The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture stochastic volatility and the leverage effect, they satisfy the following SDEs, respectively:

  1. CIR

$$dX_t=\kappa(\theta-X_t)dt +\sigma\sqrt{X}_tdW_t, X_0=x $$ Here $2\kappa\theta \geq \sigma^2$ is assumeed to gurantee the positivity of $X_t$.

  1. CEV

$$ dX_t=\mu X_t dt+\sigma X_t^{\beta}dW_t, X_0=x $$ where $\sigma>0,\beta>0$.

For $0<a$, let $$I_a=\inf\{t\geq 0: \displaystyle\int_0^t X_s ds=a\}$$ i.e, $I_a$ is the first hitting time of the integral function of $X_t$, where $X_t$ could be CIR or CEV process.

Due to the importance of the Laplace transform of $I_a$,, for example, people can find the value of Asian call option, so I am very intersted in the Laplace transform of $I_a$, i.e,

Find :
$$ \mathbb{E}_x[e^{-\lambda I_a} ] $$ where $\mathbb{E}_x=\mathbb{E}[.|X_0=x]$, and $\lambda>0$.

Adam Metzler in this paper finds the $\mathbb{E}_x[e^{-\lambda I_a}]$ when $X_t$ is a Geometric Brownian Motion, I tried to follow his approach but have not succeeded yet. In addition, I have also tried to find this in the open literature but I cannot find this. I would like to ask if some one could show me a reference or give me a hint on how to find the Laplace transform. Thank you so much for your time.


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