Shadow martingales – a stochastic mass transport approach to the peacock problem

Given a family of real probability measures (?t)t?0 increasing in convex order (a peacock) we describe systematic method to create martingale exactly fitting the marginals at any time. The key object for our approach is obstructed shadow measure peacock, generalization (obstructed) introduced [13, 46]. As input data take an (??)??[0,1] with ??(R)=? that are submeasures ?0, called parametrization ?0. Then, ? define evolution (?t?)t?0 ??=?0? across peacock by setting ?t? equal ?? (?s)s?[0,t]. We identify conditions on such this construction leads unique ?, martingale, without assumptions peacock. In case left-curtain (?lc?)??[0,1] as solution continuous-time version optimal transport problem. Furthermore, enriches knowledge Predictable Representation Property (PRP) since comes canonical Choquet representation extremal Markov martingales.