This paper investigates the so-called one-step local quasi–maximum likelihood estimator for the unit root process with GARCH~1,1! errors+ When the scaled conditional errors ~the ratio of the disturbance to the conditional standard deviation! follow a symmetric distribution, the asymptotic distribution of the estimated unit root is derived only under the second-order moment condition+ It is show...

We consider processes which have the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at −∞. We show that these processes do not have to have the distribution of standard Brownian motion in the backward direction of time, no matter which random time we take as the origin. We study the maximum and minimum r...

In one way or another, the extension of the standard Brownian motion process {B¡: t e [0,oo)} to a (Gaussian) random field {Bt: t € R+} involves a proof of the positive semi-definiteness of the kernel used to generalize p(s, 1) = cov(Bs,B¡) = s A t to multidimensional time. Simple direct analytical proofs are provided here for the cases of (i) the Levy multiparameter Brownian motion, (ii) the C...

Simple random walk and Brownian motion are two strongly interconnected mathematical concepts. They are widely involved in not only pure math, but also in many other scientific fields. In this paper I will first introduce and define some basic concepts of discrete-time random walk. Then I will construct Brownian Motion with some basic properties, and use a method called the strong approximation ...

We generalize the notion of brownian bridge. More precisely, we study a standard brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations which we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general pre...

We derive explicit forms of Markovian transition probability densities for the velocity space and phase-space Brownian motion of a charged particle in a constant magnetic field. An old-fashioned problem of the Brownian motion of a charged particle in a constant magnetic field has originated from studies of the diffusion of plasma across a magnetic field [1], [2] and nowadays, together with a fr...

We construct a Brownian path conditioned on its minimum value over a xed time interval by simple transformations of a Brownian bridge. Path transformations have proved useful in the study of Brownian motion and related processes , by providing simple constructions of various conditioned processes such as Brownian bridge, meander and excursion, starting from an unconditioned Brownian motion. As ...