Moment-matching approximations for stochastic sums in non-Gaussian Ornstein-Uhlenbeck models

In this paper, we recall actuarial and financial applications of sums of dependent random variables that follow a non-Gaussian mean-reverting process and contemplate distribution approximations. Our work complements previous related studies restricted to lognormal random variables; we revisit previous approximations and suggest new ones. We then derive moment-based distribution approximations for random sums attuned to Asian option pricing and computation of risk measures of random annuities. Various numerical experiments highlight the speed–accuracy benefits of the proposed methods

A Simple Discrete Approximation for the Renewal Function

Background: The renewal function is widely useful in the areas of reliability, maintenance and spare components inventory planning. Its calculation relies on the type of the probability density function of component failure times which can be, regarding the region of the component lifetime, modelled either by the exponential or by one of the peak-shaped density functions. For most peak-shaped distribution families the closed form of the renewal function is not available. Many approximate solutions can be found in the literature, but calculations are often tedious. Simple formulas are usually obtained for a limited range of functions only. Objectives: We propose a new approach for evaluation of the renewal function by the use of a simple discrete approximation method, applicable to any probability density function. Methods/Approach: The approximation is based on the well known renewal equation. Results: The usefulness is proved through some numerical results using the normal, lognormal, Weibull and gamma density functions. The accuracy is analyzed using the normal density function. Conclusions: The approximation proposed enables simple and fairly accurate calculation of the renewal function irrespective of the type of the probability density function. It is especially applicable to the peak-shaped density functions when the analytical solution hardly ever exists

Methodologies for Estimating Traffic Flow on Freeways Using Probe Vehicle Trajectory Data

Probe vehicle data are increasingly becoming the primary source of traffic data. As probe vehicle data become more widespread, it is imperative that methods are developed so that traffic state estimators such as flow, density, and speed can be derived from such data. In this dissertation three different methodologies are proposed for predicting traffic flow or volume on a freeway. All of the proposed methodologies exploit several different traffic flow theories in conjunction with probe vehicle data to predict traffic flow. The first methodology takes advantage of the fundamental diagram or speed-flow relationship. The relationship states that flow can be estimated when speed is known. In this case, flow is traffic volume and speed comes from probe vehicles. Flow is predicted for four different models of fundamental diagrams and is analyzed at different time aggregation intervals. Results show that of the four fundamental diagrams, Van Aerde’s Model is the best performing model with the lowest average percent error. It is also observed that flow prediction is more accurate during low speed (congestion) compared to high speed (free-flow) conditions. The second methodology exploits the shockwave theory, which pertains to the propagation of a change (discontinuity) in traffic flow. From probe vehicle trajectories, shockwave is estimated as the boundary between free-flow and congested regimes of traffic flow. After clustering the traffic regimes into free-flow and congested periods, the traffic flow during congestion is estimated using the Northwestern congested-regime fundamental diagram. From this estimation, the flow during free-flow is then predicted. Analyses show that the percent error of the predicted flow during free-flow ranges from -9 to 1%. The third methodology is the car-following approach which relies on the spacing or distance between a leader and follower which can be directly measured from the trajectories. Based on a set of known probability distributions, the position of the follower vehicle with respect to the lead vehicle is estimated given that the spacing between the two random probe vehicles is known. A framework is developed to automatically process probe trajectories to extract relevant probe data under stop-and-go traffic conditions. The model is tested based on NGSIM datasets. The results show that when vehicle spacing is small the prediction of follower position is very accurate. As spacing increases the error in predicted follower position also increases. Though there exists some estimation error, all three approaches can reasonably predict flow for freeways using probe vehicle data