Life Distribution Properties of Devices Subject to a Lévy Wear Process

Assume that a device is subject to wear. Over time the wear is assumed to be an increasing Lévy process (X_t). Suppose the device has a threshold Y with right-tail probability Ḡ. Let ζ be the failure time of the device and F̄_x be its survival probability given that X₀ = x. It is shown that life distribution properties of Ḡ are inherited as corresponding properties of F̄_x for each x ∈ R₊. Optimal replacement policies for such devices are discussed for suitably chosen cost functions when Ḡ is absolutely continuous on R₊ with a bounded failure rate.