It was noted in Chapter 6 that Brownian motion is not differentiable along its path, that is with respect to t, see property BM6. However, even just a passing familiarity with the literature on random walks and unit root tests will have alerted the reader to the use of notation that ...
Key Features: The motion is continuous but the path is nowhere differentiable, leading to a fractallike structure. It is a null recurrent symmetric random walk, meaning it returns to its starting point infinitely often, but the expected time to return is infinite.Practical Applications:...
But while the paths of the Brownian motion stochastic process are continuous, they are not differentiable. This remarkable statement is difficult to comprehend. Indeed, many elementary calculus explanations implicitly tend to assume that all continuous functions are differentiable, and if we were to be...
Although Brownian motions are continuous everywhere they are differentiable nowhere. Essentially this means that a Brownian motion has fractal geometry. This has important implications regarding the choice of calculus methods used when Brownian motions are to be manipulated. Brownian motions satisfy both ...
differentiable on SN−1 by Rademacher's theorem and to recall the represen- tation (3.2). Let F be a measurable map from [0, T ] into the compact and convex subsets of RN and let (Ω, P ) denote Brownian motion on SN−1 whose sample paths ω(t) are uniformly distributed on ...
Since the basket does not follow a geometric Brownian motion dynamics, the determination of σk,ti has to be made carefully. For this, consider a general Basket on different assets Xi with weights pi: (4.1)Bt=∑ipiXi,t. We define σB,t, its volatility at time t, by ...
An approximation of the fractional Brownian motion based on the Ornstein-Uhlenbeck process is used to obtain an asymptotic likelihood function. Two estimators of the Hurst index are then presented in the likelihood approach. The first estimator is produc
In this work, we develop Mandelbrot's idea that Weierstrass's nowhere differentiable function can be modified and randomized to approximate fractional Brownian motion (FBM). Our approach covers the convergence of processes of a more general type and allows us to consider different dependence structures...
The spectral approximation u N for all t ≥0 is a continuously differentiable func- tion with the values in X N . At t =0 it satisfies the initial condition, while at t >0 it satisfies P N du N dt +L N u N −Q =0, 11.1 Spectral Representation of Spatial Derivatives 577...
It is well knownthat Brownian motion has a modification, the sample paths of which arecontinuous almost surely, but sample paths of any modification are nowheredifferentiable. As it turns out, these facts remain true for fractional Brownianmotion.We say that a stochastic process {XðtÞ;0#...