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