\leq M| X-Y|+V \\
&\leq 2MR+V;
\end{align*}
i.e., $| v|$ is uniformly bounded on $\Omega $.  Hence,
with $R<K'$, there is a constant
$C_{n,a,b,c,\eta ,V,F}'''$ for which
\[
\big| [ c(X+\tau )-c(X)] v(X+\tau )\phi (X)\big|\leq
C'''| \tau |\phi (X),
\]
and we may choose $N$ sufficiently large so that
$L_{\rho _{\pm }}(\phi) \geq 0$ for nonnegative
$\phi \in W_{0}^{1,2}(\Omega) $.  Since $\rho _{\pm }\leq 0$ on
$\partial \Omega _{\tau }$, Lemma
\ref{est} again implies that $\rho _{\pm }\leq 0$ on $\Omega _{t}$;
that is,
\[
| v(X+\tau )-v(X)|\leq M| \tau |+N| \tau |w(X)
\]
for all $X\in \Omega _{\tau }$.  Choosing a final constant
$C_{n,a,b,c,\eta ,V,F}''''$ so that
$M+Nw(X)<C''''$ on $\Omega $, we find that $v$ satisfies the Lipschitz
condition
\[
| v(X)-v(Y)|\leq C''''| X-Y|
\]
for all $X$, $Y\in \Omega $.  By  \cite[Lemma 3.5]{L}, this implies the
desired estimate (\ref{equi_est}).
\end{proof}

We now apply these results to find weak $W^{1,2}$-solutions of \eqref{L_w}
with $\varepsilon =0$ on sufficiently small balls $\Omega =B(R) $
by taking an appropriate subsequence of the family of solutions
$v_{\varepsilon }$:

\begin{theorem}\label{W_e_sol}
Let $\Omega $, $a_{ij}$, $b_{i}$, and $c$ satisfy the
hypotheses of Lemma \ref{equi}. Then for any $f\in W^{1,\infty }(
\Omega) $ and $v_{0}\in C^{2}(\overline{\Omega }) $,
there is a unique weak $W^{1,2}$-solution $v$ to \eqref{L_w} with
$\varepsilon =0$.
\end{theorem}

\begin{proof}
For $\varepsilon \in (0,1] $, let $v_{\varepsilon }$ be the
unique weak $W^{1,2}$-solution of \eqref{L_w} given by Proposition
\ref{weak_sol}.  Then by Lemma \ref{equi}, the family
$\{ v_{\varepsilon}\} _{\varepsilon \in (0,1] }$ is uniformly bounded and
equicontinuous, and so that there exists a uniformly convergent
subsequence $v=\lim_{m}v_{\varepsilon _{m}}$.
The estimate (\ref{equi_est}) implies
also that $v\in W^{1,2}(\Omega) $ and satisfies (\ref{equi_est}) as well.
To see that $v-v_{0}\in W_{0}^{1,2}(\Omega) $, we
note that since $v-v_{0}$ is equicontinuous and equal to $0$ on
$\partial \Omega $, it follows that
 $(v-v_{0}-\frac{1}{k}) ^{+}\in W^{1,2}(\Omega) $ is compactly supported
in $\Omega $ for every integer $k>0$.  Hence,
$(v-v_{0}-\frac{1}{k}) ^{+}\in W_{0}^{1,2}(\Omega) $, and since
$(v-v_{0}-\frac{1}{k}) ^{+}\to (v-v_{0}) ^{+}$ in
$W^{1,2}(\Omega) $ (c.f. Lemma \ref{est}, part (a), it follows that
$(v-v_{0}) ^{+}\in W_{0}^{1,2}(\Omega) $.  Furthermore,
the same argument holds for $v_{0}-v$, so
$(v-v_{0}) ^{-}\in W_{0}^{1,2}(\Omega) $ and hence so does $v-v_{0}$.
Finally, to show that $L_{0,\eta }u=f$, we have for any
$\phi \in C_{0}^{\infty}(\Omega) $
\begin{align*}
-\int_{\Omega }f\phi dx
&=\int_{\Omega }\sum_{ij}a_{ij}(
 D_{j}v_{\varepsilon _{m}}) (D_{i}\phi)
-\sum_{i}b_{i}(D_{i}v_{\varepsilon _{m}}) (\phi)
-cv_{\varepsilon _{m}}\phi \\
&\quad +\varepsilon (D_{t}v) (D_{t}u) +\eta (
D_{t}v) \phi \,dx\,dt \\
&= \int_{\Omega }v_{\varepsilon _{m}}\Big[ \sum_{ij}-D_{j}(
a_{ij}D_{i}\phi) +\sum_{i}D_{i}(b_{i}\phi)  \\
&\quad -c\phi -\varepsilon _{m}D_{tt}\phi -\eta D_{t}\phi \Big] \,dx\,dt.
\end{align*}
Since the integrand is uniformly bounded we obtain from Dominated
Convergence that
\[
-\int_{\Omega }f\phi dx
=\int_{\Omega }v\Big[\sum_{ij}-D_{j}(
a_{ij}D_{i}\phi) +\sum_{i}D_{i}(b_{i}\phi)
 -c\phi -\eta D_{t}\phi \Big] \,dx\,dt
\]
and the theorem is proved.
\end{proof}

