Consider the following SDE for $z_t = (x_t,v_t)\in \mathbb{R}\times [-1,1]$:

$$dx_t=-\mu x_t dt+ v_0 v_t dt + \sigma dW_t,$$ $$dv_t=-\frac{a^2}{2} v_t dt - a\sqrt{1-v_t^2} dB_t,$$ where $\mu, v_0, a, \sigma >0$, $B_t$ and $W_t$ are independent Wiener processes.

I would like to find the stationary distribution for the process $z_t$. So far this is what I have tried. I wrote down the Fokker-Planck equation for the SDE system and attempted to solve for the steady-state distribution. However, I couldn't quite solve the resulting PDE (second order and in two variables: $x$ and $v$).


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