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.