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Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and an economical application. (arXiv:2308.14506v1 [math.OC])

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In this manuscript we consider optimal control problems of deterministic and
stochastic differential equations with delays in the state and in the control.
First we prove an equivalent Markovian reformulation on Hilbert spaces of the
state equation. Then, using the dynamic programming approach for
infinite-dimensional systems, we prove that the value function is the unique
viscosity solution of the infinite-dimensional Hamilton Jacobi Bellman
equation. Finally we apply this result to a stochastic optimal advertising
problem with delays in the state and in the control.

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