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Stochastic Processes

  • Class 60
  • Practice 0
  • Independent work 90
Total 150

Course title

Stochastic Processes

Lecture type

Elective

Course code

183365

Semester

6

ECTS

5

Lecturers and associates

Course objectives

Conditioning on a random variable; Conditioning on a sigma-field; Conditional expectation and distributions.
Sums of independent random variables. Stoping times Wald identities; Generating functions.
Random walks; Probability of ruin; Recurrent events.
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.
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.
Homogeneous Poisson processes; Memoryless property.
Poisson processes and uniform, exponential and binomial distributions; Nonhomogeneous Poisson processes; Mixed and Compound Poisson Processes; Poisson arrivals.
Midterm exam.
Basic concepts and examples; Transition probabilities and rates; Birth and death processes; Kolmogorov differential equations; Stationary state probabilities; Ergodic theorems.
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.
Basic concepts; The Erlang model, M/M/1 and M/M/c queue.
Loss systems: M/M systems; Waiting systems: M/G and G/M models; Network of queueing systems.
Introduction. Properties of Brownian motion; Multidimensional and conditional distributions; First passage times.
Transfornmations of the Brownian motion; Brownian motion with drift; White noise; Diffusion processes.
Final exam.

Required reading

(.), Neven Elezović, Stohastički procesi

Minimal learning outcomes

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