 Survival probability

For a nonnegative first-passage time $T_A$, possibly infinite, the survival probability isIt describes the Probability that absorption has not yet occurred. The tail-integral identityholds as an identity of extended nonnegative quantities, even when absorption is not almost sure or the expected time is infinite. Indeed, $T_A={\int }_0^{\infty }{1}_{\{T_A>t\}}dt$ pathwise, and Tonelli's theorem exchanges expectation and integration. For an integer-valued hitting time, the corresponding formula is ${\mathbb{E}}_x{T}_A={\sum }_{n=0}^{\infty }{\mathbb{P}}_x(T_A>n)$.

If the finite-time distribution is absolutely continuous, its first-passage-time density is $-{\partial }_{t}Q(x,t)$. Where $Q>0$, its hazard function is $-{\partial }_t\log Q$. Constant hazard $\lambda >0$ gives $Q(t)=e^{-\lambda t}$ and mean $1/\lambda$.

The limit of $Q(x,t)$ as $t\to \infty$ is the Probability of never being absorbed; a positive limit forces infinite mean. Conversely, zero limit does not guarantee a finite mean. Brownian first passage to a positive level occurs almost surely, but its survival tail is proportional to $t^{-1/2}$, whose integral diverges.