By Kardi Teknomo, PhD.

stochastic process

Share this: Google+
< Previous | Contents | Next >

What is Arithmetic Brownian Motion?

A Brownian motion with drift is called arithmetic Brownian motion or ABM. The actual model of ABM is a stochastic differential equation (SDE) of this form

This model has two parameters:

  1. 1.Drift,
  2. 2.Volatility, (sometimes it is also called diffusion coefficient)

The meaning of drift parameter is a trend or growth rate. If the drift is positive, the trend is going up over time. If the drift is negative, the trend is going down. The meaning of volatility is a variation or the spread of distribution. The value of volatility is always positive (or zero) because it is actually related to standard deviation of the distribution.

To simulate the ABM, we need to find the solution of the stochastic differential equation above. The solution can be found by the usual integration to be:

Or,

Notice that ABM contains Brownian motion (or Wiener Process). This model now has three inputs that is 2 parameters and one initial value:

  1. Drift,
  2. Volatility,
  3. Initial value of the ABM, .

The reason we use symbol  for the drift parameter and  for the volatility is because is normally distributed with mean  and variance .

< Previous | Contents | Next >

Do you have question regarding this Stochastic Process tutorial? Ask your question here

Share and save this tutorial
Add to: Del.icio.us Add to: Digg Add to: StumbleUpon Add to: Reddit Add to: Slashdot Add to: Technorati Add to: Netscape Add to: Newsvine Add to: Mr. Wong Add to: Webnews Add to: Folkd Add to: Yigg Add to: Linkarena Add to: Simpy Add to: Furl Add to: Yahoo Add to: Google Add to: Blinklist Add to: Blogmarks Add to: Diigo Add to: Blinkbits Add to: Ma.Gnolia Information

These tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi. (2017) Stochastic Process Tutorial .
http://people.revoledu.com/kardi/tutorial/StochasticProcess/