Time Of Fractional Brownian Motion Research Articles

Second-order limit laws for occupation times of fractional Brownian motion

Abstract We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

Time series characterization via horizontal visibility graph and Information Theory

Abstract Complex networks theory have gained wider applicability since methods for transformation of time series to networks were proposed and successfully tested. In the last few years, horizontal visibility graph has become a popular method due to its simplicity and good results when applied to natural and artificially generated data. In this work, we explore different ways of extracting information from the network constructed from the horizontal visibility graph and evaluated by Information Theory quantifiers. Most works use the degree distribution of the network, however, we found alternative probability distributions, more efficient than the degree distribution in characterizing dynamical systems. In particular, we find that, when using distributions based on distances and amplitude values, significant shorter time series are required. We analyze fractional Brownian motion time series, and a paleoclimatic proxy record of ENSO from the Pallcacocha Lake to study dynamical changes during the Holocene.

Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion

Let {Bt}t≥0 be a fractional Brownian motion with Hurst parameter 23<H<1. We prove that the approximation of the derivative of self-intersection local time, defined as αε=∫0T∫0tpε′(Bt−Bs)dsdt, where pε(x) is the heat kernel, satisfies a central limit theorem when renormalized by ε32−1H. We prove as well that for q≥2, the qth chaotic component of αε converges in L2 when 23<H<34, and satisfies a central limit theorem when renormalized by a multiplicative factor ε1−34H in the case 34<H<4q−34q−2.

NONPARAMETRIC COINTEGRATING REGRESSION WITH ENDOGENEITY AND LONG MEMORY

This paper explores nonparametric estimation, inference, and specification testing in a nonlinear cointegrating regression model where the structural equation errors are serially dependent and where the regressor is endogenous and may be driven by long memory innovations. Generalizing earlier results of Wang and Phillips (2009a,b,Econometric Theory25, 710–738,Econometrica77, 1901–1948), the conventional nonparametric local level kernel estimator is shown to be consistent and asymptotically (mixed) normal in these cases, thereby opening up inference by conventional nonparametric methods to a wide class of potentially nonlinear cointegrated relations. New results on the consistency of parametric estimates in nonlinear cointegrating regressions are provided, extending earlier research on parametric nonlinear regression and providing primitive conditions for parametric model testing. A model specification test is studied and confirmed to provide a valid mechanism for testing parametric specifications that is robust to endogeneity. But under long memory innovations the test is not pivotal, its convergence rate is parameter dependent, and its limit theory involves the local time of fractional Brownian motion. Simulation results show good performance for the nonparametric kernel estimates in cases of strong endogeneity and long memory, whereas the specification test is shown to be sensitive to the presence of long memory innovations, as predicted by asymptotic theory.

On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen (2005) and subsequently used by others to study the L2 and L3 moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) was introduced in Yan et al. (2008); however, the definition given there presents difficulties, since it is motivated by an incorrect application of the fractional Itô formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both Rosen (2005) and Yan et al. (2008). In the course of this endeavor we prove a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. As a further byproduct of our investigation, we also provide a small correction to an important technical second-moment bound for fBm which has appeared in the literature many times.

Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

We prove joint Holder continuity and an occupation-time formula for the self-intersection local time of fractional Brownian motion. Motivated by an occupation-time formula, we also introduce a new version of the derivative of self-intersection local time for fractional Brownian motion and prove Holder conditions for this process. This process is related to a different version of the derivative of self-intersection local time studied by the authors in a previous work.

Inverse local times of fractional Brownian motion

In this paper, we study the inverse local time of a d -dimensional fractional Brownian motion. We obtain that if L − 1 ( t ) is the inverse local time of fractional Brownian motion of Hurst index 0 < β < 1 with β d < 1 , then ∫ 0 ∞ E ( exp ( − λ L − 1 ( t ) ) ) d t = Γ ( 1 − β d ) / λ 1 − β d ( 2 π ) d / 2 . This result raises the question whether the inverse local time of a fractional Brownian motion has a stable law of parameter 1 − β d as it is the case for the standard one-dimensional Brownian motion.

Large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes

In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann-Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann-Liouville process.

Intersection Local Times of Independent Fractional Brownian Motions as Generalized White Noise Functionals

In this work we present expansions of intersection local times of fractional Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on $d$ for the existence of intersection local times in $L^2$ is derived, extending the results of D. Nualart and S. Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and more general Hurst coefficients.

Regularity of Intersection Local Times of Fractional Brownian Motions

Let \(B^{\alpha_{i}}\) be an (N i ,d)-fractional Brownian motion with Hurst index α i (i=1,2), and let \(B^{\alpha_{1}}\) and \(B^{\alpha_{2}}\) be independent. We prove that, if \(\frac{N_{1}}{\alpha_{1}}+\frac{N_{2}}{\alpha_{2}}>d\) , then the intersection local times of \(B^{\alpha_{1}}\) and \(B^{\alpha_{2}}\) exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points.One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre (J. Theor. Probab. 20:759–767, 2007), where the existence of the intersection local times of two independent (1,d)-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.

