We study relaxed optimal stochastic control problem for systems governed by forward and backward stochastic differential equations of both It and McKean-Vlasov types. The set of admissible controls is a set of measure-valued processes called relaxed controls. We assume that the control variable is allowed to enter both the drift and diffusion coefficients. We prove that these dynamical systems are in fact driven by an orthogonal continuous martingale measure, instead of a Brownian motion. Existence of optimal controls is investigated by using weak convergence techniques and Skorokhod theorem. Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of spike variation techniques. Our results will generalize the Peng's maximum principle to the class of measure valued controls. The extension to stochastic differential equations driven by relaxed Poisson random measures will also investigated.
|Effective start/end date
|1/04/20 → 1/10/22
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