Lecture (FS 2008): 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 Verena Koerting.

Problem set	Date
Problem set 1: Probability generating function, factorial moments and cumulants; Gamma distribution	2008/02/27
Problem set 2: Discrete-time 1d random walk; compound distribution; branching processes	2008/03/05
Problem set 3: Point processes; shot noise	2008/03/26
Problem set 4: Processes with independent increments; non-Gaussian white noise	2008/04/02
Problem set 5: First-passage problems and gambler's ruin	2008/04/09
Problem set 6: Dichotomic noise; Ornstein-Uhlenbeck process	2008/04/16
Problem set 7: Empirical determination of drift and diffusion coefficients; Fokker-Planck equation with periodic boundary conditions	2008/04/23
Problem set 8: Kramers' escape problem	2008/04/30
Problem set 9: Wiener process; stochastic integrals; Ito calculus	2008/05/07
Problem set 10: Chain rule for Stratonovich SDEs; product rule for Ito SDEs; harmonic oscillator with thermal noise	2008/05/14
Problem set 11: Geometric Brownian motion with time-dependent coefficients; Options on futures	2008/05/21