In previous papers, the author considered the model of anomalous diffusion, defined by stable random process on an interval with reflecting edges. Estimates of the rate convergence of this process distribution to a un...In previous papers, the author considered the model of anomalous diffusion, defined by stable random process on an interval with reflecting edges. Estimates of the rate convergence of this process distribution to a uniform distribution are constructed. However, recent physical studies require consideration of models of diffusion, defined not only by stable random process with independent increments but multivariate fractional Brownian motion with dependent increments. This task requires the development of special mathematical techniques evaluation of the rate of convergence of the distribution of multivariate Brownian motion in a segment with reflecting boundaries to the limit. In the present work, this technology is developed and a power estimate of the rate of convergence to the limiting uniform distribution is built.展开更多
Let {S t H, t ≥ 0） be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 〈 H 〈 1. Its main properties are studied. They suggest that SH lies between the ...Let {S t H, t ≥ 0） be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 〈 H 〈 1. Its main properties are studied. They suggest that SH lies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H for which SH is not a semi-martingale.展开更多
A family of tests for the presence of regression effect under proportional and non-proportional hazards models is described. The non-proportional hazards model, although not completely general, is very broad and inclu...A family of tests for the presence of regression effect under proportional and non-proportional hazards models is described. The non-proportional hazards model, although not completely general, is very broad and includes a large number of possibilities. In the absence of restrictions, the regression coefficient, β(t), can be any real function of time. When β(t) = β, we recover the proportional hazards model which can then be taken as a special case of a non-proportional hazards model. We study tests of the null hypothesis;H0:β(t) = 0 for all t against alternatives such as;H1:∫β(t)dF(t) ≠ 0 or H1:β(t) ≠ 0 for some t. In contrast to now classical approaches based on partial likelihood and martingale theory, the development here is based on Brownian motion, Donsker’s theorem and theorems from O’Quigley [1] and Xu and O’Quigley [2]. The usual partial likelihood score test arises as a special case. Large sample theory follows without special arguments, such as the martingale central limit theorem, and is relatively straightforward.展开更多
We use reflecting Brownian motion(RBM)to prove the well-known Gauss–Bonnet–Chern theorem for a compact Riemannian manifold with boundary.The boundary integrand is obtained by carefully analyzing the asymptotic behav...We use reflecting Brownian motion(RBM)to prove the well-known Gauss–Bonnet–Chern theorem for a compact Riemannian manifold with boundary.The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.展开更多
We study the uniqueness and existence of solutions of reflected G-stochastic differential equations （RGSDEs） with nonlinear resistance under an integral-Lipschitz condition of coefficients. Moreover, we obtain the c...We study the uniqueness and existence of solutions of reflected G-stochastic differential equations （RGSDEs） with nonlinear resistance under an integral-Lipschitz condition of coefficients. Moreover, we obtain the comparison theorem for RGSDEs with nonlinear resistance.展开更多
文摘In previous papers, the author considered the model of anomalous diffusion, defined by stable random process on an interval with reflecting edges. Estimates of the rate convergence of this process distribution to a uniform distribution are constructed. However, recent physical studies require consideration of models of diffusion, defined not only by stable random process with independent increments but multivariate fractional Brownian motion with dependent increments. This task requires the development of special mathematical techniques evaluation of the rate of convergence of the distribution of multivariate Brownian motion in a segment with reflecting boundaries to the limit. In the present work, this technology is developed and a power estimate of the rate of convergence to the limiting uniform distribution is built.
文摘Let {S t H, t ≥ 0） be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 〈 H 〈 1. Its main properties are studied. They suggest that SH lies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H for which SH is not a semi-martingale.
文摘A family of tests for the presence of regression effect under proportional and non-proportional hazards models is described. The non-proportional hazards model, although not completely general, is very broad and includes a large number of possibilities. In the absence of restrictions, the regression coefficient, β(t), can be any real function of time. When β(t) = β, we recover the proportional hazards model which can then be taken as a special case of a non-proportional hazards model. We study tests of the null hypothesis;H0:β(t) = 0 for all t against alternatives such as;H1:∫β(t)dF(t) ≠ 0 or H1:β(t) ≠ 0 for some t. In contrast to now classical approaches based on partial likelihood and martingale theory, the development here is based on Brownian motion, Donsker’s theorem and theorems from O’Quigley [1] and Xu and O’Quigley [2]. The usual partial likelihood score test arises as a special case. Large sample theory follows without special arguments, such as the martingale central limit theorem, and is relatively straightforward.
文摘We use reflecting Brownian motion(RBM)to prove the well-known Gauss–Bonnet–Chern theorem for a compact Riemannian manifold with boundary.The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.
基金The author would like to thank the referees for their careful reading and helpful suggestions. This work was partially supported by the China Scholarship Council （No. 201306220101）, the National Natural Science Foundation of China （Grant No. 11221061）, and the Programme of Introducing Talents of Discipline to Universities of China （No. B12023）.
文摘We study the uniqueness and existence of solutions of reflected G-stochastic differential equations （RGSDEs） with nonlinear resistance under an integral-Lipschitz condition of coefficients. Moreover, we obtain the comparison theorem for RGSDEs with nonlinear resistance.
