From small steps to big results

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:

\[{}_{t+h}p_x = {}_tp_x.(1-{}_hq_{x+t}).\qquad(3)\]

If the period \(h\) is small enough, we can express the probability of dying, \({}_hq_{x+t}\), in terms of the force of mortality, \(\mu_{x+t}\):

\[{}_hq_{x+t} = h.\mu_{x+t}+o(h)\qquad(4)\]

where the function \(o(h)\) collects second- and higher-order powers of \(h\) and, crucially, is such that:

\[\lim_{h\to0^+}\frac{o(h)}{h} = 0\qquad(5)\]

i.e. \(o(h)\) tends to zero faster than \(h\) does.  If we substitute equation (4) into equation (3) and re-arrange we get the following:

\[\frac{{}_{t+h}p_x-{}_tp_x}{h} = -{}_tp_x\mu_{x+t} + \frac{o(h)}{h}.\qquad(6)\]

We can now let \(h\to0^+\) and make use of equation (5):

\[\lim_{h\to0^+}\frac{{}_{t+h}p_x-{}_tp_x}{h} = -{}_tp_x\mu_{x+t}.\qquad(7)\]

The left-hand side of equation (7) is the definition of the first partial derivative of \({}_tp_x\) with respect to \(t\), so we have an ordinary differential equation (ODE) of degree 1 and order 1:

\[\frac{\partial}{\partial t}{}_tp_x = -{}_tp_x\mu_{x+t}.\qquad(8)\]

We are nearly there, as the solution to equation (8) is:

\[{}_tp_x = \exp\left(-\int_0^t\mu_{x+s}ds\right)+C\qquad(9)\]

where \(C\) is the constant of integration.  However, we also have a boundary condition: since the probability of dying in a time interval of length zero is zero, \({}_0p_x=1\).  From this we know that \(C=0\) in equation (9) and thus we have the result in equation (1) at the start of this posting.

 

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Stephen Richards
Stephen Richards is the Managing Director of Longevitas
Model types in Longevitas
Longevitas users can choose between seventeen types of survival model (μx) and seven types of GLM (qx). In addition there are a further seven extensions of the GLM models for qx to span multi-year data without violation of the independence assumption. Longevitas also offers non-parametric analysis, including Kaplan-Meier survival curves and traditional A/E comparisons against standard tables.