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Register a new userEstimation error for occupation time functionals of stationary Markov processes
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The approximation of integral functionals with respect to a stationary Markov process by a Riemann-sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to get a better understanding of the estimation error and to prove a general error bound. The presented approach admits general integrands and gives a unifying explanation for different rates obtained in the literature. Several examples demonstrate how the general bound can be related to well-known function spaces.
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The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional c`adl`ag process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes, such as fractional Brownian motion. The results are used to approximate occupation times and local times. For Brownian motion, the upper bounds are shown to be sharp up to a log-factor.
The approximation of integral type functionals is studied for discrete observations of a continuous It^o semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions with fractional smoothness. An explicit $L^2$-lower bound shows that already lower order quadrature rules, such as the trapezoidal rule and the classical Riemann estimator, are rate optimal, but only the trapezoidal rule is efficient, achieving the minimal asymptotic variance.
We consider the connections among `clumped residual allocation models (RAMs), a general class of stick-breaking processes including Dirichlet processes, and the occupation laws of certain discrete space time-inhomogeneous Markov chains related to simulated annealing and other applications. An intermediate structure is introduced in a given RAM, where proportions between successive indices in a list are added or clumped together to form another RAM. In particular, when the initial RAM is a Griffiths-Engen-McCloskey (GEM) sequence and the indices are given by the random times that an auxiliary Markov chain jumps away from its current state, the joint law of the intermediate RAM and the locations visited in the sojourns is given in terms of a `disordered GEM sequence, and an induced Markov chain. Through this joint law, we identify a large class of `stick breaking processes as the limits of empirical occupation measures for associated time-inhomogeneous Markov chains.
The mixed fractional Vasicek model, which is an extended model of the traditional Vasicek model, has been widely used in modelling volatility, interest rate and exchange rate. Obviously, if some phenomenon are modeled by the mixed fractional Vasicek model, statistical inference for this process is of great interest. Based on continuous time observations, this paper considers the problem of estimating the drift parameters in the mixed fractional Vasicek model. We will propose the maximum likelihood estimators of the drift parameters in the mixed fractional Vasicek model with the Radon-Nikodym derivative for a mixed fractional Brownian motion. Using the fundamental martingale and the Laplace transform, both the strong consistency and the asymptotic normality of the maximum likelihood estimators have been established for all $Hin(0,1)$, $H eq 1/2$.
We build and study a data-driven procedure for the estimation of the stationary density f of an additive fractional SDE. To this end, we also prove some new concentrations bounds for discrete observations of such dynamics in stationary regime.
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