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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.

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    $\begingroup$ It will definitely follow from the formula for the Laplace transform. I would in fact instead look at the Fourier transform, which is smooth and exponentially decaying by this formula. $\endgroup$
    – Christian Remling
    Commented Oct 13, 2017 at 16:07
  • $\begingroup$ I would have prefered to find the Fourier transform, but a closed formula does not seem to exist. Have you got a reference for inversing the Laplace transform ? $\endgroup$
    – Adrien Laurent
    Commented Oct 13, 2017 at 16:49
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    $\begingroup$ The FT is the holomorphic continuation of the Laplace transform, just use the same formula. $\endgroup$
    – Christian Remling
    Commented Oct 13, 2017 at 17:11
  • $\begingroup$ Ok! Thank you. Would you by chance know a way to derive a closed formula for the density ? $\endgroup$
    – Adrien Laurent
    Commented Oct 13, 2017 at 17:15
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    $\begingroup$ I think there probably isn't a closed form expression. In Karatzas and Shreve, Exercise 8.11, you can find an infinite series expression of the density, which in particular shows that there is a density. It converges super-exponentially. $\endgroup$
    – Nate Eldredge
    Commented Oct 13, 2017 at 17:22

2 Answers 2

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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

$T$ 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.

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Computing the inverse Laplace transform can easily be done with the following python code

http://code.activestate.com/recipes/576934/

or more accurateley (using the module mpmath)

http://code.activestate.com/recipes/576938/

I've just computed the inverse.

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