Calculus For Fractional Brownian Motion Research Articles

Optimal Hölder continuity and hitting probabilities for SPDEs with rough fractional noises

We investigate the optimal Hölder continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index 12 and spatial index H≤1/2. Using stochastic calculus for fractional Brownian motion, we prove that these systems are well-posed and the solutions are Hölder continuous. Furthermore, the optimal Hölder exponents are obtained, which is the first result, as far as we knew, on the optimal Hölder continuity of SHEs and SWEs driven by fractional Brownian sheet that is rough in space. Based on this sharp regularity, we obtain lower and upper bounds of hitting probabilities of the solutions in terms of Bessel–Riesz capacity and Hausdorff measure, respectively.

A formula of small time expansion for Young SDE driven by fractional Brownian motion

This paper shows an explicit small time expansion formula of expectation of the solution to Young SDEs driven by fractional Brownian motion H>1/2. The expansion coefficients are obtained by using Malliavin calculus for fractional Brownian motion. Furthermore, we show an analytically tractable expansion formula for the expectation of the solution to a general one-dimensional Young SDE driven by fractional Brownian motion and confirm the validity of our small time expansion through numerical experiments.

Stochastic calculus for fractional Brownian motion with Hurst exponent H>¼: A rough path method by analytic extension

The d-dimensional fractional Brownian motion (FBM for short) Bt=((Bt(1), …, Bt(d)), t∈ℝ) with Hurst exponent α, α∈(0, 1), is a d-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_{s}^{(i)}B_{t}^{(j)}]=\frac{1}{2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2\alpha})$. The long-standing problem of defining a stochastic integration with respect to FBM (and the related problem of solving stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either d or α. The case α=½ corresponds to the usual stochastic integration with respect to Brownian motion, while most computations become singular when α gets under various threshhold values, due to the growing irregularity of the trajectories as α→0. We provide here a new method valid for any d and for α>¼ by constructing an approximation Γ(ɛ)t, ɛ→0, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process Γz on the cut plane z∈ℂ∖ℝ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see [5]) but as yet a little mysterious divergence of Lévy’s area for α→¼.

Q-Fractional Brownian Motion in Infinite Dimensions with Application to Fractional Black–Scholes Market

In this article, we consider a white noise calculus for fractional Brownian motion with values in a separable Hilbert space, whereby the covariance operator Q is a kernel operator (Q-fractional Brownian motion). We introduce a fractional exponential, Q-fractional test function space , and a corresponding distribution space and the Q-fractional version of the Wick product, so that we can define a stochastic integral with respect to a fractional white noise process. Furthermore, a Q-fractional version of the Girsanov formula and a Q-fractional version of the Clark–Haussmann–Ocone theorem are proved. As applications, the solution for a stochastic evolution equation driven by fractional Brownian motion and an infinite-dimensional fractional Black–Scholes market are discussed.

An introduction to white–noise theory and Malliavin calculus for fractional Brownian motion

Fractional Brownian motion (FBM) with Hurst parameter index between 0 and 1 is a stochastic process originally introduced by Kolmogorov in a study of turbulence. Many other applications have subsequently been suggested. In order to obtain good mathematical models based on FBM, it is necessary to have a stochastic calculus for such processes. The purpose of this paper is to give an introduction to this newly developed theory of stochastic integration for FBM based on white-noise theory and (Malliavin–type) differentiation.