Decomposition and embedment of trajectories after explosion for a birth and death process

Let X = X(t),t < σ (σ is lifespan) be a birth and death process with explosion whose characteristic triple is (α, C, D). For a set $\bar M \subset \bar E = \left\{ {0, 1, 2, \cdots , \infty } \right\}$ , reserving the trajectories before the first explosion time τ and decomposing the trajectories after τ for X according to $\bar M$ we obtain a new birth and death process $_{\bar M} X = \left\{ {_{\bar M} X\left( t \right), t < _{\bar M} \tau } \right\}$ . We calculate the average lifetime after τ for $_{\bar M} X$ and corresponding characteristic triple $\left( {_{\bar M} \alpha , _{\bar M} C, _{\bar M} D} \right)_{\bar M} X$ of $_{\bar M} X$ in terms of (α,C, D) and $\bar M$ . This means that a lot of given birth and death processes can be embedded in one and the same birth and death process. If $k \in \bar E$ and $\bar M = \left\{ k \right\}$ , we decompose X into $_k X, k \in \bar E$ .

Let X = {X(t),t ＜σ} (σ is lifespan) be a birth and death process with explosion whose characteristic triple (Mα,MC,MD) of MX in terms of (α, C, D) and M. This means that a lot of given birth and death processes can be embedded in one and the same birth and death process. If κ∈ E and M = {κ},we decompose X into κX, κ∈ E.