  We introduce a new diffusion process Xt to describe asset prices within an
economic bubble cycle. The main feature of the process, which differs from
existing models, is the drift term where a mean-reversion is taken based on an
exponential decay of the scaled price. Our study shows the scaling factor on Xt
is crucial for modelling economic bubbles as it mitigates the dependence
structure between the price and parameters in the model. We prove both the
process and its first passage time are well-defined. An efficient calibration
scheme, together with the probability density function for the process are
given. Moreover, by employing the perturbation technique, we deduce the
closed-form density for the downward first passage time, which therefore can be
used in estimating the burst time of an economic bubble. The object of this
study is to understand the asset price dynamics when a financial bubble is
believed to form, and correspondingly provide estimates to the bubble crash
time. Calibration examples on the US dot-com bubble and the 2007 Chinese stock
market crash verify the effectiveness of the model itself. The example on
BitCoin prediction confirms that we can provide meaningful estimate on the
downward probability for asset prices.
