Status symbols

One of the most basic objects in a probabilistic model is the indicator \(1_{\cal A}\) of an event \(\cal A\):

\[ 1_{\cal A} = \left\{ \begin{array}{ll} 1 & \mbox{if ${\cal A}$ occurs} \\ 0 & \mbox{if ${\cal A}$ does not occur}. \end{array} \right. \]

Indicators are useful things. They allow us to switch between probability and expectation:

\[ \text{Prob}[{\cal A}] = \text{E}[1_{\cal A}] \]

and they let us to do arithmetic on 'things' that are not numeric in nature. For example, if \({\cal A}_i\) is the event 'coin comes up heads in the \(i\)th of a run of tosses' then \(\sum_i 1_{{\cal A}_i}\) is the total number of heads. Try doing that with the wordy definition.

Time introduces a new dimension. A contingent event is one that has not happened by some starting time that we label \(t=0\), and either has or has not happened by any later time \(t>0\). So we have a whole collection of events, that we may denote by \(\{ {\cal A}_t \}_{t \ge 0}\), defined by their indicators:

\[ 1_{{\cal A}_t} = \left\{ \begin{array}{ll} 1 & \mbox{if ${\cal A}$ occurs by time $t$} \\ 0 & \mbox{if ${\cal A}$ has not occurred by time $t$}. \end{array} \right. \]

In fact, these objects are so much more convenient than the 'wordy' definitions that it would be handy to have a compact notation for the probabilities that they have or have not occurred. How about:

\[\begin{eqnarray*}
{}_tq_{\cal A} & = & \mbox{Probability that ${\cal A}$ has occurred by time $t$} \\
{}_tp_{\cal A} & = & \mbox{Probability that ${\cal A}$ has not occurred by time $t$?}
\end{eqnarray*}\]

Does this look familiar? It is the standard International Actuarial Notation, agreed upon at the Third International Congress of Actuaries in Paris in 1900, and dating back to King's textbook for the Institute of Actuaries (King, 1887). The notation recognized a 'status' that exists until it is ended by the occurrence of a defined event; therefore let  \({\cal A}\) above denote the collection of events that terminate a status. Canonical examples of statuses were:

Hence the standard actuarial notation, such as:

\[\begin{eqnarray*}
{}_tq_{x} & = & \mbox{Probability that status $x$ has ended by time $t$} \\
{}_tq_{x:y} & = & \mbox{Probability that status $x\!\!:\!\!y$ has ended by time $t$} \\
{}_tp_{\overline{x:y}} & = & \mbox{Probability that status $\overline{x\!\!:\!\! y}$ has not ended by time $t$}.
\end{eqnarray*}\]

Missing in 1900 was a proper definition of the events underlying the probabilities, 'proper' meaning a mathematical rather than a wordy description. In fact, we need an indicator like that in equation (28) of Macdonald & Richards (2025), for example:

\[\begin{eqnarray*}
Y_{x}(t) & = & 1_{\{{\rm Status\ } x {\rm\ still\ exists\ at\ time\ } t^-\}} \\
Y_{x:y}(t) & = & 1_{\{{\rm Status\ } x:y {\rm\ still\ exists\ at\ time\ } t^-\}} \\
Y_{\overline{x:y}}(t) & = & 1_{\{{\rm Status\ } \overline{x:y} {\rm\ still\ exists\ at\ time\ } t^-\}}.
\end{eqnarray*}\]

If the reader objects, and says "but the definition still contains words, they have just been moved into a subscript", we answer "true, but misses the point — I can do useful things with \(Y_{x}(t)\), like integrate it over time and add up the results over a sample of lives — try doing that with just the words."

The definition of 'status'  and the standard notation was a neat idea, especially when we remember the really convoluted statuses that actuaries then had to handle, involving three, four or more lives. It encapsulated the probabilities of complicated contingent events, decades before Kolmogorov (1933) gave proper meanings to the words 'event' and `probability' and laid the foundations of stochastic processes. Actuaries, we might say, had the right status symbols, somewhat ahead of events.