Ramón Gutiérrez Jáimez

On the Karhunen-Loeve expansion for transformed processes

We discuss the influence of the transformation{X(t)}→{f(t)X(τ(t))} on the Karhunen-Loève expansion of {X(t)}. Our main result is that, in general, the Karhunen-Loève expansion of {X(t)} with respect to Lebesgues measure is transformed in the Karhunen-Loève expansion of {f(t)X(τ(t))} with respect to the measuref−2(t)dτ(t). Applications of this result are given in the case of Wiener process, Brownian bridge, and Ornstein-Uhlenbeck process.ResumenSe discute la influencia de la transformación {X(t)}→{f(t)X(τ(t))} en el desarrollo de Karhunen-Lòeve de {X(t)}. Nuestro principal resultado es que, en general, el desarrollo de Karhunen-Loève de {X(t)} respecto a la medida de Lebesgue se transforma en el desarrollo de Karhunen-Loève de {f(t)X(τ(t))} respecto a la medidaf−2(t)dτ(t). Asimismo, se dan aplicaciones de este resultado en el caso de procesos de Wiener, puente browniano y Ornstein-Uhlenbeck.

Family of pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2 ; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function 3F2(α1, α2, α3;γ1, γ2; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family.

APPROXIMATING THE NONHOMOGENEOUS LOGNORMAL DIFFUSION PROCESS VIA POLYNOMIAL EXOGENOUS FACTORS

ABSTRACT In this article we propose a methodology for building a lognormal diffusion process with polynomial exogenous factors in order to fit data that present an exponential trend and show deviations with respect to an exponential curve in the observed time interval. We show that such a process approaches a nonhomogeneous lognormal diffusion and proves that it is specially useful in the case when external information (exogenous factors) about the process is not available even though the existence of these influences is clear. An application to the global man-made emissions of methane is provided.

On the estimation of the drift coefficient in diffusion processes with random stopping times

This paper considers stochastic differential equations with solutions which are multidimensional diffusion processes with drift coefficient depending on a parametric vector ?. By considering a trajectory observed up to a stopping time, the maximum likelihood estimator for ? has been obtained and its consistency and asymptotic normality have been provedSummaryThis paper considers stochastic differential equations with solutions which are multidimensional diffusion processes with drift coefficient dependent of a parametric vector θ. By considering a trajectory observed up to a stopping time, the maximum likelihood estimator for θ has been obtained and its consistency and asymptotic normality have been proved.ResumenEn este trabajo consideramos ecuaciones diferenciales estocásticas, cuyas soluciones son procesos de difusión multidimensionales con coeficiente tendencia dependiente de un vector paramétrico θ. Considerando una trayectoria observada hasta un tiempo aleatorio, hemos obtenido el estimador de máxima verosimilitud para θ, y hemos probado su consistencia y normalidad asintótica.

A comparison theorem of Marcus-Shepp in the uniform continuity of Gaussian processes

ResumenEn este Trabajo, sobre la base de un resultado conocido de Slepian en Teoría de comparación para procesos gaussianos, se obtiene un resultado sobre modelos de continuidad uniforme para procesos gaussianos, separables, realvaluados de media cero, que genraliza un resultado de Marcus-Shepp (Lema 3.2 de [5]).

Estudio del caracter markoviano fuerte y regularidades de la solucion de ecuaciones integrales estocasticas ito generalizadas

SumarioEl objetivo de este trabajo es un estudio sobre los carateres felleriano y markoviano fuerte y las propiedades de regularidad del proceso solución de una ecuacción integral estocástica generalizada (tipo Ito), pero generalizada en el sentido de considerar una formulación en términos de procesos operadorvaluados. Esta formulación generaliza simultánea e independientemente las integrales de Cabaña y Daletsky.AbstractThe object of this work is a study about the fellerian and markovian characters and the properties of regularity of the solution process of a generalized stochastic integral equation (Ito type), understanding this generalization in the sense that its formulation is give in terms of operator-valued processes. This formulation generalized simultaneous and independently the integrals of Cabaña and Daletsky.

Sobre el caracter Felleriano de la solucion de la ecuacion de Ito generalizada

SumarioEl contenido de este trabajo es un estudio sobre el carácter felleriano de la solución de una ecuación integral estocástica, del tipo de Ito, pero generalizada en el sentido de considerar una formulación en términos de procesos Operadores-valuados. En concreto [4], dadas sus características, es del tipo “Cabaña-Daletsky” en el sentido de que generaliza a ambos tipos, independientemente considerados.Para realizar este estudio, es preciso entre otras bases, introducir el espacio muestral apropiado, que se construye por el método de Dynkin, llegándose a que es de la, forma

Proof of the conjectures of H. Uhlig on the singular multivariate beta and the Jacobian of a certain matrix transformation

\tilde \Omega = \tilde x \cdot \Omega