  This paper proposes a new stochastic model of traffic dynamics in Lagrangian
coordinates. The source of uncertainty is heterogeneity in driving behavior,
captured using driver-specific speed-spacing relations, i.e., parametric
uncertainty. It also results in smooth vehicle trajectories in a stochastic
context, which is in agreement with real-world traffic dynamics and, thereby,
overcoming issues with aggressive oscillation typically observed in sample
paths of stochastic traffic flow models. We utilize ensemble filtering
techniques for data assimilation (traffic state estimation), but derive the
mean and covariance dynamics as the ensemble sizes go to infinity, thereby
bypassing the need to sample from the parameter distributions while estimating
the traffic states. As a result, the estimation algorithm is just a standard
Kalman-Bucy algorithm, which renders the proposed approach amenable to
real-time applications using recursive data. Data assimilation examples are
performed and our results indicate good agreement with out-of-sample data.
