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\begin{document}
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\title[Ulam-Hyers Stability of Black-Scholes Equation]{Ulam-Hyers Stability of Black-Scholes Equation}
\author{Nicolaie Lungu}
\address{Technical University of Cluj-Napoca\\
Department of Mathematics\\
28 Memorandumului Street\\
400114 Cluj-Napoca, Romania}
\email{nlungu@math.utcluj.ro}
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\author{Sorina Anamaria Ciplea}
\address{Technical University of Cluj-Napoca\\
Department of Management and Technology\\
28 Memorandumului Street\\
400114 Cluj-Napoca, Romania}
\email{sorina.ciplea@cem.utcluj.ro}
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\subjclass{35L70, 45H10, 47H10}
\keywords{Black-Scholes equation, Ulam-Hyers stability, generalized Ulam-Hyers stability,
derivative financial product, Green function}

\begin{abstract}
The goal of this paper is to give a Ulam-Hyers stability result for Black-Scholes
equation, in which the unknown function is the price of a derivative financial product.
In this case we study the Ulam-Hyers stability with the Green function.
\end{abstract}

\maketitle

\section{Introduction}

The Black-Scholes equation was introduced as a model for the financial mathematics (\cite{1}).
We will consider the following equation (\cite{10}, \cite{11}):
\begin{equation}
\label{1.1}
\ds\f{\p V(s,t)}{\p t}+\ds\f{\sigma ^2 s^2}{2}\, \ds\f{\p^2 V(s,t)}{\p s^2}
+rs\ds\f{\p V(s,t)}{\p s}-rV(s,t)=F(s,t)
\end{equation}
$$\Omega =\{(s,t)\mid s\in (s_1,s_2),\ t\in (T_1,T)\},\ V\in C^2(\Omega ),$$
where $V(s,t)$ represents the price of the derivative financial product.
The independent variables $(s,t)$ are the share price of the underlying assets
and time, respectively.
The constants $\sigma $ and $r$ are the volatility of the underlying asset and the risk-free
interest rate, respectively.
This equation is of the parabolic type and it can be considered as a diffusion equation.
In what follows, we refer to this equation as BS equation.
In this case we consider the conditions (\cite{11}):

(i) Cauchy problem:
\begin{equation}
\label{1.2}
V(s,T)=\varphi (s),
\end{equation}
$\varphi (s)$ is the pay-off function of a given derivative problem at $t=T$.

(ii) The boundary conditions (Darboux):
\begin{equation}
\label{1.3}
V(s_1,t)=b_1(t),\q
V(s_2,t)=b_2(t).
\end{equation}

By the corresponding substitution (\cite{11}), we have the equation:
\begin{equation}
\label{1.4}
\ds\f{\p v(s,t)}{\p t}+\ds\f{\sigma ^2s^2}{2}\, \ds\f{\p^2 v(s,t)}{\p s^2}
+rs\ds\f{\p v(s,t)}{\p s}-rv(s,t)=h(s,t),
\end{equation}
with
\begin{equation}
\label{1.5}
v(s,T)=f(s)
\end{equation}
and homogeneous conditions:
\begin{equation}
\label{1.6}
v(s_1,t)=v(s_2,t)=0,
\end{equation}
$$\Omega =\{(s,t)\mid s\in (s_1,s_2),\ t\in (T_1,T)\},\ h\in C(\Omega ,\mathbb{R}).$$

In what follows we consider the Cauchy-Darboux problem (\ref{1.4})+(\ref{1.5})+(\ref{1.6}).

