Band 29, Heft 1

We address the least-squares estimation of the drift coefficient parameter θ = ( λ , A , B , ω p ) \theta=(\lambda,A,B,\omega_{p}) of a time-inhomogeneous Ornstein–Uhlenbeck process that is observed at high frequency, in which the discretized step size ℎ satisfies h → 0 h\to 0 . In this paper, under the conditions n ⁢ h → ∞ nh\to\infty and n ⁢ h 2 → 0 nh^{2}\to 0 , we prove the consistency and the asymptotic normality of the estimators. We obtain the convergence of the parameters at rate n ⁢ h \sqrt{nh} , except for ω p \omega_{p} at n 3 ⁢ h 3 \sqrt{n^{3}h^{3}} . To verify our theoretical findings, we do a simulation study. We then illustrate the use of the proposed model in fitting the energy use of light fixtures in one Belgium household and the stock exchange.

We apply the Monte Carlo method to solving the Dirichlet problem of linear parabolic equations with fractional Laplacian. This method exploits the idea of weak approximation of related stochastic differential equations driven by the symmetric stable Lévy process with jumps. We utilize the jump-adapted scheme to approximate Lévy process which gives exact exit time to the boundary. When the solution has low regularity, we establish a numerical scheme by removing the small jumps of the Lévy process and then show the convergence order. When the solution has higher regularity, we build up a higher-order numerical scheme by replacing small jumps with a simple process and then display the higher convergence order. Finally, numerical experiments including ten- and one hundred-dimensional cases are presented, which confirm the theoretical estimates and show the numerical efficiency of the proposed schemes for high-dimensional parabolic equations.

In this paper, we propose and investigate two new kernel regression estimators based on a bias reduction transformation technique. We study the properties of these estimators and compare them with Nadaraya–Watson’s regression estimator and Slaoui’s (2016) regression estimator. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators, and the asymptotic MISE (mean integrated squared error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations and a real Malaria dataset.

In this paper, we address a long-standing open problem in stochastic simulation: construction of a random walk on spheres (RWS) algorithm for solving a system of elasticity equations, known as the Lamé equation. Many attempts to generalize the classic probabilistic representations like the Kac formula for parabolic and scalar elliptic equations failed. A different approach based on a branching random walk on spheres (BRWS) introduced in our paper of 1995 [K. K. Sabelfeld and D. Talay, Integral formulation of the boundary value problems and the method of random walk on spheres, Monte Carlo Methods Appl. 1 1995, 1, 1–34] made little progress in solving this problem. In the present study, we further improve the BRWS algorithm by a special implementation of a branching anisotropic random walk on spheres process.