Analytical formulation for explaining the variations in traffic states: A fundamental diagram modeling perspective with stochastic parameters

Despite the simplicity and practicality of (deterministic) fundamental diagram models in highway traffic flow theory, the wide scattering effect observed in empirical data remains highly controversial, particularly for explaining traffic state variations. Owing to the analytical properties of the fundamental diagram modeling approach, in this study, we proposed an analytical and quantitative method for analyzing traffic state variations. We investigated the scattering effect in the fundamental diagram and proposed two stochastic fundamental diagram (SFD) models with lognormal and skew-normal distributions to explain the variations in traffic states. The first SFD model assumes that the scattering effect results from stochasticity in both the free-flow speed and the speed at critical density. Both random variables were assumed to follow the lognormal distribution. In the second SFD model, an integrated error term that was assumed to follow the skew-normal distribution over different density ranges was appended to the deterministic fundamental diagram. The properties of these two SFD models were analyzed and compared, and the parameters in these SFD models were calibrated using real-world loop detector data. The observed scatters from the empirical data were reproduced well by the simulated fundamental diagram model, indicating the validity of the proposed SFD models for explaining traffic state variations. Using these two analytical SFD models, we can analyze the stochastic capacity of freeways with closed forms. More importantly, the sources of stochasticity in freeway capacity can be traced in terms of randomly distributed parameters in fundamental diagram models.     

Market implied volatilities for defaultable bonds

Typically, implied volatilities for defaultable instruments are not available in the financial market since quotations related to options on defaultable bonds or on credit default swaps are usually not quoted by brokers. However, an estimate of their volatilities is needed for pricing purposes. In this paper, we provide a methodology to infer market implied volatilities for defaultable bonds using equity implied volatilities and CDS spreads quoted by the market in relation to a specific issuer. The theoretical framework we propose is based on the Merton’s model under stochastic interest rates where the short rate is assumed to follow the Hull–White model. A numerical analysis is provided to illustrate the calibration process to be performed starting from financial market data. The market implied volatility calibrated according to the proposed methodology could be used to evaluate options where the underlying is a risky bond, i.e. callable bond or other types of credit-risk sensitive financial instruments.

     

General phase transition models for vehicular traffic with point constraints on the flow

In this paper, we present a general phase transition model that describes the evolution of vehicular traffic along a one‐lane road. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the free‐flow phase and by a system of 2 conservation laws in the congested phase. The free‐flow phase is described by a one‐dimensional fundamental diagram corresponding to a Newell‐Daganzo type flux. The congestion phase is described by a two‐dimensional fundamental diagram obtained by perturbing a general fundamental flux. In particular, we study the resulting Riemann problems in the case a local point constraint on the flow of the solutions is enforced.     

Application of the fuzzy–stochastic methodology to appraising the firm value as a European call option

The valuing of a firm equity as a call option is a crucial problem in financial decision-making. There are two basic aspects that are studied; contingent claim features (payoff functions) and risk (stochastic process of underlying assets). However, non-preciseness (vagueness, uncertainty) of input data is often neglected. Thus, a combination of risk (stochastic) and uncertainty (fuzzy instruments) could be a useful approach in calculating a firm value as a call option. The Black–Scholes methodology of appraising equity as a European call option is applied. Fuzzy–stochastic methodology under fuzzy numbers (T-numbers) is proposed and described. Fuzzy–stochastic model of appraising a firm equity is proposed. Input data are in a form of fuzzy numbers and result, firm possibility-expected equity value is also determined vaguely as a fuzzy set. Illustrative example is introduced.     

