Find a density function for the endpoint of this stochastic process

$(X_t, Y_t, Z_t)$ 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.

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What is $t_f$? Also your link seems to be wrong - i'm getting: No document with DOI "10.1.1.17.8220". –  Stefan Hansen Jan 7 '13 at 7:33
$t_f$ is some positive constant - for generality, I'm leaving it variable. And link fixed (I think), thank you. –  GMB Jan 7 '13 at 7:36
Anyway, $(X_t,Y_t,Z_t)$ is distributed like $(\sqrt{t}X_1,t\sqrt{t}Y_1,\sqrt{t}Z_1)$ hence the case $t=1$ suffices. –  Did Jan 7 '13 at 7:44