m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index

Abstract

Given an integer m, a probability measure ν on [0, 1], a process X and a real function g, we define the m-order ν-integral having as integrator X and as integrand g(X). In the case of the fractional Brownian motion B H , for any locally bounded function g, the corresponding integral vanishes for all odd indices m >

Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

“…Using the same linear regression as in the proof of Theorem 4.1 in [9], we can prove that, when H < 1/2:…”

From Constructive Field Theory to Fractional Stochastic Calculus. (I) An introduction: Rough Path Theory and Perturbative Heuristics

“…The last property implies that its derivative is a (distribution-valued) stationary field. 3 The one-dimensional case is very different and much simpler, and has been treated in [27]. 4 In other words (informally at least) EB called Lévy area.…”

On Simpson’s Rule and Fractional Brownian Motion with $$H = 1/10$$ H = 1 / 10

“…This theorem uses Malliavin calculus, and applies to a random vector with components in the form of Malliavin divergence integrals. After proving Theorem 2.3, the main task in proving (3) is to verify the conditions of Theorem 2.3, which are relatively long and technical.…”

“…For a smooth function f : R → R, we take the 'Simpson's rule' Riemann sum with uniform partition, It can be shown (see [3], or Section 3.1) that this sequence of sums converges in probability when B is fBm with H > 1/10, but in general it does not converge in probability when H ≤ 1/10. In this paper, we consider the particular case of H = 1/10, and show that S S n (t) does converge weakly to a random variable.…”