arXiv subjects

Qidi Peng

Publications and source records attributed to Qidi Peng.

At least 19 recordsLinked to original sources

arXiv2026

Estimation of the Self-similarity Index of Non-stationary Increments Self-similar Processes via Lamperti Transformations

We introduce a novel method for estimating the self-similarity index of a general $H$-self-similar process with either stationary or non-stationary increments. The estimation algorithm is developed based on a modified Lamperti transformation, which transforms $H$-self-similar processes to stationary ones. As an application, we show how to use this approach to estimate the self-similarity index of fractional Brownian motion, subfractional Brownian motion, bifractional Brownian motion, and trifractional Brownian motion. Simulation study is performed to support the consistency of our estimators. Implementation in Python is publicly shared. Application on the estimation of the self-similarity index of the Nile river water level data from the year 900 to 1200 C.E..

math.ST↗

arXiv2025

Investigation and Development of the Methodologies for Simulating Self-similar Processes

This paper is devoted to the study of simulating a large class of self-similar processes. Since most current simulation approaches are limited to case-by-case studies, every existing approach has its constraints and flaws; hence a general and efficient simulation approach is in demand. Our study sheds some light in this direction. The paper's contributions are bi-fold. First, reviews and improvements are made to some existing methods for simulating specific self-similar processes. Second, we propose a novel method to simulate a general self-similar process, where we use a modified inverse Lamperti transformation to transform self-similarity to stationarity. Successful applications are made to simulate fractional Brownian motion and sub-fractional Brownian motion.

math.PR↗

arXiv2025

On a theorem of Nikol'skii

We present Bernstein lethargy theorem and examine the relationship between Bernstein lethargy theorem and reflexivity.

math.FA↗

arXiv2022

Fractional Brownian Motion: Local Modulus of Continuity with Refined Almost Sure Upper Bound and First Exit Time from One-sided Barrier

Based on an optimal rate wavelet series representation, we derive a local modulus of continuity result with a refined almost sure upper bound for fractional Brownian motion. \sloppy The obtained upper bound of the small fractional Brownian increments is of order $\mathcal O_{a.s.}\big(|h|^H\sqrt{\log\log |h|^{-1}}\big)$ as $|h|\to0$, and an upper bound of its $p$th moment is provided, for any $p>0$. This result fills the gap of the law of iterated logarithm for fractional Brownian motion, where the moments' information of the random multiplier in the upper bound is missing. With this enhanced upper bound and some new results on the distribution of the maximum of fractional Brownian motion, we obtain a new and refined asymptotic estimate of the upper-tail probability for a fractional Brownian motion to first exit from a positive-valued barrier over time $T$, as $T\to+\infty$.

math.PR↗

arXiv2021

Variable Selection and Regularization via Arbitrary Rectangle-range Generalized Elastic Net

We introduce the arbitrary rectangle-range generalized elastic net penalty method, abbreviated to ARGEN, for performing constrained variable selection and regularization in high-dimensional sparse linear models. As a natural extension of the nonnegative elastic net penalty method, ARGEN is proved to have variable selection consistency and estimation consistency under some conditions. The asymptotic behavior in distribution of the ARGEN estimators have been studied. We also propose an algorithm called MU-QP-RR-W-$l_1$ to efficiently solve ARGEN. By conducting simulation study we show that ARGEN outperforms the elastic net in a number of settings. Finally an application of S&P 500 index tracking with constraints on the stock allocations is performed to provide general guidance for adapting ARGEN to solve real-world problems.

stat.ML↗

arXiv2021

Linear Multifractional Stable Sheets in the Broad Sense: Existence and Joint Continuity of Local Times

We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $\alpha\in(0,2]$, to include both linear multifractional Brownian sheets ($\alpha=2$) and linear multifractional stable sheets ($\alpha<2$). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local H\"older condition of the local times in the set variable. Among the main results of this paper, Theorem 2.4 provides a sufficient and necessary condition for the existence of local times of LMSS; Theorem 3.1 shows a sufficient condition for the joint continuity of local times; and Theorem 4.1 proves a sharp local H\"older condition for the local times in the set variable. All these theorems improve significantly the existing results for the local times of multifractional Brownian sheets and linear multifractional stable sheets in the literature.

math.PR↗

arXiv2020

Cluster Analysis on Locally Asymptotically Self-similar Processes with Known Number of Clusters