Hölder properties of local times for fractional Brownian motions

We study the local times of fractional Brownian motions for all temporal dimensions, N, spatial dimensions d and Hurst parameters H for which local times exist. We establish a Holder continuity result that is a refinement of Xiao (Probab Th Rel Fields 109:129-157, 1997). Our approach is an adaptation of the more general attack of Xiao (Probab Th Rel Fields 109:129-157, 1997) using ideas of Baraka and Mountford (1997, to appear), the principal result of this latter paper is contained in this article.

Power variation for Gaussian processes with stationary increments

We develop the asymptotic theory for the realised power variation of the processes X = ϕ • G , where G is a Gaussian process with stationary increments. More specifically, under some mild assumptions on the variance function of the increments of G and certain regularity conditions on the path of the process ϕ we prove the convergence in probability for the properly normalised realised power variation. Moreover, under a further assumption on the Hölder index of the path of ϕ , we show an associated stable central limit theorem. The main tool is a general central limit theorem, due essentially to Hu and Nualart [Y. Hu, D. Nualart, Renormalized self-intersection local time for fractional Brownian motion, Ann. Probab. (33) (2005) 948–983], Nualart and Peccati [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. (33) (2005) 177–193] and Peccati and Tudor [G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, in: M. Emery, M. Ledoux, M. Yor (Eds.), Seminaire de Probabilites XXXVIII, in: Lecture Notes in Math, vol. 1857, Springer-Verlag, Berlin, 2005, pp. 247–262], for sequences of random variables which admit a chaos representation.

The renormalization of self intersection local times of fractional Brownian motion

We present a new approach to treat the problem of self intersection local time of a d-dimensional Fractional Brownian motion based on the property of chaotic representation and the white noise analysis. This approach could be generalized to general Gaussian processes.

Regularity of renormalized self-intersection local time for fractional Brownian motion

Let BH be a d-dimensional fractional Brownian motion with Hurst parameter H 2 (0,1). We study the regularity, in the sense of the Malliavin calculus, of the renormalized self- intersection local time

Wavelet entropy and fractional Brownian motion time series

We study the functional link between the Hurst parameter and the normalized total wavelet entropy when analyzing fractional Brownian motion (fBm) time series—these series are synthetically generated. Both quantifiers are mainly used to identify fractional Brownian motion processes [L. Zunino, D.G. Pérez, M. Garavaglia, O.A. Rosso, Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform, Fractals 12(2) (2004) 223–233]. The aim of this work is to understand the differences in the information obtained from them, if any.

Renormalized self-intersection local time for fractional Brownian motion

Let BtH be a d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). Assume d≥2. We prove that the renormalized self-intersection local time $$\ell=\int_{0}^{T}\int_{0}^{t}\delta(B_{t}^{H}-B_{s}^{H})\,ds\,dt-\mathbb{E}\biggl(\int_{0}^{T}\int_{0}^{t}\delta (B_{t}^{H}-B_{s}^{H})\,ds\,dt\biggr)$$ exists in L2 if and only if H H\geq\frac{3}{2d}$, r(ɛ)lɛ converges in distribution to a normal law N(0,Tσ2), as ɛ tends to zero, where lɛ is an approximation of l, defined through (2), and r(ɛ)=|logɛ|−1 if H=3/(2d), and r(ɛ)=ɛd−3/(2H) if 3/(2d)<H.

Weighted Local Time for Fractional Brownian Motion and Applications to Finance

A Meyer-Tanaka formula involving weighted local time is derived for fractional Brownian motion and geometric fractional Brownian motion. The formula is applied to the study of the stop-loss-start-gain (SLSG) portfolio in a fractional Black-Scholes market. As a consequence, we obtain a fractional version of the Carr-Jarrow decomposition of the European call and put option prices into their intrinsic and time values.

Grandes déviations du temps local du mouvement brownien fractionnaire

The purpose of this paper is to prove a large deviation principle for a local time of fractional Brownian motion B H for all H∈(0,1). To cite this article: E.H. Lakhel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 797–801.

Functional limit theorem for occupation times of Gaussian processes---non-critical case

where X ^ X ^ , . . . are independent copies of X 7 . We shall consider only the case when fractional Brownian motion has jointly continuous local times which means the sample paths are point recurrent case, i.e., 0 2. Suppose f{x) > 0 be a bounded integrable function on M such that / := J R d f[χ)dx φ 0. We denote by lΊ^{t,x) the local time of fractional Brownian motion X 7 ' d , then putting a = 1 ηd,

A Note on the Local Time of Fractional Brownian Motion

Asymptotic behavior of the local time at the origin of q-dimensional fractional Brownian motion is considered when the index γ approaches the critical value 1/q. It is proved that, under a suitable (temporally inhomogeneous) normalization, it converges in law to the inverse of an extremal process which appears in the extreme value theory.