Modern Stochastic Processes: Overview and Applications

MAM intensive research course for PhD students and researchers.
MAM research frontier lecture series.

Modern Stochastic Processes:  Overview and Applications

Research environment MAM, UKK, Mälardalen University  

MAM research frontier lectures series in Mathematics and Applied Mathematics

PhD course given by


Professor em Dmitrii Silvestrov

Mälardalen University/Stockholm University

Phones: +46-(0)8-164551, +46-(0)21-101667 (m)


Web page:


Course content

This doctoral course presents an overview of basic models of stochastic processes including Stationary processes, Processes with independent increments, Markov Chains, Renewal models, Gaussian processes, Diffusion processes, Stochastic integrals, Stochastic differential equations as well as basic applications of the above models.                                                                                                                       

Course will include seven (two hours each) lectures delivered in September- December 2019.


Lecture 1: Introduction to the Course and Overview of Stochastic Processes.

Lecture 1 notes (pdf 449 kB)  

Phase spaces, Time models, Finite-dimensional distributions, Spaces of trajectories,

Measures generated by stochastic processes, Separability, Stochastic continuity, Measurability.


Lecture 2:  Stationary Processes

Definitions, Spectral representations for stationary processes, Forecasting, Ergodic behaviour, Applications.

Lecture_2_notes (pdf 254 kB)


Lecture 3: Processes with Independent Increments

Wiener process, functionals of Wiener process, reflection principle, Girsanov theorem, Poisson and compound Poisson processes, Lévy processes, Lévy-Khinchine representation, Applications


Lecture 4:  Markov Chains

Discrete and continuous time Markov chains with discrete phase space, Random walks, Kolmogorov systems of equations, Ergodic behavior, Markov jump processes, Queuing models, MCMC, Applications.


Lecture 5: Renewal Models

Regenerative processes, Renewal models, Counting processes, Renewal theorem, Risk processes, Semi-Markov processes, Coupling, Applications.


Lecture 6: Gaussian and Diffusion processes

Definition, Kolmogorov equations, Gaussian processes, Ornstein-Uhlenbeck process, Brownian bridge, Fractional Brownian motion, Geometrical Brownian process, Black-Scholes formula, Applications.


Lecture 7: Stochastic Integrals and Stochastic Differential Equations

Definition, Properties of stochastic integrals, Stochastic differentials, Itô calculus,

SDE, Solutions, Linear SDE, SDE and diffusion processes, Applications.


Suplementary Materials

[1] Ross, S. Introduction to Probability Models. 12th Edition. Academic Press, 842 pp., 2019.

[2] Grimmett, G., Stirzaker, D.  Probability and Random Processes, Third Edition, Oxford University Press, 608 pp. 2001.

[3] Kovalenko, I.N., Kuznetsov, N., Yu. Shurenkov, V.M. Models of Random Processes: A Handbook for Mathematicians and Engineers, CRC-Press, 448 pp., 1996.


[4] Grimmett, G., Stirzaker, D. One Thousand Exercises in Probability, Second Edition, Oxford University Press, 488 pp., 2001.

[5] Dorogovtsev, A.Ya, Silvestrov, D.S., Skorokhod, A.V., Yadrenko, M.I. Probability Theory: Collection of Problems, AMC, 347 pp., 1997.

[6] Gusak, D., Kukush, A., Kulik, A., Mishura, Yu., Pilipenko, A. Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory (Problem Books in Mathematics), Springer, 376 pp. 2009.


Course Schedule

Lecture 1: 

September: 13 (Friday), 13:15-15:00, Room U3-083 (Hilbert room, UKK, Mälardalen University, Västerås)

Lectures 2:

November: 4, (Monday), 10:15-12:00, Room U3-083 (Hilbert room, UKK, Mälardalen University, Västerås)

Lectures 3:

November 5 (Tuesday), 10:15-12:00, Room U3-104 (Turing room, UKK, Mälardalen University, Västerås)

Lecture 4:

November 8 (Friday), 13:15-15:00, Room U3-083 (Hilbert room, UKK, Mälardalen University, Västerås)

Lecture 5, Lecture 6: 

November: 18, (Monday), 13:15-16:00, Room U3-083 (Hilbert room, UKK, Mälardalen University, Västerås)

Lecture 6, Lecture 7:

November: 19 (Tuesday), 13:15-16:00, Room U3-083 (Hilbert room, UKK, Mälardalen University, Västerås)


Contact (about PhD course examination and other practical questions connected to the lecturers):

Professor Sergei Silvestrov,, 021-101524