\section{Weak solutions in general bounded domains and solutions involving
derivatives}

We will now use the Perron process in the same manner as \cite{L} to
extend this result to a general bounded domain $\Omega $.  We begin with
the following definitions: given $f\in C^{1}(\overline{\Omega })
$ and $v_{0}\in C^{2}(\overline{\Omega }) $, we say that $u\in
C(\overline{\Omega }) $ is a subsolution of the problem
\begin{equation}
\begin{gathered}
L_{0,\eta }v =f\quad\text{in }\Omega    \\
v =v_{0}\quad\text{on }\mathcal{P}\Omega
\end{gathered} \label{L_0}
\end{equation}
if $u\leq v_{0}$ on $\mathcal{P}\Omega $ and if for any ball $B=B(R)$ with
$R<K'$, the solution $\overline{u}$ of
\begin{equation}
\begin{gathered}
L_{0,\eta }\overline{u} =f\quad\text{in }B  \\
 \overline{u} =u\quad\text{on }\partial B
\end{gathered} \label{sol_ul}
\end{equation}
satisfies $\overline{u}\geq u$ in $B$.  Supersolutions are defined
similarly by reversing the inequalities.  From the discussion
in \cite[Chapter 3.4]{L}, we see that subsolutions and supersolutions
exhibit the following properties:

\begin{lemma} \label{supersub}
Consider the problem \eqref{L_0}:
\begin{itemize}
\item[(a)]  If $u$ is a subsolution and $w$ a supersolution, then
$w\geq u$ in $\Omega $.

\item[(b)]  Let $u$ be a subsolution and assume $B(R)\subset \Omega $
with $R<K'$.  Then if $\overline{u}$ solves \eqref{sol_ul}, the function
$U$ defined by
\[
U=\begin{cases}
u& \text{on }\overline{\Omega }\backslash B \\
\overline{u}& \text{on }B
\end{cases}
\]
is another subsolution, called the lift of $u$ relative to $B$.

\item[(c)]  If $u$ and $w$ are subsolutions, then so is
$\max \{ u,w\} $.
\end{itemize}
\end{lemma}

Recall from Theorem \ref{W_e_sol} that the derivatives of the solution $v$
to \eqref{L_w} satisfy the estimate (\ref{equi_est}) of Lemma \ref{equi}.
To apply the Perron process, we need a form of this estimate that does not
make explicit use of the boundary function $v_{0}$.  Corollary 3.20 of \cite
{L} provides such a result in the case that the coefficients $b_{i}$ and
$c $ are $0$.  With some minor modifications, this estimate can be shown to
hold when $b_{i}$ and $c$ are constant, and so we state the result without
proof:

\begin{lemma} \label{equi_alt}
Let $\Omega =B(R)$ with $R<K'$, and assume $a_{ij}$, $b_{i}$, $f$,
and $F$ are as in Theorem \ref{W_e_sol} while $c\leq 0$ is
constant.  Let $w$ be the function of Lemma \ref{equi}.  Then there is a
constant $C_{n,a,b,c,\eta }$ such that if $v\in W^{1,2}(\Omega)
\cap C(\Omega) $ satisfies $L_{0,\eta }v=f$ in the weak sense
on $\Omega $, then
\[
w^{2}\sum_{i}| D_{i}v|^{2}+w^{4}|
D_{t}v|^{2}\leq C\big(\sup | v|^{2}+F\big) .
\]
\end{lemma}

