# 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?

• I think there is some confusion here, what do you mean by "higher than the one that actually occurs"? Is that empirical or another model? – emcor Sep 2 '14 at 23:11
• Thanks for getting back to me; $t_{i}$ and $t_{i+1}$ are the empirical occurrence times of two consecutive events hence, in the interval $(t_{i},t_{i+1}]$, only one event occurs. – g_puffo Sep 2 '14 at 23:32