Mathematical models of bubbles

In this paper we review recent attempts to model mathematically the price evolution of risk assets when they are undergoing bubble pricing. We consider both continuous processes and processes with jumps, and use the framework that, under a risk-neutral measure, the price process will be a strict local martingale (and not a ‘true’ martingale) when bubble pricing is present. Finally, we mention briefly the issue of causes for bubbles, and one approach to modelling them mathematically.