# Langevin method for a continuous stochastic car-following model and its stability conditions

## TL;DR

This paper introduces a stochastic car-following model using Langevin equations and an extended Cox-Ingersoll-Ross process to better reflect human perception variability and derives stability conditions that align with empirical observations.

## Contribution

It develops a novel stochastic car-following model incorporating Langevin equations and extended CIR processes, providing new stability criteria considering randomness in driver behavior.

## Key findings

- Stochastic model captures traffic instability at low speeds.
- Stability conditions account for random driver perception.
- Results align with empirical observations of traffic flow.

## Abstract

In car-following models, the driver reacts according to his physical and psychological abilities which may change over time. However, most car-following models are deterministic and do not capture the stochastic nature of human perception. It is expected that purely deterministic traffic models may produce unrealistic results due to the stochastic driving behaviors of drivers. This paper is devoted to the development of a distinct car-following model where a stochastic process is adopted to describe the time-varying random acceleration which essentially reflects the random individual perception of driver behavior with respect to the leading vehicle over time. In particular, we apply coupled Langevin equations to model complex human driver behavior. In the proposed model, an extended Cox-Ingersoll-Ross (CIR) stochastic process will be used to describe the stochastic speed of the follower in response to the stimulus of the leader. An important property of the extended CIR process is to enhance the non-negative properties of the stochastic traffic variables (e.g. non-negative speed) for any arbitrary model parameters. Based on stochastic process theories, we derive stochastic linear stability conditions which, for the first time, theoretically capture the effect of the random parameter on traffic instabilities. Our stability results conform to the empirical results that the traffic instability is related to the stochastic nature of traffic flow at the low speed conditions, even when traffic is deemed to be stable from deterministic models.