The authors model trades-through, i.e. transactions that reach at least the second level of limit orders in an order book. Using tick-by-tick data on Euronext-traded stocks, they show that a simple bivariate Hawkes process fits nicely their empirical observations of tradesthrough. The authors show that the cross-influence of bid and ask trades-through is weak.
Hawkes processes limit order book trades-through high-frequency trading microstructure