Here (\cite{11})
\begin{align}
\label{1.7}
h(s,t)
& =F(s,t)+\ds\f{s-s_1}{s_2-s_1}[r(b_2(t)-b_1(t))+b'_1(t)-b'_2(t)]\nonumber\\
& -b'_1(t)+rb_1(t)+rs\ds\f{b_1(t)-b_2(t)}{s_2-s_1}
\end{align}
and
\begin{equation}
\label{1.8}
v(s,t)=\int_{s_1}^{s_2}G(s,t;\eta )f(\eta )d\eta
+\int_t^T \int_{s_1}^{s_2}G(s,t-\tau;\eta )h(\eta ,\tau)d\eta d\tau.
\end{equation}

Further we study the problem of Ulam-Hyers stability of this equation,
because the unknown function appears here as the price of financial derivatives.

We recall that this equation can be called Black-Merton-Scholes equation and it was a subject
of the Nobel Prize in Economics in 1997.

\section{Notions and definitions}
In this paper we will present some types of Ulam stability for the Black-Scholes equation.

In 1940, on a talk given at Wisconsin University, S.M. Ulam has formulated the following problem:
"{\it Under what conditions does there exist near every approximately homomorphism
of a given metric group an homomorphism of the group?}~"
(\cite{4}, \cite{8}, \cite{9}, \cite{12}, \cite{13}, \cite{20}).
Generally, we say that a differential equation is stable (Ulam) if for every approximate
solution of the differential equation, there exists an exact solution
near it.
In this way we can study the stability of the price of a derivative financial product.
The goal of this paper is to give a stability result for Black-Scholes equation
(\cite{1}, \cite{11}).

It seems that the first paper on the Ulam-Hyers stability of partial differential equations
was written by Pr\'astaro and Rassias (\cite{15}).
For other results on the stability of differential equations and partial differential equations
we refer to (\cite{2}, \cite{3}, \cite{5}, \cite{6}, \cite{7}, \cite{14}, \cite{17}, \cite{19}).

Let $\eps >0$, $\varphi \in C(\mathbb{R}_+,\mathbb{R}_+)$ and $\varphi (0)=0$.
We consider the following inequations:
\begin{equation}
\label{2.1}
\left|\ds\f{\p u(s,t)}{\p t}+\ds\f{\sigma ^2s^2}{2}\, \ds\f{\p^2 u(s,t)}{\p s^2}
+rs\ds\f{\p u(s,t)}{\p s}-ru(s,t)-h(s,t)\right|\le \eps ,\ \forall \ (s,t)\in \Omega
\end{equation}
\begin{equation}
\label{2.2}
\left|\ds\f{\p u(s,t)}{\p t}+\ds\f{\sigma ^2s^2}{2}\, \ds\f{\p^2 u(s,t)}{\p s^2}
+rs\ds\f{\p u(s,t)}{\p s}-ru(s,t)-h(s,t,u)\right|\!\le \!\eps ,\ \forall \ (s,t)\!\in\! \Omega .
\end{equation}

\begin{definition}
\label{d2.1}
$($\cite{17}, \cite{18}$)$
The equation $(\ref{1.5})$ is Ulam-Hyers stable if there exists a real number $c_1$
such that for each solution $u$ of $(\ref{2.1})$ there exists a solution $v$ of $(\ref{1.5})$ with
\begin{equation}
\label{2.3}
|u(s,t)-v(s,t)|\le c_1\cdot \eps ,\ \forall \ (s,t)\in \Omega .
\end{equation}
\end{definition}

\begin{definition}
\label{d2.2}
The equation $(\ref{1.5})$ is generalized Ulam-Hyers stable if there exists
$\varphi \in C(\mathbb{R}_+,\mathbb{R}_+)$, $\varphi (0)=0$, continuous,
such that for each solution $u$ of $(\ref{2.2})$ there exists a solution $v$ of $(\ref{1.5})$ with
\begin{equation}
\label{2.4}
|u(s,t)-v(s,t)|\le \varphi (\eps ),\ \forall \ (s,t)\in \Omega .
\end{equation}
\end{definition}

\begin{remark}
\label{r2.1}
A function $u$ is a solution of (\ref{2.1}) if and only if there exists a function
$g\in C(\Omega )$ such that

