Optimal liquidation, acquisition and market making problems in HFT under Hawkes models for LOB

The present paper is focused on the solution of optimal control problems such as optimal acquisition, optimal liquidation, and market making in relation to the high-frequency trading market. We have modeled optimal control problems with the price approximated by the diffusion process for the general compound Hawkes process (GCHP), using results from the work of Swishchuk and Huffman. These problems have been solvedusing a price process incorporating the unique characteristics of the GCHP. The GCHP was designed to reflect important characteristics of the behaviour of real-world price processes such as the dependence on the previous process and jumping features. In these models, the agent maximizes their own utility or value function by solving the Hamilton-Jacobi-Bellman (HJB) equation and designing a strategy for asset trading. The optimal solutions are expressed in terms of parameters describing the arrival rates and the midprice process from the price process, modeled as a GCHP, allowing such characteristics to influence the optimization process, aiming towards the attainment of a more general solution. Implementations of the obtained results were carried out using real LOBster data.