Patricia Román

First-passage-time densities for time-non-homogeneous diffusion processes

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous onedimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Guti6rrez

INFERENCE IN GOMPERTZ-TYPE NONHOMOGENEOUS STOCHASTIC SYSTEMS BY MEANS OF DISCRETE SAMPLING

ABSTRACT We consider an extension of the Gompertz homogeneous diffusion process by introducing time functions (exogenous factors) that affect its trend. After obtaining its transition probability density function, the inference on the parameters of the process is obtained by considering discrete sampling of the sample paths. Finally, we apply this stochastic process to model housing price in Spain.

FORECASTING FOR THE UNIVARIATE LOGNORMAL DIFFUSION PROCESS WITH EXOGENOUS FACTORS

The forecasting problem for the univariate lognormal diffusion process with exogenous factors is studied. For point predictions we propose the use of the mode function together with the mean function and, for some particular cases, the conditional version of these functions. For the interval predictions we obtain confidence bands for the mentioned functions; taking into consideration the percentile functions, intervals containing the variable of the process, for each time, with a specific probability are also obtained.

First-passage-time location function: Application to determine first-passage-time densities in diffusion processes

A time-dependent function, namely the First-Passage-Time Location function, is introduced in the context of the study of first-passage-times. From this function, a strategy is developed in order to solve numerically the Volterra integral equation of the second kind verified by the first-passage-time densities for diffusion processes. The proposed procedure provides the advantages in the application of quadrature methods in terms of an appropriate choice of the integration step, as well as an outstanding reduction in the computational cost. Some examples are developed showing the validity of that strategy as well as the computational advantages.

SOME TIME RANDOM VARIABLES RELATED TO A GOMPERTZ-TYPE DIFFUSION PROCESS

In the context of a Gompertz-type diffusion process (associated with a particular expression of the Gompertz curve for which its limit value depends on the initial value), we consider the study of two time-random variables: the inflection time of the model and the time at which it achieves a certain percentage of the total growth. Once we have shown the difficulties in the approach of the former, we deal with it as a particular case of the second. Furthermore, because of the peculiar characteristics of the considered process, we prove that these two problems can be formulated as a first-passage-time problem through a constant boundary. Finally, we conclude with an application to the growth of rabbits, in which we obtain the density functions of the inflection time and the time at which a rabbit achieves the half of its total growth.

Approximate and generalized confidence bands for the mean and mode functions of the lognormal diffusion process

Approximate and generalized confidence bands for the mean and mode functions of the univariate lognormal diffusion process are obtained. To this end, the already existing methods for building confidence intervals for the mean of the lognormal distribution have been suitably adapted. Moreover, a new method has been proposed. The bands obtained from the above procedures are compared through a simulation study and the comparisons are made both in terms of coverage errors and average widths. This comparative study allows to choose the most appropriate confidence band for each particular case in practical situations. This is shown in an application to a real data set.

A Gompertz‐type diffusion process for the study of social phenomena

We propose a growth stochastic model associated to the Gompertz curve (bounded and S-shaped growth curve) that improves, in certain aspects, other models in the literature. Growth is relevant to many applications. For this reason, many mathematical models have been proposed. For instance, Capocelli and Ricciardi [1] proposed a diffusion process associated with the Gompertz curve, including a noise term in the differential equation associated to the deterministic model. On the other hand, Tan [2] defined the stochastic Gompertz birth-death process as that whose mean function is a Gompertz curve. The curves considered in these two works have different expressions; the essential difference between them is its limiting value. For the first one, this is independent of the initial value x0, but not for the second one. There are many practical situations in the Social Sciences in which phenomena may be modeled by dynamic variables, with continuous state space, following a Gompertz pattern growing whose limit value depends on x0. Following the methodology in [1], we present here a new diffusion process whose mean function is a Gompertz curve, a property which is not fulfilled by the above diffusion process. This has the mentioned feature, and it may well be used for forecasting aims. The resulting process is a lognormal diffusion process with an exogenous factor of the kind h(t) = e−βt, for which the known inferential results [3 and references herein] are not valid. Our work also involves an inferential study of the process as well as an application to real data concerning the salaries along several years in the different Spanish communities. This application allows one to find groups with similar behavior patterns.