Multi agent nonlinear model of the order book (Part I)

We define and consider rigorously new multiagent model of the market order book. Our approach is based on ideas of non-equilibrium statistical mechanics. The model describes different market regimes. For example, i. convergence to a stationary process in the absence of a news; ii. price and volume change in the case when new information hits the market; iii. slow price change regime under the flow of sell or buy orders. Our models is distinct from an existent models with zero intellect in two ways. First it captures interaction between the market agents. Second is possesses slow decaying correlations. The model has two additional parameters which we call slow variables. These parameters measure a sentiment about the fair price of an asset by two groups of market participants, namely bulls and bears. We explain an intuition behind basic equations, prove basic theorems and demonstrate results of numerical simulations. The standard in mathematical finance Ornstein–Uhlenbeck process with variable volatility comes out in a diffusion limit of our model. We also formulate some open questions.