We extend Kyle's (1985) model of insider trading to the case where noise trading volatility follows a general stochastic process. We determine conditions under which, in equilibrium, price impact and price volatility are both stochastic, driven by shocks to uninformed volume even though the fundamental value is constant. The volatility of price volatility appears 'excessive' because insiders choose to trade more aggressively (and thus more information is revealed) when uninformed volume is higher and price impact is lower. This generates a positive relation between price volatility and trading volume, giving rise to an endogenous subordinate stochastic process for prices.

We analyze dynamic trading by an activist investor who can expend costly effort to affect firm value. We obtain the equilibrium in closed form for a general activism technology, including both binary and continuous outcomes. Variation in parameters can produce either positive or negative relations between market liquidity and economic efficiency, depending on the activism technology and model parameters. Two results that contrast with the previous literature are that (a) the relationship between market liquidity and economic efficiency is independent of the activist's initial stake for a broad set of activism technologies, and (b) an increase in noise trading can reduce market liquidity because it increases uncertainty about the activist's trades (the activist trades in the opposite direction of noise traders) and thereby increases information asymmetry about the activist's intentions.