Brownian Motion and Lévy Processes
Wiener (1923) and Lévy (1939) mathematically gave the theoretical foundation and construction of the Brownian Motion based on the phenomenon of small particles observed by Brown (1827) and Einstein (1905). This is called the Brownian Motion or Wiener process, and is an example of a Markov process with continuous time and state space. This is now one of the most useful stochastic processes in applied sciences such as physics, economics, communication theory, biology, management science, and mathematical statistics, and recently, it is also used to make several valuable models in finance.
KeywordsBrownian Motion Optimum Policy Damage Model Wiener Process Standard Brownian Motion
- 8.Ito K, Nakagawa T (2008) Comparison of three cumulative damage models. In: Sheu SH, Dohi T (eds) Advanced reliability modeling III. McGraw-Hill, Taipei, pp 332–338Google Scholar
- 11.Sato K (2001) Basic results on Lévy processes. In: Bandorff-Nielsen O, Mikosch T, Resnick S (eds) Lévy processes: theory and applications. Birkhäuser, BostonGoogle Scholar
- 15.Satow T, Nakagawa T (1997) Replacement policies for a shock model with two kinds of damages. In: Osaki S (ed) Stochastic modeling in innovative manufacturing. Springer Lecture notes in economics and mathematical systems, vol 445. Springer, Berlin, pp 188–195Google Scholar