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

The difference between individuals and social groups in multidirectional movement

Social groups exhibit some degree of social cohesion that is more than a simple collection or aggregation of individuals. The study on behaviours of social groups is essential to gain a deeper understanding on pedestrian dynamics. In this paper, the crowd movement among individuals and social groups are studied by performing a series of multidirectional pedestrian flow experiments under laboratory condition. Interestingly, we found that the speed of individual is not always higher than that of social groups. Pedestrians try to use different strategies to arrive their destination. With different strategies, their speed and movement time show different properties. These findings may provide basis for facility design and evacuation plan

Evacuation characteristics of preschool children through bottlenecks

Pedestrian movement through bottlenecks have been widely studied from various aspects to understand the effects of bottlenecks on the pedestrian flow. However, few attentions have been paid to the movement characteristics of preschool children, who show obvious differences behaviour compared to adults due to the poor balance and understanding of danger especial under emergencies. In this study, we focus on the evacuation characteristics of preschool children through bottlenecks with laboratory experiments. From all the experiment, we do not observe clear lane formation process from the trajectories diagrams. It is found that the first arrive first out principle does not work in the situation with competition. Compared to adults, children are more likely to fall and hard to be controlled during movement, which is very dangerous in emergencies. The highest speed for the preschool children can beyond 3 m/s and is depend on the location in the crowd for each individual. For a given number of evacuees, the total evacuation time firstly decreases a linear with the increasing the bottleneck width and then keeps a constant if nobody falls down during the movement. Falling down of children will increase the evacuation time incredibly. The findings will be beneficial for the evacuation drill design in kindergarten as well as the facility design for young children

Stretchable hybrid bilayered luminescent composite based on the combination of strain-induced and triboelectrification-induced electroluminescence

High luminescence intensity from materials that are excited by external stimuli is highly desired. In this work, a stretchable hybrid luminescent composite (HLC) that has multiple luminescence modes is reported. The luminescence can be excited either by externally applied mechanical strain or by a moving object that slides against the HLC. When the HLC is deformed, such as being twisted or folded, the ZnS/Cu phosphor experiences mechanical strain that trigger the mechanoluminescence (ML) of the phosphors. Moreover, as the HLC slides against a contact object, the triboelectrification at the contact interface induces the electroluminescence of phosphor. Here, a series of internal and external factors were studied on how they influence the luminescent intensity. It is found that the luminescent intensity from the two modes can be superposed. The HLC material was used to fabricate a fiber-based luminescent device that can be driven by air flow. The overall luminescent intensity is enhanced by over 72% compared to that obtained solely from the ML. The HLC reported in this work has such potential applications as self-powered light sources and sensors as means of detecting dynamic motions and interactio