Multiple Decrement Model
- \(_tp_x^{(\tau)}=\displaystyle\prod_{j=1}^m{_tp'^{(j)}_x}=\exp\left\{-\displaystyle\int_0^t\sum_{j=1}^m\mu_{x+s}^{(j)}\ ds\right\}=1-{_tq^{(\tau)}_x}\)
- \(_tq_x^{(j)}=\displaystyle\int_0^t{_sp^{(\tau)}_x}\mu_{x+s}^{(j)}\ ds\)
- \(\ell_{x+t}=\ell_x\ {_tp^{(\tau)}_x}\)
- \(d_x^{(j)}=\ell_xq^{(j)}_x\)
Associated Single Decrement
- \(_tp'^{(j)}_x=\exp\left\{-\displaystyle\int_0^t\mu_{x+s}^{(j)}\ ds\right\}\)
- \(_tq'^{(j)}_x=\displaystyle\int_0^t{_sp'^{(j)}_x}\mu_{x+s}^{(j)}\ ds\)
Fractional Age Assumptions
Under both UDD in multiple decrement model and CF, \[p'^{(j)}_x=\left(p^{(\tau)}_x\right)^{q^{(j)}_x/q^{(\tau)}_x}\quad\longleftrightarrow\quad q^{(j)}_x=\frac{\ln p'^{(j)}_x}{\ln p^{(\tau)}_x}q^{(\tau)}_x\]
UDD in multiple decrement model
- \(_tq^{(j)}_x=tq^{(j)}\)
- \(_tp^{(\tau)}_x=1-tq^{(\tau)}_x\)
- \(_tp^{(\tau)}_x\mu_{x+t}^{(j)}=q^{(j)}_x\)