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ST908 Stochastic Calculus for Finance


  • 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

  • Itô’s formula

  • 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%).

Illustrative Bibliography:

  • Klenke, Probability theory, Springer, London, 2008.
  • D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd ed., Springer,
    Berlin, 1999.
  • I. Karatzas and S. E. Shreve, Brownian motion and Stochastic Calculus, 2nd ed., Springer,
    New York, 1991

Examination Period: January