  We study and generalize in various ways the model of rational expectation
(RE) bubbles introduced by Blanchard and Watson in the economic literature.
First, bubbles are argued to be the equivalent of Goldstone modes of the
fundamental rational pricing equation, associated with the symmetry-breaking
introduced by non-vanishing dividends. Generalizing bubbles in terms of
multiplicative stochastic maps, we summarize the result of Lux and Sornette
that the no-arbitrage condition imposes that the tail of the return
distribution is hyperbolic with an exponent mu<1. We then extend the RE bubble
model to arbitrary dimensions d and, with the renewal theory for products of
random matrices applied to stochastic recurrence equations, we extend the
theorem of Lux and Sornette to demonstrate that the tails of the unconditional
distributions follow power laws, with the same asymptotic tail exponent mu<1
for all assets. Two extensions (the crash hazard rate model and the
non-stationary growth rate model) of the RE bubble model provide ways of
reconciliation with the stylized facts of financial data. The later model
allows for an understanding of the breakdown of the fundamental valuation
formula as deeply associated with a spontaneous breaking of the price symmetry.
Its implementation for multi-dimensional bubbles explains why the tail index mu
seems to be the same for any group af assets as observed empirically. This work
begs for the introduction of a generalized field theory which would be able to
capture the spontaneous breaking of symmetry, recover the fundamental valuation
formula in the normal economic case and extend it to the still unexplored
regime where the economic growth rate is larger than the discount growth rate.
