Let $x(t)$ be an standard Wienner process defined by the SDE: $$dx(t) = \mu dt + \sigma \, dW $$
where the constants $\mu$ ,$\lambda$ and $\sigma$ are non-negative and constants.
Let be $\tau$ the first time $x$ crosses one of the two possible absorbing barriers $a$ or $b$ and it is defined by:
$$ \tau = \inf \{t>0 \mid x(t) \notin (a,b)\} $$
The distribution of $\tau$ conditional on hitting a particular absorbing barrier $a$ or $b$ for the standard Wienner process is given by (Srivastaba,2016 and Lin, 1998):
$$ \frac{d}{dt}\mathbb{P}(\tau < t \mid x(\tau) = a) = \frac{\exp\left(\frac{a\mu}{\sigma^2} - \frac{1}{2} \left(\frac{\mu}{\sigma}\right)^2 t\right) \sum\limits_{n=-\infty}^{\infty} f\left(t;a_n\right)}{P^{+}\left(x_0,\mu,\sigma,a\right)} $$
$$ \frac{d}{dt}\mathbb{P}(\tau < t \mid x(\tau) = b) = \frac{\exp\left(\frac{\mu}{\sigma^2} - \frac{1}{2}\left(\frac{\mu}{\sigma}\right)^2 t\right) \sum\limits_{n=-\infty}^\infty f(t;b_n)}{P^{-}(x_0,\mu,\sigma,b)} $$
where $a < 0 <b\text{ }$ , $\text{ }a_n = \frac{1}{\sigma} - [2n(b - a) - a]\text{ }$, $\text{ } b_n = \frac{1}{\sigma} - [2n(b - a) + b]$
and $f\left(t,a\right)$ is the density for the first time that a standard Brownian motion hits the barrier $x = a$ ,$f(t,a) = \frac{a}{\sqrt{2\pi t^3}}\exp\left(-\frac{a^2}{2t}\right)$.
Also notice $P^\pm$ does not require attention for the question below.
Now let $x(t)$ be an Ornstein Uhlenbeck process defined by the SDE: $$dx(t) = \mu \, dt -\lambda x(t) \, dt + \sigma \, dW $$
where the constants $\mu$ ,$\lambda$ $\sigma$ are non-negative and constants.
The OU process can be expressed in terms of a transformed Wienner process evolving in exponential time [$x_0 + W\left(u\left(t\right)\right)$]. Notice that $t$ is transformed by: $$ u(t) = \frac{\sigma^2\exp(2\lambda t) - 1}{2 \lambda} $$ and the boundaries of the new process [$W\left(u\left(t\right)\right)$] become time varying too (Srivastaba,2016 and Lin, 1998): $$a^{*}(t) = \left(a-\frac{\mu}{\sigma}\right)\sqrt{1 - \frac{2\lambda u(t)}{\sigma^2}} + \frac{\mu}{\sigma}$$
and
$$b^{*}(t) = \left(b-\frac{\mu}{\sigma}\right)\sqrt{1 - \frac{2\lambda u(t)}{\sigma^2}} + \frac{\mu}{\sigma}$$
Given that $u$ is a monotone increasing function, is possible to obtain the first passage time distribution of the OU process [$x_0 + W\left(u\right)$] given:
$$ \tau_{U} = \inf \{u(t)>0 \mid x(u(t)) \notin (a^*(t),b^*(t))\} $$
so that
$$ \tau = u^{-1}(\tau_{U}) $$
The question is how to do that
Here is what I have tried:
Using the chain rule it is possible to propose the following:
$$\frac{d}{dt}\mathbb{P}(\tau < t \mid x(\tau) = a) = \frac{d}{dt}\mathbb{P}(u(\tau) < u(t) \mid x(u(\tau)) = a^{*}(t)) \cdot u(t)^{'} $$
$$\frac{d}{dt}\mathbb{P}(\tau < t \mid x(\tau) = b) = \frac{d}{dt}\mathbb{P}(u(\tau) < u(t) \mid x(u(\tau)) = b^{*}(t)) \cdot u(t)^{'} $$
Notice that when performing the transformation, the drift parameter is taken away from the process $W(u)$ and is placed in the boundaries $a^{*}(t)$ and $b^{*}(t)$, so the distributions become:
$$ \frac{d}{dt}\mathbb{P}(\tau < t \mid x(\tau) = a) = \frac{\sum\limits_{n=-\infty}^{\infty} f\left(t;a_n\right)}{P^{+}\left(x_0,\mu,\sigma,a\right)}\cdot \sigma^2\,exp(2\,\lambda) $$
$$ \frac{d}{dt}\mathbb{P}(\tau < t \mid x(\tau) = b) = \frac{\sum\limits_{n=-\infty}^{\infty} f\left(t;b_n\right)}{P^{+}\left(x_0,\mu,\sigma,b\right)}\cdot \sigma^2\,exp(2\,\lambda) $$
where $a< b\text{ }$ , $\text{ }a_n = \frac{1}{\sigma} - [2n(b(t) - a(t)) - a(t)]\text{ }$, $\text{ } b_n = \frac{1}{\sigma} - [2n(b(t) - a(t)) + b(t)]$ respectively.
Regardless, I am unable to obtain proper density distributions.
References:
Srivastava et al. 2016 https://arxiv.org/abs/1508.03373
Lin,1998 https://www.sciencedirect.com/science/article/pii/S0167668798000213