Thank you for your reminding. I confused something here. \(\lambda(\delta)=Aexp(-\kappa\delta) \) is not a pdf.This model is from the paper "High-frequency trading in a limit order book "by MARCO AVELLANEDA and SASHA STOIKOV.
\(\lambda(\delta)=Aexp(-\kappa\delta) \) is acutally the exponential arriaval rates of market order, where \( \delta \) is the distance from mid price. I am finding a way to calibrate this model in practice.
In the 2006 version of this paper, the authors say
"Once we have chosen our time step \(dt\),e we obtain the value for \(A\) through the relation \(\lambda(M/2)=Aexp(\kappa M/2)dt\), which says that if we post a limit order at a distance \(M/2\) from the mid-price, this limit order is essentially a market order that will get executed with probability one in a \(dt\) time interval.
I am not clear about
(1) how to select the time interval \(dt\)
(2)how to calibrate \(A\) and \(\kappa\) from \(\lambda(\delta)=Aexp(-\kappa\delta) \) and \(\lambda(M/2)=Aexp(\kappa M/2)dt\). It seems to me that these two equations may not be consistent,because by counting the market order the former equation can be calibrated alone to get \(A\) and \(\kappa\).