Constructing Random Times with Given Survival Processes and Applications to Valuation of Credit Derivatives

Problem (P). Let (Ω ,G ,F,P) be a probability space endowed with the filtration F = (Ft)t∈R+ . Assume that we are given a strictly positive, càdlàg, (P,F)-local martingale N with N0 = 1 and an F-adapted, continuous, increasing process Λ , with Λ0 = 0 and Λ∞ = ∞, and such that Gt := Nte−Λt ≤ 1 for every t ∈ R+. The goal is to construct a random time τ on an extended probability space and a probability measure Q on this space such that: (i) Q is equal to P when restricted to F, that is, Q|Ft = P|Ft for every t ∈ R+, (ii) the Azéma supermartingale GQ t := Q(τ > t |Ft) of τ under Q with respect to the filtration F satisfies GQ t = Nte −Λt , ∀ t ∈ R+. (1)