We propose a class of stochastic processes that we call captive diffusions, which evolve within measurable pairs of càdlàg bounded functions that admit bounded right-derivatives at points where they are continuous. In full generality, such processes allow reflection and absorption dynamics at their boundaries—possibly in a hybrid manner over non-overlapping time periods—and if they are martingales, continuous boundaries are necessarily monotonic. We employ multi-dimensional captive diffusions equipped with a totally ordered set of boundaries to model random processes that preserve an initially determined rank. We run numerical simulations on several examples governed by different drift and diffusion coefficients. Applications include interacting particle systems, random matrix theory, epidemic modelling and stochastic control.

中文翻译：

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俘获扩散及其在保序动力学中的应用

我们提出了一类我们称之为俘获扩散的随机过程，它在可测量的 càdlàg 有界函数对中演化，这些有界函数在连续点上允许有界右导数。总的来说，这样的过程允许在其边界处发生反射和吸收动力学——可能在不重叠的时间段内以混合方式——如果它们是鞅，连续边界必然是单调的。我们采用配备完全有序边界集的多维俘获扩散来模拟随机过程，以保留最初确定的等级。我们对由不同漂移和扩散系数控制的几个示例进行数值模拟。应用包括相互作用的粒子系统、随机矩阵理论、流行病建模和随机控制。