## Abstract

Let (Z,κ) be a Walsh Brownian motion with spinning measure κ. Suppose μ is a probability measure on R^{n}. We first provide a necessary and sufficient condition for μ to be a stopping distribution of (Z,κ). Then if the stopped process is required to be uniformly integrable, we show that such a stopping time exists if and only if μ is balanced. Next, under the assumption of being balanced, we identify the minimal stopping times with those τ such that the stopped process Z^{τ} is uniformly integrable. Finally, we generalize Vallois’ embedding, and prove that it minimizes the expectation E[Ψ(L_{τ}^{Z})] among all the admissible solutions τ, where Ψ is a strictly convex function and (L_{t}^{Z})_{t≥0} is the local time of the Walsh Brownian motion at the origin.

Original language | English (US) |
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Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Stochastic Processes and their Applications |

Volume | 134 |

DOIs | |

State | Published - Apr 2021 |

## Keywords

- Excursion theory
- Skorokhod embedding problem
- Vallois’ embedding
- Walsh Brownian motion

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics