Forza di mortalità

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La forza di mortalità (nota anche come tasso istantaneo di mortalità) è una grandezza della matematica attuariale che si ottiene calcolando il limite per h che tende a zero del quoziente di mortalità.

Per definizione, è noto che la probabilità affinché una testa di età x muoia entro l'età x+dx è data da:

$\mu dx={\frac {l(x)-l(x+dx)}{l(x)}}=-{\frac {l(x+dx)-l(x)}{l(x)}}$

Se per ipotesi si considera $\ l(x)$ derivabile e con derivata prima continua, è possibile, commettendo un errore a meno di infinitesimi di ordine superiore a dx, approssimare l'incremento a numeratore con il differenziale della funzione $\ l(x)$ , pari a $\ l'(x)$ $\ dx$ .

Quindi:

$\mu dx=-{\frac {l'(x)}{l}}\cdot dx$

dove:

\mu =-{l'(x) \over l(x)}=-{d \over dx}\log l(x)

Il significato della seguente espressione è fondamentale: commettendo un errore a meno di infinitesimi di ordine superiore a dx, la probabilità di morire entro un intervallo dx è proporzionale a dx.

Dalla forza di mortalità si può agevolmente passare alla legge di sopravvivenza $\ l_{x}$ .

Infatti, integrando l'equazione precedente in un intervallo compatto [0,x] si ottiene:

[\log l]_{0}^{x}=-\int _{0}^{x}{\mu dt}

Sfruttando le proprietà dei logaritmi, si può riscrivere l'equazione come:

\log {\frac {l}{l_{0}}}=-\int _{0}^{x}{\mu dt}

E quindi:

l_{x}=l_{0}\mathrm {e} ^{-\int _{0}^{x}{\mu dt}}

Bibliografia

Lezioni di matematica e tecnica attuariale, R. Cultrera

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