Self-contained control theory for general controlled Hawkes processes

Develop a self-contained control theory for multidimensional Hawkes processes with general, potentially non-Markovian excitation kernels, beyond the partial results currently available for particular kernels such as the exponential kernel.

Background

The paper studies stochastic control of multidimensional Hawkes-driven jump-diffusions when the excitation kernel may have general memory dependence and may depend on the current control action. Unlike the single-exponential-kernel case, a general kernel does not generally produce a finite-dimensional Markov intensity process, which prevents direct application of classical finite-dimensional stochastic-control methods.

The authors address this difficulty by approximating the general kernel with mixtures of exponential kernels and proving convergence of the resulting Markovianized processes and value functions. They explicitly identify the development of a complete control theory for the original general-kernel Hawkes system as unresolved, noting that only partial preliminary results are available.

References

This kernel lies at the heart of the complexity of Hawkes processes, and developing a self-contained control theory for them remains an open challenge, with only partial preliminary results available so far.

— Continuous-Time Reinforcement Learning for Controlled Hawkes Jump-Diffusions  (2608.19151 - Bielecki et al., 19 Aug 2026) in Section 1, subsection “Hawkes processes: theory, applications, and control”