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Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations
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We introduce a new framework of Markovian lifts of stochastic Volterra integral equations (SVIEs for short) with completely monotone kernels. We define the state space of the Markovian lift as a separable Hilbert space which incorporates the singularity or regularity of the kernel into the definition. We show that the solution of an SVIE is represented by the solution of a lifted stochastic evolution equation (SEE for short) defined on the Hilbert space and prove the existence, uniqueness and Markov property of the solution of the lifted SEE. Furthermore, we establish an asymptotic log-Harnack inequality and some consequent properties for the Markov semigroup associated with the Markovian lift via the asymptotic coupling method.
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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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