Abstracts

We introduce a continuous model for the limit order book density with infinitesimal tick size, where the evolution of buy and sell side is described by a semilinear second-order SPDE. The mid price process defines a free boundary separating buy and sell side. Price changes are assumed to be determined by the bid-ask imbalance. Following empirical observations by Lipton, Pesavento and Sotiropoulos (2013) we allow this dependency to be nonlinear. The resulting limit order book model can be considered as a generalization of the linear stochastic Stefan problem introduced by Kim, Sowers and Zheng (2012).
In order to show existence of a solution we transform the problem into a stochastic evolution equation, where the boundary interaction leads to an additional non-Lipschitz drift. There is no chance to control this term in any reasonable Banach space, however, smoothing properties of the heat semigroup allow to make use of the mild formulation of the equation. Despite of the non-standard setting for the stochastic evolution equation, we show existence of a unique maximal mild solution of the general model; extending results of Kim, Sowers and Zheng. We show that this solution is continuous, and, up to a stopping time, solves the equation even in the analytically strong sense. Additional assumptions on the boundary interaction then yield non-explosion and global existence. Finally, we obtain that a Nagumo-type condition is sufficient for positivity of the order volume, a natural property any limit order book model should satisfy.