Apologies, I'm not very good at stochastic calculus so I'll ask the below as best I can..!
I have a particle I am trying to model which bounces around the origin, but which has a momentum component.
I'm looking to approximate it with something resembling an Ornstein Uhlenbeck process for velocity $V_t$. I started with the definition from the wiki : en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process, but I have added an additional drift term based on the particle's displacement $S_t$ from the origin:
$dV_t = -(\alpha S_t+ \beta V_t) dt +\sigma dW_t$ where $S_t$ is $\int_0^t{V_s ds}$
For simplicity's sake, assume $S_0=0$ $V_0=0$
Specifically I am interested in the stationary process of $S_t$, which means potentially not all configurations of parameters work - I wish to ignore cases where $\left|S_t\right| \to \infty$
I would like to find out:
- Necessary / sufficient conditions for $\xi$, $\theta$ such that the stationary process of $S_t$ exists
- A representation for $\mathrm{Cov}{[{S_{t_1},S_{t_2}}]}$ where $t_1 = t + u$, $t_2 = t + v$, $t \to \infty$
Any help that can point me in the right direction would be greatly appreciated!!
Edit: It would appear this might be the Langevin equation for harmonic motion in a fluid? Can someone confirm if this is correct / if an analytical solution is available for this?