Is there a random walk theory that can account for situations with more than two choices? In the article "Two-stage Dynamic Signal Detection: A Theory of Choice, Decision Time, and Confidence" from 2010 by Pleskac and Busemeyer, a random walk model is presented for situations where a discrete choice is made (that is, signal present/not present, yes/no, 1/0 et cetera). Here, if there is no time pressure involved, a threshold value is set which has to be reached before an answer is given. Contrary, if there is a time pressure involved, the value at the last time point is used to decide which answer is given, even if the threshold point hasn't been reached at that time.

I wonder whether there are any extension of this model which can incorporate situations with more than two choices (from 3 alternatives and up to a fully continius choice space). I guess you just could incorporate more dimensions (the model referred to above is two-dimensional), but if the answers are non-independent (for example, when using a rating scale), I guess the random walk function would have to be fairly complicated. That is, if strong evidence is accumulated for number 3, how much evidence should be added to 4 and 2? Should evidence be subtracted from 1 and 5?
Instead of having a single integrator with two bounds for two choices (symmetric random walk model), you can have many competing integrators each with a bound (race model). For example, see Fig 2. of Gold and Shadlen 2007 and references therein.

As for the continuous choices case, it is important to understand a limit of discrete choices can be very different from a continuous choice. For such a limit to make sense, there should be a notion of similarity between the choices, and I don't think those race models would generalize easily. A high dimensional continuous attractor dynamics is more likely to support such theory.