Any Hawkes jump-diffusion with an integrable kernel can be approximated arbitrarily well by an augmented Markov jump-diffusion, and optimal control values converge under the same approximation.
Malliavin calculus and optimal control of stochastic Volterra equations
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Markov approximation for controlled Hawkes Jump-Diffusions with general kernels
Any Hawkes jump-diffusion with an integrable kernel can be approximated arbitrarily well by an augmented Markov jump-diffusion, and optimal control values converge under the same approximation.