Most of nowadays stock exchanges work fully electronic. The so called limit order book captures (instantaneous) demand and supply in the market and performs directly the price formation. Setting up a macroscopic description of the dynamics of buy and sell side of the order book, we obtain a stochastic extension of the classical Stefan problem, where the mid price defines an inner boundary separating both sides of the order book. While existence can be shown for certain specifications of these problems, they turn out to be intractable. For further analysis, one then has to rely on different techniques such as Wong-Zakai type approximations or particle representations.
Another approach is to simplify the problem by considering a linearized class of SPDE models which we can solve in certain cases explicitly and calibrate to market data. Using established price predictors such as order flow imbalance, we then obtain a model for the mid price dynamics.