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

Attentional Encoder Network for Targeted Sentiment Classification

Targeted sentiment classification aims at determining the sentimental tendency towards specific targets. Most of the previous approaches model context and target words with RNN and attention. However, RNNs are difficult to parallelize and truncated backpropagation through time brings difficulty in remembering long-term patterns. To address this issue, this paper proposes an Attentional Encoder Network (AEN) which eschews recurrence and employs attention based encoders for the modeling between context and target. We raise the label unreliability issue and introduce label smoothing regularization. We also apply pre-trained BERT to this task and obtain new state-of-the-art results. Experiments and analysis demonstrate the effectiveness and lightweight of our model.Comment: 7 page