非线性霍克斯过程的适度偏差
Moderate Deviations for Nonlinear Hawkes Processes
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中文总结 AI 辅助
本文为非线性霍克斯过程建立完全适度区间内的样本路径适度偏差原理，通过构造泊松校正子并利用鞅方法，得到介于中心极限与大偏差尺度间的结果，并给出渐近方差响应公式。
中文摘要 AI 辅助
霍克斯过程是一种简单点过程，其强度依赖于自身历史；由此产生的动力学通常是非马尔可夫的。我们在完全适度区间内，为非线性霍克斯过程建立了样本路径适度偏差原理。由于非线性霍克斯过程不存在泊松簇表示，我们使用过去构型作为马尔可夫状态，并为中心化随机强度构造一个势函数，即泊松校正子。一个单调泊松耦合表明，校正子的加一增量是一致有界的。因此，中心化计数过程是一个具有有界跳跃的鞅与一个指数可忽略的边界项之和。可预测二次变差的指数稳定性由非线性霍克斯过程的过程级大偏差原理得出。鞅适度偏差定理随后为介于中心极限尺度与大偏差尺度之间的每一个尺度给出了结果。同样的构造给出了渐近方差的响应公式，并特别验证了在自激情形下方差支配平稳平均强度。
英文摘要
A Hawkes process is a simple point process whose intensity depends on its history; the resulting dynamics are generally non-Markovian. We establish a sample-path moderate deviation principle for a nonlinear Hawkes process in the full moderate regime. Since a Poisson cluster representation is unavailable for nonlinear Hawkes processes, we use the past configuration as a Markov state and construct a potential, or Poisson corrector, for the centered stochastic intensity. A monotone Poisson coupling shows that the add-one increment of the corrector is uniformly bounded. The centered counting process is consequently the sum of a martingale with bounded jumps and an exponentially negligible boundary term. Exponential stabilization of the predictable quadratic variation follows from the process-level large deviation principle for nonlinear Hawkes processes. The martingale moderate deviation theorem then yields the result for every scale between the central-limit and large-deviation scales. The same construction gives a response formula for the asymptotic variance and, in particular, verifies that the variance dominates the stationary mean intensity in the self-exciting case.
发表机构
- Fudan University（复旦大学）
- Florida State University（佛罗里达州立大学）
机构由 AI 辅助整理，请以论文原文为准。
