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

Abstract

Given a family of real probability measures ${({\mathrm{\mu }}_t)}_{t\ge 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in [13, 46]. As input data we take an increasing family of measures ${({\mathrm{\nu }}^{\mathit{\alpha }})}_{\mathit{\alpha }\in [0,1]}$ with ${\mathrm{\nu }}^{\mathit{\alpha }}(\mathbb{R})=\mathit{\alpha }$ that are submeasures of ${\mathrm{\mu }}_0$, called a parametrization of ${\mathrm{\mu }}_0$. Then, for any α we define an evolution ${({\mathrm{\eta }}_t^{\mathit{\alpha }})}_{t\ge 0}$ of the measure ${\mathrm{\nu }}^{\mathit{\alpha }}={\mathrm{\eta }}_0^{\mathit{\alpha }}$ across our peacock by setting ${\mathrm{\eta }}_t^{\mathit{\alpha }}$ equal to the obstructed shadow of ${\mathrm{\nu }}^{\mathit{\alpha }}$ in ${({\mathrm{\mu }}_s)}_{s\in [0,t]}$. We identify conditions on the parametrization ${({\mathrm{\nu }}^{\mathit{\alpha }})}_{\mathit{\alpha }\in [0,1]}$ such that this construction leads to a unique martingale measure π, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization ${({\mathrm{\nu }}_{\text{lc}}^{\mathit{\alpha }})}_{\mathit{\alpha }\in [0,1]}$ we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem.

Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.