Now, given a bounded domain $\Omega $, a function $f\in W^{1,\infty }(
\Omega) $, and $v_{0}\in C^{2}(\overline{\Omega }) $, let
$S$ be the set of all subsolutions $u$ of \eqref{L_0}.  The Perron process
gives that $v(X)=\sup_{u\in S}u(X)$ defines an element of
$C(\Omega) $ that satisfies $L_{0,\eta }v=f$ in the weak sense, though we
cannot characterize its behavior at the boundary in the same way that we
could the weak $W^{1,2}$-solutions.  A proof that $v$ is a weak solution
follows:

\begin{theorem}\label{perron}
Let $\Omega $ be a bounded domain, and let $a_{ij}$, $b_{i}$,
and $c$ satisfy the hypotheses of Lemma \ref{equi_alt}.
Given any $f\in W^{1,\infty }(\Omega) $ and
$v_{0}\in C^{2}(\overline{\Omega }) $, let $S$ be the set of
all subsolutions of \eqref{L_0} and
define $v(X)=\sup_{u\in S}u(X)$.  Then $v\in C(\Omega) $ and
$v $ satisfies $L_{0,\eta }v=f$ in the weak sense on $\Omega $.
\end{theorem}

\begin{proof}
First, note from Lemma \ref{est} that
$u_{0}=-\frac{1}{\eta }\|f\|_{\infty }t-\|v_{0}\|_{\infty }$
is a subsolution and $-u_{0}$ is a
supersolution, hence $v$ is well-defined and bounded.  To show that $v$ is
a weak solution, let $X=(x,t)\in \Omega $ and $R<K'$ be such that
$B_{X}(R) \subset \Omega $.  Fix $X_{1}=(x,t+R/8)$ and let
$\{ u_{m}\} \subset S$ be a sequence for which
$u_{m}(X_{1})\to v(X_{1})$.
Let $w_{m}=\max \{u_{m},u_{0}\} $ so that the $w_{m}$ are increasing,
and define $W_{m}$ to be the lift of $w_{m}$ relative to $B_{X}(R) $.
By Lemma \ref{equi_alt}, there is a subsequence $W_{m_{k}}$ such that
$W_{m_{k}}$ converges uniformly to a solution $w$ of $L_{0,\eta }w=f$
in $B_{X}(\frac{R}{2}) $.  That $w(X_{1})=v(X_{1})$ is clear; we now
claim that $w=v$ for $Y$ sufficiently near $X$.

Indeed, let $X_{2}\in B_{X}(\frac{R}{8}) $, and choose a
sequence $\{ u_{m}'\} \subset S$ for which
$u_{m}'(X_{2})\to v(X_{2})$. Let
$w_{m}'=\max\{ u_{m}',w_{m}\} $ so that $w_{m}'$ is an
increasing sequence for which
$w_{m}'(X_{1})\to v(X_{1}) $ and $w_{m}'(X_{2}) \to v(X_{2}) $.
Let $W_{m}'$ be the lift of
$w_{m}'$ relative to $B_{X}(\frac{R}{4}) $, and let
$W_{m_{k}}'$ be a subsequence which converges uniformly to a
solution $w'$ of
$L_{0,\eta }w=f$ in $B_{X}(\frac{R}{8} ) $.
Then $w'\geq w$ in $B_{X}(\frac{R}{8}) $
and $w'(X_{2}) =v(X_{2}) $.  However,
$w'(X_{1}) =w(X_{1}) $, so by the strong
maximum principle it follows that $w'=w$ in
$B_{X}(\frac{R}{8}) $, and in particular
$w(X_{2}) =w'(X_{2}) =v(X_{2}) $.  Since $X_{2}$ was an arbitrary
element of $B_{X}(\frac{R}{8}) $, it follows that $w=v$ in
$B_{X}(\frac{R}{8}) $.  Thus, $L_{0,\eta }v=f$ for functions
$\phi \in C_{0}^{\infty }$ with support contained in
$B_{X}(\frac{R}{8}) $.  Since $X\in \Omega $ was chosen arbitrarily,
we can show that $L_{0,\eta }v=f$ in the weak sense for any
$\phi \in C_{0}^{\infty }(\Omega) $ by taking an appropriate partition
of unity, and the theorem is proved.
\end{proof}

