Approximation and optimality necessary conditions in relaxed stochastic control problems
type: Item
instance of: scholarly article
title: Approximation and optimality necessary conditions in relaxed stochastic control problems
zbMATH Open document ID: 1119.49027
DOI: 10.1155/JAMSA/2006/72762
author: Brahim Mezerdi
author: Boualem Djehiche
author: Seïd Bahlali
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2007-09-10T00:00:00Z
full work available at URL: https://eudml.org/doc/53388
review text: Summary: 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 Pontryagin type, extending the well-known Peng stochastic maximum principle to the class of measure-valued controls.
Mathematics Subject Classification ID: 49K45
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 5189223
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: Peng maximum principle
zbMATH Keywords: maximum principle of Pontryagin type
MaRDI profile type: Publication
cites work: Q4213420
cites work: Q5612829
cites work: Q4742671
cites work: Martingale measures and stochastic calculus
cites work: Compactification methods in the control of degenerate diffusions: existence of an optimal control
cites work: Existence of an Optimal Markovian Filter for the Control under Partial Observations
cites work: Q4145540
cites work: Sur la généralisation de la notion de commande d'un système guidable
cites work: Q3724215
cites work: On the Existence of Optimal Controls
cites work: Q3959169
cites work: Q3908230
cites work: Existence results for optimal stochastic controls
cites work: Q3788861
cites work: Q4735893
cites work: Necessary conditions for optimality for a diffusion with a non-smooth drift
cites work: Approximation in optimal control of diffusion processes
cites work: Tightness of probabilities on C([0,1];\(S_ p\)) and D([0,1];\(S_ p\))
cites work: Necessary conditions for optimality in relaxed stochastic control problems
cites work: Adapted solution of a backward stochastic differential equation
cites work: A General Stochastic Maximum Principle for Optimal Control Problems
cites work: Q3747436
cites work: Q4255599
cites work: Q5524051
cites work: Stochastic Near-Optimal Controls: Necessary and Sufficient Conditions for Near-Optimality
description: scientific article; zbMATH DE number 5189223