A Stochastic Version of Kyle's Lambda

I assume that a market maker selects a buy and a sell price at her exchange to maximize profits. Up to first-order approximation, this problem is equivalent to a social planner’s problem and the problem of maximizing trader participation in the exchange. The optimal market maker rule has several important properties. First, price impact − commonly known as Kyle’s Lambda − declines in trade volume. That is, the market maker dampens price movements on high volume days and amplifies price movements on low volume days. Second, Lambda is a random variable because it is measurable with respect to trade volume. Consequently, equilibrium prices are characterized by endogenous stochastic volatility and heavy tails. My equilibrium concept does not require markets to clear per period; instead, the resulting price distribution must match that of initial trader beliefs