# Stochastic processes 2

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M5
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Brownian motion:
Basic properties, existence, path properties, natural filtration, first hitting time, Markov properties, strong Markov property, reflection principle, associated processes (running supremum process, Brownian bridge etc.),
Continuous time martingales:
Filtrations, stopping times, stopping theorems,
uniform integrability, maximal inequalities, convergence of martingales.
Stochastic integral:
Stochastic integral wrt Brownian motion,
Itô isometry, continuous semimartingales, local martingales, quadratic variation and covariation, stochastic integral wrt continuous semimartingales, Itô's formula, Girsanov Theorem, representation of martingales.

S. Resnick: Adventures in Stochastic Processes, Birkhäuser Boston, 2002.
I. Karatzas, S. E. Shreve: Brownian Motion and Stochastic Calculus, 2nd Edition, Springer, 2005.
M. Yor, D. Revuz: Continuous Martingales and Stochastic Calculus, 2nd Edition, Springer, 2004
J. M. Steele: Stochastic Calculus and Financial Applications, Springer,
New York, 2001.

Objectives and competences

This course is an introduction to the theory of stochastic processes in continuous time with continuous sample paths. It rigorously treats Brownian motion as a basic example and building block, introduces martingales in continuous time, stochastic calculus and Ito's formula.

Intended learning outcomes

Knowledge and understanding:
Mathematical tools for rigorous treatment and applications of stochastic processes.
Application:
Basic tools for modelling in many branches of
Mathematics and its applications.
Reflection:
The contents of the course help in retrospect to deepen the understanding of the concepts of probability, dependence and time.
Transferable skills:
The skills acquired are transferable to other areas of mathematical modelling, in particular it is immediately applicable to financial models.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (examination, oral, coursework, project):
written exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Mihael Perman:
PERMAN, Mihael, PITMAN, Jim, YOR, Marc. Size-biased sampling of Poisson processes and excursions. Probability theory and related fields, ISSN 0178-8051, 1992, 92, no. 1, str. 21-39. [COBISS-SI-ID 12236377]
PERMAN, Mihael, WELLNER, Jon A. On the distribution of Brownian areas. Annals of applied probability, ISSN 1050-5164, 1996, let. 6, št. 4, str. 1091-1111. [COBISS-SI-ID 7101017]
Janez Bernik:
BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Realizing irreducible semigroups and real algebras of compact operators. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2008, vol. 348, no. 2, str. 692-707. [COBISS-SI-ID 14899289]
BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Positivity and matrix semigroups. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2011, vol. 434, iss. 3, str. 801-812. [COBISS-SI-ID 15745625]
BERNIK, Janez, MARCOUX, Laurent W., RADJAVI, Heydar. Spectral conditions and band reducibility of operators. Journal of the London Mathematical Society, ISSN 0024-6107, 2012, vol. 86, no. 1, str. 214-234. [COBISS-SI-ID 16357721]