Lecture (FS 2009): Random processes: Theory and applications from physics to finance

Random phenomena are of paramount importance in all areas of science. In the theory of random or stochastic processes, the random evolution of time-dependent quantities is considered. Prominent examples are the Brownian motion of a particle immersed in a liquid, the evolution of a chemical reaction, the motion of charge carriers in an electrical circuit, and the price fluctuations of a security on the stock market.

This lecture starts with a concise introduction to the basic concepts of probability theory and random processes. Markov processes as particularly important class of stochastic processes are discussed and their applications to various fields is presented.

Topics

• Basics of probability theory
• Random processes: General concepts
• Markov processes:
  • Master equation
  • Fokker-Planck equation
  • Stochastic differential equations
• Mathematical finance

Time and place

Wednesday, 10:15-12:00h, Department of Physics, room 4.1

Exercise classes

Exercise classes are held every Wednesday, 16:00-17:30h in room 4.1. The problem set which has been handed out the week before is discussed and solved together with the assistant Robert Andrzej Zak.

Per problem set, one problem has to be handed in and will be graded. Required for obtaining the credit points are 50% of the points of those problems and regular participation in the exercise class.

Problem set	Date
Problem set 1: Birthday paradox; Baye's theorem; Transformation of variables	2009/02/25
Problem set 2: Discrete-time random walk; Poisson distribution; Probability generating function, facorial moments and cumulants	2009/03/11
Problem set 3: Branching processes; Compound distribution; Box-Muller algorithm	2009/03/18
Problem set 4: Shot noise; Sinusoidal process with random amplitude and phase; Cauchy-Schwarz inequality	2009/03/25
Problem set 5: Markov Chains; Metropolis-Hastings algorithm	2009/04/01
Problem set 6: First-passage problems and gambler's ruin	2009/04/08
Problem set 7: Dichotomic noise; Wiener process	2009/04/15
Problem set 8: Fokker-Planck equation; No perpetuum mobile of the second kind	2009/04/22
Problem set 9: Ornstein-Uhlenbeck process; Method of characteristics	2009/04/29
Problem set 10: Ito's Lemma; Generalized stochastic integrals; Correlation formula for the Ito integral	2009/05/06
Problem set 11: Noisy harmonic oscillator; Numerical integration of SDEs; Chain rule for Stratonovich SDE	2009/05/13
Problem set 12: Programming exercises	2009/05/20 & /27