How do the general multiple state probability properties reduce to the multiple decrement case?
1. Just to reinforce the fundamentals, in a multiple decrement probability model, we have
begin{eqnarray*}
_t p_x^{j0}=0, ~ j=1,ldots,n text{ and } _t p_x^{jj}=1, ~ j=1,ldots,n.
end{eqnarray*}
2. For state (i=0) (alive or active), transition probabilities are equal to occupancy probabilities. That is,
begin{eqnarray*}
~ _t p_x^{00} = ~ _t p_x^{overline{00}} .
end{eqnarray*}
This is because it is not possible to return to state 0 after having left it (all other states are absorbing).
As a consequence, we may express
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).
3. The Chapman-Kolmogorov equations reduce to
begin{eqnarray*}
_{m+n} p_x^{0j} &=& sum_{k=0}^n ~ _m p_x^{0k} times ~_n p_{m+x}^{kj}
= ~ _m p_x^{00} times ~ _n p_{m+x}^{0j} + ~ _m p_x^{0j} .
end{eqnarray*}
4. Kolmogorov’s forward equation reduces to
begin{eqnarray*}
frac{partial}{partial t} ~ _t p_x^{0j}=
sum_{k=0, k ne j}^n
left(
_t p_x^{0k} mu_{x+t}^{kj}
– ~ _t p_x^{0j} mu_{x+t}^{jk}
right)
= ~ _t p_x^{00} mu_{x+t}^{0j}.
end{eqnarray*} Thus, we may write
begin{eqnarray*}
_t p_x^{0j}=
int_0^t ~ _s p_x^{00} mu_{x+s}^{0j} ~ ds .
end{eqnarray*}