  Motivated by problems in behavioural finance, we provide two explicit
constructions of a randomized stopping time which embeds a given centered
distribution $\mu$ on integers into a simple symmetric random walk in a
uniformly integrable manner. Our first construction has a simple Markovian
structure: at each step, we stop if an independent coin with a state-dependent
bias returns tails. Our second construction is a discrete analogue of the
celebrated Az\'ema-Yor solution and requires independent coin tosses only when
excursions away from maximum breach predefined levels. Further, this
construction maximizes the distribution of the stopped running maximum among
all uniformly integrable embeddings of $\mu$.
