$(X_t, Y_t, Z_t)$ is a three-dimensional stochastic process described as follows:
$X_t$ is a Brownian Motion.
$Y_t = \int_0^t X_s ds$
$Z_t = \inf_{s \in [0, t]} X_s$
I would like to find a density function for the value of the process at time $t_f$. Can this be done, despite the fact that $Z_t$ makes the process non-Markov?
Helpful information: the transition density function for $(X_t, Y_t)$ is well known. See the intro to this paper. The challenge is adding $Z_t$ into the mix.