One-level limit order book models with memory and variable spread

We propose a new model for the level I of a Limit Order Book (LOB), which incorporates the information about the standing orders at the opposite side of the book after each price change and the arrivals of new orders within the spread. Our main result gives a diffusion approximation for the mid-price process. To illustrate the applicability of the considered framework, we also propose a feasible method to compute several quantities of interest}, such as the distribution of the time span between price changes and the probability of consecutive price increments conditioned on the current state of the book. The proposed method is used to develop an efficient simulation scheme for the price dynamics, which is then applied to assess numerically the accuracy of the diffusion approximation.