Optimal fits of diffusion constants from single-time data points of Brownian trajectories

Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe interactions between a particle and its environment but the price to be paid is the absence of ensemble averaging and a consequent lack of statistics. Here we address the benchmark question of how to accurately extract the diffusion constant of one single Brownian trajectory. We analyze a class of estimators based on weighted functionals of the square displacement. For a certain choice of the weight function these functionals provide the true ensemble averaged diffusion coefficient, with a precision that increases with the trajectory resolution.

On ergodic least-squares estimators of the generalized diffusion coefficient for fractional Brownian motion

We analyse a class of estimators of the generalized diffusion coefficient for fractional Brownian motion Bt of known Hurst index H, based on weighted functionals of the single time square displacement. We show that for a certain choice of the weight function these functionals possess an ergodic property and thus provide the true, ensemble-averaged, generalized diffusion coefficient to any necessary precision from a single trajectory data, but at expense of a progressively higher experimental resolution. Convergence is fastest around H ? 0.30, a value in the subdiffusive regime.

First passages in bounded domains: When is the mean first passage time meaningful?

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(ω) distribution of the random variable ω=τ1/(τ1+τ2), which is a measure for how similar the first passage times τ1 and τ2 are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.

Comment on the existence of a long range correlation in the geomagnetic disturbance storm time (Dst) index

Very recently (Banerjee et al. in Astrophys. Space, doi:1007/s10509-011-0836-1, 2011) the statistics of geomagnetic Disturbance storm (Dst) index have been addressed, and the conclusion from this analysis suggests that the underlying dynamical process can be modeled as a fractional Brownian motion with persistent long-range correlations. In this comment we expose several misconceptions and flaws in the statistical analysis of that work. On the basis of these arguments, the former conclusion should be revisited.

Unit Root Analysis of Traffic Time Series in Toll Highways

Concession contracts in highways often include some kind of clauses (for example, a minimum traffic guarantee) that allow for better management of the business risks. The value of these clauses may be important and should be added to the total value of the concession. However, in these cases, traditional valuation techniques, like the NPV (net present value) of the project, are insufficient. An alternative methodology for the valuation of highway concession is one based on the real options approach. This methodology is generally built on the assumption of the evolution of traffic volume as a GBM (geometric Brownian motion), which is the hypothesis analyzed in this paper. First, a description of the methodology used for the analysis of the existence of unit roots (i.e., the hypothesis of non-stationarity) is provided. The Dickey-Fuller approach has been used, which is the most common test for this kind of analysis. Then this methodology is applied to perform a statistical analysis of traffic series in Spanish toll highways. For this purpose, data on the AADT (annual average daily traffic) on a set of highways have been used. The period of analysis is around thirty years in most cases. The main outcome of the research is that the hypothesis that traffic volume follows a GBM process in Spanish toll highways cannot be rejected. This result is robust, and therefore it can be used as a starting point for the application of the real options theory to assess toll highway concessions.

Estudio de opciones reales para decisiones de inversión en un proyecto minero de nueva ejecución

El objetivo principal es desarrollar la metodología de opciones reales para evaluar la posible puesta en marcha de un proyecto minero. Para esto, el proyecto se divide en dos partes: En la primera parte, con carácter teórico se analizan las inversiones desde el punto de vista tradicional, comparando la problemática de estas valoraciones en ambientes de incertidumbre y flexibilidad operativa. Se analizan las opciones financieras y se comparan con las opciones reales, en cuanto a similitudes y problemáticas. Se desarrollan también los procesos estocásticos que afectan a las variables del proyecto de inversión. Se explican además, las metodologías para el cálculo de las opciones reales, incluido el cálculo de la volatilidad de las mismas. En una segunda parte, se estudia el yacimiento aurífero de Corcoesto, para el cual se realiza la simulación del plan de negocio según las características necesarias para la explotación, donde los ingresos se modelizan mediante un movimiento geométrico browniano para simular el comportamiento del precio de la onza de oro. Se elige un desarrollo de árboles binomiales para estimar el valor futuro del proyecto, a la vez que se establece un intervalo de precios de la opción para adquirir el proyecto minero. Este intervalo estará determinado por las incertidumbres del proyecto calculadas según las metodologías de Copeland y Antikarov, y Heraht y Park. Abstract This project is aimed mainly to develop real options theory to assess a mining project start-up. The project is divided in two documents: The first document with theorical content, investments are analyzed from the clasical point of view, comparing the advantages and disadvantages of this appraisal in high uncertainity and operational flexibility conditions. Financial options are analyzed and compared to real options, in both similarities and problematics. Stochastical process that affect the project variables are also developed. Methods for estimating real options value, including the methods for volatility estimation are commented. In the second document, the Corcoesto gold deposit has been studied. A bussines plan simulation has been maked according to the characteristics of the extraction, where incomes have been simulated with a geometrical Brownian movement to estimate the gold onze behaviour. The binomial tree method has been generated to study the future project value, as well as a range of option prices, for adquiring the mine project. This interval is determined by the project uncertainity calculated with the theories from Copeland and Antikarov and Herath and Park

Empirical analysis of traffic volume for the application of the options theory to highway concessions

*************************************************************************************** EL WCTR es un Congreso de reconocido prestigio internacional en el ámbito de la investigación del transporte que hasta el 2010 publicaba sus libros de abstracts con ISBN. Por ello consideramos que debería seguir teníendose en cuenta para los indicadores de calidad ******************************************************************************************* Investment projects in the field of transportation infrastructures have a high degree of uncertainty and require an important amount of resources. In highway concessions in particular, the calculation of the Net Present Value (NPV) of the project by means of the discount of cash flows, may lead to erroneous results when the project incorporates certain flexibility. In these cases, the theory of real options is an alternative tool for the valuation of concessions. When the variable that generates uncertainty (in our case, the traffic) follows a random walk (or Geometric Brownian Motion), we can calculate the value of the options embedded in the contract starting directly from the process followed by that variable. This procedure notably simplifies the calculation method. In order to test the hypothesis of the evolution of traffic as a Geometric Brownian Motion, we have used the available series of traffic in Spanish highways, and we have applied the Augmented Dickey-Fuller approach, which is the most widely used test for this kind of study. The main result of the analysis is that we cannot reject the hypothesis that traffic follows a Geometric Brownian Motion in the majority of both toll highways and free highways in Spain.