

%Jin Wang syllabus
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\centerline{\Large Oral Qualifying Exam Syllabus} \vspace {10pt}
\centerline{\large Jin Wang}

\vspace{10pt}\large Committee: Ren-Raw Chen, Paul Feehan, Zheng-chao
Han, Daniel Ocone

\vspace{10pt}
\section{Major topic: Math Finance}
\vskip-15pt

\subsection{Brownian Motion}
\begin{itemize}
%{\setlength{\parsep}{5pt} \setlength{\itemsep}{1pt}\slshape}
\item Scaled random walk
\item Brownian motion: definition, flitration,
martingale property
\item First-order variation, quadratic variation
%\item First passage time distribution
\item Reflection principle
\end{itemize}

\vspace{-15pt}

\subsection{Stochastic Calculus}
\begin{itemize}
\item Ito's integral: construction, properties
\item Ito formula
\item Black-Scholes-Merton equation
\item Put-call parity
%\item Brownian bridge
\end{itemize}

\vspace*{-15pt}

\subsection{Risk-Neutral Pricing}
\begin{itemize}
\item Girsanov's theorm
\item Risk neutral measure, pricing under the risk-neutral measure
\item Martingale representation theorem
\item Fundermental theorem of asset pricing
%\item Forwards and futures
\end{itemize}

\vspace*{-15 pt}

\subsection{Connecions with Partial Differential Equations}
\begin{itemize}
\item The Markov property
\item Feynman-Kac theorems
\item Derivation of the Black-Scholes PDE
%\item Interest rate modesls

\end{itemize}

\vspace*{-15pt}
\subsection{Exotic Options}
\begin{itemize}
%\item Maximum of Brownian motion with drift
\item Knock-out barrir options
\item Lookback option
\item Asian option
\end{itemize}


%\vspace*{-15pt}
%\subsection{American Derivative Securities}
%\begin{itemize}
%\item Stopping times
%\item Perpetual American put
%\item Finite-expiration American put
%\item American call
%\end{itemize}
%\vspace*{-15pt}
%\subsection{Change of Numeraire}
%\begin{itemize}
%\item Numeraire
%\item Forward mearures
%\end{itemize}
%\vspace*{-15pt}

\subsection{Volatility Models}
\begin{itemize}
\item Implied volatility, volatility smile
\item Local volatility
\item Relating implied volatility with local volatility
\item Heston model
\end{itemize}

\subsection{Jump Processes}
\begin{itemize}
\item Poisson process, Poisson random measure
\item Compound Poisson process, jump measure of compound Poisson
process
\item Levy process, Levy measure
\item Levy-Ito decomposition
\item Levy-Khinchin representation
\end{itemize}

% PDE
\section{Minor topic: Partial Differential Equations}
\vspace{-10pt}


\subsection{Heat Equation}
\begin{itemize}
\item Derivation of the fundamental solution
\item Maximum principle, uniqueness
\item Energy methods
\end{itemize}

\vspace{-15pt}
\subsection{Second Order Elliptic Equations}
\begin{itemize}
\item Definition of weak solution
\item Existence of weak solutions via Lax-Milgram theorem
\item Weak maximum priciple
\item Hopf Lemma
\item Strong maximum principle

\end{itemize}

\vspace{-15pt}
\subsection{Second Order Parabolic Equations}
\begin{itemize}
\item Definition of the weak solutions
\item Existence of weak solutions, Galerkin approximations
\item Weak maximum principle
\item Harnack's inequality
\item Strong maximum principle
\end{itemize}

\vspace{-15pt}
\begin{thebibliography}{9}
\bibitem%
{Shreve}
Steven E. Shreve
{\em Stochastic Calculus for Finance II},
Springer, 2003.

\bibitem%
{Gatheral} {Jim Gatheral} {\em Case Studies in Financial Modelling
course notes}, Courant Institute, 2005

\bibitem%
{Cont} {Rama Cont} {\em Financial Modelling with Jump Processes},
Chapman \& Hall/CRC, 2003

\bibitem%
{Evans} L.C. Evans, {\em Partial Differential Equations}, AMS
Providence, 1998.


\end{thebibliography}

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