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Optimal control problem for reflected stochastic differential equation: existence

Ayoub Laayoun, Badr Missaoui

Abstract

We establish the existence of both optimal relaxed controls and strict optimal controls for systems driven by Reflected Stochastic Differential Equations RSDEs. Our approach is based on weak convergence techniques for the associated RSDEs in the uniform convergence topology, along with an appropriate Skorokhod representation theorem.

Optimal control problem for reflected stochastic differential equation: existence

Abstract

We establish the existence of both optimal relaxed controls and strict optimal controls for systems driven by Reflected Stochastic Differential Equations RSDEs. Our approach is based on weak convergence techniques for the associated RSDEs in the uniform convergence topology, along with an appropriate Skorokhod representation theorem.

Paper Structure

This paper contains 9 sections, 9 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem, notations and assumptions
  3. Strict control
  4. Relaxed control
  5. The main results
  6. Proof of the main results
  7. Proof of theorem \ref{['theo1']}
  8. Proof of Corollary \ref{['theo2']}
  9. Proof of Theorem \ref{['approximatio1']}

Key Result

Theorem 3.1

Under assumptions $\textbf{(A.1)-(A.3)}$, the relaxed control problem has an optimal solution.

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Theorem 3.1
  • Corollary 3.1
  • Remark 3.1
  • Example 3.1: see page 86 in Kushner-exp
  • Theorem 3.2
  • Lemma 4.1
  • ...and 10 more