由Lévy过程驱动的加权自排斥扩散的长时间行为和统计推断
Long Time Behavior and Statistical Inference of the Weighted Self-Repelling Diffusion Driven by Lévy Process
摘要: 假设是一个跳有界且界限为1的Lévy过程，生成三元组为。在本文中，我们考虑了由Lévy过程驱动的线性自排斥扩散方程，其中，和。这类过程是一类自交互扩散过程。我们研究了当t趋于无穷时解的长时间行为，发现它具有一种循环收敛性，这在此前的研究中尚未有类似的结论。进一步的，当w=0时在连续观测情况下，通过最小二乘法给出了方程参数的估计。我们证明了的估计量具有强相合性，并讨论了它的渐近分布。
Abstract: Let  be a Lévy process with jumps bounded by 1 and generating triplet . In this paper, as an attempt we consider the linear self-repelling diffusion driven by a Lévy process, , where  and the parameter . This process is similar to a type of self-interacting diffusion process. This paper studies the long time behaviour of the solution as t tends to infinity, and we find that it exhibits a cyclic convergence property, for which similar conclusions have not appeared in previous studies. In addition, when w=0, by using least squares method, we establish the strong consistency of the estimate  and discuss its asymptotic distribution under the consecutive observation.
文章引用：鲁蕴涵, 闫理坦. 由Lévy过程驱动的加权自排斥扩散的长时间行为和统计推断[J]. 应用数学进展, 2024, 13(3): 991-1001. https://doi.org/10.12677/aam.2024.133093

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