SHADOWS AND BARRIERS

• 成果类型: Article
• 署名作者: Brueckerhoff, Martin; Huesmann, Martin
• 署名单位: University of Munster
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1981
• 发表日期: 2024
• 页码: 960-985
• 关键词: optimal transport; skorokhod

摘要

In this article, we show an intimate connection between two objects in probability theory, which received some attention in the last years: shadows of measures and barrier solutions to the Skorokhod embedding problem (SEP). The shadow of a measure mu in the measure. is the key object in the construction of the left-curtain coupling and its siblings in martingale optimal transport by Beiglbock and Juillet (Ann. Probab. 44 (2016) 42-106; Trans. Amer. Math. Soc. 374 (2021) 4973-5002). Many prominent solutions to the SEP are first hitting times of barriers in certain phase spaces, that is, they are of the form inf{t >= 0 : (X-t, B-t) is an element of R} for some closed set R, an increasing processes X and Brownian motion B. We show that the property that a solution to the SEP is of barrier type can be characterized in terms of the shadow. This characterization 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.