Central limit theorems for multiple Skorohod integrals

In this paper, we prove a central limit theorem for a sequence of iterated Shorohod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted Hermite variations of the fractional Brownian motion are discussed.Comment: 32 pages; major changes in Sections 4 and

Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion II

This is the published version, also available here: http://dx.doi.org/10.1214/ECP.v18-2840.The purpose of this paper is to provide a complete description the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter H=1/6

Multifractality of Deutschemark/US Dollar Exchange Rates

This paper presents the first empirical investigation of the Multifractal Model of Asset Returns ("MMAR"). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Holder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Ito) diffusions and jump-diffusions by allowing a multiplicity of Holder exponents. This paper investigates multifractality in Deutschemark/US Dollar currency exchange rates. After finding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark/US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths, and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in financial data, along with more intensive study of estimation techniques and inference procedures.Multifractal model of asset returns, multifractal process, compound stochastic process, trading time, time deformation, scaling laws, multiscaling, self-similarity, self-affinity

Limits of sub-bifractional Brownian noises

Let $S^{H, K} = \{S^{H, K}_t, t\geq 0\}$ be the sub-bifractional Brownian motion (sbfBm) of dimension 1, with indices $H\in (0, 1)$ and $K\in (0, 1].$ We primarily prove that the increment process generated by the sbfBm $\left\{S^{H, K}_{h+t}-S^{H, K}_h, t\geq 0\right\}$ converges to $\left\{B^{HK}_t, t\geq 0\right\}$ as $h\rightarrow \infty$, where $\left\{B^{HK}_t, t\geq 0\right\}$ is the fractional Brownian motion with Hurst index $HK$. Moreover, we study the behavior of the noise associated to the sbfBm and limit theorems to $S^{H, K}$ and the behavior of the tangent process of sbfBm

Multifractality of Deutschemark/US Dollar Exchange Rates

This paper presents the first empirical investigation of the Multifractal Model of Asset Returns (“MMAR”). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Hölder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Itô) diffusions and jump-diffusions by allowing a multiplicity of Hölder exponents. This paper investigates multifractality in Deutschemark/US Dollar currency exchange rates. After finding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark/US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths, and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in financial data, along with more intensive study of estimation techniques and inference procedures

Bifurcation and Chaos in Fractional-Order Systems

This book presents a collection of seven technical papers on fractional-order complex systems, especially chaotic systems with hidden attractors and symmetries, in the research front of the field, which will be beneficial for scientific researchers, graduate students, and technical professionals to study and apply. It is also suitable for teaching lectures and for seminars to use as a reference on related topics