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arXiv 1408.6471math.PR
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分数阶扩散问题Euler近似格式的收敛速率和渐近误差分布

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

Yaozhong Hu, Yanghui Liu, David Nualart

备注：
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)
发表信息：
Annals of Applied Probability 2016, Vol. 26, No. 2, 1147-1207
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摘要：

对于分数布朗运动驱动的随机微分方程，已知已有的（朴素）Euler格式具有收敛速率$H>\frac{1}{2}$.由于Euler方程的极限$H\rightarrow\frac{1}{2}$对应于由标准布朗运动驱动的Stratonovich Euler方程，并且朴素Euler格式是经典Euler格式在$H=\frac{1}{2}$下的推广，因此朴素Euler格式在$H\rightarrow\frac{1}{2}$下的收敛速度变差.本文介绍一种新的（修正的Euler）逼近格式，它更接近于求解Stratonovich偏微分方程的经典Euler格式，并且具有收敛速率$γ_n^{-1}$，其中当$H<\frac{3}{4}$时为$γ_n=n^{2H-{1}/2}$，当$H=\frac{3}{4}$时为$γ_n=n/\sqrt{\log n}$，当$H>\frac{3}{4}$时为$γ_n=n$.此外，我们还研究了误差波动的渐近行为。更精确地说，如果$\{X_t,0\le t\le T\}$是一个fBm驱动的线性规划的解决方案，如果$\{X_t^n,0\le t\le T\}$是它的近似得到的新的修改的欧拉格式，那么我们证明了$γ_n(X^n-X)$稳定收敛到一个线性规划的解决方案，由矩阵值布朗运动，当$H\in(\frac{1}{2},\frac{3}{4}]$。在$H>\frac{3}{4}$的情况下，我们显示$n(X^n_t-X_t)$的$L^p$收敛，和极限过程被确定为矩阵值Rosenblatt过程驱动的线性矩阵的解决方案。并给出了该格式的弱收敛速度。我们还将我们的方法应用到朴素欧拉方案。

英文摘要：

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 Itô 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 $γ_n^{-1}$, where $γ_n=n^{2H-{1}/2}$ when $H<\frac{3}{4}$, $γ_n=n/\sqrt{\log n}$ when $H=\frac{3}{4}$ and $γ_n=n$ 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 $γ_n(X^n-X)$ 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.

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