• C1;
  • E3;
  • G12

We study rational bubbles in a standard linear asset price model. We first consider a class of bubble processes driven by multiplicative i.i.d. shocks. We show that a bubble process in this class either diverges to infinity with probability one, converges to zero with probability one, or keeps fluctuating forever with probability one, depending on investors' “confidence” in expected bubble growth. We call a bubble process having the last property “recurrent.” We develop sufficient conditions for a bubble process to be recurrent when it is driven by non-i.i.d. shocks, when the risk-free interest rate is not constant, and when the process is driven by non-i.i.d. shocks and the risk-free interest rate is not constant. In the last case we demonstrate via simulation that there can be a prolonged period in which both the bubble and the interest rate stay close to zero.