## Stochastic Processes – Winter Term 2017/18

On this webpage, you find all information on my course on **Stochastic Processes** held in the winter term 2017/18 at the University of Trier.

**General Information**

The **lectures** take place on

**Tuesday, 10:15 – 11:45 in HS 5.****Thursday, 12:15 – 13:45 in HS 3.**

The **exercise classes** take place on

**Monday, 10:15 – 11:45 in E 45.**

Exercise sheets are generally uploaded every Thursday and are due on Thursday at 13:45 of the following week. You can submit your solutions in groups of 2-3 in the post box E14 on the ground floor of building E.

**Oral Exams**

To qualify for the final exam, you are required to attain at least 50% of the points on the exercise sheets and be able to demonstrate in the exercise classes that you have fully understood the solutions that your group has handed in. The dates for the exams are

**Friday, February 09, 2018****Friday, May 04, 2018**— May the fourth/force be with you!

Please register for the exam both with Ms. Karpa (E 111) as well as through Porta.

**Lecture Notes and Slides**

The current version of the lecture notes can be downloaded here:

**Stochastic Processes Lecture Notes**(Version: January 31)

The slides presented during the lectures can be downloaded here:

**Slides 01**: The Multivariate Normal Distribution**Slides 02**: Conditional Expectation**Slides 03**: Uniform Integrability and Vitali’s Theorem

**Exercise Sheets**

The exercise sheets can be downloaded here:

**Exercise Sheet 01**: The Product Sigma-Field, Independent Increments, and the Random Walk.**Exercise Sheet 02**: Filtrations, Stopping Times, and Hitting Times.**Exercise Sheet 03**: Stopping Times, Hitting Times, the Stopped Process, and Brownian Motion.**Exercise Sheet 04**: Galmarino’s Test, Gaussian Processes, and Hölder Continuity.**Exercise Sheet 05**: Multivariate Normal Distribution, Modifications and Indistinguishability.**Exercise Sheet 06**: Complete Filtrations, Modifications vs. Indistinguishability, Continuity of Fractional Brownian Motion, and Quadratic Variation of Brownian Motion.**Exercise Sheet 07**: Conditional Expectation, Submartingales under Change of Filtration, Martingale Property of the Random Walk, and Counterexamples of Martingales.**Exercise Sheet 08**: Optional Stopping, Geometric Brownian Motion, Doubling Strategy, and Total Variation.**Exercise Sheet 09**: Brownian Martingales, Local Martingales, the Doob Decomposition, and Optional Stopping via Stochastic Integration.**Exercise Sheet 10**: Lévy’s Upward-Downward Theorem, Kolmogorov’s 0/1 Law, and the Strong Law of Large Numbers Revisited.**Exercise Sheet 11**: Stopping Times under Complete Filtrations, Distribution of the Renewal Process, and Sums of Lévy Processes.**Exercise Sheet 12**: Hitting Times of Brownian Motion, the Reflection Principle, and Distribution of the Running Maximum.

**Simulations**

The simulations presented during the lectures can be downloaded here:

**PathVsRV.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**.

**Martingale.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.