14,004 research outputs found
Malliavin calculus for difference approximations of multidimensional diffusions: truncated local limit theorem
For a difference approximations of multidimensional diffusion, the truncated
local limit theorem is proved. Under very mild conditions on the distribution
of the difference terms, this theorem provides that the transition
probabilities of these approximations, after truncation of some asymptotically
negligible terms, possess a densities that converge uniformly to the transition
probability density for the limiting diffusion and satisfy a uniform
diffusion-type estimates. The proof is based on the new version of the
Malliavin calculus for the product of finite family of measures, that may
contain non-trivial singular components. An applications for uniform estimates
for mixing and convergence rates for difference approximations to SDE's and for
convergence of difference approximations for local times of multidimensional
diffusions are given.Comment: 34 page
Optimal Collocation Nodes for Fractional Derivative Operators
Spectral discretizations of fractional derivative operators are examined,
where the approximation basis is related to the set of Jacobi polynomials. The
pseudo-spectral method is implemented by assuming that the grid, used to
represent the function to be differentiated, may not be coincident with the
collocation grid. The new option opens the way to the analysis of alternative
techniques and the search of optimal distributions of collocation nodes, based
on the operator to be approximated. Once the initial representation grid has
been chosen, indications on how to recover the collocation grid are provided,
with the aim of enlarging the dimension of the approximation space. As a
results of this process, performances are improved. Applications to fractional
type advection-diffusion equations, and comparisons in terms of accuracy and
efficiency are made. As shown in the analysis, special choices of the nodes can
also suggest tricks to speed up computations
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