Appendix H
Discrete-Time Hazard
(Source: Cameron and Trivedi [1])
The starting point is to define the discrete-time hazard function as the probability of transaction at discrete time 
given survival to time 
:
where the superscript d denotes discrete, and where 
, an adjustment made because formally 
equals 
rather than 
.
The discrete-time survivor function is obtained recursively from the hazard function as
(H.2) 
For example, 
equals the probability of no transition at time 
times the probability of no transition at time 
conditional on surviving to just before 
, so that 
. The function 
is a decreasing step function with steps at 
.
The discrete-time cumulative hazard function is
(H.3) 
Using Equation H.1, we have that the discrete probability that the spell ends at 
is 
.