Approximation and optimality necessary conditions in relaxed stochastic control problems

Abstract

We consider a control problem where the state variable is a solution of a stochastic differential equation (SDE) in which the control enters both the drift and the diffusion coefficient. We study the relaxed problem for which admissible controls are measure-valued processes and the state variable is governed by an SDE driven by an orthogonal martingale measure. Under some mild conditions on the coefficients and pathwise uniqueness, we prove that every diffusion process associated to a relaxed control is a strong limit of a sequence of diffusion processes associated to strict controls. As a consequence, we show that the strict and the relaxed control problems have the same value function and that an optimal relaxed control exists. Moreover we derive a maximum principle of the Pontriagin type, extending the well-known Peng stochastic maximum principle to the class of measure-valued controls.

On Relaxed Stochastic Optimal Control for Stochastic Differential Equations Driven by G-Brownian Motion

“…Proof : Since, due to the aggregation property of Lemma (2.5), under every lP ∈ P, the G-sdes (4.2) and (4.3) become standard sdes driven by a continuous true martingale B, the proof of this result follows from Bahlali et al (2006) or Bahlali et al (2014). We sketch it here for completeness.…”

The maximum principle in optimal control of systems driven by martingale measures

“…The relaxed optimal control in this general case is shown to be Markovian. See also Bahlali et al (2006) for an alternative proof of the existence of an optimal relaxed control based on Skorokhod selection theorem. The second advantage of the use of relaxed controls is that it is a generalization of the strict control problem, in the sense that both control problems have the same value function.…”

“…a controlled diffusion coefficient has been treated in Bahlali et al (2006), by using Ekeland's variational principle and an approximation scheme, by using the first and second order adjoint processes. Let us point out that a different relaxation has been used in Bahlali (2008); Ahmed and Charalambous (2013), where the drift and diffusion coefficient have been replaced by their relaxed counterparts.…”

“…Our method differs from the one used in Bahlali et al (2006), in the sense that we don't use neither the approximation procedure nor Ekeland's variational principle. We use a spike variation method directly on the relaxed optimal control.…”

“…See Bahlali et al (2006). Journal home page: www.jafristat.net ; www.projecteuclid.org/as Corollary 1.…”