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Actuaries denote with \({}_tp_x\) the probability that a life alive aged exactly \(x\) years will survive a further \(t\) years or more. The most basic result in survival analysis is the following relationship with the instantaneous mortality hazard, \(\mu_x\):
\[{}_tp_x = e^{-H_x(t)}\qquad(1)\]
where \(H_x(t)\) is the integrated hazard:
\[H_x(t) = \int_0^t\mu_{x+s}ds\qquad(2).\]
Simulation and survival
In an earlier post we discussed how a survival model was directly equivalent to assuming future lifetime was a random variable. One consequence of this is that survival models make it quick and simple to simulate a policyholder's future lifetime for the purposes of ICAs and Solvency II.