Fill Probabilities in a Limit Order Book with State-Dependent Stochastic Order Flows

This paper studies the fill probabilities of limit orders placed at different price levels in a limit order book. These probabilities play a central role in execution optimization, as limit orders are not guaranteed to be executed and inherently involve a trade-off between execution cost and execution risk. We model the limit order book within a general state-dependent stochastic framework, representing its dynamics as a collection of interacting queuing systems while incorporating key stylized market features. Within this framework, we derive semi-analytical expressions for several quantities of interest under state-dependent order flows, including the probability of a mid-price change, the fill probabilities of orders placed at the best quotes, and those of orders placed deeper in the book before the opposite best quote moves. While the framework can be extended to even deeper price levels, the corresponding fill probabilities are typically negligible. We validate the proposed model through extensive numerical experiments using real foreign exchange spot market data. The results demonstrate that the model remains tractable while capturing essential order book dynamics, and that the derived expressions achieve good accuracy in estimating fill probabilities.