Lecture Notes
Probability Theory I (in German)
- Foundations of Probability Theory
- Why Probability Theory?
- Modeling Random Experiments
- Rules of Calculus for Probability Measures
- Abstract Probability Spaces and Random Variables
- Examples of Distributions
- Real-Valued Random Variables
- Cumulative Distribution Functions of Real-Valued Random Variables
- Densities and the Radon-Nikodým Theorem
- Expectation, Variance, and Correlation
- Characteristic Functions and Higher-Order Moments
- Conditional Probability and Independence
- Conditional Probability
- Independence of Events and Random Variables
- Product Spaces and Joint Distributions
- Joint Distribution of Independent Random Variables
- Sums of Independent Random Variables and Convolutions
- Types of Convergence and Fundamental Limit Theorems
- The Weak Law of Large Numbers
- Convergence in Probability and Almost Sure Convergence
- The Strong Law of Large Numbers
- Convergence in Distribution
- The Central Limit Theorem
- Stochastic Processes
- Examples of Stochastic Processes
- Filtrations and Conditional Expectations
- Stopping Times and Hitting Times
- Brownian Motion
- Brownian Motion and Gaussian Processes
- Path Properties of Brownian Motion
- The Kolmogorov Consistency Theorem
- The Kolmogorov-Centsov Continuity Theorem
- Existence of Brownian Motion
- Martingales
- Definition and Examples
- Discrete Time Martingales
- Paths of Continuous Martingales
- Martingale Convergence
- Upcrossings and Submartingale Limits
- Uniform Integrability and Vitali’s Theorem
- The Martingale Convergence Theorem
- Continuous Time Martingales
- Stochastic Processes
- Examples of Stochastic Processes
- Filtrations, Stopping Times, and Hitting Times
- Brownian Motion
- Definition of Brownian Motion
- Gaussian Processes
- Path Properties of Brownian Motion
- The Kolmogorov Consistency Theorem
- The Kolmogorov-Centsov Continuity Theorem
- Existence of Brownian Motion
- Martingales
- Definition and Examples
- Discrete Time Martingales
- Paths of Continuous Martingales
- Martingale Convergence
- Upcrossings and Submartingale Limits
- The Martingale Convergence Theorem
- Continuous Time Martingales
- Cadlag Modifications
- Levy Processes
- Definition and Examples
- Properties of Levy Processes
- The Strong Markov Property
- Characterization of Brownian Motion and Poisson Process
- The Levy-Ito Decomposition
- Financial Markets, Derivatives, Arbitrage
- An Overview of Mathematical Finance
- Basic Financial Securities
- Absence of Arbitrage and Put-Call Parity
- Finite Financial Market Models
- Financial Markets and Stochastic Processes
- Filtrations and the Dynamic Resolution of Information
- Trading Strategies and the Wealth Process
- Risk Neutral Pricing and Fundamental Theorems
- Arbitrage and the First Fundamental Theorem
- Risk Neutral Pricing in Finite Market Models
- Completeness and the Second Fundamental Theorem
- Incomplete Markets and Superhedging
- Characterization of the Set of Arbitrage Free Prices
- Superhedging Duality and the Optional Decomposition
- American Options and Optimal Stopping
- Optimal Stopping Problems and the Snell Envelope
- Risk Neutral Pricing of American Options
- Stochastic Analysis of Brownian Motion
- Continuous Time Limit of the CRR Model
- Brownian Motion and Square-Intergable Martingales
- Stochastic Integration with Respect to Brownian Motion
- Ito Processes and Ito’s Formula
- Risk Neutral Pricing in the Black-Scholes Model
- The Black-Scholes Model
- Absence of Arbitrage and Efficient Strategies
- Option Pricing and the Black-Scholes Formula Revisited
- PDE Pricing and the Feynman-Kac Representation
- Semimartingales and Stochastic Integration
- Finite Variation Processes and Lebesgue-Stieltjes Integration
- Square-Integrable Martingales and Local Martingales
- Quadratic Variation of Local Martingales
- Semimartingales and Stochastic Integration
- Ito’s Formula for Semimartingales
- Option Pricing in Semimartingale Markets
- Financial Markets and Trading Strategies
- Arbitrage and Equivalent Martingale Measures
- Risk Neutral Pricing and Complete Markets
- Standard Brownian Market Models
- The Fundamental Theorem of Standard Markets
- Levy’s Characterization, Girsanov’s Theorem, and Arbitrage
- Martingale Representation and Completeness
- PDE Pricing in Diffusion Markets
- Stochastic Differential Equations
- Stochastic Volatility and Interest Rate Theory
- Implied Volatility and Volatility Surfaces
- Dupire’s Local Volatility Model
- An Overview of Stochastic Volatility Models
- Fixed Income Markets and Types of Interest Rates
- Short Rate Models
- Instantaneous Forward Rates in the HJM Framework
- Optimal Investment and Stochastic Control
- Investor Preferences and Utility Functions
- Formulation of Merton’s Optimal Investment Problem
- Stochastic Optimal Control and Dynamic Programming
- The Merton Problem with Power Utility
Stochastic Control and Optimization (in German)
- Stochastic Optimization Problems
- Optimal Stopping: American Options
- Classical Stochastic Control: Optimal Fishing
- Singular Control: Control of a Space Ship
- Impulse Control: Optimal Harvesting
- Stochastic Analysis
- Martingale Theory
- Stochastic Differential Equations
- Classical Stochastic Control
- Controlled Stochastic Differential Equations
- The Optimization Criterion
- Martingale Optimality and the Bellman Principle
- The Hamilton-Jacobi-Bellman Equation
- Application: Optimal Fishing
- Application: The Merton Problem
- Viscosity Solutions
- A Non-Differentiable Value Function
- Definition of Viscosity Solutions
- Alternative Definition in terms of Sub- and Superdifferentials
- Uniqueness and Ishii’s Lemma
- The Stochastic Perron’s Method
- What is the Stochastic Perron’s Method
- The Comparison Theorem
- Stochastic Supersolutions
- Stochastic Subsolutions
- The Viscosity Property of the Value Function