Discount factor with compound Poisson process

Question

I am trying to solve the following: I would like to compute the discount factor where the short rate is defined by a compound Poisson process. Consider the discount factor from $0$ to $T$ denoted by $P(t,T)$. Let $r_t$ be a compound Poisson process modelling the short rate. Its arrival time has a constant intensity $\lambda$, its jumps are denoted by $\delta_i$ and the jumps occur at time $T_i$. The jumps are i.i.d and independent from the intensity. The discount factor is defined by: $P(t,T)=\mathbb{E}\left[exp(-\int_t^T r_u du \right]$. Let $N(t)$ be the number of jumps between $0$ and $T$. In this case, the area is: $\sum_{i=0}^{N(t)} \delta_i (T - T_i ) $, introducing $\delta_0 = r_0$ and $T_0 = 0$. Now to compute the expectation, I should use the total probability theorem,i.e. conditioning on the number of jumps: $P(t,T) = \sum_{k} \mathbb{E} \left[- exp( \sum_{i=0}^{k} \delta_i (T - T_i ) )\right] P(N(t)=k)$ Do you have any clues how I can go further? Many thanks for the help. Gigio