 Study

# Computing

## Stochastic Processes

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

### Course title

Stochastic Processes

Elective

183365

6

5

### 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.