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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?

Thanks in advance!

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1 Answer 1

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 $$ \mathscr Af(x) = a(x)f'(x)+\frac12 f''(x) $$

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Thank you! Can you also point out some ways or references that may help me solve the sdes, and determine their attraction regions? –  Ethan Mar 29 '13 at 18:23
    
@Ethan: Oksendal's "SDE: Introduction with applications" is a std reference –  Ilya Mar 29 '13 at 18:25
    
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

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