No abstract

Given a family of real probability measures (µt) t≥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 (ν α ) α∈[0,1] with ν α (R) = α that are submeasures of µ0, called a parametrization of µ0. Then, for any α we define an evolution (η α t ) t≥0 of the measure ν α = η α 0 across our peacock by setting η α t equal to the obstructed shadow of ν α in (µs) s∈[0,t] . We identify conditions on the parametrization (ν α ) α∈[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 (ν α lc ) α∈[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.

Given a family of real probability measures (µt) t≥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 [12,45]. As input data we take an increasing family of measures (ν α ) α∈[0,1] with ν α (R) = α that are submeasures of µ0, called a parametrization of µ0. Then, for any α we define an evolution (η α t ) t≥0 of the measure ν α = η α 0 across our peacock by setting η α t equal to the obstructed shadow of ν α in (µs) s∈[0,t] . We identify conditions on the parametrization (ν α ) α∈[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 (ν α lc ) α∈[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.

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 ≥ 0 : (Xt, Bt) ∈ R} for some increasing process (Xt) t≥0 and a barrier R if and only if there exists a time-change (T l ) l≥0 such that for all l ≥ 0 the equationis satisfied, i.e. the distribution of Bτ on the event that the Brownian motion is stopped after T l is the shadow of the distribution of BT 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.