The course will provide the students with rigorous introduction to the theory of stochastic calculus and its applications in finance. It will start from stochastic processes in continuous time. Next, the Brownian motion process will be introduced and analyzed. The reflection principle will be used to derive important properties of the Brownian motion process. The concept of a continuous-time martingale will be introduced, and several properties of martingales proved. The concept of the stochastic integral will be introduced. The quadratic variation process will be defined. Stochastic differential equations will be discussed. Ito formula, integration by parts, and change of the order of integration will be presented. Integration with respect to a martingale will be covered as well. Further topics include the Girsanov theorem, Brownian martingale representation, and the Feynman-Kac equation. Geometric Brownian motion will be used to construct a continuous-time model of the evolution of a stock price, and all mathematical concepts and results will be illustrated with applications to finance. In particular, the BlackScholes option pricing theory will be developed and the Feynman-Kac partial differential equation will be specialized to finance.
- Créateur de cours: Boubakeur Labed