2. UDD in the MDT

Another way to build a multi-decrement table from the associated single decrement tables (and vice-versa) is to assume uniform distribution of decrements (UDD) in the multi-decrement table (MDT).
Under this assumption, for integer age (x) and fraction (t) (( 0 leq t lt 1)), we have
begin{eqnarray*}
_t q_x^{(j)} = t times q_x^{(j)} .
end{eqnarray*}
If the (UDD) assumption holds for all decrements, then
begin{eqnarray*}
~ _t q_x^{(tau)} = 1 – t times q_x^{(tau)} .
end{eqnarray*}Recall Kolmogorov’s forward equation (
frac{partial}{partial t} ~ _t q_x^{(j)} = frac{partial}{partial t} ~ _t p_x^{0j}
= ~ _t p_x^{00} mu_{x+t}^{0j}= ~ _t p_x^{(tau)} mu_{x+t}^{(j)}). Thus, we may express the transition force as
begin{eqnarray*}
mu_{x+t}^{(j)} = frac{q_x^{(j)}}{1 – t times q_x^{(tau)}} .
end{eqnarray*}
With ( q_x^{prime(j)} = 1- exp left{ – int_0^1 mu_{x+t}^{(j)} ~ dt right} ), we have
begin{eqnarray*}
– ln(1-q_x^{prime(j)}) &=& int_0^1 mu_{x+t}^{(j)} ~ dt \
&=& int_0^1 frac{q_x^{(j)}}{1 – t times q_x^{(tau)}} ~ dt ~=~ frac{q_x^{(j)}}{q_x^{(tau)}} (-ln(1- q_x^{(tau)})) .
end{eqnarray*}
Thus, as with the constant transition force, we have
begin{eqnarray*}
1- q_x^{prime(j)} &=& left( p_x^{(tau)}right)^{q_x^{(j)}/q_x^{(tau)}} .
end{eqnarray*}

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