Embedding measures into multi-dimensional stochastic processes and rough paths

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A classic problem in stochastic analysis is the Skorokhod embedding problem: given a Brownian motion and a distribution on the reals, the task is to stop the Brownian trajectories such that it matches the given distribution. Despite the abstract formulation, this problem has found many applications including mathematical finance, statistics, and functional limit theorems. More recently, it played a central part in combining ideas from optimal transport and martingale theory. Skorokhod's initial question makes immediately sense for multi-dimensional martingales (and with some reformulations also for classes of more general stochastic processes). While some existence results are known, the literature gets very quickly sparse when it comes to concrete constructions of such stopping times. An attractive approach to the multidimensional Skorokhod embedding is the recent martingale optimal transport theory ("A. Cox, M. Beiglbock and M. Huesmann. "Optimal transport and Skorokhod embedding", Inv. Math. May 2017,2: 327-400). Martingale optimal transport leads to proofs of existence and optimality of solutions to Skorokhod embeddings even when the underlying process is multidimensional, but the proofs are not-constructive and typically do not help to actually construct the stopping time for a given target distribution. The goal of this research proposal is to produce new approaches to the Skorokhod multidimensional embedding problem that can lead to a concrete construction of such stopping times; a further aim is to extend the existing theory to be able to cover examples that arise in rough path theory and to explore connections with optimal martingale transport and other new applications. This project falls within the EPSRC Statistics and applied probability research area. Paul Gassiat from University Paris-Dauphine will be involved as a collaborator.As a first step, we will revisit the so-called Root solution of the Skorokhod embedding: Root showed that for one-dimensional Brownian motion, the stopping time can be realized as the hitting time of a subset of time-space. In this case, recent work has shown that this subset of time-space can be computed as the free boundary of a parabolic partial differential equation (see, A. M. G. Cox, J.Wang. "Root's barrier: Construction, optimality, and applications to variance options". The Annals of Applied Probability, 23(3):859-894, 2013; P. Gassiat, H. Oberhauser, and G. dos Reis. "Root's barrier, viscosity solutions of obstacle problems and reflected FBSDEs." Stochastic Processes and their Applications 125.12 (2015): 4601-4631.) or alternatively as the solution of an integral equation, (see P. Gassiat, A. Mijatovic and H Oberhauser. "An integral equation for Root's barrier and the generation of Brownian increments.", The Annals of Applied Probability 25.4 (2015): 2039-2065). In fact, abstract potential theoretic arguments show that Root type solutions hold for a large class of multidimensional Markov processes and these arguments can be reinterpreted in terms of recent advances in Skorokhod embeddings.This project falls within the EPSRC Mathematical Analysis research area.