### Stochastic Processes

#### Learning Outcomes

1. Understand the basic principles of stochastic processes
2. Learn to distinguish between stochastic processes according to their properties.
3. Understand the characteristic lack of memory in different cases
4. To interpret the behavior of the process in accordance with the theoretical laws.
5. Determine the probability of prominent events related to stochastic processes.
6. Learn to model simple problems using stochastic techniques.
7. Apply stochastic techniques in the analysis of various systems.

#### Forms of Teaching

Lectures

Partial e-learning

Independent assignments

#### Week by Week Schedule

1. Conditioning on a random variable; Conditioning on a sigma-field; Conditional expectation and distributions.
2. Sums of independent random variables; Stoping times Wald identities; Generating functions.
3. Random walks; Probability of ruin; Recurrent events.
4. Foundations and examples; construction of Markov chains; Transition probabilities and the Chapman-Kolmogorov equation; Stopping times and strong Markov property; Absorbing states; transient and recurrent states; Branching processes.
5. Limit theorems and stationary distributions; State classifications; Ergodic theorems; Finite-dimensional distributions of processes; Moments; correlation and covariation functions; Classes of processes: Markov, homogenous Markov, weak/strong stationary, independent increment processes; Transition and density matrix and Chapman-Kolmogorov equation for Markov processes.
6. Homogeneous Poisson processes; Memoryless property.
7. Poisson processes and uniform, exponential and binomial distributions; Nonhomogeneous Poisson processes; Mixed and Compound Poisson Processes; Poisson arrivals.
8. Midterm exam.
9. Basic concepts and examples; Transition probabilities and rates; Birth and death processes; Kolmogorov differential equations; Stationary state probabilities; Ergodic theorems.
10. Renewal Functions; Excess life, current life and total life; Strong laws of large numbers; Recurrence times; Terminating renewal processes; Stationary renewal processes; Alternating Renewal Processes.
11. Basic concepts; The Erlang model, M/M/1 and M/M/c queue.
12. Loss systems: M/M systems; Waiting systems: M/G and G/M models; Network of queueing systems.
13. Introduction; Properties of Brownian motion; Multidimensional and conditional distributions; First passage times.
14. Transfornmations of the Brownian motion; Brownian motion with drift; White noise; Diffusion processes.
15. Final exam.

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#### Literature

(.), Neven Elezović, Stohastički procesi,

#### General

ID 183365
Summer semester
5 ECTS
L3 English Level
L3 e-Learning
60 Lectures
0 Exercises
0 Laboratory exercises
0 Project laboratory