This work compares the performance of all existing 2-CUSUM stopping rules used in the problem of sequential detection of a change in the drift of a Brownian motion in the case of two-sided alternatives. As a performance measure an extended Lorden criterion is used. According to this criterion the optimal stopping rule is an equalizer rule. This paper compares the performance of the modified drift harmonic mean 2-CUSUM equalizer rules with the performance of the best classical 2-CUSUM equalizer rule whose threshold parameters are chosen so that equalization is achieved. This comparison is made possible through the derivation of a closed-form formula for the expected value of a general classical 2-CUSUM stopping rule.

In this paper we establish relationships between four important concepts: (a) hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c) Schrödinger's equation with linear potential, and (d) heat equation problems with moving boundary. We relate (a) and (b) by means of Girsanov's theorem, which suggests a strategy to extend our ideas to problems in R n and general diffusions. This approach also leads to (c) because we may relate, through a Feynman-Kac representation, functionals of a Bessel bridge with two Schrödinger-type problems. Finally, the relationship between (c) and (d) suggests a possible link between Generalized Airy processes and their hitting times.

This work deals with first hitting time densities of Ito processes whose local drift can be modeled in terms of a solution to Burgers equation. In particular, we derive the densities of the first time that these processes reach a moving boundary. We distinguish two cases: (a) the case in which the process has unbounded state space before absorption, and (b) the case in which the process has bounded state space before absorption. The reason as to why this distinction has to be made will be clarified.Next, we classify processes whose local drift can be expressed as a linear combination to solutions of Burgers equation. For example the local drift of a Bessel process of order 5 can be modeled as the sum of two solutions to Burgers equation and thus will be classified as of class B 2 . Alternatively, the Bessel process of order 3 has a local drift that can be modeled as a solution to Burgers equation and thus will be classified as of class B 1 . Examples of diffusions within class B 1 , and hence those to which the results described within apply, are: Brownian motion with linear drfit, the 3D Bessel process, the 3D Bessel bridge, and the Brownian bridge.

In this article, we extend the traditional GARCH(1,1) model by including a functional trend term in the conditional volatility of a time series. We derive the main properties of the model and apply it to all agricultural commodities in the Mexican CPI basket, as well as to the international prices of maize, wheat, swine, poultry, and beef products for three different time periods that implied changes in price regulations and behavior: before the North American Free Trade Agreement (NAFTA; 1987-1993 ), post-NAFTA (1994 -2005, and commodity supercycle (2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014). The proposed model seems to adequately fit the volatility process and, according to heteroscedasticity tests, also outperforms the ARCH(1) and GARCH(1,1) models, some of the most popular approaches used in the literature to analyze price volatility. Our results show that, consistent with anecdotal evidence, price volatility trends increased from the period

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