Referring to Dempster-Shafer theory, we introduce a bivariate random walk enforcing Markovianity and time-homogeneity under a pessimistic view towards ambiguity. This is done through a suitable family of joint t-step transition belief functions, generalizing the product of two independent binomial transition probabilities, where ambiguity is expressed by a parameter. Given a real-valued function of the pair at a fixed time horizon, we define the dynamic lower and upper Value-at-Risk (VaR), generated by the corresponding dynamic p-box.