We analyze the characteristic features of jam formation on a circular
one-lane road. We have applied an optimal velocity model including stochastic noise, where cars are treated as moving and interacting particles. The motion of N cars is described by the system of 2N stochastic differential equations with multiplicative white noise. Our system of cars behaves in qualitatively different ways depending on the values of control parameters c (dimensionless density), b (sensitivity parameter characterising the fastness of relaxation), and α (dimensionless noise intensity). In analogy to the gas-liquid phase transition in supersaturated vapour at low enough temperatures, we observe three different regimes of traffic flow at small enough values of b < bcr. There is the free flow regime (like gaseous phase) at small densities of cars, the coexistence of a jam and free flow (like liquid and gas) at
intermediate densities, and homogeneous dense traffic (like liquid phase) at large densities. The transition from free flow to congested traffic occurs when the homogeneous solution becomes unstable and evolves into the limit cycle. The opposite process takes place at a different density, so that we have a hysteresis effect and phase transition of the first order. A phase transition of second order, characterised by critical exponents, takes place at a certain critical density c = ccr.
Inclusion of the stochastic noise allows us to calculate the distribution of headway distances and time headways between the successive cars, as well as the distribution of jam (car cluster) sizes in a congested traffic.
We focus on the stochastic description of the stock price dynamics. Thereby we concentrate on the Heston model and the Hull-White model. We derive the stationary probability density distribution of the variance of both models in the case of zero correlation coefficient. These distributions are used to calculate solutions for the logarithmic returns of the stock price for short time lags. Furthermore we compare the received results with numerical simulations. In addition we apply the solutions of both models to the German tick-by-tick Dax data. The data are from May 1996 to December 2001. We use the probability density distributions of the logarithmic returns, calculated out of the data, and fit these distributions to the theoretical distributions.
We present a comparison of nucleation in an isothermal-isochoric container with traffic congestion on a one-lane freeway. The analysis is based, in both cases, on the probabilistic description by stochastic master equations. Further we analyze the characteristic features of traffic breakdowns. To describe this phenomenon we apply the stochastic model regarding the jam emergence to the formation of a large car cluster on the highway.