Pedro Mota

In this paper we construct a new set of basis functions for the numerical solution of nonhomogeneous heat conduction problems with Dirichlet boundary conditions and null initial data. These functions can be seen as Newtonian potentials of plane waves for the heat equation and satisfy a null initial condition. Density results for adapted waves will be established and several numerical simulations will be presented in order to discuss the accuracy and feasibility of the proposed method. An application of the method for heat problems with non null initial temperature will also be discussed.

We study the exchange rate behavior, mainly as to the presence of asymmetry in the shocks to conditional variance. Particularly, we investigate if the presence of that asymmetric response is specific to a marked behavior of the currency, appreciation/Dutch disease/depreciation, and if it appears masked when taking long non-homogeneous periods. Taking Mozambique Metical bilateral exchange rate against South Africa Rand, a major trading partner, we identify specific movements in defined sub-periods, where the most recent has the Dutch disease under scrutiny. Our results point out that asymmetry emerges especially when the currency is depreciating, while it is masked when considering larger periods that combine differences in currency behavior.

In the current literature we can find mainly two approaches to the
SDE regime switching modeling. The traditional one, the externally induced
regime switching diffusions is described by the switching being derived from
a separate continuous time Markov process, with a finite, or denumerable,
state space { indexing the regimes { the random times of the regime switches
being exactly the jump times of the finite valued Markov process. There is a
first alternative approach in which the regime switching occurs whenever the
trajectory enters in some prescribed region on the state space; the regions we
consider will be mainly open intervals defined by unknown thresholds for the
trajectories; thresholds that, in principle, should also be estimated. In this
approach the partitioning of the the state space is already defined in the drift
and volatility of the SDE. In a second alternative approach the switching occurs
in a random way but at some random times defined when the trajectories hit
some prescribed thresholds, that again, must be estimated. We may designate
these two alternative approaches as auto-induced regime switching diffusions
as there is no external noise source to force the switching occurrence. We prove
a generalization of an existence result of the existence of auto-induced regime
switching SDE solutions for irregular coefficients and a result that encompasses
some of the cases of both externally and auto-induced regime switching SDE
solutions.