\begin{remark} \label{rmk10}\rm
We observe that proofs of Theorem \ref{perron} with more general conditions
on the coefficients of $L_{0,\eta }$ are known, c.f.
\cite[Theorem 9.1]{LSU}.  However, the scheme given above for the
constant-coefficient case
is relatively straightforward and is all we require for the present
discussion.
\end{remark}

We may apply this result to obtain solutions when $f$ is a certain type of
distribution.  Let $\Omega $ be a bounded, convex domain with smooth
boundary, and let $f\in \mathcal{D}'(\Omega) $ be of
the form $f=D_{\alpha }g$ in the sense of distributions, where
$g\in C(\overline{\Omega }) $.  Observe that if the coefficients
$a_{ij}$,
$b_{i}$, and $c$ are constant, then $L_{0,\eta }$ makes sense as a continuous
map on the space $\mathcal{D}'(\Omega) $.  We give
the following existence result as a corollary to Theorem \ref{perron}:

\begin{corollary} \label{D_sol}
Let $\Omega $, $f$ be as above and assume that $a_{ij}$,
$b_{i}, $ and $c$ satisfy the hypotheses of Lemma \ref{equi_alt}.  Then
there is a $w\in \mathcal{D}'(\Omega) $ for which
$L_{0,\eta }w=f$.
\end{corollary}

\begin{proof}
Given $\phi \in \mathcal{D}(\Omega) $, we have for the action
of $f$ on $\phi :$
\[
(f,\phi) =(-1) ^{| \alpha |}\int_{\Omega }gD_{\alpha }\phi \,dx\,dt.
\]
Since $g\in C(\overline{\Omega }) $ and $\Omega $ is convex with
a smooth boundary, we may integrate by parts to obtain
\[
(f,\phi) =(-1) ^{| \beta |
}\int_{\Omega }GD_{\beta }\phi \,dx\,dt
\]
where $G\in C^{1}(\overline{\Omega }) $ and $\beta _{i}=\alpha
_{i}+1$.  Now, let $v\in C(\Omega) $ be the weak solution of
$L_{0,\eta }v=G$ on $\Omega $ from Theorem \ref{perron} and define
$w\in \mathcal{D}'(\Omega) $ by
\[
(w,\phi) =(-1) ^{| \beta |}\int_{\Omega }vD_{\beta }\phi \,dx\,dt.
\]
A straightforward calculation shows that $L_{0,\eta }w=f$ in the sense of
distributions, and the result follows.
\end{proof}

\section{Classical solutions defining distributions at their boundary}

As mentioned in the Introduction, there has been an increasing interest
in studying classical solutions to various differential equations whose
boundary values define distributions in the sense of (\ref{lim}).
Much of the work in this area has focused on differential equations
arising from operator semigroups, such as the heat equation
\cite{MAT2,CK,K} and the Hermite heat equation (\cite{D}).  The
characterizations take the form of growth conditions on solutions $u$ to
these equations defined on
$\mathbb{R}^n\times (0,T)$.  Motivated by these results, we consider
in this section sufficient growth conditions on classical solutions
to parabolic equations on
$\mathbb{R}^n\times (0,T)$ whose boundary values define distributions
of the form $\sum_{| \alpha |\leq M}D_{\alpha }(g_{\alpha}) $,
where each $g_{\alpha }\in C(\mathbb{R}^n) $ is bounded.
Our approach is based on \cite[Theorem 2.4]{CK},
which characterizes the growth of smooth solutions to the heat
equation with boundary values in the space of infra-exponentially
tempered distributions.

