Journal article: This paper shows that it is possible to extend the scope of the existence of rational bubbles when uncertainty is introduced associated with rank-dependent expected utility. This RDU assumption can be viewed as a transformation of probabilities depending on the pessimism/optimism of the agent. The results show that pessimism favors the existence of deterministic bubbles, when optimism may promote the existence of stochastic bubbles. Moreover, under pessimism, the RDU assumption may generate multiple bubbly equilibria. The RDU assumption also leads to new conditions ensuring the (absence of) Paretooptimality of the competitive equilibrium without bubbles. These conditions still govern the existence of bubbles.