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\begin{document}
\title{Oral Qualifying Exam Syllabus}
\author{Eduardo Osorio}
\date{September 25, 2006}
\maketitle
\begin{enumerate}
  \item[{\bf 1.}] {\bf Stochastic Calculus and Financial Mathematics}
   \begin{enumerate}
    \item Probability Theory and Stochastic Processes
     \begin{enumerate}
      \item Continuity Theorem and Central Limit Theorem
      \item Filtrations and Stopping Times
      \item Martingales
       \begin{itemize}
        \item Doob's Martingale Inequality
        \item Doob's Optional Stopping Time Theorem
       \end{itemize}
     \end{enumerate}
    \item Stochastic Calculus
     \begin{enumerate}
      \item Brownian Motion
       \begin{itemize}
        \item Scaled Random Walks
        \item Construction of Brownian Motion
        \item Quadratic Variation
        \item Markov Property
        \item Reflection Principle
       \end{itemize}
      \item Construction of the Ito Integral
      \item Ito-Doeblin Formula
      \item Multivariable Stochastic Calculus
      \item Levy's Characterization Theorem
      \item Girsanov's Theorem
      \item Martingale Representation Theorem
      \item Stochastic Differential Equations
       \begin{itemize}
        \item Existence and Uniqueness Theorems
        \item Weak and Strong Solutions
       \end{itemize}
      \item The Markov Property
       \begin{itemize}
        \item The Markov Property for Ito Diffusions
        \item The Strong Markov Property for Ito Diffusions
       \end{itemize}

     \end{enumerate}
    \item Risk Neutral Pricing
     \begin{enumerate}
      \item Risk-Neutral Measure
      \item Fundamental Theorems Of Asset Pricing
     \end{enumerate}
    \item Connections with Partial Differential Equations
     \begin{enumerate}
      \item The Generator of an Ito Diffusion
      \item Kolmogorov's Backward Equation and The Resolvent
      \item The Feynman-Kac Formula
      \item Black-Scholes-Merton Equation
     \end{enumerate}
    \item Options
     \begin{enumerate}
      \item European Options
      \item Exotic Options
      \item American Derivative Securities
     \end{enumerate}
    \item Jump Processes
     \begin{enumerate}
      \item Poisson Process
      \item Compound Poisson Process
      \item Jump Processes and Their Integrals
      \item Stochastic Calculus for Jump Processes
      \item Change of Measure
      \item Pricing in a Jump Model
     \end{enumerate}
   \end{enumerate}
  \item[{\bf 2.}] {\bf Partial Differential Equations}
   \begin{enumerate}
    \item Heat Equation
     \begin{enumerate}
      \item Fundamental Solution, Cauchy Problem and Non-Homogeneous
            Case.
      \item Solutions for the Bounded Domain Case
      \item Mean-Value Formula
      \item Properties of Solutions
       \begin{itemize}
        \item Strong Maximum Principle, Uniqueness
        \item Regularity
        \item Local Estimates for Solutions
       \end{itemize}
      \item Energy Methods
       \begin{itemize}
        \item Uniqueness
        \item Backwards Uniqueness
       \end{itemize}
     \end{enumerate}
    \item Transform Methods
     \begin{enumerate}
      \item Fourier Transform
      \item Laplace Transform
     \end{enumerate}
    \item Second Order Parabolic Equations
     \begin{enumerate}
      \item Definitions
       \begin{itemize}
        \item Parabolic Equations
        \item Weak Solutions
       \end{itemize}
      \item Existence of Weak Solutions
       \begin{itemize}
        \item Galerkin Approximations
        \item Energy Estimates
        \item Existence and Uniqueness
       \end{itemize}
      \item Regularity
      \item Maximum Principles
       \begin{itemize}
        \item Weak Maximum Principle
        \item Harnack's Inequality
        \item Strong Maximum Principle
       \end{itemize}
     \end{enumerate}
   \end{enumerate}
\end{enumerate}
{\bf References:}

[1] Steven E. Shreve: Stochastic Calculus for Finance II, Springer,
2004.

[2] Bernt {\O}ksendal: Stochastic Differential Equations,
Springer-Verlag, 2003.

[3] Ioannis Karatzas, Steven E. Shreve: Brownian Motion and
Stochastic Calculus, Springer-Verlag, 1988.

[4] D.W. Stroock, S.R.S. Varadhan: Multidimensional Diffusion
Processes, Springer-Verlag, 1979.

[5] Lawrence C. Evans: Partial Differential Equations, AMS
Providence, 1998.

[6] Fritz, John: Partial Differential Equations, Springer-Verlag,
1982.
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