TY - JOUR

T1 - A martingale analysis of first passage times of time-dependent Wiener diffusion models

AU - Srivastava, Vaibhav

AU - Feng, Samuel F.

AU - Cohen, Jonathan D.

AU - Leonard, Naomi Ehrich

AU - Shenhav, Amitai

N1 - Funding Information:
We thank the editor, Philip Smith, and the referees for helpful comments, especially regarding this article's exposition and discussion. We thank Ryan Webb for the reference to Smith (2000) and the associated code. We thank Phil Holmes and Patrick Simen for helpful comments and discussions. This work was supported in part by the C.V. Starr Foundation (AS), the Princeton University Insley-Blair Pyne Fund, ONR grant N00014-14-1-0635, ARO grant W911NF-14-1-0431 (VS and NEL), and NIH Brain Initiative grant U01-NS090514 (VS).

PY - 2017/4/1

Y1 - 2017/4/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jmp.2016.10.001

DO - 10.1016/j.jmp.2016.10.001

M3 - Article

C2 - 28630524

AN - SCOPUS:85016461016

VL - 77

SP - 94

EP - 110

JO - Journal of Mathematical Psychology

JF - Journal of Mathematical Psychology

SN - 0022-2496

ER -