  We introduce a new definition of speculative bubbles in discrete-time models
based on the discounted stock price losing mass at some finite drop-down under
an equivalent martingale measure. We provide equivalent probabilistic
characterisations of this definition and give examples of discrete-time
martingales that are speculative bubbles and those that are not. In the
Markovian case, we provide sufficient analytic conditions for the presence of
speculative bubbles. We also show that the existence of speculative bubbles is
directly linked to the existence of a non-trivial solution to a linear Volterra
integral equation of the second kind involving the Markov kernel. Finally, we
show that our definition of speculative bubbles in discrete time is consistent
with the strict local martingale definition of speculative bubbles in
continuous time in the sense that a properly discretised strict local
martingale in continuous time is a speculative bubble in discrete time.
