Lecture Notes

Stochastic Processes
  1. Stochastic Processes
    1. Examples of Stochastic Processes
    2. Filtrations, Stopping Times, and Hitting Times
  2. Brownian Motion
    1. Definition of Brownian Motion
    2. Gaussian Processes
    3. Path Properties of Brownian Motion
    4. The Kolmogorov Consistency Theorem
    5. The Kolmogorov-Centsov Continuity Theorem
    6. Existence of Brownian Motion
  3. Martingales
    1. Definition and Examples
    2. Discrete Time Martingales
    3. Paths of Continuous Martingales
  4. Martingale Convergence
    1. Upcrossings and Submartingale Limits
    2. The Martingale Convergence Theorem
    3. Continuous Time Martingales
    4. Cadlag Modifications
  5. Levy Processes
    1. Definition and Examples
    2. Properties of Levy Processes
    3. The Strong Markov Property
    4. Characterization of Brownian Motion and Poisson Process
    5. The Levy-Ito Decomposition
  1. Financial Markets, Derivatives, Arbitrage
    1. An Overview of Mathematical Finance
    2. Basic Financial Securities
    3. Absence of Arbitrage and Put-Call Parity
  2. Finite Financial Market Models
    1. Financial Markets and Stochastic Processes
    2. Filtrations and the Dynamic Resolution of Information
    3. Trading Strategies and the Wealth Process
  3. Risk Neutral Pricing and Fundamental Theorems
    1. Arbitrage and the First Fundamental Theorem
    2. Risk Neutral Pricing in Finite Market Models
    3. Completeness and the Second Fundamental Theorem
  4. Incomplete Markets and Superhedging
    1. Characterization of the Set of Arbitrage Free Prices
    2. Superhedging Duality and the Optional Decomposition
  5. American Options and Optimal Stopping
    1. Optimal Stopping Problems and the Snell Envelope
    2. Risk Neutral Pricing of American Options
  6. Stochastic Analysis of Brownian Motion
    1. Continuous Time Limit of the CRR Model
    2. Brownian Motion and Square-Intergable Martingales
    3. Stochastic Integration with Respect to Brownian Motion
    4. Ito Processes and Ito’s Formula
  7. Risk Neutral Pricing in the Black-Scholes Model
    1. The Black-Scholes Model
    2. Absence of Arbitrage and Efficient Strategies
    3. Option Pricing and the Black-Scholes Formula Revisited
    4. PDE Pricing and the Feynman-Kac Representation
  1. Semimartingales and Stochastic Integration
    1. Finite Variation Processes and Lebesgue-Stieltjes Integration
    2. Square-Integrable Martingales and Local Martingales
    3. Quadratic Variation of Local Martingales
    4. Semimartingales and Stochastic Integration
    5. Ito’s Formula for Semimartingales
  2. Option Pricing in Semimartingale Markets
    1. Financial Markets and Trading Strategies
    2. Arbitrage and Equivalent Martingale Measures
    3. Risk Neutral Pricing and Complete Markets
  3. Standard Brownian Market Models
    1. The Fundamental Theorem of Standard Markets
    2. Levy’s Characterization, Girsanov’s Theorem, and Arbitrage
    3. Martingale Representation and Completeness
    4. PDE Pricing in Diffusion Markets
    5. Stochastic Differential Equations
  4. Stochastic Volatility and Interest Rate Theory
    1. Implied Volatility and Volatility Surfaces
    2. Dupire’s Local Volatility Model
    3. An Overview of Stochastic Volatility Models
    4. Fixed Income Markets and Types of Interest Rates
    5. Short Rate Models
    6. Instantaneous Forward Rates in the HJM Framework
  5.  Optimal Investment and Stochastic Control
    1. Investor Preferences and Utility Functions
    2. Formulation of Merton’s Optimal Investment Problem
    3. Stochastic Optimal Control and Dynamic Programming
    4. The Merton Problem with Power Utility
  1. Stochastic Optimization Problems
    1. Optimal Stopping: American Options
    2. Classical Stochastic Control: Optimal Fishing
    3. Singular Control: Control of a Space Ship
    4. Impulse Control: Optimal Harvesting
  2. Stochastic Analysis
    1. Martingale Theory
    2. Stochastic Differential Equations
  3. Classical Stochastic Control
    1. Controlled Stochastic Differential Equations
    2. The Optimization Criterion
    3. Martingale Optimality and the Bellman Principle
    4. The Hamilton-Jacobi-Bellman Equation
    5. Application: Optimal Fishing
    6. Application: The Merton Problem
  4. Viscosity Solutions
    1. A Non-Differentiable Value Function
    2. Definition of Viscosity Solutions
    3. Alternative Definition in terms of Sub- and Superdifferentials
    4. Uniqueness and Ishii’s Lemma
  5. The Stochastic Perron’s Method
    1. What is the Stochastic Perron’s Method
    2. The Comparison Theorem
    3. Stochastic Supersolutions
    4. Stochastic Subsolutions
    5. The Viscosity Property of the Value Function