24 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
Exact asymptotic for distribution densities of Levy functionals
A version of the saddle point method is developed, which allows one to
describe exactly the asymptotic behavior of distribution densities of Levy
driven stochastic integrals with deterministic kernels. Exact asymptotic
behavior is established for (a) the transition probability density of a
real-valued Levy process; (b) the transition probability density and the
invariant distribution density of a Levy driven Ornstein-Uhlenbeck process; (c)
the distribution density of the fractional Levy motion.Comment: Revised versio
Parameter estimation for non-stationary Fisher-Snedecor diffusion
The problem of parameter estimation for the non-stationary ergodic diffusion with Fisher-Snedecor invariant distribution, to be called Fisher-Snedecor diffusion, is considered. We propose generalized method of moments (GMM) estimator of unknown parameter, based on continuous-time observations, and prove its consistency and asymptotic normality. The explicit form of the asymptotic covariance matrix in asymptotic normality framework is calculated according to the new iterative technique based on evolutionary equations for the point-wise covariations. The results are illustrated in a simulation study covering various starting distributions and parameter values