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Fractional Brownian motion (fBm) extends classical Brownian motion by introducing dependence between increments, governed by the Hurst parameter $H\in (0,1)$. Unlike traditional Brownian motion, the increments of an fBm are not independent.…
Consider an estimation of the Hurst parameter $H\in(0,1)$ and the volatility parameter $\sigma>0$ for a fractional Brownian motion with a drift term under high-frequency observations with a finite time interval. In the present paper, we…
The characteristic feature of semi-selfsimilar process is the invariance of its finite dimensional distributions by certain dilation for specific scaling factor. Estimating the scale parameter $\lambda$ and the Hurst index of such processes…
We focus on fast-slow systems involving both fractional Brownian motion (fBm) and standard Brownian motion (Bm). The integral with respect to Bm is the standard Ito integral, and the integral with respect to fBm is the generalised…
In this paper, we establish existence and uniqueness of strong solutions for a stochastic differential equation driven by an additive noise given by the sum of two correlated fractional Brownian sheets with different Hurst parameters. Our…
The geometry of the multifractional Brownian motion (mBm) is known to present a complex and surprising form when the Hurst function is greatly irregular. Nevertheless, most of the literature devoted to the subject considers sufficiently…
Consider the fractional Brownian Motion (fBM) $B^H=\{B^H(t): t \in [0,1] \}$ with Hurst index $H\in (0,1)$. We construct a probability space supporting both $B^H$ and a fully simulatable process $\hat B_{\epsilon}^H $ such that $$\sup_{t\in…
We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ and multiplicative noise component $\sigma$. When…
In order to understand the impact of random influences at physical boundary on the evolution of multiscale systems, a stochastic partial differential equation model under a fast random dynamical boundary condition is investigated. The…
The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H> 1/2. The integral is defined initially on the processes that are "piecewise" predictable on a…
Within the rough path framework we prove the continuity of the solution to random differential equations driven by fractional Brownian motion with respect to the Hurst parameter $H$ when $H \in (1/3, 1/2]$.
Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price…
We consider the problem of Hurst index estimation for solutions of stochastic differential equations driven by an additive fractional Brownian motion. Using techniques of the Malliavin calculus, we analyze the asymptotic behavior of the…
We study the two-dimensional fractional Brownian motion with Hurst parameter $H>{1/2}$. In particular, we show, using stochastic calculus, that this process admits a skew-product decomposition and deduce from this representation some…
In this article, we study the stability of solutions to 3D stochastic primitive equations driven by fractional noise. Since the fractional Brownian motion is essentially different from Brownian motion, lots of stochastic analysis tools are…
We introduce a class of Gaussian processes with stationary increments which exhibit long-range dependence. The class includes fractional Brownian motion with Hurst parameter H>1/2 as a typical example. We establish infinite and finite past…
In this paper we show the existence and uniqueness of a solution for a stochastic differential equation driven by an additive noise which is the sum of two fractional Brownian motions with different Hurst parameters. The proofs are based on…
The fractional Brownian motion can be considered as a Gaussian field indexed by $(t,H)\in {\mathbb{R}_{+}\times (0,1)}$, where $H$ is the Hurst parameter. On compact time intervals, it is known to be almost surely jointly H\"older…
In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a H\"older continuous function with H\"older exponent in $(1/3,1/2)$. Our stochastic integral is a…
We construct and study branching fractional Brownian motion with Hurst parameter $H\in(1/2,1)$. The construction relies on a generalization of the discrete approximation of fractional Brownian motion (Hammond and Sheffield, Probability…