Can response time be incorporated into signal detection theory? In signal detection theory, one typically uses "signal" and "no signal" responses to analyze the data (that is, the analysis is based on a discrete choice for each trial, generating the four possible outcomes "hit", "miss", "false alarm" and "correct rejection").

Is there any standard way to also incorporate response time into the signal detection analysis?
I'd suggest checking out the Linear Ballistic Accumulator (Donkin et al., 2011) model for a scenario like this. While LBA can be used to model any number of alternatives in a speeded choice task, to model signal detection you'd want to model just two accumulators, one for the "signal" response and one for the "no signal" response. With this scenario, response bias will manifest as a difference between the accumulators in terms of their distance from starting point to criterion, while discriminability will manifest as the mean drift rate (which will probably be the same across the two accumulators). 

References

Donkin, C., Brown, S., &amp; Heathcote, A. (2011). Drawing conclusions
from choice response time models: A tutorial using the linear
Ballistic accumulator. Journal of Mathematical Psychology, 55, 140-151.