A probability model of the influence of the order book on the price process

The Limit Order Book model is considered, with buy and sell orders arriving as two independent Cox processes. It includes the price impact model built on the basis of a physical model of perfectly elastic collision. Price is treated as a particle of some mass, moving along a straight line without friction. The incoming buy orders and outgoing sell orders hit the price giving it additional momentum in one direction, while incoming sell orders and outgoing buy orders do the same in the opposite direction. A functional limit theorem for the price process is obtained at a high intensity of incoming order flow, which allows approximation by some Lévy process.