## Extension of some feature of SDE Ornstein-Uhlenbeck type

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Hi everyone,

I am looking for some ideas (or references) in order to get an explicit SDE (if it exists) which would have a stylised property extending in some sense the mean-reversion property of SDE of Ornstein-Uhlenbeck type.

More formally, is it possible to have a $n$-means reverting process defined by an SDE ?

I imagine this SDE would have the form like $dS_t=f_1(S_t,t)dt+...+f_n(S_t,t)dt+\sigma dW_t$

where $f_i$'s are such that if $S_t$ is closed to i-th mean $m_i$ then it stays closed to this point with high probability.

I am sorry to not define the necessary concepts more clearly but as I am only looking for ideas (or refernces) on this, I rather define some intuitive concept than a fully formal framework in order not to close any possibility.

Thank's for the time spend reading those lines

PS : I would like to avoid the n states regime switching technology if possible

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mathworks.com/access/helpdesk/help/toolbox/econ/… – Steve Huntsman Jan 19 2010 at 18:17

## 1 Answer

I believe your notation is redundant. Let $g = f_1 + ... + f_n$. Correct me if I'm wrong, but you just want a function $g$ so that $dS(t) = g(s(t),t)dt + \sigma dW_t$ has the behavior you specify.

It's not clear exactly what properties you want. One possibility is that you can just let $S$ be a Brownian motion in a potential function $\Phi$ with $n$ local minimums. In that case, you don't need $g$ to depend on $t$: $g = -\Phi'$. You may want the potential function to be approximated by the potential well of an Ornstein-Uhlenbeck process near each minimum.

This would mean S would stay near each minimum for a while, but it would leak out eventually with probability 1. You can increase the potential with time to force the process to stay near the local minimum with positive probability. However, this would no longer resemble a fixed Ornstein-Uhlenbeck process near each minimum.

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Douglas Zare Thank you for your interest  You are right we can set a function $g$ but I thought that $g$ could have the form I specified with $f_i(t,S_t)$ not far from $a_i(m_i-S_t)$ when $S_t$ is close to $m_i$ and approximately null when far, which seems close to what you suggest in your second paragraph. Second, I am still looking for the "right" criteria to establish a clear sense to the multi-mean reverting property, and your answer is helpful for this goal to be acheived, I will try to propose candidates for both the definition of the property and for $g$. Best Regards  – The Bridge  Jan 20 2010 at 10:19
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