- This is a core module for the MSc in Mathematical Finance.
- Not available to undergraduate students.
- PhD students interested in taking the module should consult the lecturer.
30 hours of lectures and 10 hours of tutorials
1 Conditional expectations
- Elementary conditional expectations
- Measure-theoretic conditional expectations
- Properties of conditional expectations
2 Martingale Theory
- Stochastic processes and filtrations
- Martingales, submartingales, and supermartingales
- Discrete stochastic integral
- Stopping times and stopping theorem
- Martingale convergence theorems
- Applications to Finance (option pricing in complete markets)
3 Markov Processes
- Markov processes and Markov property
- Strong Markov property
4) Brownian motion and continuous local martingales
- Definition and fundamental properties of Brownian
- Quadratic variation
- Continuous local martingales and semimartingales
5) Stochastic calculus
Integration with respect to local martingales
Finite variation processes and Lebesgue-Stieljes integration
Integration with respect to semimartingales
Lévy’s characterisation of Brownian motion
Stochastic exponentials and Novikov’s condition
- Girsanov’s theorem
- Ito representation theorem
- Feynman-Kac formula
- Applications to Finance (Black Scholes model)
6) Stochastic differential equations
- Strong solutions and Lipschitz-theory
- Examples (OU-processes, CIR processes, etc.)
- 2-hour exam in January (80%), class tests (20%).
- Klenke, Probability theory, Springer, London, 2008.
- D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd ed., Springer,
- I. Karatzas and S. E. Shreve, Brownian motion and Stochastic Calculus, 2nd ed., Springer,
New York, 1991
Examination Period: January