Construction Of Brownian Motion Research Articles

On a nonstandard Brownian motion and its maximal function

This article uses Radically Elementary Probability Theory (REPT) to prove results about the Wiener walk (the radically elementary Brownian motion) without the technical apparatus required by stochastic integration. The techniques used replace measure-theoretic tools by discrete probability and the rigorous use of infinitesimals. Specifically, REPT is applied to the results in Palacios (The American Statistician, 2008) to calculate certain expectations related to the Wiener walk and its maximal function. Because Palacios uses mostly combinatorics and no measure theory his results carry over through REPT with minimal changes. The paper also presents a construction of the Wiener walk which is intended to mimic the construction of Brownian motion from “continuous” white noise. A brief review of the nonstandard model on which REPT is based is given in the Appendix in order to minimize the need for previous exposure to the subject.

Black Scholes Option Pricing Model Brownian Motion Approach

Brownian motion has become one of the fundamental building blocks of modern quantitative finance. The mathematical theory of Brownian motion has been applied in contexts ranging far beyond the movement of particles in fluids. Until recently, stock market researchers have confronted the same problem. While they can chart the path of the market on a minute by minute basis it is very hard for them to observe who buys, who sells and how demand and supply affects price fluctuations. There exist many researchers about how the behavior of different investors makes the option price movement in a stock market. The purpose of this paper is to construct the black Scholes option pricing model in the stock markets by using of Brownian motion approach. The main ambition of this study is fourfold: 1) First we begin our approach to construction of Brownian motion from the simple symmetric random walk. 2) Next we introduce the Black – Scholes option pricing model with stock price movement by using of Geometric Brownian motion. 3) Then we extent this Brownian motion approach in the stock market and 4) Finally we construct the model for the generalization based on the deformation of the standard Brownian motion and Black Scholes pricing formula. And this paper will end with conclusion.

Self-regulating processes

We construct functions and stochastic processes for which a functional relation holds between amplitude and local regularity, as measured by the pointwise or local Holder exponent. We consider in particular functions and processes built by extending Weierstrass function, multifractional Brownian motion and the Levy construction of Brownian motion. Such processes have recently proved to be relevant models in various applications. The aim of this work is to provide a theoretical background to these studies and to provide a first step in the development of a theory for such self-regulating processes.

A characterisation of, and hypothesis test for, continuous local martingales

We give characterisations for Brownian motion and continuous local martingales, using the crossing tree, which is a sample-path decomposition based on first-passages at nested scales. These results are based on ideas used in the construction of Brownian motion on the Sierpinski gasket (Barlow and Perkins 1988). Using our characterisation we propose a test for the continuous martingale hypothesis, that is, that a given process is a continuous local martingale. The crossing tree gives a natural break-down of a sample path at different spatial scales, which we use to investigate the scale at which a process looks like a continuous local martingale. Simulation experiments indicate that our test is more powerful than an alternative approach which uses the sample quadratic variation.

A Haar-like Construction for the Ornstein Uhlenbeck Process

The classical Haar construction of Brownian motion uses a binary tree of triangular wedge-shaped functions. This basis has compactness properties which make it especially suited for certain classes of numerical algorithms. We present a similar basis for the Ornstein-Uhlenbeck process, in which the basis elements approach asymptotically the Haar functions as the index increases, and preserve the following properties of the Haar basis: all basis elements have compact support on an open interval with dyadic rational endpoints; these intervals are nested and become smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number. Thus the expansion in our basis, when evaluated at a dyadic rational, terminates in a finite number of steps. We prove the covariance formulae for our expansion and discuss its statistical interpretation.

Strong approximation of continuous local martingales by simple random walks

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

Martingale decomposition of Dirichlet processes on the Banach space C0[0, 1

We prove that for a given symmetric Dirichlet form of type g( u, v) = ∫ E 〈 A( z)∇ u( z), ∇ v( z)〉 hμ(d z) with E = C 0[0, 1] and H = classical Cameron-Martin space the corresponding diffusion process (under P μ ) can be decomposed into a forward and a backward E-valued martingale. The construction of the martingale is direct and explicit since it is based on a modification of Levy's construction of Brownian motion. Applications to prove tightness of laws of diffusions of the above kind are given.

On homogeneous chaos

AbstractThis paper discusses the Wiener–Itô chaos decomposition of an L2 function φ over Wiener space, and is concerned in particular with the identification of the integrands ƒn in the chaos decompositionFirst these are identified as Radon–Nikodým derivatives. Two elementary non-standard proofs of the Wiener–Itô chaos decomposition are given, based on Anderson's construction of Brownian motion and Itô integration.

The Haar-function construction of brownian motion indexed by sets

The Haar-function construction of Brownian Motion is obtained when the process is indexed by families of ‘smooth’ subsets of the k-dimensional cube. As the uniform limit of signed-measures, the resultant process is a Gaussian process with continuous paths with respect to the Hausdorff metric. A class of related Gaussian set functions defined on these subsets is also introduced.

The continuum limit of a classical 3-component, one-dimensional Heisenberg model is Brownian motion on the surface of a sphere

We show by explicit computation that when the temperature T is chosen as a function of the lattice spacing ε so that the correlation length stays fixed at one as ε→0, then the correlation functions of a classical one-dimensional Heisenberg model converge as ε→0 to the correlation functions of Brownian motion on the surface of a sphere. We remark that this provides a ’’statistical mechanical’’ construction of Brownian motion on the surface of a sphere.