Embedding of Walsh Brownian motion

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