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Interpreting Integrated Intensity Between Consecutive Events

Article Quant Q&A · Author: g_puffo
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Summary

The document asks how to interpret the integral of a stochastic process's intensity over the interval between two consecutive arrivals. It contrasts the usual interpretation of integrated intensity as an expected event count over a fixed time window with a claim from irregularly spaced financial data modeling that a value above one indicates the fitted intensity underestimates arrivals.

The central issue is that the interval endpoints are event times, so the realized count between consecutive events is fixed at one, while the integrated intensity is a time-rescaled waiting interval used in model diagnostics. The question raises a useful distinction between expected counts over deterministic intervals and compensator increments evaluated at random event times. It does not provide a resolution or empirical evidence, so readers would need further probability theory context to assess the cited interpretation.

Key ideas

  • Integrated intensity over a fixed time interval represents the expected event count under suitable process assumptions.
  • Between consecutive arrival times, the observed event count is one by construction.
  • A compensator increment at random arrival times should not be interpreted exactly like an expected count over a fixed window.
  • The document raises a diagnostic interpretation question but does not resolve it.

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# Intensity Function of Stochastic Processes


# Intensity Function of Stochastic Processes












I'm fitting some financial data to a model based on a stochastic process and evaluating the fit of it by looking at the compensator. However, I cannot understand well what does it mean to take the integral of the intensity function associated to a stochastic process. My understanding is that if $\lambda(t)$ is the intensity function of a stochastic process, then $ \int_0^T\lambda(t)dt$ tells me the expected number of occurrences in the time interval $(0,T)$. For example, if I have a Poisson process with parameter $\lambda$ then in a time interval of 3 units of time we expect to have $3\lambda$ arrivals. Following this logic, the compensator $$\Lambda_{(t_i,t_{i+1})}=\int_{t_i}^{t_{i+1}}\lambda(t)dt$$ should simply tell me the expected number of occurences in the time interval $(t_i,t_{i+1})$. My problem starts now: if $t_{i}$ and $t_{i+1}$ are two consecutive arrival times and if $\Lambda_{(t_i,t_{i+1})}>1$ it should mean that the expected number of arrivals predicted by my process is higher than the one that actually occurs (hence, that the process over-estimates the nuber of occurences in the time interval).But, according to "Modelling Irregularly Spaced Financial Data" by Hautsch, when $\Lambda_{(t_i,t_{i+1})}>1$ it actually means that the intensity function under-estimates the number of arrivals in the time interval which "fits" with the idea that the compensator actually tells me the expected inter-arrival time. Any suggestions?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.