The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$.


I define $T=\inf \{t>0,|W|(t)= 1\}$. One can find some useful informations about this random variables like its expectation 1, variance $\frac{2}{3}$ or Laplace transform $\mathbb{E}[e^{-xT}]=\frac{1}{\cosh(\sqrt{2x})}$ (see Revuz and Yor).


How can I know if $T$ has a density ? Can I inverse the Laplace transform easily (numerically could be enough) ? Any additional information on $T$ would be appreciated.



Does the hitting time of +1/-1 of a Brownian motion posess a density?
The law of $T$ (and more generally the law of hitting times of Bessel processes) has been studied extensively in the literature, in particular by Marc Yor. In the survey


[PROBABILITY LAWS RELATED TO THE JACOBI THETA AND
RIEMANN ZETA FUNCTIONS, AND BROWNIAN EXCURSIONS](http://www.ams.org/journals/bull/2001-38-04/S0273-0979-01-00912-0/S0273-0979-01-00912-0.pdf)


$T$ is denoted by $C\_1$ and several properties of the density are listed in the Table 1, page 443. 
Interestingly, the Mellin transform of $T$ is closely related to the Riemann zeta function.


