This paper begins to explore a rigorous introduction to probability theory using ideas from algebra, measure theory, and other areas. We start with a basic explanation of terms and ideas and then move onto an attempt at explaining Brownian motion.

Brownian motion of particles affects many branches of science. We report on the Brownian motion of micrometer-sized beads of glass held in air by an optical tweezer, over a wide range of pressures, and we measured the instantaneous velocity of a Brownian particle. Our results provide direct verification of the energy equipartition theorem for a Brownian particle. For short times, the ballistic ...

Fundamental to many applications in financial engineering is the normal (Gaussian) distribution. It is the building block for simulating such basic stochastic processes as Brownian motion and geometric Brownian motion. In this section, we will go over algorithms for generating univariate normal rvs and learn how to use such algorithms for constructing sample paths of Brownian motion and geometr...

Fundamental to many applications in financial engineering is the normal (Gaussian) distribution. It is the building block for simulating such basic stochastic processes as Brownian motion and geometric Brownian motion. In this section, we will go over algorithms for generating univariate normal rvs and learn how to use such algorithms for constructing sample paths of Brownian motion and geometr...

Langevin equation for a nano-particle suspended in a laminar fluid flow was analytically studied. The Brownian motion generated from molecular bombardment was taken as a Wiener stochastic process and approximated by a Gaussian white noise. Euler-Maruyama method was used to solve the Langevin equation numerically. The accuracy of Brownian simulation was checked by performing a series of simulati...

This paper seeks to study standard Brownian motion and some of its properties. We construct this stochastic process and demonstrate a few properties including continuity and non-differentiability.

in this paper, the effect of the brownian term in natural convection of cuo-water nanofluid inside a partially filled porous cavity, with internal heat generation has been studied. it is assumed that the viscosity and thermal conductivity of nanofluid consists of a static part and a brownian part of which is a function of temperature and the volume fraction of nanofluid. because of internal hea...

in this paper, we introduce an efficient method based on haar wavelet to approximate a solutionfor the two-dimensional linear stochastic fredholm integral equation. we also give an example to demonstrate the accuracy of the method.