Stochastic Processes

mathematics

elective courses

Main topics: Gaussian processes; Poisson processes; The theory of Markov processes; Diffusion processes and their relation to stochastic differential equations; Weak and strong solutions of stochastic differential equations; Processes with independent increments.

1. Gaussian processes, stationary processes (1 lecture)

2. Poisson Process, generalized Poisson process (1 lecture)

3. Markov processes and operator semigroups, strong Markov property,

reflection principle, continuous time Markov chains. (5 lectures)

4. Diffusion processes and their relation to stochastic differential

equations (2 lectures)

5. Weak and strong solutions of stochastic differential equations.

Stroock-Varadhan Theorem on existence of a weak solution (sketch of the

proof), Yamada-Watanabe Theorem (with the proof) (4 lectures)

6. Processes with independent increments (Levy processes, stable

processes) (2 lectures)

1. I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag 1997.

2. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag 1999.

3. R. Schilling. L. Partzsch, Brownian Motion. An Introduction to Stochastic Processes. De Gruyter 2014.

4. A.D. Wentzell, Lectures on the theory of stochastic processes. PWN 1980

1. Knows the basic properties of the Wiener process and the Poisson process.

2. He knows the concept of the Markov process and is able to illustrate it with examples. He understands the concept of strong ownership

Markov and knows how to apply it.

3. Understands the basic relationships of Markov processes with the theory of semigroups.

4. Knows the concept of Markov chain with continuous time.

5. Understands the concept of the diffusion process, knows the Feynman-Kac formula and their relations with partial equations.

6. Can solve the Dirichlet problem using probabilistic methods.

Final grade will be based on students’ performance during the semester and final exam