Dealing with the inventory risk: solution with drift

I'm implementing the solution with drift from "Dealing with the inventory risk" from Gueant, Lehalle and Tapia. I'm using the link https://arxiv.org/pdf/1105.3115.pdf as reference.

I can reproduce the standard solution with no issues, and then reproduce the surfaces in the paper. I'm trying now to implement the solution with drift, which is very similar to the standard solution, with a new component beta being added to the matrix M.

However when looking at the answer, I see a plus sign that is counterintuitive to me. This first appears in the proposition 4:

However the signs in the new matrix M doesn't exactly match the propositon 4:

Real apologies if I'm making a pointless case here. I don't have a formal training in mathematics but this supposed inconsistency really bugged me and I had no one else to ask.

Moreover, I would like to ask a few more questions:

1 - How should u (since beta = kappa * u) be properly normalized?

2 - Lehalle commented in another post that https://stanford.edu/class/msande448/2018/Final/Reports/gr5.pdf is a good basis of an implementation. However, in the incorporation of trading signals, it seems intuitive to me that the markov chain which is basis for the execution policy should be updated to incorporate the change of signals to perhaps trigger a requote (and thus avoid adverse selection). I read https://deanstreetlab.github.io/papers/papers/High%20Frequency%20Trading/High%20Frequency%20Market%20Making%20-%20Optimal%20Quoting.pdf which more or less provides a solution, although the framework is very different from "Dealing with the inventory risk". Are there papers around this topic? If no, which execution strategies would be optimal?

• Regarding my first point: from what I've been simulating, it seems that it should a plus sign and not a minus sign in every occurrence of beta. Commented Feb 8, 2023 at 20:02