I am trying to calculate the following expectation : $$\mathbf{E}^{\mathbf{Q}}\left[ e^{-\int_{t_0}^{\tau} r_s ds} \mathbf{1}_{\{\tau < T\}} \mathscr{F}_{t_0}\right]$$
where $\tau$ is the first jump of a "Poisson" process with stochastic intensity $(\lambda_t)_t$. This implies that $\mathbf{Q}\left[ \left. \{ \tau\in [t,t+dt] \} \right| \tau > t \textrm{ and } \mathscr{F}_t\right] = \lambda_t dt$, at least when $\lambda$ si "deterministic".
The stochastic process $(r_t)_t$ is independent of the process $(\lambda_t)_t$.
(Both previous processes are adapted to a filtration $\left( \mathscr{F}_t \right)$ defined on a probability space with probability measure $\mathbf{Q}$. I suppose that $\tau > t_0$ and that $\tau$ is $\mathscr{F}_{t_0}$-measurable.)
I did the following calculation : \begin{equation}\begin{split} \mathbf{E}^{\mathbf{Q}}\left[ e^{-\int_{t_0}^{\tau} r_s ds} \mathbf{1}_{\{\tau < T\}}\right] & = \mathbf{E}^{\mathbf{Q}}\left[ \mathbf{E}^{\mathbf{Q}}\left[ \left. e^{-\int_{t_0}^{\tau} r_s ds} \mathbf{1}_{\{\tau < T\}}\right| \mathscr{F}_{t_0}\right] \right]\\ & = \mathbf{E}^{\mathbf{Q}}\left[\mathbf{1}_{\{\tau < T\}} \mathbf{E}^{\mathbf{Q}}\left[ \left. e^{-\int_{t_0}^{\tau} r_s ds} \right| \mathscr{F}_{t_0}\right] \right] \textrm{as $\tau$ is $\mathscr{F}_{t_0}$-measurable}\\ & = \mathbf{E}^{\mathbf{Q}}\left[\mathbf{1}_{\{\tau < T\}} P(t_0,\tau) \right] \end{split}\end{equation} where the last equality is based on the fact that $$\mathbf{E}^{\mathbf{Q}}\left[ \left. e^{-\int_{t_0}^{\tau} r_s ds} \right| \mathscr{F}_{t_0} \right] = P(t_0,\tau)\;\;(*),$$ where $P(t_0,t)$ is defined by $$P(t_0,t) = \mathbf{E}^{\mathbf{Q}}\left[ \left. e^{-\int_{t_0}^{t} r_s ds} \right| \mathscr{F}_{t_0} \right].$$ I can't succeed in proving (*) though.
Moreover, I would like to be prove that $\mathbf{E}^{\mathbf{Q}}\left[\mathbf{1}_{\{\tau < T\}} P(t_0,\tau) \right] = \mathbf{E}^{\mathbf{Q}}\left[ \left. e^{-\int_{t_0}^{\tau} (r_s + \lambda_s) ds} \right| \mathscr{F}_{t_0} \right]$