Erd\'elyi-Kober Fractional Diffusion

The aim of this Short Note is to highlight that the {\it generalized grey Brownian motion} (ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erd\'elyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as {\it Erd\'elyi-Kober fractional diffusion}. The ggBm is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of self-similar processes with stationary increments and it depends on two real parameters: $0 < \alpha \le 2$ and $0 < \beta \le 1$. It includes the fractional Brownian motion when $0 < \alpha \le 2$ and $\beta=1$, the time-fractional diffusion stochastic processes when $0 < \alpha=\beta <1$, and the standard Brownian motion when $\alpha=\beta=1$. In the ggBm framework, the Mainardi function emerges as a natural generalization of the Gaussian distribution recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.Comment: Accepted for publication in Fractional Calculus and Applied Analysi

Short note on the emergence of fractional kinetics

In the present Short Note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non-Markovian master equations. Particle trajectories are assumed to be solely Markovian and described by the Continuous Time Random Walk model. But, as a consequence of the complexity of the medium, each trajectory is supposed to scale in time according to a particular random timescale. The link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed. In particular, when a stationary distribution is considered, the timescale distribution is uniquely determined as a function related to the fundamental solution of the space-time fractional diffusion equation. In contrast, when the non-stationary case is considered, the timescale distribution is no longer unique. Two distributions are here computed: one related to the M-Wright/Mainardi function, which is Green's function of the time-fractional diffusion equation, and another related to the Mittag-Leffler function, which is the solution of the fractional-relaxation equation

Distance, bank heterogeneity and entry in local banking markets

We examine the determinants of entry into Italian local banking markets during the period 1991-2002 and build a simple model in which the probability of branching in a new market depends on the features of both the local market and the potential entrant. Our econometric findings show that, all else being equal, banks are more likely to expand into those markets that are closest to their pre-entry locations. We also find that large banks are more able to cope with distance-related entry costs than small banks. Finally, we show that banks have become increasingly able to open branches in distant markets, probably due to the advent of information and communication technologies.entry, barriers to entry, local banking markets, geographical distance.

The M-Wright function in time-fractional diffusion processes: a tutorial survey

In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the M-Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast type.Comment: 32 pages, 3 figure

Agglomeration within and between regions: Two econometric based indicators

We propose two indexes to measure the agglomeration forces acting within and between different regions. Unlike the existing measures of agglomeration, our model-based indexes allow for simultaneous treatment of both aspects. Local plant diffusion in a given industry is modelled as a spatial error components process (SEC). Maximum likelihood inference on model parameters is dealt with, including the problem of data censoring. The statistical properties of standard agglomeration indexes in the data environment provided by our SEC model are then treated. Finally, our methodology is applied to Italian census data for both manufacturing and service industries.agglomeration, spatial autocorrelation, spatial error components model

Switching costs in local credit markets

Switching costs are a key determinant of market performance. This paper tests their existence in the corporate loan market in which they are likely to play a central role because of the complexity of contracts and informational problems. Using very detailed data at bank-firm level on four Italian local credit markets we empirically show that firms tend to iterate their choice of the main bank over time. This inertia is not related to unobserved and time invariant preferences of firms across banks and can be attributed to the existence of switching costs. We also offer evidence that banks price discriminate between new and old borrowers by charging lower interest rates to the former in order to cover part of the switching costs. The discount is about 44 basis points, equal to 7 per cent of the average interest rate. These results prove robust to a number of other potential identification drawbacks.switching costs, local credit markets, price discrimination, lending relationships