14 Survival and hazard semigroups
On this page · Permalink · Source line 1583
Let \(A\) be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states; eventual absorption is a separate hypothesis. Let \(p_0\) be an initial transient-state distribution. Define
The exit-rate row vector is
Theorem 14.1: Semigroup survival identities
For \(t\ge0\), the finite-time exit density and, wherever \(S(t)>0\), the hazard are
Consequently, \(S\) is nonincreasing and
whenever \(S(t)>0\). The density may be defective: its total mass is \(1-\lim_{t\to\infty}S(t)\). It is a probability density when absorption is eventual, so that this limit is zero.
Read proof · Visual guide: A zero initial hazard can still lead to certain exit · Read statement and proof side by side · Permalink · Source line 1596 · Claim record · Lean coverage · Code · Worked examples
Proof of Theorem 14.1
Differentiate:
The hazard identity is the definition of conditional instantaneous exit rate. Integrating \((\log S)'=-h\) gives the final formula.
Back to statement · Read statement and proof side by side · Permalink · Source line 1612 · Claim record · Lean coverage · Code · Worked examples
