Limits for weighted p-variations and likewise functionals of fractional diffusions with drift
Limits for weighted p-variations and likewise functionals of fractional diffusions with drift
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DOI：
10.1016/j.spa.2006.05.016
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发表时间：
2007-03
影响因子：
1.4
通讯作者：
J. León;C. Ludeña
中科院分区：
文献类型：
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作者：
J. León;C. Ludeña
设Xt是由分数布朗运动BtH驱动的扩散方程的路径解，其中Hurst常数H>1/2，扩散系数σ（t，x）.考虑这个解的连续增量，ΔXi=Xi/n−X（i−1）/n。使用解Xt的圆柱近似，我们的主要结果得出，如果1/2<H<3/4，则如果Z是与BH无关的标准正态随机变量，则过程[公式：见正文]弱收敛于W（CH，p <$0 t σp（s，Xs）ds）as n→∞其中W是与B和CH无关的Wiener过程，p是一个依赖于H和p的常数。在p-变分的地方，我们可以考虑满足几乎乘法结构的函数，例如偶数多项式或绝对值多项式。通过考虑离散样本X1的二阶增量，我们得到了整个区间1/2<H<1的类似结果。最后，我们证明了在没有漂移的情况下收敛是稳定的，并利用这个结果讨论了分数阶扩散方程弱解的弱收敛。
Let Xtbe the pathwise solution of a diffusion driven by a fractional Brownian motion BtHwith Hurst constant H>1/2 and diffusion coefficient σ(t,x). Consider the successive increments of this solution, ΔXi=Xi/n−X(i−1)/n. Using a cylinder approximation for the solution Xt, our main result yields that if 1/2<H<3/4 then, if Z is a standard normal random variable which is independent of BH, the process [Formula: see text] converges weakly to W(CH,p∫0tσp(s,Xs)ds) as n→∞ where W is a Wiener process which is independent of BHand CH,pis a constant which depends on H and on p. In the place of p-variations we may consider functions that satisfy an almost multiplicative structure such as even polynomials or polynomials of absolute values. By considering second order increments of the discrete sample Xiwe obtain analogous results for the whole interval 1/2<H<1. Finally, we show convergence is stable in the absence of drift and use this result to discuss weak convergence for weak solutions of the fractional diffusion equation.