Geometric Brownian motion

[Bill the Catuary]

Does anyone know the answer to the following?

http://actuary.ca/phpBB/viewtopic.php?t=40523

Your help would be much appreciated!
Thank you!

[Michael Davlin]

For the processes that incorporate time dependency into the a and b functions, does it follow that the logarithm of the underlying variable follows a generalized Wiener process (as for the strict definition of GBM)?
I think your question can most easily be answered if you attack it in the opposite direction. If y = e^x and x follows the Ito process:

dx = a(x, t)dt + b(x, t)dz

then an application of Ito's Lemma proves that:

dy = y [(a(ln y, t) + 1/2 b^2(ln y, t))dt + b(ln y, t)dz].

While exactuary is correct about terminology (geometric brownian motion is defined to have constant drift and volatility), the above SDE for y is certainly a generalization of GBM.

Working from the other direction, if x = ln y, and:

dy = a(y, t)dt + b(y, t)dz,

then:

dx = (a(e^x, t) - 1/2 b^2(e^x, t))dt + b(e^x, t)dz].

The confusing thing about this is the fact that, for SDEs, dy / y != d ln y; those expressions are equivalent only for ODEs. In general, exponentiating a general Ito process results in something that looks like a generalized GBM with an upward drift adjustment, while taking the log of a general Ito process results in another general Ito process (which looks nothing like a GBM) with a downward drift adjustment.

And, yes, there are interest rate models in use which assume time varying drift and volatility.