Simulation of jump times of Cox process with CIR intensity

I want to sample the jump times of a Cox process where the intensity $\lambda(t)$ is given by a CIR process, i.e. $$\mathrm{d}\lambda(t)=\kappa(\theta-\lambda(t))\mathrm{d}t+\sigma\sqrt{\lambda (t)}\mathrm{d}W(t), \quad \lambda(0)=\lambda_0.$$ The distribution of the time of the first jump $\tau_1:=\inf\left\{t\Big\vert N\left(\int_0^t\lambda(s)\mathrm{d}s\right)=1\right\}$ can be computed as follows: $$\mathbb{P}\left(\tau_1\geq t\right) = \mathbb{P}\left(N\left(\int_0^t\lambda(s)\mathrm{d}s\right)=0\right) \\ =\mathbb{E}\left(\mathbb{E}\left(\mathbf{1}_{N\left(\int_0^t\lambda(s)\mathrm{d}s\right)=0}\Bigg\vert\lambda(s), 0\leq s\leq t\right)\right)\\ =\mathbb{E}\left(\exp\left(-\int_0^t\lambda(s)\mathrm{d}s\right)\right)\\ =\mathcal{L}(1)$$ with the Laplace-transform $\mathcal{L}(\mu)$ of $\int_0^t\lambda(s)\mathrm{d}s$ evaluated at 1. This function is known from the bond pricing literature and is given by $$\mathcal{L}(\mu)= \exp\left(\phi_\mu(t)-\lambda(0)\psi_\mu(t)\right)$$ where $$\phi_\mu(t) = \frac{2\kappa\theta}{\sigma^2}\log\left(\frac{2\gamma\mathrm{e}^{\frac{t(\gamma+\kappa)}{2}}}{\gamma-\kappa+\mathrm{e}^{\gamma t}(\gamma+\kappa)}\right)$$ and $$\psi_\mu(T)=\frac{2\mu\left(\mathrm{e}^{\gamma t}-1\right)}{\gamma-\kappa+\mathrm{e}^{\gamma t}(\gamma+\kappa)}$$ with $\gamma=\sqrt{\kappa^2+2\sigma^2\mu}$. Hence, I can easily derive the density of $\tau_1$ and may sample from it by some acceptance-rejection or inversion method. The problem is now the following: The density clearly involves $\lambda(0)$ and hence for the simulation of $\tau_2$ I would need to know $\lambda_{\tau_1}$ to proceed. I haven't found anything about how to sample $\lambda_{\tau_1}$ conditional on $\tau_1$ on the internet. Is there an algorithm to exactly sample the jump times of the Cox process with CIR intensity?

• @ibf_1994 why is the time of the first jump defined that way? I have read the Lando 1998 paper, but the intuition is still not clear – dleal Sep 4 '18 at 13:37