Probability Theory II

The lecture covers the following topics:

Stochastic Processes
Examples of stochastic processes, filtrations and conditional expectations, stopping times and hitting times.

Brownian Motion
Brownian motion and Gaussian processes, path properties of Brownian motion, the Kolmogorov consistency theorem, the Kolmogorov-Centsov continuity theorem, existence of Brownian motion.

Definition and examples, discrete time martingales, paths of continuous martingales.

Martingale Convergence
Upcrossings and submartingale limits, uniform integrability and Vitali’s theorem, the martingale convergence theorem, continuous time martingales.

Markov Chains
Markov property and transition probabilities, the strong Markov property, classification of states, invariant measures and stationary distributions, convergence to the stationary distribution.



PathsVsRV.m: Interpreting a stochastic process as a family of random variables vs. interpreting a stochastic process as a path-valued random variable.

WhiteNoise.m: Simulation of a white noise process constructed from standard normal random variables.

RandomWalk.m: Simulation of a classical random walk.

MarkovChain.m: Simulation of a Markov chain switching between two states with a given transition probability.

RenewalProcess.m: Simulation of a renewal process constructed from exponential random variables.

HittingTime.m: Simulation of the hitting time of a closed set by a continuous process.

HittingTimeContinuousOpen.m: Simulation of the hitting time of an open set by a continuous process highlighting the need for the right continuity of the filtration.

StoppedProcess.m: Simulation of a stochastic process stopped at a stopping time.

BrownianMotion.m: Simulation of a Brownian motion.

BrownianBridge.m: Simulation of a Brownian motion and the corresponding Brownian bridge.

OUProcess.m: Simulation of an Ornstein-Uhlenbeck process for various choices of the mean reversion speed and volatility.

FractionalBrownianMotion.m: Simulation of a fractional Brownian motion for various choices of the Hurst index. Requires the function fbm1d.

Martingle.m: Brownian motion as an example of a martingale.

MartingaleCounterexample.m: Brownian bridge as an example of a process which is not a martingale.

OptionalStopping.m: The Optional Stopping Theorem in action. By exiting a fair game at a bounded stopping time, you cannot make gains with positive probability without running the risk of losing money as well.

UpcrossingBrownianMotion.m: Upcrossings of a Brownian motion.

UpcrossingFractionalBrownianMotion.m: Upcrossings of a fractional Brownian motion. Requires the function fbm1d.