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

Abstract

Given a family of real probability measures $({\mu }_t{)}_{t\ge 0}$ 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 \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $({\nu }^{\alpha }{)}_{\alpha \in [0,1]}$ with ${\nu }^{\alpha }(\mathbb{R})=\alpha$ that are submeasures of ${\mu }_0$, called a parametrization of ${\mu }_0$. Then, for any $\alpha$ we define an evolution $({\eta }_t^{\alpha }{)}_{t\ge 0}$ of the measure ${\nu }^{\alpha }={\eta }_0^{\alpha }$ across our peacock by setting ${\eta }_t^{\alpha }$ equal to the obstructed shadow of ${\nu }^{\alpha }$ in $({\mu }_s{)}_{s\in [0,t]}$. We identify conditions on the parametrization $({\nu }^{\alpha }{)}_{\alpha \in [0,1]}$ such that this construction leads to a unique martingale measure $\pi$, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization $({\nu }_{\text{lc}}^{\alpha }{)}_{\alpha \in [0,1]}$ 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.

Final version

References in corpus (2)

• On A Mixture Of Brenier and Strassen Theorems
• Fitting Martingales To Given Marginals