We study in detail and explicitly solve the version of Kyle's model introduced in \mbox{%DIFAUXCMD 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 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 \mbox{%DIFAUXCMD BB
We study in detail and explicitly solve the version of Kyle's model introduced in a specific case in \mbox{%DIFAUXCMD 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 final payoff is Bernoulli distributed as in BB. We show in the second part that the signal of the market makers use in the general case is a time-changed version of the one that they would have used had the final payoff 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. 
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\mbox{%DIFAUXCMD BB
a specific case in \mbox{%DIFAUXCMD BB
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finaly
final
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that
of
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use if the final pay-off
have used had the final payoff
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Also, differently from \mbox{%DIFAUXCMD BB