On the equilibrium of insider trading under information acquisition with long memory

In this paper, we study a general model of insider trading with a random deadline, in which an insider has some information acquisition on a risky asset at cost and noise traders trade with a process of stochastic volatil-ity, both types of agents trading with some memory of their corresponding histories described by some kinds of fractional Brownian motions (FBMs) with Hurst parameters in (21, 1) respectively. With semimartingales approximat-ing to FBMs, a closed form of market equilibrium consisting of instantaneous precision, linear trading rate and semi-strong pricing rule is obtained; and by product, a market equilibrium when the insider possesses the complete informa-tion on the asset before trading is also given. It shows that in the equilibrium, the price process converges almost surely to the fundamental value of the risky asset as time goes to infinite, and market depth is a semimartingale while not a martingale. As Hurst parameters both in insider's information flow and noise trades tend to 21, our results can converge to those respectively when both the insider and noise traders have no memory. Finally, some simulations are given to illustrate how informational efficiency, price informativeness or market liq-uidity vary with trading time, insider's cost, Hurst parameters or approximate factors of both noise traders' memory and insider's memory.