(i) $|g(s,t)|\le \eps ,\ \forall \ (s,t)\in \Omega $;

\medskip
(ii) $\ds\f{\p u(s,t)}{\p t}+\ds\f{\sigma ^2s^2}{2}\, \ds\f{\p^2 u(s,t)}{\p s^2}
+rs\ds\f{\p u(s,t)}{\p s}-ru(s,t)=h(s,t)+g(s,t)$.
\end{remark}

\begin{remark}
\label{r2.2}
A function $u$ is a solution of (\ref{2.2}) if and only if there exists a function
$g\in C(\Omega )$ such that

(i) $|g(s,t)|\le \eps ,\ \forall \ (s,t)\in \Omega $;

\medskip
(ii) $\ds\f{\p u(s,t)}{\p t}+\ds\f{\sigma ^2s^2}{2}\, \ds\f{\p^2 u}{\p s^2}
+rs\ds\f{\p u(s,t)}{\p s}-ru(s,t)=h(s,t)+g(s,t)$.
\end{remark}

\section{Ulam-Hyers stability of equation BS}

Here we will present some results of Ulam-Hyers stability for the equation BS.

\begin{theorem}
\label{t3.1}
We suppose that:

(i) $\Omega $ is a bounded domain and $G$ is the Green function for the BS equation;

(ii) $h\in C(\ov{\Omega }),\ f\in C(s_1,s_2)$;

(iii) $\ds\int_t^T \ds\int_{s_1}^{s_2}|G(s,t-\tau;\eta )|d\eta d\tau\le q<1,\ \forall \ (s,t)\in \Omega $.

\noindent
Then:

(a) the problem $(1.5)+(1.6)$ has a unique solution;

(b) the equation BS, $(\ref{1.5})$, is Ulam-Hyers stable.
\end{theorem}

\noindent
{\it Proof.}
(a) This is a well known result, consequence of Banach principle (\cite{16}).

(b) Let $u$ be a solution of the inequation (\ref{2.1}).
Let $v$ be the unique solution of the problem (\ref{1.5})+(\ref{1.6}).
From Remark \ref{r2.1} and the condition (iii) we have that
$$|u(s,t)-v(s,t)|\le \left|\int_{s_1}^{s_2}G(s,t;\eta )f(\eta )d\eta
+\int_t^T \int_{s_1}^{s_2}G(s,t-\tau;\eta )h(\eta ,\tau)d\eta d\tau\right.$$
$$+\int_t^T \int_{s_1}^{s_2}G(s,t-\tau;\eta )g(\eta ,\tau)d\eta d\tau
-\int_{s_1}^{s_2}G(s,t;\eta )f(\eta )d\tau$$
$$\left.-\int_t^T \int_{s_1}^{s_2}G(s,t-\tau;\eta )h(\eta ,\tau)d\eta d\tau\right|$$
$$\le \int_t^T \int_{s_1}^{s_2}|G(s,t-\tau;\eta )|\cdot |g(\eta ,\tau)|d\eta d\tau\le q\cdot \eps .$$
So, the equation (\ref{1.5}) is Ulam-Hyers stable.

\section{Generalized Ulam-Hyers stability of nonlinear BS equation}

In this paragraph we will consider the nonlinear BS equation.
Let $\Omega $ be the domain considered above.

In what follows, we consider the mixed problem (Cauchy-Darboux) (\cite{11}):
\begin{equation}
\label{4.1}
\ds\f{\p v(s,t)}{\p t}+\ds\f{\sigma ^2 s^2}{2}\, \ds\f{\p^2 v(s,t)}{\p s^2}
+rs\ds\f{\p v(s,t)}{\p s}-rv(s,t)=h(s,t,v),
\end{equation}
\begin{equation}
\label{4.2}
\ba{l}
v(s,T)=f(s),\medskip \\
v(s_1,t)=v(s_2,t)=0.
\ea
\end{equation}

\begin{theorem}
\label{t4.1}
Let us consider the equation $(\ref{4.1})$ and the inequation $(\ref{2.2})$.
Let $G$ be the Green function corresponding to BS equation.

