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Necessary and Sufficient Optimality Conditions for Relaxed and Strict Control of Forward-Backward Doubly SDEs with Jumps Under Full and Partial Information

Authors
  • Al-Hussein, AbdulRahman1
  • Gherbal, Boulakhras2
  • 1 Qassim University, 51452, Buraydah, Saudi Arabia , 51452 (Saudi Arabia)
  • 2 University of Mohamed Khider, Biskra, 07000, Algeria , Biskra (Algeria)
Type
Published Article
Journal
Journal of Systems Science and Complexity
Publisher
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Publication Date
Dec 01, 2020
Volume
33
Issue
6
Pages
1804–1846
Identifiers
DOI: 10.1007/s11424-020-9013-3
Source
Springer Nature
Keywords
License
Yellow

Abstract

This paper derives necessary and sufficient conditions of optimality in the form of a stochastic maximum principle for relaxed and strict optimal control problems with jumps. These problems are governed by multi-dimensional forward-backward doubly stochastic differential equations (FBDSDEs) with Poisson jumps and has firstly relaxed controls, which are measure-valued processes, and secondly, as an application, the authors allow them to have strict controls. The FBDSDEs with jumps are fully-coupled, the forward and backward equations work in different Euclidean spaces in general, the backward equation is Markovian, and the control problems are considered under full information or partial information in terms of σ-algebras that provide such information. The formulation of these equations as well as performance functionals are given in abstract forms to allow the possibility to cover most of the applications available in the literature. Moreover, coefficients of such equations are allowed to depend on control variables.

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