### 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.

**Martingales**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.

#### Resources

- Lecture Notes: Probability Theory II

#### 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**.