## Stochastic Processes – Winter Term 2018/19

On this webpage, you find all information pertaining to my course on **Stochastic Processes** held in the winter term 2018/19 at the University of Trier.

**News**

**Exercise Sheet 03**is available for download below. Please submit your solutions by Tuesday, November 13.**Exercise Sheet 04**is available for download below. Please submit your solutions by Tuesday, November 20.

**General Information**

The **lectures** take place on

**Monday, 12:15 – 13:45 in B 13.****Tuesday, 14:15 – 15:45 in E 10.**

The **exercise classes** will be held by **Jonas Jakobs** and take place on

**Tuesday, 16:15 – 17:45 in HS 7.**

Exercise sheets are generally uploaded every Tuesday and are due on Tuesday 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 exam dates will be published here at a later point.

**Lecture Notes and Slides**

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

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

The slides presented during the lectures can be found here:

The video presented in the lecture can be watched here:

**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**: Hitting Times of White Noise Processes, Measurability of the Stopped Process, Optional Times, and Suprema and Infima of Stopping Times.**Exercise Sheet 04**: Invariance of Brownian Motion, Brownian Bridge, the Ornstein-Uhlenbeck Process, and Hölder Continuity.

**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**.