Thiele's differential equation

Imagine watching a policy's reserve continuously, second by second, instead of only once a year. At every instant three things tug on the reserve: interest is being earned on it, premiums are flowing in, and there is a constant tiny risk that a claim must be paid out. Thiele's differential equation is the exact rule for how the reserve changes at each instant under these three forces — the continuous-time version of the year-by-year reserve recursion.

Precisely, for a continuously paid policy the equation states that the rate of change of the reserve V(t) equals interest earned plus premium received minus the expected cost of paying out the difference between the benefit and the current reserve to anyone who dies right then. In symbols, dV/dt = delta * V(t) + P(t) - mu(x+t) * (b(t) - V(t)), where delta is the force of interest, P(t) the premium rate, mu the force of mortality, and b(t) the benefit. The last term is the instantaneous 'cost of insurance': for each death the insurer must top the reserve up to the full benefit, and that net-amount-at-risk (b - V) costs mu per unit time.

Thiele's equation, published by the Danish actuary T. N. Thiele in 1875, is the theoretical backbone of modern life-contingencies and the natural language for multi-state Markov models, where one such equation is written per state. In practice reserves are still usually computed by the discrete recursion or by numerically integrating Thiele, but the equation gives the clearest conceptual picture of WHY a reserve moves: it separates the three drivers (interest, premium, mortality risk) into three clean terms, exposing the engine inside every reserve.