Shadows and barriers

Abstract

In this article, we show an intimate connection between two objects in probability theory, which received some attention in the last years: shadows of measures and barrier solutions to the Skorokhod embedding problem (SEP). The shadow of a measure μ in the measure ν is the key object in the construction of the left-curtain coupling and its siblings in martingale optimal transport by Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106; Trans. Amer. Math. Soc. 374 (2021) 4973–5002). Many prominent solutions to the SEP are first hitting times of barriers in certain phase spaces, that is, they are of the form $inf\{\mathit{t}\ge 0:({\mathit{X}}_{\mathit{t}},{\mathit{B}}_{\mathit{t}})\in \mathcal{R}\}$ for some closed set $\mathcal{R}$, an increasing processes X and Brownian motion B.

We show that the property that a solution to the SEP is of barrier type can be characterized in terms of the shadow. This characterization allows us to construct new families of barrier solutions that naturally interpolate between two given barrier solutions. We exemplify this by an interpolation between the Root embedding and the left-monotone embedding.