Multidimensional Fractional Brownian Motion Research Articles

Limit Theorem for Self-intersection Local Time Derivative of Multidimensional Fractional Brownian Motion

Limit Theorem for Self-intersection Local Time Derivative of Multidimensional Fractional Brownian Motion

A queueing model with ON/OFF sources: approximation and stationarity

Fractional Brownian motion approximation of queueing networks has been studied extensively. In the existing results related to this topic, the Hurst parameter of multidimensional fractional Brownian motion is only a constant H . However, just as pointed out by many scholars and practitioners, various Hurst parameters may be more appropriate. On the other hand, as a multivariate extension of fractional Brownian motion, operator fractional Brownian motion has operator self-similarity, and the dependence structure across the components of it is determined by the Hurst matrix. Moreover, it has also many potential applications in queueing theory. Inspired by these facts, we consider a queueing network with ON/OFF sources, and show that the workload process can be approximated by a reflected operator fractional Brownian motion under a heavy traffic condition. With this fact, it is important to consider stationarity. However, except for some special cases, there is no literature related to this topic. In our work, we construct an explicit stationary process associated with a two-dimensional reflected operator fractional Brownian motion.

Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions

We develop a new framework for error analysis on stochastic numerical schemes, with the rough path theory and stochastic backward error analysis. Based on our approach, we prove that the almost sure convergence rate of the modified Milstein scheme for stochastic differential equations driven by multiplicative multidimensional fractional Brownian motion with Hurst parameter H∈(14,12) is (2H−12)− for sufficiently smooth coefficients, which is optimal in the sense that it is consistent with the best probable convergence rate of implementable approximations of the Lévy area of fractional Brownian motion. Our result gives a positive answer to the conjecture proposed in [12] for the case H∈(13,12), and reveals for the first time that numerical schemes constructed by a second-order Taylor expansion converge for the case H∈(14,13].

Hurst exponent estimation of fractional surfaces for mammogram images analysis

The work presents a methodology to precise simulation and parameter estimation of multidimensional fractional Brownian motion (fBm). The simulation approach uses circulant embedding algorithm and solution of Poisson equation, while generalizing it to multiple dimensions. For estimation, a method using Wishart distribution and maximum likelihood is presented and verified on simulated data. Unlike approximate methods for generating multidimensional fBm and its Hurst exponent estimation, this approach shows unbiased results for all processes with short memory and majority of cases with long memory. The methodology is applied to mammography screening images to find significant differences between benign and cancerous breast lumps.

Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<frac{1}{2}. Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.

Crank–Nicolson scheme for stochastic differential equations driven by fractional Brownian motions

We study the Crank–Nicolson scheme for stochastic differential equations (SDEs) driven by a multidimensional fractional Brownian motion with Hurst parameter H>1/2. It is well known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank–Nicolson scheme achieves a convergence rate of n−2, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes, the corresponding Crank–Nicolson scheme for m-dimensional SDEs has a slower rate than for one-dimensional SDEs. Precisely, we shall prove that when the fBm is one-dimensional and when the drift term is zero, the Crank–Nicolson scheme achieves the convergence rate n−2H, and when the drift term is nonzero, the exact rate turns out to be n−12−H. In the general multidimensional case the exact rate equals n12−2H. In all these cases the asymptotic error is proved to satisfy some linear SDE. We also consider the degenerated cases when the asymptotic error equals zero.

A Bismut–Elworthy–Li formula for singular SDEs driven by a fractional Brownian motion and applications to rough volatility modeling

In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multidimensional fractional Brownian motion with Hurst parameter H <1/2. Singular here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of the δ price sensitivity of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDE's. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed local time variational calculus.

A general drift estimation procedure for stochastic differential equations with additive fractional noise

In this paper we consider the drift estimation problem for a general differential equation driven by an additive multidimensional fractional Brownian motion, under ergodic assumptions on the drift coefficient. Our estimation procedure is based on the identification of the invariant measure, and we provide consistency results as well as some information about the convergence rate. We also give some examples of coefficients for which the identifiability assumption for the invariant measure is satisfied.

