Christoph Belak

Stochastic Processes

The lecture covers the following topics:

Stochastic processes
Examples, filtrations, stopping times, hitting times.

Brownian motion
Definition, Gaussian processes, path properties, Kolmogorov’s consistency theorem, Kolmogorov-Centsov continuity theorem.

Martingales
Definition and examples, discrete time martingale theory, path properties of continuous martingales.

Martingale convergence
Upcrossings and submartingale limits, martingale convergence, continuous time martingales, cadlag modifications.

Levy processes
Definition and examples, properties, strong Markov property, characterization of Brownian motion and Poisson process, Levy-Ito decomposition.

Simulations

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.

PoissonProcess.m: Simulation of a Poisson process.

CompoundPoisson.m: Simulation of a compound Poisson process and underlying Poisson process.