  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.
