  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.