We suppose that:

(i) $h\in C(\Omega )$ and there exists $l_h>0$ with
$$l_h\int_t^T \int_{s_1}^{s_2}|G(s,t-\tau;\eta )|d\eta d\tau\le q<1$$
and a comparison function $\varphi :\mathbb{R}_+\to \mathbb{R}_+$ such that
$$|h(s,t,u)-h(s,t,v)|\le l_h\varphi (|u-v|).$$
Then

(a) the Cauchy-Darboux problem $(\ref{4.1})+(\ref{4.2})$ has a unique solution $v$;

(b) if the function $\psi :\mathbb{R}_+\to \mathbb{R}_+$, $\psi (z)=z-\varphi (z)$,
is strictly increasing and onto or bijective, the problem $(\ref{4.1})+(\ref{4.2})$
is generalized Ulam-Hyers stable.
\end{theorem}

\noindent
{\it Proof.}
(a) This result is a consequence of Banach theorem.

(b) Let $u$ be a solution of the inequality (\ref{2.2}) and $v$ the unique solution
of the problem (\ref{4.1})+(\ref{4.2}).
From the above conditions we have
$$|u(s,t)-v(s,t)|
=\left|\int_{s_1}^{s_2} G(s,t;\eta )f(\eta )d\eta
+\int_t^T \int_{s_1}^{s_2} G(s,t-\tau;\eta )h(s,t,u)d\eta d\tau\right.$$
$$+\int_t^T \int_{s_1}^{s_2} G(s,t-\tau;\eta )g(\eta ,\tau)d\eta d\tau
-\int_{s_1}^{s_2} G(s,t;\eta )f(\eta )d\eta $$
$$\left.-\int_t^T \int_{s_1}^{s_2} G(s,t-\tau;\eta )h(\eta ,\tau,v)d\eta d\tau\right|$$
$$=\left|\int_t^T \int_{s_1}^{s_2} G(s,t-\tau;\eta )h(s,t,u)d\eta d\tau
+\int_t^T \int_{s_1}^{s_2} G(s,t-\tau;\eta )g(\eta ,\tau)d\eta d\tau\right.$$
$$\left.-\int_t^T \int_{s_1}^{s_2} G(s,t-\tau;\eta )h(s,t,v)d\eta d\tau\right|$$
$$\le \int_t ^T \int_{s_1}^{s_2} |G(s,t-\tau;\eta )|\cdot |h(s,t,u)-h(s,t,v)|d\eta d\tau$$
$$+\int_t^T \int_{s_1}^{s_2} |G(s,t-\tau;\eta )|\cdot |g(\eta ,\tau)|d\eta d\tau$$
$$\le \int_t ^T \int_{s_1}^{s_2} |G(s,t-\tau;\eta )|l_h \varphi (|u-v|)d\eta d\tau
+\int_t^T \int_{s_1}^{s_2} |G(s,t-\tau;\eta )|\cdot |g(\eta ,\tau)|d\eta d\tau,$$
then we have
$$|u(s,t)-v(s,t)|\le \varphi (|u(s,t)-v(s,t)|)+\ds\f{\eps }{l_h}$$
and
$$\psi (|u(s,t)-v(s,t)|)\le \ds\f{\eps }{l_h},$$
therefore we have
$$|u(s,t)-v(s,t)|\le \psi ^{-1}\left(\ds\f{\eps }{l_h}\right).$$
So the equation (\ref{4.1}) is generalized Ulam-Hyers stable.


\begin{thebibliography}{99}
\bibitem{1}
Black, F., and  Scholes, M.S.,
{\it The pricing of options and corporate liabilities},
Journal of Political Economics, 81, 1973, 637-654.

