From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index $${{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}$$

Stochastic Calculus for Fractional Brownian Motion I. Theory

This paper describes some of the results in [5] for a stochastic calculus for a fractional Brownian motion with the Hurst parameter in the interval (1/2, 1). Two stochastic integrals are defined with explicit expressions for their first two moments. Multiple and iterated integrals of a fractional Browinian motion are defined and various properties of these integrals are given. A square integrab...

Fractional Brownian motion: stochastic calculus and applications

Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter H ∈ (0, 1) called the Hurst index. In this note we will survey some facts about the stochastic calculus with respect to fBm using a pathwise approach and the techniques of the Malliavin calculus. Some applications in turbulence and finance will be discussed. Math...

Stochastic calculus with respect to fractional Brownian motion

— Fractional Brownian motion (fBm) is a centered selfsimilar Gaussian process with stationary increments, which depends on a parameter H ∈ (0, 1) called the Hurst index. In this conference we will survey some recent advances in the stochastic calculus with respect to fBm. In the particular case H = 1/2, the process is an ordinary Brownian motion, but otherwise it is not a semimartingale and Itô...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi–martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.