We begin with the following:  let $L$ be an operator of the form
\[
Lu=\sum_{ij}a_{ij}D_{ij}u+\sum_{i}b_{i}D_{i}u+cu
\]
where $a_{ij}$, $b_{i}$, and $c$ belong to
$C^{\infty }(\mathbb{R}^n) $ with bounded derivatives.
 Our interest lies in the behavior
of solutions $u(x,t)$ to the problem
\begin{equation}
Lu-u_{t}=0  \label{L_t}
\end{equation}
defined on
$\mathbb{R}^n\times (0,T)$.  Our first lemma concerns the existence
of a ``suitable'' function $v\in C_{0}^{\infty }(\mathbb{R}) $
that we will need in the proof.

\begin{lemma} \label{v}
Let $M\geq 0$ be an integer and $T>0$.  There is a function
$v\in C_{0}^{\infty }(\mathbb{R}) $ with
$\mathop{\rm supp}(v) \subset [ 0,\frac{T}{2}] $ for which
$v=\frac{t^{M}}{M!}$ on $(0,\frac{T}{4}) $
and $v^{(M+1) }=\delta +w$ in the sense of distributions, where
$w\in C^{\infty }(\mathbb{R}) $ with
$\mathop{\rm supp}(w) \subset [ \frac{T}{4},\frac{T}{2}] $.
\end{lemma}

\begin{proof}
Define the function
\[
f=\begin{cases}
\frac{t^{M}}{M!}&\text{for }t>0  \\
0&\text{for }t\leq 0,
\end{cases}
\]
and let $\alpha \in C^{\infty }(\mathbb{R}) $ be such that
$\alpha (t)=1$ for $t<\frac{5T}{16}$ and $\alpha(t)=0 $ for
$t>\frac{7T}{16}$.  Then $v=\alpha f$ is the desired function.
\end{proof}

Now, given a classical solution $u(x,t)$ to (\ref{L_t}), we are interested
in studying the behavior of $u$ on test functions
$\phi \in \mathcal{D}(\Omega)$ in the sense of (\ref{lim}).
This is done by using the function $v$ of Lemma \ref{v}
in conjunction with the operator $L$ to
``split'' the integral of (\ref{lim}) into two manageable parts:

\begin{proposition} \label{L_suff}
Let $u(x,t)$ be a smooth solution to the parabolic equation
\eqref{L_t} on $\mathbb{R}^n\times (0,T)$ such that $|u(x,t)|\leq Ct^{-M}$ for some integer
$M\geq 0$.  Then, for any $\phi \in \mathcal{D}(\mathbb{R}^n) $,
we have
\[
\lim_{t\to 0^{+}}\int_{\mathbb{R}^n}u(x,t)\phi (x)dx
=\sum_{| \alpha |\leq 2M+2}g_{\alpha }D_{\alpha }\phi
\]
where each $g_{\alpha }$ is continuous and bounded.  In particular, the
operation
\[
g(\phi )=\lim_{t\to 0^{+}}\int_{\mathbb{R}^n}u(x,t)\phi (x)dx
\]
defines an element of $\mathcal{D}'(\mathbb{R}^n) $.
\end{proposition}

