Date: Wednesday, October 28, 2020
Location: Passcode: 790109 https://umich.zoom.us/j/95407665241 Virtual (4:00 PM to 5:00 PM)
Title: Embedding of Walsh Brownian motion
Abstract: Let $(Z,\kappa)$ be a Walsh Brownian motion with spinning measure $\kappa$. Suppose $\mu$ is a probability measure on $\mathbb{R}^n$. We first provide a necessary and sufficient condition for $\mu$ to be a stopping distribution of $(Z,\kappa)$. Then if the stopped process is required to be uniformly integrable, we show that such a stopping time exists if and only if $\mu$ is \emph{balanced}. Next, under the assumption of being balanced, we identify the minimal stopping times with those $\tau$ such that the stopped process $Z^{\tau}$ is uniformly integrable. Finally, we generalize Vallois' embedding, and prove that it minimizes the expectation $\mathbb{E}[\Psi(L^Z_{\tau})]$ among all the admissible solutions $\tau$, where $\Psi$ is a strictly convex function and $(L_t^Z)_{t \geq 0}$ is the local time of the Walsh Brownian motion at the origin.
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Speaker: Xin Zhang
Institution: UM
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