By the method of change measures, the moderate deviations for the Bessel clock ∫t0ds/xs（v） is studied, where （Xt（v）, t ≥0） is a squared Bessel process with index v 〉 0. Xs The rate function can be given explicitly. Furthermore, the functional moderate deviations for the Bessel clock are obtained.
The goal of this paper is to study large deviations for estimator and score function of some time inhomogeneous diffusion process. Large deviation in the non-steepness case with explicit rate functions is obtained by using parameter-dependent change of measure.
Let {β（s）,s ≥ O} be the standard Brownian motion in R^d with d ≥ 4 and let ｜Wr（t）｜ be the volume of the Wiener sausage associated with {β（s）, s ≥ O} observed until time t. Prom the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for ｜Wr（t）｜ - e｜Wr（t）｜ in this case.
We consider the comparison theorem of one-dimensional stochastic differential equation with non-Lipschitz diffusion coefficient. Considering the two one-dimensional stochastic differential equations as a two-dimensional equation,we present a necessary condition such that comparison theorem holds by viscosity solution approach.
ZHAO Shoujiang,GAO Fuqing School of Mathematics and Statistics,Wuhan University,Wuhan 430072,Hubei,China
Under the framework of white noise analysis, the existence of scattering solutions to the abstract dynamical φ4^4 wave equations in terms of generalized operators （see Section 3 below） is proven via a combination of the characterization for the symbol of generalized operators and the classical scattering results. In addition, some properties （Poincare invariance and irreducibility） of the solutions are discussed.
