Stochastic Processes

A variety of stochastic processes. Short definitions or explanations of each process are written in each file.

Contents

• Brownian motion (Wiener process)

  • Law of the iterated logarithm
  • Arc-sine law of Lévy
  • Passage time and Maximum
  • Brownian bridge

• Bessel process

• Geometric Brownian motion

• Simple random walk

  • Recurrence of random walks
  • Law of the iterated logarithm

• Perturbed random walk

  • Periodic
  • on random media

• Reinforced random walk

  • Pólya's urn

• Poisson process

  • compensated Poisson process

• Compound Poisson process

  • Cramér-Lundberg Risk Process
  • The M/G/1 queue

• Renewal process

  • Path property

• Random matrix

  • Dyson model

Requirements

• Pipenv

References

• Y. Funaki: Probability theory. Asakura-shoten, 2004. (Japanese)
• I. Karatzas, S. E. Shreve: Brownian Motion and Stochastic Calculus. 2nd Edition, Springer, 1997.
• H. Kuo: Introduction to Stochastic Integration. Springer, 2006.
• A. E. Kyprianou: Fluctuations of Lévy Processes with Applications. 2nd Edition, Springer, 2014.
• https://www.math.kyoto-u.ac.jp/ja/event/seminar/probability

External Links

• Brownian motion
• Geometric Brownian motion
• Random walk
• Poisson process
• Bessel process