We conduct cluster analysis on a class of locally asymptotically self-similar stochastic processes, which includes multifractional Brownian motion as a representative. When the true number of clusters is supposed to be known, a new covariance-based dissimilarity measure is introduced, from which we obtain the approximately asymptotically consistent clustering algorithms. In simulation studies, clustering data sampled from multifractional Brownian motions with distinct functional Hurst parameters illustrates the approximated asymptotic consistency of the proposed algorithms. Clustering global financial markets' equity indexes returns and sovereign CDS spreads provides a successful real world application.

stat.ML↗

arXiv2019

Series Representation of Jointly S$\alpha$S Distribution via Symmetric Covariations

We introduce the notion of symmetric covariation, which is a new measure of dependence between two components of a symmetric $\alpha$-stable random vector, where the stability parameter $\alpha$ measures the heavy-tailedness of its distribution. Unlike covariation that exists only when $\alpha\in(1,2]$, symmetric covariation is well defined for all $\alpha\in(0,2]$. We show that symmetric covariation can be defined using the proposed generalized fractional derivative, which has broader usages than those involved in this work. Several properties of symmetric covariation have been derived. These are either similar to or more general than those of the covariance functions in the Gaussian case. The main contribution of this framework is the representation of the characteristic function of bivariate symmetric $\alpha$-stable distribution via convergent series based on a sequence of symmetric covariations. This series representation extends the one of bivariate Gaussian.

math.ST↗

arXiv2019

Some Developments in Clustering Analysis on Stochastic Processes

We review some developments on clustering stochastic processes and come with the conclusion that asymptotically consistent clustering algorithms can be obtained when the processes are ergodic and the dissimilarity measure satisfies the triangle inequality. Examples are provided when the processes are distribution ergodic, covariance ergodic and locally asymptotically self-similar, respectively.

stat.ML↗

arXiv2019

Covariance-based Dissimilarity Measures Applied to Clustering Wide-sense Stationary Ergodic Processes

We introduce a new unsupervised learning problem: clustering wide-sense stationary ergodic stochastic processes. A covariance-based dissimilarity measure together with asymptotically consistent algorithms is designed for clustering offline and online datasets, respectively. We also suggest a formal criterion on the efficiency of dissimilarity measures, and discuss of some approach to improve the efficiency of our clustering algorithms, when they are applied to cluster particular type of processes, such as self-similar processes with wide-sense stationary ergodic increments. Clustering synthetic data and real-world data are provided as examples of applications.

stat.ML↗

arXiv2018

A General Class of Multifractional Processes and Stock Price Informativeness

We introduce a general class of stochastic processes driven by a multifractional Brownian motion (mBm) and study the estimation problems of their pointwise Hölder exponents (PHE) based on a new localized generalized quadratic variation approach (LGQV). By comparing our suggested approach with the other two existing benchmark estimation approaches (classic GQV and oscillation approach) through a simulation study, we show that our estimator has better performance in the case where the observed process is some unknown bivariate function of time and mBm. Such multifractional processes, whose PHEs are time-varying, can be used to model stock prices under various market conditions, that are both time-dependent and region-dependent. As an application to finance, an empirical study on modeling cross-listed stocks provides new evidence that the equity path's roughness varies via time and the stock price informativeness properties from global stock markets.

q-fin.MF↗

arXiv2018

Representation Theorems of $\mathbb{R}$-trees and Brownian Motions Indexed by $\mathbb R$-trees

We provide a new representation of an $\mathbb R$-tree by using a special set of metric rays. We have captured the four-point condition from these metric rays and shown an equivalence between the $\mathbb R$-trees with radial and river metrics, and these sets of metric rays. In stochastic analysis, these graphical representation theorems are of particular interest in identifying Brownian motions indexed by $\mathbb R$-trees.

