Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^{o} SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $\gamma_n^{-1}$, where $\gamma_n=n^{2H-{1}/2}$ when $H<\frac{3}{4}$, $\gamma_n=n/\sqrt{\log n}$ when $H=\frac{3}{4}$ and $\gamma_n=n$ if $H>\frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n,0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $\gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.Comment: Published at http://dx.doi.org/10.1214/15-AAP1114 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A new boundary of the mapping class group

Based on the action of the mapping class group on the space of measured foliations, we construct a new boundary of the mapping class group and study the structure of this boundary. As an application, for any point in Teichmuller space, we consider the orbit of this point under the action of the mapping class group and describe the closure of this orbit in the Thurston compactification and the Gardiner-Masur compactification of Teichmuller space. We also construct some new points in the Gardiner-Masur boundary of Teichmuller space.Comment: To appear in Acta Mathematica Sinica, English Serie

A NEW COMPACTIFICATION OF TEICHMÜLLER SPACE

We construct a new compactification of Teichmüller space. We prove that this new compactification is finer than the Gardiner–Masur compactification of Teichmüller space and the action of the mapping class group on Teichmüller space extends continuously to this new compactification. We also construct some special points in the new boundary. The construction of the new compactification is based on the Hubbard-Masur theorem, which states that there is an one-to-one corresponding between holomorphic differentials and measured foliations

A Novel Segmentation Approach Combining Region- and Edge-Based Information for Ultrasound Images

Ultrasound imaging has become one of the most popular medical imaging modalities with numerous diagnostic applications. However, ultrasound (US) image segmentation, which is the essential process for further analysis, is a challenging task due to the poor image quality

Identifying ceRNA Networks Associated With the Susceptibility and Persistence of Atrial Fibrillation Through Weighted Gene Co-Expression Network Analysis

Background: Atrial fibrillation (AF) is the most common arrhythmia. We aimed to construct competing endogenous RNA (ceRNA) networks associated with the susceptibility and persistence of AF by applying the weighted gene co-expression network analysis (WGCNA) and prioritize key genes using the random walk with restart on multiplex networks (RWR-M) algorithm.Methods: RNA sequencing results from 235 left atrial appendage samples were downloaded from the GEO database. The top 5,000 lncRNAs/mRNAs with the highest variance were used to construct a gene co-expression network using the WGCNA method. AF susceptibility- or persistence-associated modules were identified by correlating the module eigengene with the atrial rhythm phenotype. Using a module-specific manner, ceRNA pairs of lncRNA–mRNA were predicted. The RWR-M algorithm was applied to calculate the proximity between lncRNAs and known AF protein-coding genes. Random forest classifiers, based on the expression value of key lncRNA-associated ceRNA pairs, were constructed and validated against an independent data set.Results: From the 21 identified modules, magenta and tan modules were associated with AF susceptibility, whereas turquoise and yellow modules were associated with AF persistence. ceRNA networks in magenta and tan modules were primarily involved in the inflammatory process, whereas ceRNA networks in turquoise and yellow modules were primarily associated with electrical remodeling. A total of 106 previously identified AF-associated protein-coding genes were found in the ceRNA networks, including 16 that were previously implicated in the genome-wide association study. Myocardial infarction–associated transcript (MIAT) and LINC00964 were prioritized as key lncRNAs through RWR-M. The classifiers based on their associated ceRNA pairs were able to distinguish AF from sinus rhythm with respective AUC values of 0.810 and 0.940 in the training set and 0.870 and 0.922 in the independent test set. The AF-related single-nucleotide polymorphism rs35006907 was found in the intronic region of LINC00964 and negatively regulated the LINC00964 expression.Conclusion: Our study constructed AF susceptibility- and persistence-associated ceRNA networks, linked genetics with epigenetics, identified MIAT and LINC00964 as key lncRNAs, and constructed random forest classifiers based on their associated ceRNA pairs. These results will help us to better understand the mechanisms underlying AF from the ceRNA perspective and provide candidate therapeutic and diagnostic tools

Evolutionary Analysis of Structural Protein Gene VP1 of Foot-and-Mouth Disease Virus Serotype Asia 1

Foot-and-mouth disease virus (FMDV) serotype Asia 1 was mostly endemic in Asia and then was responsible for economically important viral disease of cloven-hoofed animals, but the study on its selection and evolutionary process is comparatively rare. In this study, we characterized 377 isolates from Asia collected up until 2012, including four vaccine strains. Maximum likelihood analysis suggested that the strains circulating in Asia were classified into 8 different groups (groups I–VIII) or were unclassified (viruses collected before 2000). On the basis of divergence time analyses, we infer that the TMRCA of Asia 1 virus existed approximately 86.29 years ago. The result suggested that the virus had a high mutation rate (5.745 × 10−3 substitutions/site/year) in comparison to the other serotypes of FMDV VP1 gene. Furthermore, the structural protein VP1 was under lower selection pressure and the positive selection occurred at many sites, and four codons (positions 141, 146, 151, and 169) were located in known critical antigenic residues. The remaining sites were not located in known functional regions and were moderately conserved, and the reason for supporting all sites under positive selection remains to be elucidated because the power of these analyses was largely unknown