Martingales Associated to Peacocks Using the Curtain Coupling

We consider right-continuous peacocks, that is families of real probability measures (μt)t that increase in the convex order. Given a sequence of time partitions we associate the sequence of martingales that are Markovian, constant on the partition intervals [tk, tk+1[, and designed in a way that the transition kernels at times tk+1 are the curtain couplings of marginals μtk and μtk+1 . We study the limit “curtain” processes obtained when the mesh of the partition goes to zero and ask for existence, uniqueness and relevancy with respect to the original data. We provide a proof of existence of such limits for the finite-dimensional convergence. Under certain additional regularity assumptions, we prove that there is a unique limit curtain process and it is a Markovian martingale. We first consider peacocks consisting of uniform measures that we study by hand. It enters in and represents a class of peacocks studied with general techniques in a parallel work by Henry-Labordère, Tan and Touzi [11]. We obtain the same type of results for the limit curtain processes associated to a class of analytic discrete peacocks. Finally, modifying the previous peacocks we construct one discrete and one continuous (counter)example that generate limit curtain processes that are non-Markovian martingales. Since the seminal article [18] by Hans G. Kellerer it is exactly known what are the families (μt)t of real probability measures that can coincide with (Law(Xt))t when Xt is a martingale. A simple application of the conditional Jensen inequality reveals a necessary condition: The functions t 7→ μt have to be nondecreasing in convex order. Indeed if φ is a convex function ∫ φdμs = E(φ(Xs)) ≤ E(φ(Xt)) = ∫ φdμt. The opposite inclusion is true and it is the first conclusion of Kellerer Theorem. The second aspect is central in Kellerer’s article as attests its title: Markov-Komposition und eine Anwendung auf Martingale (Markov composition and an application to martingales). In fact the martingale (Xt)t associated to μt may be chosen to be Markovian and this statement completes Kellerer Theorem. Theorem (Kellerer, 1972). Let (μt)t be a family of integrable probability measures on R. The following conditions are equivalent 2010 Mathematics Subject Classification. 60E15, 60G44, 60B10 (primary), and 49M25, 60G48, 60J25 (secondary).