Let $N_t$ denote a doubly stochastic Poisson (Cox) process with intensity process $\lambda_t$. Lando (1998) defines the time to first jump $\tau$ as:
$$\tau = \inf\{ t: \int_{0}^{t} \lambda_u du \geq E_1 \} $$
where $E_1$ is a unit exponential random variable. Why is this a good definition for a time to first jump? I can't figure it out yet.
Further, I will redefine $\tau$ to be just the "time for first jump" as I would normally imagine it to be. Then $P(\tau > t) = P(N_t = 0)$. Then, given that the distribution of a Cox process is:
$$ P(N_t - N_s = k) = \frac{(\int_s^t \lambda_u du)^k}{k!}e^{(-\int_s^t \lambda_u du)} $$
If we set $s=0$ and $k = 0$, then:
$$P(\tau > t) = P(N_t = 0) = e^{(-\int_s^t \lambda_u du)} $$
Why is it wrong to just solve the integral above? I would imagine that maybe it can't be solve for all cases, but maybe for some notable ones.