发表机构
Dresden Technical University; Wrocław University of Science and Technology（德累斯顿工业大学; 弗罗茨瓦夫科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究在仿射重置下多项式过程的矩，证明其可通过矩阵指数或矩阵更新方程计算，并涵盖多种现有模型。
AI 中文摘要
我们考虑多项式过程，即保持多项式且不增加次数的连续时间马尔可夫过程，并为其添加重置机制。重置意味着在独立更新过程的跳跃时刻，多项式过程被瞬时重置到一个新状态，该状态可以是确定性的、随机的或依赖于状态的。我们证明，在特定类型的重置（“仿射重置”）下，所得过程的矩仍然易于处理，并可通过矩阵指数（对于泊松重置时间）或矩阵更新方程（对于更一般的重置时间）来计算。我们的框架涵盖了许多现有文献中已考虑的模型，例如具有确定性增长或Lévy流入的增长-崩溃过程，以及重置到原点、重置到固定分布或按固定比例乘法的布朗运动。
英文摘要
We consider polynomial processes, i.e., continuous-time Markov processes that preserve polynomials without increasing degree, and augment them with resetting. Resetting means that at the jump times of an indepen- dent renewal process, the polynomial process is instantaneously reset to a new state, which can be chosen deterministically, randomly or in state-dependent fashion. We show that under certain types of resets ('affine resetting') the moments of the resulting process remain tractable and can be calculated by matrix exponentials (for Poissonian reset times) or by a matrix renewal equa- tion (for more general reset times). Our framework encompasses many models that have been considered in the existing literature, such as growth-collapse processes with deterministic growth or with Lévy inflow, and Brownian motion with resets to the origin, to a fixed distribution or by multiplying by a fixed proportion.
