Distributional Properties of CUSUM Stopping Times and Stopped Processes

摘要

Let {Z(t), t ≥ 0} be a stochastic process with stationary independent increments and let T_1 = inf {t ≥ 0 : Z(t) - min_(0≤s≤t), Z(s) ≥ h}, h > 0. Under suitable conditions on Z(t), we obtain the joint Laplace transform of T_1 and Z(T_1) by deriving a formula for ψ_1(α, β) = E exp(αZ(T_1) - βT_1), where β > 0 and a is a suitable number in some interval containing zero. Furthermore, if T_2 is T_2 for Z~*(t) = -Z(t), then ψ_1(α, β) leads to a formula for ψ_2(α, β) = E exp(αZ(T_2) - βT_2). Moreover, if T = min(T_1, T_2), then a formula for ψ_1(α, β) = E exp(αZ(T) - βT) is also given in terms of ψ_1(α, β) and ψ_2(α, β). Our results provide quite direct and simple derivations of several results due to Taylor and others. Many of the isolated examples with detail theories of their own are unified by the given generalization. A number of examples are discussed.