I have a stochastic process $X_{t}$ that has probability $p(t)$ that $a < X_{t} < b$ with $p(t)$ a decreasing function. I want to find the random variable $D_{t}$ which describes the duration $X_{t}$ has been within $a$ and $b$ up until $t$. Let $Y_{t} \sim$ Bernoulli($p(t)$) denote whether $X_{t}$ is in the interval then $D_{t}$ can be given by: $$D_{t} = \int_0^t Y_{u} \ du$$
Is there any way I can recover the PDF or CDF of $D_{t}$? I've tried by taking the limit of the Poisson-Binomial distribution/infinite convolutions but haven't had any luck. My intuition tells me that $$\mathbb{E}[D_{t}] = \int_0^t p(u) \ du$$ but I'd really like to have the PDF/CDF.
Update:
Thanks to comments, I realize I need to add additional information:
I also have that the joint density for two points in time $t_{1} > t_{0}$ is given by $$f_{X_{t_{1}},X_{t_{0}}}(x_{t_{1}},x_{t_{0}}) = f_{X_{t_{1}-t_{0}}}(x_{t_{1}}-x_{t_{0}}) \cdot f_{X_{t_{0}}}(x_{t_{0}})$$ and more generally the joint density for $n$ points in time is $$f_{X_{t_{n-1}},\dots,X_{t_{0}}}(x_{t_{n-1}},\dots,x_{t_{0}}) = f_{X_{t_{0}}}(x_{t_{0}})\prod_{i=1}^{n-1} f_{X_{t_{i}-t_{i-1}}}(x_{t_{i}}-x_{t_{i-1}})$$
