𝜀-maximum principle for stochastic delayed control problems with jump-diffusion processes and application to social network-based advertising model

Abstract

Stochastic control of delayed systems is a challenging research area. This paper investigates the near-optimal control problem for systems described by stochastic delayed differential equations with jumps, where both the delayed state and control influence the drift, diffusion and jump-diffusion terms, under a non-convex control domain. We establish necessary and sufficient conditions for near-optimality by employing the near-maximum condition on the [Formula: see text]-function, which extends the Hamiltonian function in an integral sense. The main results rely on Ekeland’s variational principle, stability properties of the state, and the first and second adjoint processes associated with the control variable. Finally, to demonstrate the applicability of our theoretical findings, we analyze a near-optimal advertising expenditure strategy aimed at minimizing the cost function in a delayed advertising model that incorporates social network effects.