math.MG↗

arXiv2018

Bernstein Lethargy Theorem and Reflexivity

In this paper, we prove the equivalence of reflexive Banach spaces and those Banach spaces which satisfy the following form of Bernstein's Lethargy Theorem. Let $X$ be an arbitrary infinite-dimensional Banach space, and let the real-valued sequence $\{d_n\}_{n\ge1}$ decrease to $0$. Suppose that $\{Y_n\}_{n\ge1}$ is a system of strictly nested subspaces of $X$ such that $\overline Y_n \subset Y_{n+1}$ for all $n\ge1$ and for each $n\ge1$, there exists $y_n\in Y_{n+1}\backslash Y_n$ such that the distance $\rho(y_n,Y_n)$ from $y_n$ to the subspace $Y_n$ satisfies $$ \rho(y_n,Y_n)=\|y_n\|. $$ Then, there exists an element $x\in X$ such that $\rho(x,Y_n)=d_n$ for all $n\ge1$.

math.FA↗

arXiv2018

Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces

This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is a non-increasing sequence of non-negative numbers tending to 0, then for any $c\in(0,1]$ we can find $x_{c} \in X$, such that the distance $ρ(x_{c}, Y_n)$ from $x_{c}$ to $Y_n$ satisfies $$ c d_n \leq ρ(x_{c},Y_n) \leq 4c d_n,~\mbox{for all $n\in\mathbb N$}. $$ We prove the above inequality by first improving Borodin (2006)'s result for Banach spaces by weakening his condition on the sequence $\{d_n\}$. The weakened condition on $d_n$ requires refinement of Borodin's construction to extract an element in $X$, whose distances from the nested subspaces are precisely the given values $d_n$.

math.FA↗

arXiv2016

Subspace Condition for Bernstein's Lethargy Theorem

In this paper, we consider a condition on subspaces in order to improve bounds given in the Bernstein's Lethargy Theorem (BLT) for Banach spaces. Let $d_1 \geq d_2 \geq \dots d_n \geq \dots > 0$ be an infinite sequence of numbers converging to $0$, and let $Y_1 \subset Y_2 \subset \dots\subset Y_n \subset \dots \subset X$ be a sequence of closed nested subspaces in a Banach space $X$ with the property that $\overline{Y}_{n}\subset Y_{n+1}$ for all $n\ge1$. We prove that for any $c \in (0,1]$, there exists an element $x_c \in X$ such that $$ c d_n \leq ρ(x_c, Y_n) \leq \min (4, \tilde{a}) c\, d_n. $$ Here, $ρ(x, Y_n)= \inf \{ ||x-y||: \,\,y\in Y_n\}$, $$\tilde{a} =\sup_{i\ge1}\sup_{\left \{ q_{i} \right \}}\left \{ a_{n_{i+1}-1}^{-3}\right \}$$ where the sequence $\{a_n\}$ is defined as: for all $ n \geq 1 $, $$ a_n = \inf_{l \geq n} \, \inf_{q \in \langle q_l, q_{l+1},\dots \rangle} \frac{ρ(q,Y_l)}{||q||} $$ in which each point $q_n$ is taken from $Y_{n+1} \setminus Y_{n}$, and satisfies $\inf\limits_{n\ge1} a_n > 0$. The sequence $\{n_i\}_{i\ge1}$ is given by %Theorem \ref{100}, $\{n_i\}$ satisfying (\ref{ni}) and $n_{i}\leq n<n_{i+1}$. $$ n_1=1;~n_{i+1}= \min \left \{ n\ge1 : \frac{d_n}{{a_{n}^{2}}} \leq d_{n_{i}}\right \},~i\geq 1. $$

math.FA↗

arXiv2016

Estimation of the Pointwise Hölder Exponent of Hidden Multifractional Brownian Motion Using Wavelet Coefficients

We propose a wavelet-based approach to construct consistent estimators of the pointwise Hölder exponent of a multifractional Brownian motion, in the case where this underlying process is not directly observed. The relative merits of our estimator are discussed, and we introduce an application to the problem of estimating the functional parameter of a nonlinear model.

math.PR↗

arXiv2016

A New Algorithm to Simulate the First Exit Times of a Vector of Brownian Motions, with an Application to Finance

We provide a new methodology to simulate the first exit times of a vector of Brownian motions from an orthant. This new approach can be used to simulate the first exit times of dimension higher than two. When at least one Brownian motion has non-zero drift, the joint density function of the first exit times in N dimensions needs to be known, or approximated. However, when the drifts are all zero, a simpler simulation algorithm is obtained without using the joint density function.

math.PR↗

arXiv2015

A Representation Theorem for Smooth Brownian Martingales - New Example

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a "frozen path". The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, which we only highlight in this article, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.

math.PR↗