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A recently proposed alternative to multifractional Brownian motion (mBm) with random Hurst exponent is studied, which we refer to as It\^o-mBm. It is shown that It\^o-mBm is locally self-similar. In contrast to mBm, its pathwise regularity…
Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter $H \in (0, 1)$ called the Hurst index. The use of time-changed processes in modeling often requires the…
This paper presents a new approach to the analysis of mixed processes \[X_t=B_t+G_t,\qquad t\in[0,T],\] where $B_t$ is a Brownian motion and $G_t$ is an independent centered Gaussian process. We obtain a new canonical innovation…
Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these…
This paper derives the asymptotic behavior of $$\mathbb{P} \{ \int\limits_0^\infty \mathbb{I}\Big(B_H(s)-c_1s>q_1u, B_H(s)-c_2s>q_2u\Big)ds>T_u\},\quad u \to \infty,$$ where $B_H$ is a fractional Brownian motion, $c_1,c_2,q_1,q_2>0,\ H \in…
We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not exceeding a fixed positive level over a large time interval $(0,T)$, $T\gg 1$. The probability $p(T)$ of such event is frequently a regularly varying function at…
Sub-fractional Brownian motion is a process analogous to fractional Brownian motion but without stationary increments. In \cite{GGL1} we proved a strong uniform approximation with a rate of convergence for fractional Brownian motion by…
In this paper we introduce a definition of a multi-dimensional fractional Brownian motion of Hurst index $H \in (0, 1)$ under volatility uncertainty (in short G-fBm). We study the properties of such a process and provide first results about…
We consider stochastic integration with respect to fractional Brownian motion (fBm) with $H < 1/2$. The integral is constructed as the limit, where it exists, of a sequence of Riemann sums. A theorem by Gradinaru, Nourdin, Russo & Vallois…
Fractional Brownian motion (fBm) is an important scale-invariant Gaussian non-Markovian process with stationary increments, which serves as a prototypical example of a system with long-range temporal correlations and anomalous diffusion.…
We study the asymptotic properties of an estimator of Hurst parameter of a stochastic differential equation driven by a fractional Brownian motion with $H > 1/2$. Utilizing the theory of asymptotic expansion of Skorohod integrals introduced…
A multivariate fractional Poisson process was recently defined in Beghin and Macci (2016) by considering a common independent random time change for a finite dimensional vector of independent (non-fractional) Poisson processes; moreover it…
We study the density of the signature of fractional Brownian motions with parameter $H>1/4$. In particular, we prove existence, smoothness, global Gaussian upper bounds and Varadhan's type asymptotics for this density. A key result is that…
In this work we present different results concerning the signature and the cubature of fractional Brownian motion (fBm). The first result regards the rate of convergence of the expected signature of the linear piecewise approximation of the…
The pointwise maximum of two independent and identically distributed isotropic fractional Brownian fields (with Hurst parameter $H<1/2$) is observed in a family of points in the unit square $\mathbf{C}=(-1/2,1/2]^{2}$. We assume that these…
The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and…
There is much confusion in the literature over Hurst exponents. Recently, we took a step in the direction of eliminating some of the confusion. One purpose of this paper is to illustrate the difference between fBm on the one hand and…
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…
We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal entries are independent one-dimensional H\"older continuous Gaussian processes of order gamma in (1/2,1). Using the stochastic calculus with…
We study the persistence probability for some discrete-time, time-reversible processes. In particular, we deduce the persistence exponent in a number of examples: first, we deal with random walks in random sceneries (RWRS) in any dimension…