Shadows and Barriers

We show an intimate connection between solutions of the Skorokhod Embedding Problem which are given as the first hitting time of a barrier and the concept of shadows in martingale optimal transport. More precisely, we show that a solution $τ$ to the Skorokhod Embedding Problem between $μ$ and $ν$ is of the form $τ= \inf \{t \geq 0 : (X_t,B_t) \in \mathcal{R}\}$ for some increasing process $(X_t)_{t \geq 0}$ and a barrier $\mathcal{R}$ if and only if there exists a time-change $(T_l)_{l \geq 0}$ such that for all $l \geq 0$ the equation $\mathbb{P}[B_τ \in \cdot , τ\geq T_l] = \mathcal{S}^ν(\mathbb{P}[B_{T_l} \in \cdot , τ\geq T_l])$ is satisfied, i.e.\ the distribution of $B_τ$ on the event that the Brownian motion is stopped after $T_l$ is the shadow of the distribution of $B_{T_l}$ on this event in the terminal distribution $ν$. This equivalence 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.

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