### 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.

#### Resources

- Lecture Notes: Stochastic Processes
- Slides: Multivariate Normal Distribution (in German)
- Slides: Conditional Expectation (in German)
- Slides: Uniform Integrability (in German)
- Video: Pollen Grains in Water — Brownian Motion

#### 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.