\begin{proof}
We define $\widetilde{u}(x,t)$ on
$\mathbb{R}^n\times (0,\frac{T}{2}) $ by
\[
\widetilde{u}(x,t)=\int_{\mathbb{R}}u(x,t+s)v(s)ds.
\]
 From the bounds on $u$ and $v$ and their derivatives, we may take the
derivative under the integral sign to conclude that $\widetilde{u}$
satisfies (\ref{L_t}) on
$\mathbb{R}^n\times (0,\frac{T}{2})$.  In particular, since the
derivative $D_{t}$ commutes with $L$, we have that
$L^{k}\widetilde{u}=(D_{t}) ^{k}\widetilde{u}$ for all integers
$k\geq 0$.  Now, for $\phi \in C_{0}^{\infty }(\mathbb{R}^n) $,
consider
\[
\int_{\mathbb{R}^n}\widetilde{u}(x,t)\phi (x)dx
=\int_{\mathbb{R}^n}\int_{\mathbb{R}}u(x,t+s)v(s)\phi (x)\,ds\,dx.
\]
Observe that we may reverse the order of integration and differentiate
under the integral sign to obtain
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}}\int_{\mathbb{R}^n}[ (-L) ^{M+1}u]
(x,t+s)v(s)\phi (x)\,dx\,ds\\
&=\int_{\mathbb{R}^n}\int_{\mathbb{R}}[ (-D_{t}) ^{M+1}u] (x,t+s)v(s)
\phi (x)\,ds\,dx.
\end{aligned}\label{int_1}
\end{equation}
For the left hand side of (\ref{int_1}), we may integrate by parts
to obtain
\[
\int_{\mathbb{R}}\int_{\mathbb{R}^n}u(x,t+s)v(s)[ (L^{\ast })
^{M+1}\phi ] (x)\,dx\,ds
\]
where $L^{\ast }$ is the operator
\begin{align*}
L^{\ast }u &=-\sum_{ij}(
D_{ij}a_{ij}u+D_{i}a_{ij}D_{j}u+D_{i}a_{ij}D_{i}u+a_{ij}D_{ij}u) \\
&\quad +\sum_{i}(D_{i}b_{i}u+b_{i}D_{i}u) -cu.
\end{align*}
As for the right hand side of (\ref{int_1}), integrating by parts yields
\begin{align*}
&\int_{\mathbb{R}^n}\int_{\mathbb{R}}u(x,t+s)v^{(M+1) }(s)
\phi (x)\,ds\,dx \\
&=\int_{\mathbb{R}^n}u(x,t)\phi (x)dx+\int_{\mathbb{R}^n}\int_{
\mathbb{R}}u(x,t+s)w(s)\phi (x)\,ds\,dx.
\end{align*}
Substituting these two results into (\ref{int_1}), we obtain
\begin{align*}
\int_{\mathbb{R}^n}u(x,t)\phi (x)dx
&=\int_{\mathbb{R}}\int_{\mathbb{R}^n}u(x,t+s)v(s)[ (L^{\ast })
^{M+1}\phi ] (x)\,dx\,ds \\
&\quad -\int_{\mathbb{R}^n}\int_{\mathbb{R}}u(x,t+s)w(s)
\phi (x)\,ds\,dx.
\end{align*}
Thus, we find in the limit as $t\to 0^{+}$, that
\begin{align*}
\lim_{t\to 0^{+}}\int_{\mathbb{R}^n}u(x,t)\phi (x)dx
&=\int_{\mathbb{R}^n}(\int_{\mathbb{R}}u(x,s)v(s)ds)
[ (L^{\ast }) ^{M+1}\phi ] (x)dx\\
&\quad-\int_{\mathbb{R}^n}(\int_{\mathbb{R}}u(x,s)w(s)ds) \phi (x)dx.
\end{align*}
Since the integrals in parentheses give continuous, bounded functions
of $x$, the result follows.
\end{proof}

\begin{remark}\label{heat_necess} \rm
In the case that $L$ is the Laplacian $\Delta $, then the
growth condition can be shown to be necessary in some sense.  Indeed, let
$g\in \mathcal{D}'(\mathbb{R}^n) $ have the form
\[
(g,\phi) =\sum_{| \alpha |\leq 2M+2}\int_{\mathbb{R}^n}g_{\alpha }
(x)D_{\alpha }\phi (x)dx
\]
where the $g_{\alpha }$ are continuous and bounded.  We define
\[
u(x,t)=(g_{y},E_{t}(x-y))
\]
on
$\mathbb{R}^n\times (0,\infty )$.  It can be shown
(c.f. \cite{AEO}) that $u(x,t)$ is a smooth solution to the heat
equation on $\mathbb{R}^n\times (0,\infty )$ and satisfies
\[
\lim_{t\to 0^{+}}\int_{\mathbb{R}^n}u(x,t)\phi (x)dx=(g,\phi)
\]
for every $\phi \in \mathcal{D}(\mathbb{R}^n) $.  Furthermore,
each term $((g_{\alpha }) _{y},(D_{\alpha })_{y}E_{t}(x-y))$ appearing
in $(g_{y},E_{t}(x-y)) $ is of the form
\begin{align*}
&(-\sqrt{4t}) ^{| \alpha |}\int_{\mathbb{R}^n}g_{\alpha }(y)
H_{\alpha }(\frac{x-y}{2\sqrt{t}})E_{t}(x-y)dy \\
&=C_{\alpha }t^{-| \alpha |/2}\int_{\mathbb{R}
^n}g_{\alpha }(x-2z\sqrt{t})H_{\alpha }(z) e^{-|
z|^{2}}dz
\end{align*}
where $H_{\alpha }$ is the Hermite polynomial of order $\alpha $.
It follows that $| u(x,t)|\leq Ct^{-M-1}$ for some constant
$C$ depending on the $g_{\alpha }$, $M$, and the dimension $n$.
We do not know if this can be sharpened to become
$| u(x,t)|\leq Ct^{-M}$.
\end{remark}

