# Embedding measures into multi-dimensional stochastic processes and rough paths

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

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.

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.

## People |
## ORCID iD |

Harald Oberhauser (Primary Supervisor) | |

Christina Zou (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509711/1 | 01/10/2016 | 30/09/2021 | |||

1941799 | Studentship | EP/N509711/1 | 01/10/2017 | 30/09/2020 | Christina Zou |

Description | My research is focussing on constructing solutions to the so-called Skorokhod embedding problem. The problem can be described as follows: Given a stochastic process, we fix a target probability distribution, i.e. a law stating which values are attained with which probability. The task is finding a random time T such that the random values, which are attained when we cut the process at time T, are distributed according to the given target distribution. Although a vast amount of literature exists, most of them only deal with the proof of existence and optimality, but not the construction. The goal of this research project is producing new approaches to the Skorokhod embedding problem that can lead to concrete constructions of such stopping times. So far, we have achieved the construction of one type of solutions, the so-called Root solution, for a large class of stochastic processes, which includes multi-dimensional Brownian motion, stable Levy processes and rough paths. The Root solution is realised as the first hitting time of a set which can be determined by a general free boundary characterisation. The preprint was submitted in June 2019 and is still under review. |

Exploitation Route | The motivation for current work on the Skorokhod embedding problem comes from option pricing in mathematical finance. The Root solution, in particular, is used for the pricing American options. The Root solution turns out to be the solution to maximise over concave functions, i.e. if we have constructed the stopping region then the average over any concave function evaluated at the stopped time points the maximal. Then if the stochastic process models the evolution of stock prices then the stopping region give the optimal region to exercise the option in order to maximise the payoff. |

Sectors | Financial Services, and Management Consultancy |

Description | Collaboration with Paul Gassiat from Paris Dauphine University |

Organisation | Paris Dauphine University |

Country | France |

Sector | Academic/University |

PI Contribution | The first part of my PhD project was done in collaboration with Paul Gassiat from Paris Dauphine University. During his visit in Oxford April 2018 and my visit in Paris October 2018, we have discussed the proofs and extensions of our joint paper. Afterwards, following our discussions, I have worked on the numerical simulations required for our paper, adapting our result to relevant examples and a further extension of the main result. |

Collaborator Contribution | Paul provided key ideas for generalising our result and for proofs in the final version of our paper and we have jointly worked on most of the main parts of the paper. |

Impact | Preprint: Gassiat, Paul, Harald Oberhauser, and Christina Z. Zou. "A free boundary characterisation of the Root barrier for Markov processes." arXiv preprint arXiv:1905.13174 (2019). The collaboration is not multidisciplinary. |

Start Year | 2018 |