Strong Existence and Higher Order Fréchet Differentiability of Stochastic Flows of Fractional Brownian Motion Driven SDEs with Singular Drift

In this paper we present a new method for the construction of strong solutions of SDE’s with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter $$H<\frac{1}{2}$$ . Furthermore, we prove the rather surprising result of the higher order Frechet differentiability of stochastic flows of such SDE’s in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a “local time variational calculus”. We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.

The First Exit Time of Fractional Brownian Motion from a Parabolic Domain

We study the first exit time of a multi-dimensional fractional Brownian motion from unbounded domains. In particular, we are interested in the upper tail of the corresponding distribution when the domain is parabola-shaped.

Smoothness of self-intersection local time of multidimensional fractional Brownian motion

In this paper, we study the existence, Hölder continuity and fractional smoothness of the self-intersection local time with respect to fractional Brownian motion:where BH is a d-dimensional fractional Brownian motion with Hurst index . We also consider the case of intersection local time for two independent multidimensional fractional Brownian motions.

Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion

We study the existence and uniqueness of the backward stochastic variational inequalities driven by \(m\)-dimensional fractional Brownian motion with Hurst parameters \(H_k\) (\(k=1,\ldots m\)) greater than \(1/2\). The stochastic integral used throughout the paper is the divergence type integral.

Mixed Stochastic Differential Equations: Existence and Uniqueness Result

In this paper we establish an existence and uniqueness result for solutions of multidimensional, time-dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter \(H>1/2\) and a multidimensional standard Brownian motion under a weaker condition than the Lipschitz one.

Fractional Gaussian fields: A survey

We discuss a family of random fields indexed by a parameter $s\in\mathbb{R} $ which we call the fractional Gaussian fields, given by \[\mathrm{FGF}_{s}(\mathbb{R} ^{d})=(-\Delta)^{-s/2}W, \] where $W$ is a white noise on $\mathbb{R}^{d}$ and $(-\Delta)^{-s/2}$ is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter $H=s-d/2$. In one dimension, examples of $\mathrm{FGF}_{s}$ processes include Brownian motion ($s=1$) and fractional Brownian motion ($1/2<s<3/2$). Examples in arbitrary dimension include white noise ($s=0$), the Gaussian free field ($s=1$), the bi-Laplacian Gaussian field ($s=2$), the log-correlated Gaussian field ($s=d/2$), Levy’s Brownian motion ($s=d/2+1/2$), and multidimensional fractional Brownian motion ($d/2<s<d/2+1$). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines. We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the $\mathrm{FGF}_{s}$ with $s\in(0,1)$ can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic $2s$-stable Levy process.

Derivative for self-intersection local time of multidimensional fractional Brownian motion

Let be a fractional Brownian motion taking values in with Hurst index . In this paper, we consider the self-intersection local time and its derivative in the spatial variable . In particular, we introduce the so-called integrated quadratic covariation and show that the Bouleau-Yor type identityholds for some suitable .

On the eigenvalue process of a matrix fractional Brownian motion

We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal entries are independent one-dimensional Hölder continuous Gaussian processes of order γ∈(1/2,1). Using the stochastic calculus with respect to the Young integral we show that these eigenvalues do not collide at any time with probability one. When the matrix process has entries that are fractional Brownian motions with Hurst parameter H∈(1/2,1), we find a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding matrix fractional Brownian motion. A new generalized version of the Itô formula for the multidimensional fractional Brownian motion is first established.

Remark on logically constant self-similar processes

Let X = {X(t), t∈ℝ+} be as self-similar processes with index α>0. We show that if X is locally constant and ℙ{X(1)=0}=0, then the law of X(t) is absolutely continuous. We discuss applicants of this result to homogeneous functions of a multidimensional fractional Brownian motion.

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.

Numerical Schemes for Rough Parabolic Equations

This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0,1) perturbed by a non-linear rough signal. It is the continuation of Deya (Electron. J. Probab. 16:1489–1518, 2011) and Deya et al. (Probab. Theory Relat. Fields, to appear), where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H>1/3.

A construction of the rough path above fractional Brownian motion using Volterra’s representation

This note is devoted to construct a rough path above a multidimensional fractional Brownian motion $B$ with any Hurst parameter $H\in(0,1)$, by means of its representation as a Volterra Gaussian process. This approach yields some algebraic and computational simplifications with respect to [Stochastic Process. Appl. 120 (2010) 1444--1472], where the construction of a rough path over $B$ was first introduced.