# bounded time integral of Ornstein Uhlenbeck process

With $dF_t=-kF_tdt+\sigma dW_t$ an Ornstein-Uhlenbeck process, I define $Q_t$ as: $Q_t=\int_0^\infty F_t \mathbb{1}_{-L\leq Q_t\leq L} dt$.

Intuitively, $F_t$ is a flow and $Q_t$ is a quantity, that we do not authorize to grow to more than $L$.

I am interested in the stationary distribution of Q. I can write the Fokker-Planck equation as:

$0=\frac{\partial}{\partial F}[k F p + \frac{\sigma^2}{2} \frac{\partial p}{\partial F}] - \frac{\partial}{\partial Q}[F p]$ with $p(F,Q)$

$0 = \frac{\partial}{\partial F}[k F b + \frac{\sigma^2}{2} \frac{\partial b}{\partial F}] + f P|_{Q=L}$ and the same equation for $Q=-L$

where $p$ is the density on $(F,Q)\in \mathbb{R} \times [-L,L]$ and b is a density on $F\in \mathbb{R}^+$

But unfortunately I am stuck there and not sure how to get -even approximate- ways to look at this. I am only interested in the marginal on $Q$ (the marginal on $F$ is trivial since $F$ does not depend on $Q$, so you just get the pdf of the OU process), but not sure how to approach this. Any help would be useful.

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could u clarify how you define $Q_t$? The way it is written does not make sense since you define it as an integral from zero to infinity and $t$ is a variable in your integrand (i.e. $dt$). –  Alexey Feb 22 at 14:49