If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be $$ \tau = \inf\{ u > 0 : X_u \geq B \}. $$ Can we express the Laplace transform (or its CDF) of $\tau$ in terms of $m$?

In particular, I am interested in the hitting time of the integrated CIR process $$ X_t = \int_0^t V_s \ ds $$ where $$ dV_t = (\alpha V_t + \beta) dt + \gamma \sqrt{V_t} dW_t. $$ The Laplace transform of $X_t$ is known in closed-form in this case, and given by $$ m_t(s) = \mathbb{E} e^{-s X_t} = \left( \frac{e^{-\alpha t/2}}{\cosh(Pt/2)-\frac{\alpha}{P}\sinh(Pt/2)} \right)^{2\beta/\gamma^2} \exp\left(-\frac{s V_0}{P} \frac{2 \sinh(Pt/2)}{\cosh(Pt/2)-\frac{\alpha}{P} \sinh(Pt/2)}\right) $$ where $P = \sqrt{\alpha^2 + 2 \gamma^2 s}$.

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    $\begingroup$ You may be able to use Dynkan's formula and then numerically integrate the resulting ODE. I doubt there is a closed form solution. $\endgroup$
    – user9403
    Oct 21, 2015 at 13:58

1 Answer 1


The Laplace transform of the integrated process CIR process is given by, see e.g. Dufresne (2001). you can download it

  1. The integrated square-root process
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    $\begingroup$ Nice paper. But how does this relate to the Laplace transform of the hitting time? $\endgroup$
    – Hans
    Jan 10, 2017 at 9:08

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