论文标题
通过分数演算的稳定过程的Meyer-Itô公式
A Meyer-Itô Formula for Stable Processes via Fractional Calculus
论文作者
论文摘要
一维严格的$α$稳定过程的无限发电机可以表示为（左右）riemann-liouville分数$α$的加权总和，并且在对称稳定过程中获得了分数laplacian。使用这种关系，我们计算了Lizorkin空间上无穷小发电机的倒数，如果$α\在（0,1）$中，我们可以从中恢复潜力，如果$α\ in（1,2）$，则可以从（0,1）$（0,1）$中恢复潜力。无限发电机的倒数是根据（左右）riemann-liouville分数积分$α$的线性组合表示的。然后，人们可以说出一类功能，这些功能在应用于严格稳定的过程中时会提供半明星，并说明具有非零（职业）本地时间术语的Meyer-Itô定理，从而提供了Tsukada（2019）给出的田中公式的概括。该结果用于找到$ | x_t-x_t-x |^γ$的Doob-meyer（或半木马）分解，$ x $又经常稳定的索引$α$和$γ\ in（α-1，α）$的$ qurtister稳定过程，将Engelbert和Kurenok（2019）的工作推广到不可或缺的案例中。
The infinitesimal generator of a one-dimensional strictly $α$-stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $α$ and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if $α\in (0,1)$ and the recurrent potential if $α\in (1,2)$. The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order $α$. One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada (2019). This result is used to find a Doob-Meyer (or semimartingale) decomposition for $|X_t - x|^γ$ with $X$ a recurrent strictly stable process of index $α$ and $γ\in (α-1,α)$, generalizing the work of Engelbert and Kurenok (2019) to the asymmetric case.
