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\large
\begin{center} \ \textbf{Oral Qualifying Exam Syllabus}
\\ \
\\
\large Ming Shi \\ April 21, 2006 \end{center}

\normalsize \begin{center}Committee: Richard Falk, Paul Feehan,
Daniel Ocone and Ted Petrie \end{center}

\renewcommand{\labelenumi}{\Roman{enumi}.}
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\begin{enumerate}  \large \item \textbf{ Major Topic: Stochastic Calculus and Financial Application}

\begin{enumerate}

\item \textbf{Brownian Motion and Martingale:}

\begin{enumerate}
\item Brownian Motion \item Scaled Symmetric Random Walk \item
Quadratic Variation \item Markov Property  \item Reflection
Principle \item First Passage Time Distribution
\end{enumerate}

\item \textbf{Stochastic Calculus:}
\begin{enumerate}
\item It\^{o}'s Integral \item It\^{o}-Doeblin Formula \item
Derivation of Black-Scholes-Merton Equation \item Put-Call parity
\item Multi-variable Stochastic Calculus
\end{enumerate}

\item \textbf{Risk-Neutral Pricing:}
\begin{enumerate}
\item Change of probability measure \item Girsanov's Theorem \item
Martingale Representation Theorem \item Pricing under Risk Neutral
Measure \item Derivation of Black-Scholes-Merton Formula \item
First and Second Fundamental Theorem of Asset Pricing
\end{enumerate}

\item \textbf{Connections with Partial Differential Equations:}
\begin{enumerate}
\item Stochastic Differential Equations \item The Markov Property
of Solutions \item Feynman-Kac Theorem
\end{enumerate}

\item \textbf{Exotic Options:}
\begin{enumerate}
\item Knock-out Barrier Options \item Lookback Options \item Asian
Options

\end{enumerate}

\item \textbf{Jump Processes:}
\begin{enumerate}
\item Poisson Process and Compensated Poisson Process \item
Compound Poisson Process \item Jump measure, Poisson random
measure \item L\'{e}vy process \item L\'{e}vy-It\^{o}
Decomposition \item L\'{e}vy-Khinchin formula
\end{enumerate}


\end{enumerate}

\bigskip
  \large \item \textbf{Minor Topic: Numerical Analysis}

\begin{enumerate}
\item \textbf{Polynomial Approximation:}

\begin{enumerate}
\item Lagrange interpolation \item Newton interpolation \item
Piecewise polynomial approximation \item Some error results
\end{enumerate}


\item \textbf{Numerical Quadrature}
\begin{enumerate}
\item Midpoint, Trapezoidal and Simpson's rules \item Derivation
and error formulas \item Basic results of Gaussian quadrature
formulas
\end{enumerate}

\item \textbf{Numerical methods for ordinary differential
equations}
\begin{enumerate}
\item Derivation and error estimates for one-step methods (e.g.,
Euler's method) \item Linear Multistep Methods (examples of explicit and
implicit methods) \item Consistency, stability, and convergence of multi-step methods
\end{enumerate}


\item \textbf{Finite Difference methods for partial differential
equations}
\begin{enumerate}
\item Laplace's equation: 5 point difference scheme \item Discrete
maximum principle, Existence and uniqueness of the approximate
solution \item Stability and error estimates
\end{enumerate}

\item \textbf{Finite Element methods for elliptic partial
differential equations}
\begin{enumerate}
\item Standard variational formulation of second order elliptic
boundary value problems \item Ritz-Galerkin approximation schemes
and simple error analysis \item Construction of finite element
subspace: dimension of the spaces, basis functions, degrees of
freedom, barycentric coordinates \item Affine families of finite
elements \item Error estimates for piecewise linear interpolation

\end{enumerate}
\end{enumerate}
\end{enumerate}

\begin{thebibliography}{}

\bibitem[1]{SHR}
S.E. Shreve \emph{Stochastic Calculus for Finance II}, Springer,
2003.

\bibitem[2]{APP}
D. Applebaum, \emph{L\'{e}vy processes and stochastic calculus}
Cambridge University Press, 2004.

\bibitem[3]{Kar}
R. Cont, P. Tankov, \emph{Financial modelling with jump processes}
Chapman \& Hall/CRC, c2004.

\bibitem[4]{Dur}
R. Durrett, \emph{Probability: Theory and Examples}, 3rd
Ed.,Duxbury Press.

\bibitem[5]{Falk1}
R. Falk, \emph{Lecture notes of Numerical Solution of PDE
(Math575), Spring 2006}

\bibitem[6]{Falk2}
R. Falk, \emph{Lecture notes of Numerical Analysis (Math573), Fall
2005}

\end{thebibliography}

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