What is a Brownian motion? Properties of Brownian motion?

AnswerBot
1y
Brownian motion is the random motion of particles in a fluid due to collisions with other particles.Brownian motion was first observed by Robert Brown in 1827.It is named after the botanist Robert Bro...
see more
Shillpa G Bharatt
6y

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion process or Brownian motion due to its historical connection with the physical process known as Brownian movement or Brownian motion originally observed by Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, and physics.

The Wiener process {\displaystyle W_{t}}is characterised by the following properties:[1]

1.    {\displaystyle W_{0}=0}{\displaystyle W}has independent increments: for every {\displaystyle t>0,} the future increments {\displaystyle W_{t+u}-W_{t},}{\displaystyle u\geq 0,}, are independent of the past values {\displaystyle W_{s}}{\displaystyle s<t.}

2.    {\displaystyle W}has Gaussian increments: {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean {\displaystyle 0}and variance {\displaystyle u}{\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).}

3.    {\displaystyle W} has continuous paths: With probability {\displaystyle 1}{\displaystyle W_{t}}is continuous in {\displaystyle t}.

The independent increments means that if 0 ≤ s1 < t1 ≤ s2 < t2 then Wt1−Ws1 and Wt2−Ws2 are independent random variables, and the similar condition holds for n increments.

Help your peers!
Add answer anonymously...
Deutsche Bank Data Analyst Interview Questions
Stay ahead in your career. Get AmbitionBox app
qr-code
Helping over 1 Crore job seekers every month in choosing their right fit company
65 L+

Reviews

4 L+

Interviews

4 Cr+

Salaries

1 Cr+

Users/Month

Contribute to help millions
Get AmbitionBox app

Made with ❤️ in India. Trademarks belong to their respective owners. All rights reserved © 2023 Info Edge (India) Ltd.

Follow us
  • Youtube
  • Instagram
  • LinkedIn
  • Facebook
  • Twitter