Research in psychology and neuroscience has successfully modeled decision making as a process of noisy evidence accumulation to a decision bound. While there are several variants and implementations of this idea, the majority of these models make use of a noisy accumulation between two absorbing boundaries. A common assumption of these models is that decision parameters, e.g., the rate of accumulation (drift rate), remain fixed over the course of a decision, allowing the derivation of analytic formulas for the probabilities of hitting the upper or lower decision threshold, and the mean decision time. There is reason to believe, however, that many types of behavior would be better described by a model in which the parameters were allowed to vary over the course of the decision process. In this paper, we use martingale theory to derive formulas for the mean decision time, hitting probabilities, and first passage time (FPT) densities of a Wiener process with time-varying drift between two time-varying absorbing boundaries. This model was first studied by Ratcliff (1980) in the two-stage form, and here we consider the same model for an arbitrary number of stages (i.e. intervals of time during which parameters are constant). Our calculations enable direct computation of mean decision times and hitting probabilities for the associated multistage process. We also provide a review of how martingale theory may be used to analyze similar models employing Wiener processes by re-deriving some classical results. In concert with a variety of numerical tools already available, the current derivations should encourage mathematical analysis of more complex models of decision making with time-varying evidence.

中文翻译：

---

时间相关维纳扩散模型首次通过时间的鞅分析


心理学和神经科学的研究已经成功地将决策建模为噪声证据积累到决策界限的过程。虽然这个想法有多种变体和实现，但大多数模型都利用两个吸收边界之间的噪声积累。这些模型的一个共同假设是，决策参数（例如累积率（漂移率））在决策过程中保持固定，从而可以推导达到决策阈值上限或下限的概率的分析公式，以及平均决策时间。然而，我们有理由相信，许多类型的行为可以通过允许参数在决策过程中变化的模型来更好地描述。在本文中，我们使用鞅理论推导了两个时变吸收边界之间具有时变漂移的维纳过程的平均决策时间、命中概率和首次通过时间（FPT）密度的公式。 Ratcliff (1980) 首先以两阶段形式研究了该模型，在这里我们考虑任意数量阶段（即参数恒定的时间间隔）的相同模型。我们的计算可以直接计算相关多阶段过程的平均决策时间和命中概率。我们还回顾了如何通过重新推导一些经典结果，使用鞅理论来分析采用维纳过程的类似模型。与现有的各种数值工具相结合，当前的推导应该鼓励对具有时变证据的更复杂的决策模型进行数学分析。