General Introduction to Stochastic Processes

This course, "General Introduction to Stochastic Processes," provides a comprehensive overview of fundamental concepts and applications in fields such as finance, insurance, and queuing theory. It covers basic notions of continuous time stochastic processes, including continuous time martingales and stopping times, which are essential for optimal decision-making. Students will explore renewal processes that model event timing, as well as the Poisson process, characterized by its memoryless property for modeling random occurrences over fixed intervals. The course explores compound Poisson processes, addressing both the time and size of events pertinent to risk management. Furthermore, it analyzes covariation and quadratic variation, essential for comprehending process fluctuations, and concludes with Itô's formula for jump diffusion processes.

Brownian Motion and Stochastic Calculus

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.