Microscopic modeling and control logic for incident-responsive automatic vehicle movements in single-automated-lane highway systems

Automatic response to lane-blocking incidents is a critical issue in the field of automated highway systems (AHS). Accordingly, this paper presents a microscopic vehicular control methodology for automatic-control (AC) vehicular movements in response to lane-blocking incidents in the AHS environment. The embedded traffic control logic is based on the basic safety requirements for automatic-control lane traffic maneuvers responding to lane-blocking incidents in the single-automated-lane AHS environment. Accordingly, respective automated vehicular control models are proposed to deal with AC vehicles moving in three corresponding sequential phases, i.e., (1) AC platoon approaching the incident site from the blocked lane, (2) mandatory lane changing and mixed car following in the adjacent lane, and (3) AC platoon reforming downstream from the incident site in the blocked automated lane. Using a microscopic simulation model which embeds these proposed models, preliminary tests are conducted to investigate the relative performance of the proposed method in various traffic flow and control scenarios. The resulting numerical results, including simplified sensitivity analyses, indicate that the proposed microscopic traffic control logic permits regulating automatic-control vehicular movements in response to the effects of lane-blocking incidents on traffic flows either in control-free lanes or in the automatic-control lanes. Implications of the results and some findings are discussed for further research.     

On the coupling of steady and adaptive velocity grids in vehicular traffic modelling

This work deals with the derivation and the analysis of a new mathematical model for vehicular traffic along a one-way road obtained by the coupling of a uniform and an adaptive discretization of the velocity variable in the framework of the kinetic theory. Interactions are modelled by stochastic games where the output of interactions depends on the local density and is not linearly additive.     

Reconstructing freeway travel times with a simplified network flow model alternating the adopted fundamental diagram

The present study summarises the travel time reconstruction performance of a network flow model by explicitly analysing the adopted fundamental diagram relation under congested and un-congested traffic patterns. The incorporated network flow model uses a discrete meso-simulation approach in which the anisotropic property of traffic flow and the uniform acceleration of vehicle packets are explicitly considered. The flow performances on link-route dynamics have been derived by reasonably alternating the adopted two-phase, i.e., congested and un-congested, fundamental relation of traffic flow. The linear speed–density relation with the creeping speed assumption is substituted with the triangular flow–density relation in order to investigate the performance of the network flow model in varying flow patterns. Applying the anisotropic mesoscopic model, the measure of travel time is obtained as a link performance from a simplified dynamic network loading process. Travel time reconstruction performance of the network flow model is sought considering the actual measures that are obtained by a probe vehicle, in addition to reconstructions by a macroscopic network flow model. The main improvements on travel time reconstruction process are encountered in terms of the computation load within the explicit analyses by the alternation of adopted two-phase fundamental diagram. Although the accuracies of the flow model with the adoption of two different fundamental diagrams are hard to differentiate, the computational burden of the simulation process by the triangular fundamental diagram is found to be considerably different.     

Piecewise linear concave dynamical systems appearing in the microscopic traffic modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are uniquely defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue is introduced, and the ergodic theory is used to define a growth rate. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the average growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of roads with a junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions.     

Un modelo de difusión anisótropa para el estudio del tráfico urbano

In order to design or redesign urban transportation networks, the employment of mathematical models is very useful for predicting the effects of possible modifications of implementing. Such models allow the determination of vehicular flows and travel times for every link of the network from the knowledge of its inherent features and the corresponding traffic demand. They are based on a phenomenological law of the social collective behavior of the drivers called Wardrop principle. It is an optimization problem, in general, very demanding from the computational point of view.In order to accelerate the computation process, in this paper, a continuum model for the urban traffic is proposed. The fundamental assumption behind this theory is that the variation of network properties is small in close regions when compared with the full system. Accordingly, it is possible to use continuous functions for representing travel times or vehicular flows. Essentially, the problem is formulated as a system of non-linear anisotropic diffusion (differential) equations that can be conveniently solved by means of the finite element method. The efficiency of the proposed model is studied by means of a comparison with results obtained with the classical optimization approach. As shown, the results are similar although the computation times are significantly reduced.     

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.     

