KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

Why Should We Analyse Convergence Using the Distribution Dynamics Approach?

- Paper first received, September 2007; in final form, September 2008 The convergence hypothesis has stimulated heated debate within the growth literature. The present paper compares the two most commonly adopted empirical approaches the regression approach and the distribution dynamics approach and argues that the former fails to uncover important features of the dynamics that might characterise the convergence process. In particular, the empirical section highlights the interpretational advantages stemming from the use of stochastic kernels to capture the evolution of the entire cross-sectional income distribution. Incidentally, comparison between the results obtained from alternative sets of Italian regions suggests that the use of administrative regions may lead to ambiguous results. Keywords: Distribution Dynamics, Stochastic Kernel, -convergence, Regions JEL classification: C14; C20; O40; O52; R10

Kernel convergence and biholomorphic mappings in several complex variables

We deal with kernel convergence of domains inℂnwhich are biholomorphically equivalent to the unit ballB. We also prove that there is an equivalence between the convergence on compact sets of biholomorphic mappings onB, which satisfy a growth theorem, and the kernel convergence. Moreover, we obtain certain consequences of this equivalence in the study of Loewner chains and of starlike and convex mappings onB.

Kernel convergence of hyperbolic components

We study families $G(\lambda,\cdot)$ of entire functions that are approximated by a sequence of families $G_n(\lambda,\cdot)$ of entire functions, where $\lambda\in\C$ is a parameter. In order to control the dynamics, the families are assumed to be of the same constant finite type. In this setting we prove the convergence of the hyperbolic components in parameter space as kernels in the sense of Carathéodory.