We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exists a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given.

Approximation of Fractional Brownian Motion by Martingales

Let $Z^{H}$ be the Rosenblatt process with the representation$$Z^H_t=\int_0^t\int_0^tL^H(t,s,r)dB_{s}dB_{r},$$where $B$ is a standard Brownian motion, $\frac12<H<1$ and $L^H$ is a given kernel. By reviewing the kernel $L^H$ we construct its approximation of multiple Wiener integrals of the form $$\int_{0}^{t}\int_{0}^{t}\left\{k_{1}(sr)^{-\frac{1}{2}H} +k_{2}(s\vee r)^{\frac{1}{2}H}(s\wedge r)^{-\frac{1}{2}H}|s-r|^{H-1}\right\}dB_{s}dB_{r}, \;\;k_1,k_2\geq 0.$$We find an optimal approximation of $Z^{H}$ via calculating accurately the values of $k_1,k_2$.

https://projecteuclid.org/download/pdf_1/euclid.ecp/1465320938

Approximating the Rosenblatt process by multiple Wiener integrals

We obtain a lower bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval. The distances between fractional Brownian motion and some subspaces of Gaussian martingales are compared. The upper and lower bounds are obtained for the constant in the representation of a fractional Brownian motion in terms of the Wiener process.

https://www.ams.org/tpms/2011-83-00/S0094-9000-2012-00838-7/S0094-9000-2012-00838-7.pdf

A bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval

In this paper, we consider the optimal approximation of a Rosenblatt process based on semimartingales of the form where (y1, y2)↦a(y1, y2) is a square integrable process and B is a standard Brownian motion. We show that there exists a unique semimartingale closest to Rosenblatt process if a is deterministic.

Approximation of the Rosenblatt process by semimartingales

In this paper, we obtain two approximations in law of the complex fractional Brownian motion by processes constructed from a Poisson process and a Levy process, respectively.

https://link.springer.com/content/pdf/10.1186/s13662-018-1892-4.pdf

Weak convergence of the complex fractional Brownian motion

We find an approximation in the space L∞([0, T ];L2(Ω)) of a fractional Brownian motion by martingales of the form ∫ t 0 a(s) dWs, where W is a Wiener process, a(s) is a power function with a negative index, that is a(s) = k · s−α where k > 0, α = H − 1/2, and H is the index of fractional Brownian motion.

/pdf/approximation-of-fractional-brownian-motion-by-wiener-3dpv34gmti.pdf

Approximation of fractional Brownian motion by Wiener integrals

Fractional Brownian Motion: Approximations and Projections

We study the problem of approximation of a fractional Brownian motion with the help of Gaussian martingales that can be represented as the integrals with respect to a Wiener process and with nonrandom integrands being “similar” to the kernel of the fractional Brownian motion. The “similarity” is understood in the sense that an integrand is the value of the kernel at some point. We establish analytically and evaluate numerically the upper and lower bounds for the distance between the fractional Brownian motion and the space of Gaussian martingales.

Oksana Banna

Papers

Approximation of fractional Brownian motion by Wiener integrals

Approximation of Fractional Brownian Motion by Martingales

Fractional Brownian Motion: Approximations and Projections

A bound for the distance between fractional Brownian motion and the space of Gaussian martingales on an interval

The distance between fractional Brownian motion and the subspace of martingales with “similar” kernels