Stochastic nonlocal traffic flow models with Markovian noise
We extend our recently introduced stochastic nonlocal traffic flow model to more general random perturbations, including Markovian noise derived from a discretized Jacobi-type stochastic differential equation. Invoking a deterministic stability estimate, we show that the arising random weak entropy solutions are measurable, ensuring that quantities such as the expectation are well-defined. We show that the proposed Jacobi-type noise is of particular interest as it ensures interpretability, preserves boundedness, and significantly alters the stochastic realizations compared to the previous white noise approach. Moreover, we introduce a local solution operator which provides information on the local effect of the noise and utilize it to derive a mean-value hyperbolic nonlocal PDE, which serves as a proxy for the mean value of the exact solution. The quality of this proxy and the impact of the noise process are analyzed in several simulation studies.