A martingale analysis of first passage times of time-dependent Wiener diffusion models

TL;DR

This paper develops martingale-based formulas to analyze the first passage times, hitting probabilities, and decision times of time-dependent Wiener diffusion models, extending classical results to multistage, variable-parameter scenarios.

Contribution

It introduces a martingale approach to derive analytical expressions for decision metrics in multistage Wiener processes with time-varying parameters, advancing decision modeling.

Findings

Derived formulas for mean decision time and hitting probabilities.

Extended classical Wiener process results to multistage, time-varying models.

Provided tools for analyzing complex, dynamic decision processes.

Abstract

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.