We use the acronym (UDD) for “uniform distributions of decrements”. Note that this is within the year, not over the entire age range. Under this assumption, for integer age (x) and fraction (t) (( 0 leq t lt 1)), we have
begin{eqnarray*}
_t p_x^{0j} = t times p_x^{0j}
end{eqnarray*}
For each decrement, exits are uniformly spaced over the year.
Now, if the (UDD) assumption holds for all decrements, then we may write
begin{eqnarray*}
~ _t p_x^{00} &=& 1 – sum_{j=1}^n ~ _t p_x^{0j} = 1 – sum_{j=1}^n t times p_x^{0j} \
&=& 1 – t times p_x^{0bullet} .
end{eqnarray*}
With Kolmogorov’s forward equation, we have
begin{eqnarray*}
frac{partial}{partial t} ~ _t p_x^{0j}= p_x^{0j}
= ~ _t p_x^{00} mu_{x+t}^{0j} .
end{eqnarray*}
Thus, we may express the transition force as
begin{eqnarray*}
mu_{x+t}^{0j} = frac{p_x^{0j}}{1 – t times p_x^{0bullet}} .
end{eqnarray*}