Introducing the Product Integral

(Feb 26, 2018)

Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:

\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]

Stephen covered the derivation of this in a previous blog, but I want to look more closely at the right-hand side of equation (1).  In particular, we can find an entirely different representation of \({}_tp_x\) as a product integral, which leads to many insights in survival models.

Recall how the integral in equation (1) is constructed.  Choose a partition of the interval \([0,t]\), that is some sequence \(\Delta_1,\Delta_2,\ldots,\Delta_n\) of non-overlapping sub-intervals that exactly cover the interval.  Define…

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Tags: survival models, survival probability, force of mortality, product integral

From small steps to big results

(Feb 1, 2018)

In survival-model work there is a fundamental relationship between the \(t\)-year survival probability from age \(x\), \({}_tp_x\), and the force of mortality, \(\mu_x\):

\[{}_tp_x = \exp\left(-\int_0^t\mu_{x+s}ds\right).\qquad(1)\]

Where does this relationship come from?  We start by extending the survival time by an amount, \(h\), and look at the \((t+h)\)-year survival probability:

\[{}_{t+h}p_x = {}_tp_x.{}_hp_{x+t}\qquad(2)\]

which is simply to say that in order to survive \((t+h)\) years, you first need to survive \(t\) years and then you need to survive a further \(h\) years.  Of course, surviving \(h\) years is the same as not dying in \(h\) years, so equation (2) can be written thus:

\[{}_{t+h}p_x…

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Tags: survival probability, force of mortality, differential equation

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