### 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…

### 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…