Random walks 
Definition, Markovianity, temporal and spatial invariance, random walks with reflecting and absorbing barriers, reflection principle, ballot theorem, distribution of the maximum, hitting time theorem, first and second arc sine law, random walks and generating functions, short introduction on Black–Scholes model.    

Brownian motion 
Definition and existence, Brownian motion as a limit of a simple random walk, path properties of Brownian motion, Brownian motion as a strong Markov process, transience and recurrence. 

Branching processes 
Definition, expectation and variance of the population size, geometric branching, probability of extinction of the population.   
   

Markov chains  
Definition, homogeneous Markov chains, transition matrix, examples of Markov chains, classification of states, classification of chains, stationary distribution and limit theorem, chains with finitely many states, short introduction on Monte Carlo method, 
MCMC and search engine algorithms. 

Poisson processes 
Definition and main properties.