Stochastics

Part A: Probability

• Probability axioms, conditional probability and independence and probability on countable sets.
• Construction of probability measures, in particular on R and R^d.
• Random variables and theory of integration.
• Independence random variables and sum of independent random variables.
• Convergence of random variables (almost sure, in probability, L^p) and laws of large numbers.
• Weak convergence and central limit theorems.
• Conditional expectation and martingales.

Part B: Stochastics

• Simulation of random variable and Monte-Carlo methods.
• Finite and countable state space Markov chains, stationary distribution, convergence, recurrence and transient behavior. Monte-Carlo Markov chains.
• Continuous time Markov chains, poisson processes and queueing processes.
• Martingales.
• Random walks and Brownian motion.

References (Probability)

• Cinlar, Probability and Stochastics
• Dudley, Real Analysis and Probability
• Durrett, Probability: Theory and Examples
• Jacod and Protter, Probability Essentials
• Resnick, Probability Path
• Rosenthal, A first look at rigorous probability
• Shiryaev, Probability

References (Stochastics)

• Lawler, Introduction to Stochastic Processes
• Resnick, Adventures in Stochastic processes
• Durrett, Essentials of Stochastic Processes
• Ross, Introduction to Probability Models
• Madras, Lectures on Monte-Carlo Methods
• Rubinstein and Kroese, Simulation and the Monte Carlo Method
• Levin, Peres and Wilmer, Markov chain and mixing times
• Hoel, Port and Stone, Introduction to Stochastic Processes
• Ross, Stochastic Processes
• Karlin and Taylor, A first course in Stochastic Processes