148 research outputs found
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 , it is known that the existing
(naive) Euler scheme has the rate of convergence . Since the limit
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 , the convergence
rate of the naive Euler scheme deteriorates for . In
this paper we introduce a new (modified Euler) approximation scheme which is
closer to the classical Euler scheme for Stratonovich SDEs for ,
and it has the rate of convergence , where
when , when
and if . Furthermore, we study the
asymptotic behavior of the fluctuations of the error. More precisely, if
is the solution of a SDE driven by a fBm and if
is its approximation obtained by the new modified Euler
scheme, then we prove that converges stably to the solution
of a linear SDE driven by a matrix-valued Brownian motion, when
. In the case , we show the
convergence of , 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
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