Hitting Times for Integrated CIR Processes and Laplace Transforms

Summary

The document asks whether the Laplace transform of an increasing process can be used to obtain the distribution or Laplace transform of its first passage time to a fixed threshold. It focuses on the integrated Cox–Ingersoll–Ross process, formed by accumulating a square-root diffusion over time. The process dynamics and a closed-form expression for the Laplace transform of the accumulated value are provided, with the CIR parameters and initial state appearing in that expression.

The included response does not derive the hitting-time distribution or explain how to invert or otherwise use the transform for that purpose. It only points to a reference for the integrated square-root process transform, which is already stated in the question. Thus the document is mainly a mathematical problem statement and a starting point for research. A key limitation is that knowing the transform of the process at a fixed time does not, by itself in this exchange, establish a method for finding the first-passage-time law.

Key ideas

• The integrated CIR process accumulates the values of a square-root diffusion through time.
• The document supplies a closed-form Laplace transform for the integrated process at a fixed time.
• It asks how to derive the first time the accumulated process reaches a specified threshold.
• The provided response cites a source for the integrated process transform but does not solve the hitting-time question.
• A fixed-time transform and a first-passage-time distribution are distinct quantities.

Tags

# Distribution of hitting time of the integrated CIR process


# Distribution of hitting time of the integrated CIR process












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

## Answer by user16891 (score -1)

https://quant.stackexchange.com/a/18988

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

- The integrated square-root process

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.