We study in detail and explicitly solve the version of Kyle's model introduced in \cite{BB} where the trading horizon is given by an exponentially distributed random time. The first part of the paper is devoted to the analysis of time-homogeneous equilibria using tools from the theory of one-dimensional diffusions. It turns out that such an equilibrium is only possible if the finaly payoff is Bernoulli distributed as in \cite{BB}. We show in the second part that the signal that the market makers use in the general case is a time-changed version of the one that they would use if the final pay-off had a Bernoulli distribution. In both cases we characterise explicitly the equilibrium price process and the optimal strategy of the informed trader. Contrary to the original Kyle model it is found that the reciprocal of market's depth, i.e. Kyle's lambda, is a uniformly integrable supermartingale. While Kyle's lambda is a potential, i.e. converges to $0$, for the Bernoulli distribured final payoff, its limit in general is different than $0$. Also, differently from \cite{BB}, we give an in-depth analysis of the admissible pricing rules for the market makers using techniques from the progressive enlargement of filtrations.
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