Brownian Motion Process

Sivaprasad Madhira and Shailaja Deshmukh ()
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Sivaprasad Madhira: Savitribai Phule Pune University
Shailaja Deshmukh: Savitribai Phule Pune University

Chapter Chapter 9 in Introduction to Stochastic Processes Using R, 2023, pp 487-545 from Springer

Abstract: Abstract This chapter considers a continuous time continuous state space Markov process known as Brownian motion or Wiener process. After tracing its history in Sect. 1, Brownian motion is defined in Sect. 2 and some of its properties are discussed. Using the continuity property of the sample paths and reflection principle, distributions of the maximum and minimum of a Wiener process over a bounded time interval are derived in Sect. 3. There are many variations and extensions of a Wiener process. Sections 4 and 5 present two extensions, namely the Brownian bridge and geometric Brownian motion, respectively. In Sect. 6, we briefly introduce some more variations including the Ornstein-Uhlenbeck process. R codes used for illustrations are given in Sect. 7.

Date: 2023
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DOI: 10.1007/978-981-99-5601-2_9

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