Geometric Brownian motion

by Nikolai V. Shokhirev

Up: Stochastic Labs
Pseudorandom numbers
-  Probability transformation
Diffusion, Wiener process
Geometric Brownian motion
-  Ornstein-Uhlenbeck process

Geometric Brownian motion

This process is very popular in financial engineering and quantitative analysis.

Stochastic Differential Equation (SDE)

The above verbal definition can be presented in the following compact form

dy = μ y dt + σ y dWt

By substituting z = ln y it can be reduced to diffusion with modified drift:

dz = d ln y = (μ - ½ σ2) dt + σ dWt

A formal solution is:

y = y0 exp[(μ - ½ σ2) t + σ Wt]


Fokker-Planck Equation

The corresponding PDE obviously has the solution:

Performing the transformation to the original variable we have

Using this density function it can be shown that the average value of y exponentially depends on time:

<y> = y0 exp(μ t)

This result can be easy obtained directly from SDE.


Stochastic simulation program


Geometric Brownian motion

Feel free to download and play (at your own risk). I am adding several new features, please check later.


Stochastic simulation program.

Some references

ABC Tutorials | Data Processing | Indirect Measurements | NMR Tutorials

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©Nikolai V. Shokhirev, 2007-2008