Under this assumption, transition forces are constant within the year,
begin{eqnarray*}
mu_{x+t}^{0j} = mu_{x}^{0j} , text{ for } 0 leq t lt 1 .
end{eqnarray*}
Recall that
begin{eqnarray*}
~ _t p_x^{00} &=& exp left{- int_0^t sum_{j=1}^n mu_{x+s}^{0j} ~ds
right}
= exp left{- int_0^t mu_{x+s}^{0bullet} ~ds right},
end{eqnarray*}
where (mu_{x}^{0bullet}= sum_{j=1}^n mu_{x}^{0j}) is the total force of transition out of state 0 at age (x). Thus, under the constant force assumption, we have
begin{eqnarray*}
~ _t p_x^{00} &=& exp left{-t times mu_{x}^{0bullet}right},
end{eqnarray*}
and, taking the limit as (t rightarrow 1), we have (p_x^{00} = e^{-
mu_{x}^{0bullet}}). Thus, we can write
begin{eqnarray*}
~ _t p_x^{00} &=& left{p_x^{00}right}^t .
end{eqnarray*}
Further,
begin{eqnarray*}
_t p_x^{0j} &=& int_0^t ~ _s p_x^{00} mu_{x+s}^{0j} ~ ds =
mu_{x}^{0j} int_0^t ~ e^{- s mu_{x}^{0bullet}} ~ ds \
&=& mu_{x}^{0j} frac{1-e^{- tmu_{x}^{0bullet}}}{mu_{x}^{0bullet}}~=~ frac{mu_{x}^{0j}}{mu_{x}^{0bullet}} (1- (p_x^{00})^t)
end{eqnarray*}
Again, taking the limit as (t rightarrow 1), we have
begin{eqnarray*}
frac{ p_x^{0j}}{1- p_x^{00}} = frac{ p_x^{0j}}{p_x^{0bullet}}=
frac{mu_{x}^{0j}}{mu_{x}^{0bullet}} .
end{eqnarray*}
Thus, we may write
begin{eqnarray*}
_t p_x^{0j} = frac{ p_x^{0j}}{p_x^{0bullet}} left(1- (p_x^{00})^tright) .
end{eqnarray*}