Under the section of Poisson jump processes, it is said that
We may also define functions on discontinuous stochastic processes. Let h be the jump intensity. The Poisson process model for jumps is that the probability of one jump in the interval [t, t + Δt] is hΔt plus higher order terms.
How can I know what the higher order terms are if we are just given the probability density being $h$? Is jump intensity (probability density) not accurate/sufficient?
Let's say the probability inside $(Δt)$ is
Therefore the probability density is
$h$, as we keep to first order only.
Therefore probability density hides some information from probability, so why do we still use probability density as it is inaccurate?