Insider Trading with a Random Deadline under Partial Observations: Maximal Principle Method

For a revised model of Caldentey and Stacchetti (Econometrica, 2010) in continuous-time insider trading with a random deadline which allows market makers to observe some information on a risky asset, a closed form of its market equilibrium consisting of optimal insider trading intensity and market liquidity is obtained by maximum principle method. It shows that in the equilibrium, (i) as time goes by, the optimal insider trading intensity is exponentially increasing even up to infinity while both the market liquidity and the residual information are exponentially decreasing even down to zero; (ii) the more accurate information observed by market makers, the stronger optimal insider trading intensity is such that the total expect profit of the insider is decreasing even go to zero while both the market liquidity and the residual information are decreasing; (iii) the longer the mean of random time, the weaker the optimal insider trading intensity is while the more both the residual information and the expected profit are, but there is a threshold of trading time, half of the mean of the random time, such that if and only if after it the market liquidity is increasing with the mean of random time increasing.