2016/12/05 by Mathias Beiglboeck, Beiglboeck, Mathias, Manu Eder +5
Economics, Econometrics and Finance · Mathematics · #60G42 #60G44 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1612.01488
openalex publication_date 2016/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We adapt ideas and concepts developed in optimal transport (and its martingale variant) to give a geometric description of optimal stopping times of Brownian motion subject to the constraint that the distribution of the stopping time is a given probability. The methods work for a large class of cost processes. (At a minimum we need the cost process to be measurable and adapted. Continuity assumptions can be used to guarantee existence of solutions.) We find that for many of the cost processes one can come up with, the solution is given by the first hitting time of a barrier in a suitable phase space. As a by-product we recover classical solutions of the inverse first passage time problem / Shiryaev's problem.