# Thiele differential equation

Consider an $n$ year term life insurance, with sum insured $S$ and level premium $P$ per time unit, issued at time $0$ to an $x$ years old person. Denote by $\mu_y$ the force of mortality at age $y$ and by $\delta$ the force of interest. If the insured is still alive at time $t \in [ 0 , n )$, then the insurer must provide a reserve, $V _ { t }$, which by statute is the mean value of future discounted benefits less premiums. Splitting into payments before and after time $t + d t$ leads to

\begin{equation} \tag{a1} V _ { t } = \mu _ { x + t} d t S - P d t + \end{equation}

\begin{equation*} + ( 1 - \mu _ { x + t }d t ) e ^ { - \delta d t } V _ { t + d t } + o ( d t ), \end{equation*}

from which one obtains that $V _ { t }$ is the solution to

\begin{equation} \tag{a2} \frac { d } { d t } V _ { t } = P + \delta V _ { t } - \mu _ { x + t} ( S - V _ { t } ), \end{equation}

subject to the condition $V _ { n } = 0$.

This is the celebrated Thiele differential equation, proclaimed "the fundament of modern life insurance mathematics" in the authoritative textbook [a1], and named after its inventor Th.N. Thiele (1838–1910). It dates back to 1875, but was published only in 1910 in the obituary on Thiele by J.P. Gram [a2], and appeared in a scientific text [a7] only in 1913.

As is apparent from the proof sketched in [a1], Thiele's differential equation is a simple example of a Kolmogorov backward equation (cf. Kolmogorov equation), which is a basic tool for determining conditional expected values in intensity-driven Markov processes. Thus, today there exist Thiele differential equations for a variety of life insurance products described by multi-state Markov processes and for various aspects of the discounted payments, e.g. higher order moments and probability distributions. The technique is an indispensable constructive device in theoretical and practical life insurance mathematics and also forms the basis for numerical procedures, see [a8].

Thiele was Professor of Astronomy at the University of Copenhagen from 1875, cofounder and Director (actuary) of the Danish life insurance company Hafnia from 1872, and first president of the Danish Actuarial Society founded in 1901. In 52 written works (three monographs; [a12], [a11], [a13]) he made contributions (a number of them fundamental) to astronomy, mathematical statistics, numerical analysis, and actuarial mathematics. Biographical/bibliographical accounts are given in [a3], [a4], [a5], [a6], [a9], [a10].

#### References

[a1] | A. Berger, "Mathematik der Lebensversicherung" , Springer Wien (1939) |

[a2] | J.P. Gram, "Professor Thiele som aktuar" Dansk Forsikringsårbog (1910) pp. 26–37 |

[a3] | A. Hald, "T.N. Thiele's contributions to statistics" Internat. Statist. Rev. , 49 (1981) pp. 1–20 |

[a4] | A. Hald, "A history of mathematical statistics from 1750 to 1930" , Wiley (1998) |

[a5] | J.M. Hoem, "The reticent trio: Some little-known early discoveries in insurance mathematics by L.H.F. Oppermann, T.N. Thiele, and J.P. Gram" Internat. Statist. Rev. , 51 (1983) pp. 213–221 |

[a6] | "Leading personalities in statistical science" N.L. Johnson (ed.) S. Kotz (ed.) , Wiley (1997) |

[a7] | N.R. Jørgensen, "Grundzüge einer Theorie der Lebensversicherung" , G. Fischer (1913) |

[a8] | R. Norberg, "Reserves in life and pension insurance" Scand. Actuarial J. (1991) pp. 1–22 |

[a9] | R. Norberg, "Thorvald Nicolai Thiele, statisticians of the centuries" , Internat. Statist. Inst. (2001) |

[a10] | T. Schweder, "Scandinavian statistics, some early lines of development" Scand. J. Statist. , 7 (1980) pp. 113–129 |

[a11] | T.N. Thiele, "Theory of observations" , Layton, London (1903) (Danish edition 1897 (Reprinted in: Ann. Statist. 2 (1931), 165-308)) |

[a12] | T.N. Thiele, "Elementær Iagttagelseslære" , Gyldendal, Copenhagen (1897) |

[a13] | T.N. Thiele, "Interpolationsrechnung" , Teubner (1909) |

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Thiele differential equation.

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