基于超过程的粗糙CIR过程与Feller随机测度方法

中文摘要 AI 辅助

Feller随机测度通过赋予Feller扩散（即CIR过程）记忆性而将其推广。它们作为近乎不稳定的Hawkes过程的标度极限出现，并包含粗糙CIR过程及其超粗糙和不连续变体。我们证明每个Feller随机测度都是一个Dawson-Watanabe超过程的占据测度，该超过程的空间运动是一个被杀的Lévy子ordinator，并在分支时间上积分。这是Hawkes-Oakes簇表示的连续统类比。它产生了存在性、参数稳定性以及由显式噪声驱动的随机方程。该噪声的结构取决于核在原点是否具有原子。若无原子，噪声是由测度的分布函数进行时间变换的布朗运动。这导致了一个鞅刻画以及关于密度及其Hölder正则性的精确结果。若有原子，噪声是由该分布函数的补偿器进行时间变换的补偿逆高斯过程，且该测度是纯原子的。

英文摘要

Feller random measures generalize the Feller diffusion (the CIR process) by giving it memory. They arise as the scaling limits of nearly unstable Hawkes processes, and include the rough CIR process and its hyper-rough and discontinuous relatives. We show that every Feller random measure is the occupation measure of a Dawson--Watanabe superprocess whose spatial motion is a killed Lévy subordinator, integrated over the branching time. This is a continuum analogue of the Hawkes--Oakes cluster representation. It yields existence, stability in the parameters, and stochastic equations driven by an explicit noise. The structure of this noise depends on whether the kernel has an atom at the origin. Without an atom, the noise is a Brownian motion time-changed by the distribution function of the measure. This leads to a martingale characterization and to sharp results on densities and their Hölder regularity. With an atom, the noise is a compensated inverse-Gaussian process time-changed by the compensator of that distribution function, and the measure is purely atomic.

发表机构

• The University of Texas at Austin（德克萨斯大学奥斯汀分校）

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