# Are these two some kinds of generalized Ornstein–Uhlenbeck processes?

An Ornstein–Uhlenbeck process is $$d X_t = (\mu - X_t) dt + d W_t$$

We try to build a model using some generalized Ornstein–Uhlenbeck processes.

1. The first one is $$d X_t = \exp(-|X_t- \mu|) (\mu - X_t) dt + d W_t$$ where we hope $\exp(-|X_t- \mu|)$ will reduce the speed of $X_t$ approaching $\mu$, as $X_t$ comes closer to $\mu$.

2. Furthermore, since an O-U sde has a attractor $\mu$, we tempt to generalize the above sde to have more than one attractors $$d X_t = \sum_{i=1}^3 \exp(-|X_t- \mu_i|) (\mu_i - X_t) dt + d W_t$$

I have little idea about these two generalized Ornstein–Uhlenbeck processes. So may I ask here if there are some references on them? Do they have weak or strong solutions? What are their generators like? How are their attraction regions like and decided?

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I've never seen these processes, but you can quite easily come up with generator since $$\mathrm dX_t = a(X_t)\mathrm dt + \mathrm dW_t$$ has a generator $$\mathscr Af(x) = a(x)f'(x)+\frac12 f''(x)$$
Hi @Ilya, thanks! I saw that your reply is based on Thoerem 7.3.3 of Oksendal's SDE book. If $\mu_i$'s are changed to depend on $t$, is $X_t$ still Markovian, and what is its generator like? –  Ethan Apr 3 '13 at 3:37
@Ethan: in such case, $X_t$ is time-inhomogeneous Markovian, and thus you have to rather consider a process $(t,X_t)$ and an operator on functions $f(t,x)$. Only the term $\frac{\partial f}{\partial t}\mu(x,t)$ appears additionally –  Ilya Apr 3 '13 at 11:25
thank you! (1) For the SDE $dX_t=(μ(t)−X_t)dt+dW_t$ in my post, why is it a time-inhomogeneous Markovian, and does "only the term $\frac{∂f}{∂t}μ(x,t)$ appears additionally" mean that $\mathcal A f(t,x) = (\mu(t) - X_t) \frac{∂f}{∂x}(t,x) + \frac{1}{2} \frac{∂^2f}{∂x^2}(t,x) + \frac{∂f}{∂t}μ(x,t)$? (2) For SDE $dX_t, b(t, X_t) dt + \sigma(t, X_t) dW_t$, is $X_t$ time-inhomogeous Markovian? What is its generator like? Do you have some references on those cases (it seems to me that Oksendal hasn't discussed those cases in his SDE book)? Tha –  Ethan Apr 3 '13 at 12:31