This paper addresses the optimal market making problem in electronic markets where market order flows exhibit clustering and long memory, modeled by general Hawkes processes with completely monotone kernels. The market maker controls bid and ask spreads to maximize expected utility from terminal wealth and intermediate rewards, while managing inventory risk. The problem is formulated as a stochastic control problem, and the associated Hamilton-Jacobi-Bellman (HJB) equation is derived. Due to the infinite-dimensional nature of the state space (intensity processes are functions), the author proves existence and uniqueness of a viscosity solution to the HJB equation on a locally compact domain. The optimal spreads are characterized in feedback form, depending on the value function and the current intensities.

To make the solution tractable, the paper proposes approximating the completely monotone kernel by a sum of exponentials, which yields a finite-dimensional Markovian representation. This allows numerical approximation of the optimal control via standard methods for finite-dimensional HJB equations. For high-dimensional cases, a probabilistic representation using branching processes is employed. The theoretical results include convergence of the approximated controls to the true optimal control as the kernel approximation improves. Numerical experiments illustrate the impact of order flow persistence on optimal spreads, showing that the market maker adjusts spreads dynamically in response to clustered order flow.