Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.     

Persistence probabilities for stationary increment processes

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion; random walks in random sceneries; random processes in Brownian scenery; and the Matheron–de Marsily model in ${\mathbb{Z}}^2$ with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of the optimal order in general and improved lower bounds (compared to previous literature) for many specific processes.     

Asymptotically good quasi-cyclic codes of fractional index

Generalizing the quasi-cyclic codes of index $1\frac{1}{3}$ introduced by Fan et al., we study a more general class of quasi-cyclic codes of fractional index generated by pairs of polynomials. The parity check polynomial and encoder of these codes are obtained. The asymptotic behaviours of the rates and relative distances of this class of codes are studied by using a probabilistic method. We prove that, for any positive real number $\delta$ such that the asymptotic GV-bound at $\frac{k+l}{2}\delta$ is greater than $\frac{1}{2}$, the relative distance of the code is convergent to $\delta$, while the rate is convergent to $\frac{1}{k+l}$. As a result, quasi-cyclic codes of fractional index are asymptotically good.     

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2,1)

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator Q, provided that $H\in (1/2,1)$ and $\mathrm{tr}(Q)$ is sufficiently small.     

The maximum rate of convergence for the approximation of the fractional Lévy area at a single point

In this note we study the approximation of the fractional Lévy area with Hurst parameter $H>1/2$, considering the mean square error at a single point as error criterion. We derive the optimal rate of convergence that can be achieved by arbitrary approximation methods that are based on an equidistant discretization of the driving fractional Brownian motion. This rate is $n^{-2H+1/2}$, where $n$ denotes the number of evaluations of the fractional Brownian motion, and is obtained by a trapezoidal rule.     

Asymptotics for high-dimensional covariance matrices and quadratic forms with applications to the trace functional and shrinkage

We establish large sample approximations for an arbitrary number of bilinear forms of the sample variance–covariance matrix of a high-dimensional vector time series using ${?}_1$-bounded and small ${?}_2$-bounded weighting vectors. Estimation of the asymptotic covariance structure is also discussed. The results hold true without any constraint on the dimension, the number of forms and the sample size or their ratios. Concrete and potential applications are widespread and cover high-dimensional data science problems such as tests for large numbers of covariances, sparse portfolio optimization and projections onto sparse principal components or more general spanning sets as frequently considered, e.g. in classification and dictionary learning. As two specific applications of our results, we study in greater detail the asymptotics of the trace functional and shrinkage estimation of covariance matrices. In shrinkage estimation, it turns out that the asymptotics differ for weighting vectors bounded away from orthogonality and nearly orthogonal ones in the sense that their inner product converges to 0.     

Fractional partial differential equations with boundary conditions

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\mathrm{\Omega })$ and $L_1(\mathrm{\Omega })$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes on bounded domains and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax–Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.     

On the non-commutative fractional Wishart process

We investigate the process of eigenvalues of a fractional Wishart process defined by $N=B^{?}B$, where B is the matrix fractional Brownian motion recently studied in [18]. Using stochastic calculus with respect to the Young integral we show that, with probability one, the eigenvalues do not collide at any time. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in (1/2,1)$, we derive a stochastic differential equation in the Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally, a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the non-commutative fractional Wishart process, which constitutes the family of fractional dilations of the free Poisson distribution.     

Numerical simulation for the three-dimension fractional sub-diffusion equation

Fractional sub-diffusion equations have been widely used to model sub-diffusive systems. Most algorithms are designed for one-dimensional problems due to the memory effect in fractional derivative. In this paper, the numerical simulation of the 3D fractional sub-diffusion equation with a time fractional derivative of order $\alpha$ $(0<\alpha <1)$ is considered. A fractional alternating direction implicit scheme (FADIS) is proposed. We prove that FADIS is uniquely solvable, unconditionally stable and convergent in $H^1$ norm by the energy method. A numerical example is given to demonstrate the efficiency of FADIS.