Modern Stochastics: Theory and Applications

The aim of this work is to establish essential properties of spatial birth-and-death processes with general birth and death rates on ${\mathbb{R}^{\mathrm{d}}}$. Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over ${\mathbb{R}^{\mathrm{d}}}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator.

The present work constitutes the second part of a two-paper project that, in particular, deals with an in-depth study of effective techniques used in econometrics in order to make accurate forecasts in the concrete framework of one of the major economies of the most productive Italian area, namely the province of Verona. It is worth mentioning that this region is indubitably recognized as the core of the commercial engine of the whole Italian country. This is why our analysis has a concrete impact; it is based on real data, and this is also the reason why particular attention has been taken in treating the relevant economical data and in choosing the right methods to manage them to obtain good forecasts. In particular, we develop an approach mainly based on vector autoregression where lagged values of two or more variables are considered, Granger causality, and the stochastic trend approach useful to work with the cointegration phenomenon.

This work is the first part of a project dealing with an in-depth study of effective techniques used in econometrics in order to make accurate forecasts in the concrete framework of one of the major economies of the most productive Italian area, namely the province of Verona. In particular, we develop an approach mainly based on vector autoregressions, where lagged values of two or more variables are considered, Granger causality, and the stochastic trend approach useful to work with the cointegration phenomenon. Latter techniques constitute the core of the present paper, whereas in the second part of the project, we present how these approaches can be applied to economic data at our disposal in order to obtain concrete analysis of import–export behavior for the considered productive area of Verona.