Probability and Stochastic Processes

Recommended Prerequisites

Basic course in probability with notions of stochastic processes

Teaching Methods

The classes are essentially expository. Problems are proposed extending or complementing the subjects studied.

Learning Outcomes

The main goal of this curricular unit is to give students the ability to deal with dependent random variables, a solid background on the fundations of stochastic processes and convergence in distribution with a special emphasis on functional results.

Work Placement(s)

No

Syllabus

Probabilitys: condicioning and condicional expectation, discrete time martingales and limit theorems.

Fundamentals of stochastic processes: infinite dimensional product spaces, Kolmogorov's existence Theorem.

Probability in metric spaces, the special cases of C[0,1], Lp and Hilbert. Central Limite Theorems (CLT) and invariance principles, Brownian movement and Brownian bridge. Dependent variables, extensions of CLT and invariance principles.

Topics to be chosen according to time availability: Markov chains and dependence properties, Markov processes, diffusion processes, continuous time martingales.

Assessment Methods

Assessment
Exam (50-60%) and Midterm exam (40-50%). Assessement might also include some contribution from home problem solving.: 100.0%

Bibliography

P. Billinglsey, Probability and Measure, Wiley,1986.

P. Billingsley, Convergence of probability measures, Wiley, 1999.

D. Williams, Probability with Martingales, Cambridge Press, 1991.

S. Varadhan, Probability Theory, Amer. Math. Soc., 2001.

S. Varadhan, Stochastic Processe, Amer. Math. Soc., 2007.