2006/09/12 by Alexander M. G. Cox, Cox, Alexander M. G., Jan Obłój +2 · 1 citation
Mathematics · Physics and Astronomy · #28A80 #60G40 #60G42 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:28A80 #msc:60G40 #msc:60G42
paper · pdf · doi:10.48550/arxiv.math/0609330
arxiv created 2006/09/12 · openalex publication_date 2006/09/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Skorokhod Embedding problem is well understood when the underlying process is a Brownian motion. We examine the problem when the underlying is the simple symmetric random walk and when no external randomisation is allowed. We prove that any measure on Z can be embedded by means of a minimal stopping time. However, in sharp contrast to the Brownian setting, we show that the set of measures which can be embedded in a uniformly integrable way is strictly smaller then the set of centered probability measures: specifically it is a fractal set which we characterise as an iterated function system. Finally, we define the natural extension of several known constructions from the Brownian setting and show that these constructions require us to further restrict the sets of target laws.