Question about the likelihood function for a semi-Markov model

Basically I have an observed state sequence and (continuous) transition times, and I'd like to obtain a likelihood function for this data.

The semi-Markov model is similar to the normal Markov chain, except the time durations spent in each state are chosen from a specific distribution. The parameters are: the state transition probabilities a_ij, the initial state distribution pi and the holding time distribution density for state i f_i(t).

I feel the likelihood should be something like this, but I've seen references where they use the cumulative distribution function instead of the probability density f_i(t).

Using the cumulative density makes sense when the observation times are arbitrary, but I'm not sure it's appropriate when you know the transition times and want to do MLE or something.

Which should it be?