\begin{remark} \label{heat_necess_2} \rm
In view of Remark \ref{heat_necess}, consider the case
that $b_{i}$ and $c$ are all $0$, and the matrix $a_{ij}$ is constant
and satisfies the condition
\[
\sum_{ij}a_{ij}x_{i}x_{j}\geq \lambda | x|^{2}
\]
where $\lambda >0$.  Based on the discussion of
\cite[Lemma 8.9.1]{KRY}, we can find a nonsingular matrix $A_{ij}$
for which $AaA^{T}=I$.  From Proposition \ref{L_suff}, we see
that if $u$ is smooth, solves $Lu=u_{t}$
and satisfies $|u(x,t)|$ $\leq Ct^{-m}$, then $u(x,t)$
defines a distribution of the form
$g=\sum_{| \alpha |\leq 2m+2}g_{\alpha }D_{\alpha }$ where each
$g_{\alpha }$ is continuous and bounded.  Conversely, given such
$g_{\alpha }$ we define the distributions
\[
v_{\alpha }=\sum\det (A)
(A_{k_{1}^{1},1}\dots A_{k_{\alpha _{1}}^{1},1}\dots
A_{k_{1}^n,n}\dots A_{k_{\alpha _{n}}^n,n})
D_{k_{1}^{1}\dots k_{\alpha _{1}}^{1}\dots k_{1}^n
\dots k_{\alpha _{n}}^n}g_{\alpha },
\]
where the summation is taken from
$k_{1}^{1},\dots k_{\alpha _{1}}^{1},\dots
k_{1}^n,\dots k_{\alpha _{n}}^n=1$ to $n$,
as determined by the chain rule.
Then each $v_{\alpha }$ satisfies
the conditions of Remark \ref{heat_necess},and so there are smooth
solutions $u_{\alpha }$ of the heat equation on
$\mathbb{R}^n\times (0,\infty )$ for which
$u_{\alpha }(0,t)=v_{\alpha }$ in the
sense of (\ref{lim}) and
$| u_{\alpha }(x,t)|\leq Ct^{-N} $ for some nonnegative integer $N$.
Then, defining $v_{\alpha}(x,t)=u_{\alpha }(Ax,t)$, we see that
$v_{\alpha }$ is a smooth solution to (\ref{L_t}) on
$\mathbb{R}^n\times (0,\infty )$ with $| v(x,t)|\leq Ct^{-N}$,
and a straightforward calculation yields
\[
\lim_{t\to 0^{+}}\int_{\mathbb{R}^n}v(x,t)\phi (x)dx=(g_{\alpha },\phi) .
\]
Hence, the conclusion of Remark \ref{heat_necess} is also valid for such
operators $L$.
\end{remark}

\subsection*{Acknowledgments}
The authors are especially grateful to the anonymous referees for their
careful reading of the manuscript and the fruitful remarks.
This work has been partially supported by ADVANCE - NSF, and by Minigrant
College of Arts and Sciences, NMSU.

\begin{thebibliography}{99}
\bibitem{AEO} J. Alvarez, M. Eydenberg, and H. Obiedat;
  The Action of
Operator Semigroups in the Topological Dual of the Beurling-Bj\"{o}rck
Space. \emph{Journal of Mathematical Analysis and Applications} 339 (2008)
405-418.

