In dynamic traffic assignment (DTA) models, it seems relevant to consider the uncertainty inherent to motorist route choices. Particularly, choices on realistic transport networks are mostly made using motorists' perceived costs of all routes from their origins to their destinations. We present an approach to address stochastic DTA based on nested cost operators, where motorists choose according to the perceived costs of the remaining trip, namely, from current position to destination. We integrate the Markovian traffic equilibrium by Baillon and Cominetti with the DTA formulation by Addison and Heydecker obtaining an arc-based stochastic DTA model, which we denote in short as ABSDTA. The resulting approach accommodates overlapping routes, respecting costs' correlation as well as the first-in-first-out rule. We present a solution method for discrete time periods, computational results on an illustrative network, including sensitivity analyses of the parameters, and comparisons with a previous suitable stochastic DTA model from the literature.