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Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of…

We consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic…

In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter $H>\frac 12$. Under some assumptions on the drift, we show that there is a unique solution, which has…

We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ with volatility coefficient $\sigma$ that is at least $\gamma$-H\"older continuous for…

Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do…

We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved…

We are interested in existence of solutions to the $d$-dimensional equation \begin{equation*} X_t=x_0+\int_0^t b(X_s)ds + B_t, \end{equation*} where $B$ is a (fractional) Brownian motion with Hurst parameter $H\leqslant 1/2$ and $b$ is an…

We consider a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$. We give an approximation result in a modulus type distance, up to the second order, by means of a sequence of rough…

We analyze the effect of additive fractional noise with Hurst parameter $H > \frac{1}{2}$ on fast-slow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet,…

Let $B^H = \{ B^H(t), t\in\mathbb{R}^N \}$ be an $(N,d)$-fractional Brownian sheet with Hurst index $H=(H_1,\dotsc,H_N)\in (0,1)^N$. The main objective of the present paper is to study the Hausdorff dimension of the image sets $B^H(F+t)$,…

For fractional Brownian motion with Hurst parameter H the Berman constant is defined. In this paper we consider a general random field (rf) Z that is a spectral rf of some stationary max-stable rf X and derive the properties of the…

The fractional Brownian motion (fBm) extends the standard Brownian motion by introducing some dependence between non-overlapping increments. Consequently, if one considers for example that log-prices follow an fBm, one can exploit the…

In this study, we develop a new theory of estimating Hurst parame- ter using conic multivariate adaptive regression splines (CMARS) method. We concentrate on the strong solution of stochastic differentional equations (SDEs) driven by…

We study the numerical approximation of SDEs with singular drifts (including distributions) driven by a fractional Brownian motion. Under the Catellier-Gubinelli condition that imposes the regularity of the drift to be strictly greater than…

In this note, we prove an $L^p$ uniform approximation of the fractional Brownian motion with Hurst exponent $0 < H < \frac{1}{2}$ by means of a family of continuous-time random walks imbedded on a given Brownian motion. The approximation is…

In this paper, we focus on mean-field anticipated backward stochastic differential equations (MF-BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H>1/2. First, the existence and uniqueness of this new type of…

In this article, we study a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter H in (1/4; 1/2). Towards this end, we apply Doss-Sussmann representation of the solution and an…

Herein we develop a dynamical foundation for fractional Brownian Motion. A clear relation is established between the asymptotic behaviour of the correlation function and diffusion in a dynamical system. Then, assuming that scaling is…

We prove a central limit theorem for an additive functional of the $d$-dimensional fractional Brownian motion with Hurst index $H\in(\frac{1}{1+d},\frac{1}{d})$, using the method of moments, extending the result by Papanicolaou, Stroock and…

In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter…