English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a stochastic differential equation(SDE) driven by a fractional Brownianmotion(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 drivenby standard Brownian motion, and the naive Euler scheme is the extension of theclassical Euler scheme for It\^{o} SDEs for $H=\frac{1}{2}$, the convergencerate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. Inthis paper we introduce a new (modified Euler) approximation scheme which iscloser 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 theasymptotic 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 Eulerscheme, then we prove that $\gamma_n(X^n-X)$ converges stably to the solutionof 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 thesolution of a linear SDE driven by a matrix-valued Rosenblatt process. The rateof weak convergence is also deduced for this scheme. We also apply our approachto the naive Euler scheme.

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