The course aims to give an introduction to the theory of stochastic processes with special emphasis on applications and examples. On successful completion of this module the students should become familiar with some of the most known classes of stochastic processes (such as martingales, markov processes, diffusion processes) and to acquire both the mathematical tools and intuition for being able to describe systems with randomness evolving in time in terms of a probability model and to analyze it charcterizing some of its properties.
Stochastic Processes: definition, finite dimensional distributions, stationarity, sample path spaces, construction, examples.
Filtrations, stopping times, conditional expectation.
Markov processes: definition, main properties and examples. Birth and death processes.
Poisson process with applications on queueing models.
Martingales: definition, main properties and examples. Branching processes.
Brownian motion: definition, construction and main properties.
Brownian Bridge, Geometric Brownian Motion, Ornstein-Uhlenbeck process.
Ito integral and stochastic differential equations. Applications and examples.
P. Billingsley, Probability and measure. John Wiley and Sons.
G. Grimmett, D. Stirzaker, Probability and random Processes. Oxford University Press.
B. Oksendal, Stochastic Differential Equations. Springer-Verlag.