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Existence of optimal controls for stochastic Volterra equations
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We provide sufficient conditions that guarantee the existence of relaxed optimal controls in the weak formulation of stochastic control problems for stochastic Volterra equations (SVEs). Our study can be applied to rough processes that arise when the kernel appearing in the controlled SVE is singular at zero. The existence of relaxed optimal policies relies on the interaction between integrability hypotheses on the kernel and growth conditions on the running cost functional and the coefficients of the controlled SVEs. Under classical convexity assumptions, we can also deduce the existence of optimal strict controls.
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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.
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