by Nikolai V. Shokhirev
Up: Stochastic Labs
- Pseudorandom numbers
- Probability transformation
- Diffusion, Wiener process
- Geometric Brownian motion
- Ornstein-Uhlenbeck process
This process is very popular in financial engineering and quantitative analysis.
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]
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.
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.
©Nikolai V. Shokhirev, 2007-2008