Academic year 2015/2016
- Course ID
- Teaching staff
- Prof. Laura Sacerdote
Prof. Goran Peskir
Prof. Federico Polito
- 1st year
- Teaching period
- Second semester
- D.M. 270 TAF B - Distinctive
- Course disciplinary sector (SSD)
- MAT/06 - probabilita' e statistica matematica
- Class Lecture
- Type of examination
- Written and oral
- Good knowledge of Probability and Analysis
Sommario del corso
The course is aimed at giving the students the skills to use diffusion processes to represent different realities of practical interest. The student should use the different techniques for carrying out the analysis of the models. The student will demonstrate both the ability of self-study of advanced topics, connected to the content of the course, and the ability to collaborate. Students should also use the software Mathematica to perform some assigned simulations.
Results of learning outcomes
At the end of the course, students will know several important methods to study stochastic models of applied interest. They will know some of the important classes of stochastic processes and will be able to study their main functional and features.
Lessons (48 hours, 6 CFU) are given in lecture rooms.
Learning assessment methods
During the course homeworks are assigned. Solution of these exercises is part of the final exam. Teamwork is allowed for this part of the work. Exam is oral. Students that do not make homeworks will solve exercises immediately before the oral exam.
The evaluation of homeworks is valid only for the Summer exam session. From September session students are required to solve exercises immediately before the oral exam.
Brownian Motion: Markov property, existence of the Brownian motion; maximum and first passage time distribution; arcosine law; iterated logarithm law; Reflected Brownian motion; Heat equation and Brownian motion; multidimensional Brownian motion.
Stationary Processes: mean square distance; autoregressive processes; ergodic theory and stationary processes; Gaussian processes
Diffusion Processes: differential equations associated with some functionals of the process; backward and forward equations; stationary measures; boundary classification for regular diffusion processes; conditioned diffusion processes; spectral representation of the transition density for a diffusion; diffusion processes and stochastic differential equations; jump-diffusion processes; first passage time problems for diffusion processes
Additionally, the course includes a module of 16 hours taught by visiting professor Goran Peskir (University of Manchester, UK) on the following topics:
Scale function, speed measure, killing, creation. Infinitesimal
generator. Additive functionals and time changes. Local time. The
Ito-Tanaka formula. A change-of-variable formula with local time on
curves and surfaces. Local time-space calculus.
Boundary classification for 1D diffusion processes: Exit, entrance,
regular (reflecting, sticky, elastic), natural boundaries.
MLS functionals and PIDE problems. Mayer functional and Dirichlet
problem. Lagrange functional and Dirichlet/Poisson problem.
Supremum functional and Neumann problem. MLS functionals and Cauchy
problem. Connection with the Kolmogorov backward equation.
Applications: (i) Optimal stopping (free-boundary problems); Optimal
stochastic control (the Hamilton-Jacobi-Bellman equation).
Suggested readings and bibliography
Karlin, Taylor. "A first Course in Stochastic Processes", Academic Press.
Karlin, Taylor. "A second Course in Stochastic Processes", Academic Press.
Mörters, Peres. "Brownian Motion", Cambridge University Press.
Kannan. "An introduction to stochastic processes", North Holland.
Peskir, Shiryaev. "Optimal stopping and free-boundary problems", Lectures in Mathematics, ETH Zurich, Birkhauser (2006).
Days Time Classroom