\bibitem{2}
Brzdek, J., Popa, D., Xu, B.,
{\it The Hyers-Ulam stability of nonlinear recurrences},
J. Math. Anal. Appl., {\bf 335}(2007), 443-449.

\bibitem{3}
C\^ampean, D.S., Popa, D.,
{\it Hyers-Ulam stability of Euler's equation},
Appl. Math. Lett., {\bf 24}(2011), 1539-1543.

\bibitem{4}
Hyers, D.H.,
{\it On the stability of the linear functional equation},
Proc. Math. Acad. Sci. USA, {\bf 27}(1941), 222-242.

\bibitem{5}
Hyers, D.H., Isac, G., Rassias, Th.M.,
{\it Stability of Functional Equations in Several Variables},
Birkh\"auser, Basel, 1998.

\bibitem{6}
Jung, S.-M.,
{\it Hyers-Ulam stability of linear partial differential equations of first order},
Appl. Math. Lett., {\bf 22}(2009), 70-74.

\bibitem{7}
Lungu, N., Ciplea, S.A.,
{\it Ulam-Hyers-Rassias stability of pseudoparabolic partial differential equations},
Carpathian J. of Math., {\bf 31}(2015), no. 2, 233-240.

\bibitem{8}
Lungu, N., Popa, D.,
{\it Hyers-Ulam stability of a first order partial differential equation},
J. Math. Anal. Appl., {\bf 385}(2012), 86-91.

\bibitem{9}
Lungu, N., Popa, D.,
{\it On the Ulam-Hyers stability of first order partial differential equations},
Carpathian J. of Math., {\bf 28}(2012), no. 1, 77-82.

\bibitem{10}
Melnikov, M.Y., Melnikov, Y.A.,
{\it Construction of Green's functions for the Black-Scholes equation},
Electronic Journal of Differential Equations, {\bf 153}(2007), 1-14.

\bibitem{11}
Melnikov, Y.A., Melnikov, M.Y.,
{\it Green's Functions, Construction and Applications},
De Gruyter Studies in Mathematics, Berlin/Boston, {\bf 42}(2012).

\bibitem{12}
Popa, D.,
{\it Hyers-Ulam-Rassias stability of linear recurrence},
J. Math. Anal. Appl., {\bf 309}(2005), 591-597.

\bibitem{13}
Popa, D.,
{\it Hyers-Ulam stability of the linear recurrence with constant coefficients},
Advances in Difference Equations, {\bf 2}(2005), 101-107.

\bibitem{14}
Popa, D., Ra\c{s}a, I.,
{\it On Hyers-Ulam stability of the linear differential equation},
J. Math. Anal. Appl., {\bf 381}(2011), 530-537.

\bibitem{15}
Pr\'astaro, A., Rassias, Th.M.,
{\it Ulam stability in geometry PDE's},
Nonlinear Func. Anal. Appl., {\bf 8}(2)(2003), 259-278.

\bibitem{16}
Rus, I.A.,
{\it Principii \c{s}i aplica\c{t}ii ale teoriei punctului fix},
Ed. Dacia, Cluj-Napoca, 1979.

\bibitem{17}
Rus, I.A., Lungu, N.,
{\it Ulam stability of a nonlinear hyperbolic partial differential equation},
Carpathian J. Math., {\bf 24}(2008), no. 3, 403-408.

\bibitem{18}
Rus, I.A.,
{\it Remarks on Ulam stability of the operatorial equations},
Fixed Point Theory, {\bf 10}(2009), no. 2, 305-320.

\bibitem{19}
Rus, I.A.,
{\it Ulam stability of ordinary differential equations},
Stud. Univ. Babe\c{s}-Bolyai Math., {\bf 54}(2009), no. 4, 125-133.

\bibitem{20}
Ulam, S.M.,
{\it A Collection of Mathematical Problems},
Interscience, New York, 1960.

\end{thebibliography}



\end{document}