(in Polish) Wstęp do procesów stochastycznych

elective courses

(in Polish) Rachunek Prawdopodobieństwa I, Rachunek Prawdopodobieństwa II

Introduction of basic concepts of the theory of stochastic processes. Definition and properties of the Poisson process and the Wiener process. Preliminary information about Markov processes and continuous time martingales.

1. Stochastic processes - basic definitions. Processes with independent increments. Brownian motion and Poisson process, properties of their trajectories. Non-differentiability of trajectories of Brownian motion. Construction of the Brownian motion using the Haar functions, construction of the Poisson process.

2. Elements of the general theory of processes: finite-dimensional distributions, information about the consistency conditions and the theorem about the existence of a process (without proof). Checking the consistency conditions for selected processes.

3. Kolmogorov's theorem on the existence of a continuous modification of a process.

4. Markov processes - basic definitions. Markov property of the Brownian motion. Strong Markov property for Brownian motion. The reflection principle and the distribution of a supremum of a Brownian motion.

5. Gaussian processes. Properties of the covariance function. Brownian motion as a Gaussian process. Fractional Brownian motion. The Ornstein-Uhlenbeck process.

6. Stopping times, filtrations. Uniform integrability - as needed.

7. Martingales with a continuous time, theorems: Doob optional sampling theorem, Doob's inequality, theorem on convergence.

8. Martingales related to Brownian motion, exit times from the sphere / times to enter the sphere, recurrence and transcience.

1. R. Schilling, Lothar Partzsch, Brownian Motion: An Introduction to Stochastic Processes Walter de Gruyter, 2012.

I. Knowledge.

1. Knows the basic concepts of modern theory of stochastic processes.

2. Knows the definition and basic properties of the Poisson process and Brownian motion.

3. Has knowledge of the basics of Markov processes and martingales with a continuous time.

II. Skills.

1. Is able to investigate stochastic processes in terms of their properties.

2. Is able to use basic theorems about continuous time martingales.

3. Is able to verify and apply Markov property of a given processes.

III. Social competence.

Can present in a comprehensible language the basic concepts of the theory of stochastic processes and present their examples.