# Extension of some feature of SDE Ornstein-Uhlenbeck type

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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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.
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 '10 at 10:19