This article derives a general class of intrinsic rational bubble solutions in a Lucas-type asset pricing model. I show that the rational bubble component of the price–dividend ratio can evolve as a geometric random walk without drift, such that the mean of the bubble growth rate is zero. Driftless bubbles are part of a continuum of equilibrium solutions that satisfy a period-by-period no-arbitrage condition. I also derive a near-rational solution in which the agent's forecast rule is under-parameterised. The near-rational solution generates intermittent bubbles and other behaviour that is quantitatively similar to that observed in long-run US stock market data.