  In this paper we further extend the optimal bubble riding model proposed by
Tangpi and Wang by allowing for price-dependent entry times. Agents are
characterized by their individual entry threshold that represents their belief
in the strength of the bubble. Conversely, the growth dynamics of the bubble is
fueled by the influx of players. Price-dependent entry naturally leads to a
mean field game of controls with common noise and random entry time, for which
we provide an existence result. The equilibrium is obtained by first solving
discretized versions of the game in the weak formulation and then examining the
measurability property in the limit. In this paper, the common noise comes from
two sources: the price of the asset which all agents trade, and also the
exogenous bubble burst time, which we also discretize and incorporate into the
model via progressive enlargement of filtration.
