## 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.

**News**

**Exercise Sheet 05**is now available for download below. Please submit your solutions by Thursday, November 23, 13:45.**Exercise Sheet 06**is now available for download below. Please submit your solutions by Thursday, November 30, 13:45.

**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 E45.**

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 the post box E14 on the ground floor of building E. To qualify for the final exam, you are required to attain at least 50% of the points and be able to demonstrate in the exercise classes that you have fully understood the solutions that your group has handed in. You can submit in groups of 2-3.

**Lecture Notes**

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

**Stochastic Processes Lecture Notes**(Version: November 01)

The slides presented during the lectures can be downloaded here:

**Slides 01**: The Multivariate Normal Distribution

**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.

**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**.