'Mean Aversion' in Stochastic Differential Equations

[Apogee]

I had a brief question regarding SDEs. Typically, I've seen models like the Ornstein-Uhlenbeck process that generally revert back to the mean over time. However, I've been trying to find a stochastic differential equation/process that avoids the mean, such as a sharp increase followed by a sharp decrease. What examples are like this and/or how would one derive something like this?

 

[Matterwave]

Do you mean perhaps a 2-peaked or multi-peaked distribution of some sort? http://en.wikipedia.org/wiki/Bimodal_distribution

 

[Apogee]

Precisely. Is there any SDE/SP that represents a bimodal/trimodal/etc. distribution?

 

[Matterwave]

Actually none come to my mind right now, although I'm sure there are.

One example of perhaps an SDE that does not fall to a mean is geometric (or arithmetic even) Brownian motion, which is basically a superposition of Brownian motion on top of geometric growth, often used to model stocks.

 

[Apogee]

Could I just take geometric Brownian motion and instead of a Wiener process that is normally or lognormally distributed, use a Wiener process that is bimodally distributed?

 

[Matterwave]

I am not familiar with any Wiener process than the regular Brownian motion...which is normally distributed...o.o

 

[Apogee]

Hahaha. Then just a random process that is bimodally distributed.