\bibitem{AMM1} P. Amster, C. Averbuj, P. de N\'{a}poli, and M. C. Mariani;
 A Black-Scholes Option Pricing Model with Transaction Costs.  \emph{
Journal of Mathematical Analysis and Applications} 303 (2005) 688-695.

\bibitem{AMM2} P. Amster, C. Averbuj, P. de N\'{a}poli, and M. C. Mariani;
 A Parabolic Problem Arising on Financial Mathematics.  To appear in \emph{
Nonlinear Analysis.  }

\bibitem{BBF} H. Berestycki, J. Busca, and I. Florent;  Computing the
Implied Volatility in Stochastic Volatility Models.  \emph{Communications
on Pure and Applied Mathematics} 10 (2004) 1352-1373.

\bibitem{BS} F. Black and M. Scholes;  The Pricing of Options and
Corporate Liabilities.  \emph{Journal of Political Economy} 81 (1973),
637-659.

\bibitem{CK} S.-Y. Chung and D. Kim.  Distributions with Exponential
Growth and a Bochner-Schwartz Theorem for Fourier Hyperfunctions.  \emph{
Publ. RIMS, Kyoto University} 31 (1995) 829-845.

\bibitem{D} B. Dhungana;  Mehler Kernel Approach to Tempered
Distributions.  \emph{Tokyo Journal of Mathematics} 29 (2006) 283-293.

\bibitem{DO} G. C. Dong;  Nonlinear Partial Differential Equations of
Second Order.  \emph{Translations of Mathematical Monographs, AMS} 95
(1991).

\bibitem{K} D. Kim;  Hyperfunctions and Heat Kernel Method. \emph{
Microlocal Analysis and Complex Fourier Analysis.  }World-Scientific (2002)
149-165.

\bibitem{HES} S. L. Heston;  A Closed-Form Solution for Options with
Stochastic Volatility with Applications to Bond and Currency Options.
\emph{Review of Financial Studies} 6 (1993) 327.

\bibitem{[H]} L. H\"{o}rmander; \emph{The Analysis of Linear Partial
Differential Operators I}, 2nd ed. Springer-Verlag (1990).

\bibitem{GT} D. Gilbarg and N. Trudinger; Elliptic Partial Differential
Equations of Second Order, 2nd ed. \emph{Grundlehren der Mathematischen
Wissenschaften} 224 (1983).

\bibitem{KRY} N. V. Krylov;  Lectures on Elliptic and Parabolic Equations
in H\"{o}lder Spaces.  \emph{Graduate Studies in Mathematics, AMS} 12
(1996).

\bibitem{LSU} O. A. Lady\v{z}enskaja, V. A. Solonnikov, and N. N. Ura\'{l}
ceva;  Linear and Quasi-linear Equations of Parabolic Type. \emph{
Translations of Mathematical Monographs, AMS} 23 (1967).

\bibitem{L} G. M. Lieberman;  \emph{Second Order Parabolic Differential
Equations}.  World Scientific (1996).

\bibitem{MAT1} T. Matsuzawa; A Calculus Approach to Hyperfunctions, I.
\emph{Nagoya Mathematics Journal} 108 (1987) 53-66.

\bibitem{MAT2} T. Matsuzawa; A Calculus Approach to Hyperfunctions, II.
\emph{Transactions of the AMS} 313 (1989) 619-654.

\bibitem{Me} R. Merton; \emph{Continuous-Time Finance}. Blackwell,
Cambridge (2000).

\bibitem{SCH} L. Schwartz;  \emph{Th\'{e}orie des Distributions}, 3rd.
ed.  Hermann, Paris (1966).

\bibitem{T} S. Thangavelu; Lectures on Hermite and Laguerre Expansions.
 \emph{Mathematical Notes }42, Princeton Unversity Press (1993).

\bibitem{W} P. Wilmott, J. Dewynne, and S. Howison;
  \emph{Option Pricing}. Oxford Financial Press (2000).

\end{thebibliography}

\end{document}