At the moment I am trying to understand constinuous-time Markov chains, where the jump rate is not a constant, but a function in time and position. Let $Q_t$ denote the Markov chain, $T_1<T_2< \dots$ the sequence of jump times, such that the random variables $T_1, T_2-T_1, \dots , T_{k+1}-T_k,\dots$ are independent, and $\sigma(t,x)$ the jump rate. Then the random variables $T_{k+1}-T_k$ are not exponentially distributed (as in easier cases), and one has to find the density function of the waiting times.
In the book "Continuous-time Markov chains and Applications" from G. George Yin and Qing Zhang they describe the density function of $T_1$ with $$\rho_{T_1}(t_1)= \mathbf{1}_{\{t_1>0\}}\exp\left(-\int_0^{t_1}\sigma(s,x)\,\text{d}s \right)\cdot \sigma(t_1,x). $$
My question now would be the following: Is there something one could say about the expected value or the variance of $T_1$? The definition of the expected value would give
$$ \mathbb{E}(T_1)= \int_{R}t_1\cdot\exp\left(-\int_0^{t_1}\sigma(s,x)\,\text{d}s \right)\cdot \sigma(t_1,x)\,\text{d}t_1$$
which seems rather unpleasant to compute.