Application of averaging techniques to traffic flow theory

Real traffic data are very versatile and are hard to explain with the so‐called standard fundamental diagram. A simple microscopic model can show that the heterogeneity of traffic results in a reduced mean flow and that the reduction is proportional to the density variance. Standard averaging techniques allow us to evaluate this reduction without having to describe the complex microscopic interactions. Using a second equation for the variance results in a two‐dimensional hyperbolic system that can be put in conservative form. The Riemann problem is completely solved in the case of a parabolic fundamental diagram, and the solutions are compared with the famous second‐order Aw–Rascle–Zhang model in a simulation of lane reduction. Adding a diffusion term results in entropy production, and the diffusive model is studied as well. Finally, a numerical scheme is used and converges to the analytical solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical Approximation of Uncertain Nonlinear Systems

We address qualitative characteristics of dynamical systems and their approximation using set–valued numerical methods, where we aim at robust results. In this spirit, model uncertainties are incorporated into the problem formulation as outcomes of an external stochastic process. The systems under consideration are described in terms of finite state Markov chains. An example serves to illustrate the procedure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling traffic flow with a time‐dependent fundamental diagram

We model traffic flow with a time‐dependent fundamental diagram. A time‐dependent fundamental diagram arises naturally from various factors such as weather conditions, traffic jam or modern traffic congestion managements, etc. The model is derived from a car‐following model which takes into account the situation changes over the time elapsed time. It is a system of non‐concave hyperbolic conservation laws with time‐dependent flux and the sources. The global existence and uniqueness of the solution to the Cauchy problem is established under the condition that the variation in time of the fundamental diagram is bounded. The zero relaxation limit of the solutions is found to be the unique entropy solution of the equilibrium equation. Copyright © 2004 John Wiley & Sons, Ltd.     

基于内生参考点的交通网络均衡模型

应用累积前景理论研究了随机交通网络中出行者有限理性的路径选择行为,选择期望 超额出行时间作为参考点,反映出行者同时考虑出行时间的可靠性和不可靠性,建立了基于累积前景理论的随机网络均衡及其等价的变分不等式模型,运用基于连续平均法的启发式算法求解,并给出算例验证了该模型和算法,最后分析了有限理性的假设和内生的参考点对出行者路径选择行为和随机网络均衡的影响．     

Incorporating crossed classification credibility into the Lee–Carter model for multi-population mortality data

Recent developments in actuarial literature have shown that credibility theory can serve as an effective tool in mortality modelling, leading to accurate forecasts when applied to single or multi-population datasets. This paper presents a crossed classification credibility formulation of the Lee–Carter method particularly designed for multi-population mortality modelling. Differently from the standard Lee–Carter methodology, where the time index is assumed to follow an appropriate time series process, herein, future mortality dynamics are estimated under a crossed classification credibility framework, which models the interactions between various risk factors (e.g. genders, countries). The forecasting performances between the proposed model, the original Lee–Carter model and two multi-population Lee–Carter extensions are compared for both genders of multiple countries. Numerical results indicate that the proposed model produces more accurate forecasts than the Lee–Carter type models, as evaluated by the mean absolute percentage forecast error measure. Applications with life insurance and annuity products are also provided and a stochastic version of the proposed model is presented.     

An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

Optimal mean–variance investment and reinsurance problems for the risk model with common shock dependence

In this paper, we study the optimal investment–reinsurance problems in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Under the criterion of mean–variance, two cases are considered: One is the optimal mean–variance problem with bankruptcy prohibition, i.e., the wealth process of the insurer is not allowed to be below zero at any time, which is solved by standard martingale approach, and the closed form solutions are derived; The other is the optimal mean–variance problem without bankruptcy prohibition, which is discussed by a very different method—stochastic linear–quadratic control theory, and the explicit expressions of the optimal results are obtained either. In the end, a numerical example is given to illustrate the results and compare the values in the two cases.     

基于不确定测度及积分的民航空管运行风险决策模型研究

以往民航空管安全管理中,大多对单一类型威胁或差错进行评估以管理风险,忽略了威胁与差错之间的交互作用,难以准确反映出不同类型风险之间的相关性。本文提出了基于不确定测度的空管运行风险决策模型,即将不确定积分Choquet积分作为风险函数,在由威胁和差错构成的集合的概念下,由Shapley值作为校正函数,重新评估各个威胁和差错的风险值。本文通过应用上述风险决策模型,结合案例得出了考虑交互作用的空管运行修正风险值,该风险值与单一风险值相比更贴近空管运行实际情况,因而对风险决策更具指导意义。