# MAI0125

Stochastic Processes/

Stokastiska processer

The course will be given in November 2014.

**Number of credits:** 8 hp

**Examiner:** Mikhail Lifshits

**Course literature:** Mikhail Lifshits, Stochastic Processes by Example, World Scientific, 2014.

**Course contents:** This doctoral/master course covers a rigorous mathematical framework basics of stochastic processes and main models including Gaussian processes, stationary processes, processes with independent increments, random measures and related stochastic integrals. Application examples are also included.

1. Basic notions for stochastic processes. Definition, finite-dimensional distributions, Gaussian processes, distributions of stochastic processes, separability, stochastic continuity. Main types of stochastic processes.

2. Gaussian processes. Expectation and covariance as elements determining Gaussian process. Examples.

3. Random measures and related stochastic integrals. Gaussian white noise. Poisson random measure. Isometric property of integrals. Integral representations of some processes.

4. Stationary Processes. Definition, spectral representation, differentiability criteria, law of large numbers. Processes with stationary increments.

5. Compound Poisson random variables. Definition and representation as a random sum. Limits of compound Poisson distributions. Infinite divisibility. Levy-Khinchin representation. Stable distributions.

6. Levy processes. Integral representation via Poisson measure. Stable Levy processes.

7. Convergence of Stochastic Processes (spaces of trajectories C and D, functional limit theorems, examples of continuous functionals, convergence in C, convergence in D). Invariance principle, Donsker theorem for empirical distribution function.

8. Application example: "telecom" service system. Setting, parametrization, main characteristics and limit theorems.

**Organisation:** 10 lectures (2*45 min).

**Examination:** Weekly home assignments.

**Prerequisites:** Probability theory at the graduate level (e.g., MAI0074 Probability theory part 1).

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Last updated: 2022-11-14