Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $P = \sqrt{\alpha^2 + 2 \gamma^2 s}$.

share|improve this question
Maybe this question should be asked on MSE or even on mathoverflow. – user8 Jul 1 '13 at 8:35
You may be able to use Dynkan's formula and then numerically integrate the resulting ODE. I doubt there is a closed form solution. – user9403 Oct 21 '15 at 13:58

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
share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.