## 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 11**is available for download below. Please submit your solutions by Tuesday, January 22.**Exercise Sheet 12**is available for download below. Please submit your solutions by Tuesday, January 29. This is the last exercise sheet for this semester!

**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 are

**February 18, 2019**(Monday) and**March 29, 2019**(Friday).

Please **register in time** for the exam both with Ms. Karpa (E 113) 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 14)

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.**Exercise Sheet 05**: The Multivariate Normal Distribution and Quadratic Variation of Brownian Motion.**Exercise Sheet 06**: Nowhere Monotonicity of Brownian Motion, Non-Measurability of the Set of Continuous Functions, Existence of Independent Sequences, and Modifications vs. Indistinguishability.**Exercise Sheet 07**: Complete Filtrations, Modifications vs. Indistinguishability, Continuity of Fractional Brownian Motion, and Projection Property of Conditional Expectation.**Exercise Sheet 08**: Submartingales under Change of Filtration, Martingale Property of the Random Walk, Counterexamples of Martingales, and Geometric Brownian Motion.**Exercise Sheet 09**: Brownian Martingales, Total Variation, the Doob Decomposition, and Optional Stopping via Stochastic Integration.**Exercise Sheet 10**: Lévy’s Downward and Upward Theorems, Kolmogorov’s 0/1 Law, and the Strong Law of Large Numbers.**Exercise Sheet 11**: Stopping Times under Complete Filtrations, Distribution of the Renewal Process, and Sums of Independent Lévy Processes.**Exercise Sheet 12**: Hitting Times of Brownian Motion, the Reflection Principle, and the 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.