Is jump intensity inaccurate in describing random process?

Question

In https://en.wikipedia.org/wiki/It%C3%B4's_lemma

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? Is jump intensity not accurate/sufficient?

For example, the higher order terms are $h(Δt)^3$, but the jump intensity hides that, so why do we still use jump intensity?

Answer 1

Jump intensity is sufficient to describe a Poisson process. The intuitive reason is because the Poisson process is memoryless. For a tiny fraction of time, there is a tiny probability $h\Delta t$ of jumping, and if it doesn't jump, the next instant of time is another independent trial. So only the leading order in $\Delta t$ matters since only the limit $\Delta t\rightarrow 0$ matters.

You can say things about longer intervals of time, but they're uniquely implied by the infinitessimal description. For instance, the number of jumps in a time period $T$ is Poisson distributed with mean $hT.$

I should add that this is not only true for Poisson processes. For instance, modulated poisson processes and jump diffusion also involve an instantaneous jump intensity... it just can change or fluctuate randomly in time. The key is that it gives a momentary infinitessimal probability of jumping (i.e. an instantaneous rate).

Answer 2

You can actually derive the higher order terms. Say that the jump intensity is $h·Δt+C·Δt^2+O(Δt^3)$. Now subdivide into $N$ partial intervals of equal length. Then the probability in each of the small intervals is $h·Δt/N+C·Δt^2/N^2+O(Δt^3/N^3)$. The probability of no jump in all of the sub-intervals is $$ \bigl(1-h·Δt/N-C·Δt^2/N^2-O(Δt^3/N^3)\bigr)^N $$ which converges to $e^{-h·Δt}$ independent of the original higher order terms. This then just tells us that the higher order terms are those of the series expansion of the exponential, $$ 1-e^{-h·Δt}=h·Δt-\frac12h^2·Δt^2+\frac16h^3·Δt^3\mp… $$