On the Root Solution to the Skorokhod Embedding Problem Given Full Marginals

Abstract

This paper examines the Root solution of the Skorokhod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Obłój, and Touzi [9]. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE.

From Bachelier to Dupire via Optimal Transport

“…Henry-Labordere, Tan, and Touzi [27] as well as Br 'uckerhoff, Huesmann, and Juillet [14] provide continuous time versions of the shadow coupling (originally introduced in [8]). Richard, Tan, and Touzi [52] give a continuous time version of the Root solution to the Skorokhod embedding problem. In a slightly different but related direction, Boubel and Juillet [12] consider a continuum of marginals on the real line that do not satisfy an order condition and construct a canonical Markov-process matching these marginals.…”

On the Continuity of the Root Barrier

“…Gassiat et al [2019] extended the free boundary problem approach to sufficiently regular Markov processes including discontinuous Lévy processes. On the other hand, Cox et al [2019] and Beiglböck et al [2020] generalized the formulation of the Skorokhod embedding problem with one target measure µ to a problem with a sequence of target measures (µ i ) N i=1 , and Richard et al [2020] even to a continuum of target measures (µ i ) i∈ [0,1] . But, to the best of our knowledge, the literature lacks general regularity results on the barrier function r.…”

A free boundary characterisation of the Root barrier for Markov processes

“…One of the most intuitive solutions is due to Root [57]: for a one-dimensional Brownian motion and μ,ν in convex order, there exists a space-time subset -the so-called Root barrier -such that its hitting time by (t, X t ) solves SEP(X , μ, ν). More recently, connections with obstacle PDEs [23,28,33,34,38], optimal transport [3,[5][6][7]20,37,39,40], and optimal stopping [23,25] and extensions to the multi-marginal case [4,22,56] have been developed. However, already for multi-dimensional Brownian motion much less is known about solutions to SEP(X , μ, ν), see for example work of Falkner [30] that highlights some of the difficulties that arise in the multi-dimensional Brownian case.…”