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Robust pointwise second order necessary conditions for singular stochastic optimal control with model uncertainty

Guangdong Jing

TL;DR

This work develops robust second-order necessary conditions for singular stochastic optimal control under model uncertainty. By combining convex variational methods with Malliavin calculus (including the Clark-Ocone formula) and a minimax framework, it derives an integral-type second-order inequality and establishes a pointwise version under additional regularity and monotonicity assumptions. A common reference measure for uncertainty is obtained, enabling tractable, verifiable conditions that excise nonoptimal singular controls. An illustrative example demonstrates the practical discrimination power of the results, highlighting their value in robust controller design under ambiguity.

Abstract

We study the singular stochastic optimal control problem with model uncertainty, where the necessary conditions determined by the corresponding maximum principle are trivial. Robust integral form and pointwise second order necessary optimality conditions under certain compactness conditions are derived. Both the drift and diffusion terms are control dependent but the control region are assumed to be convex. The convex variational method is employed, because linear structure is essential in deriving the weak limit of uncertainty measures. Other main technical ingredients in obtaining the integral type conditions are compact analysis and minimax theorem, while for the pointwise ones it is Clark-Ocone formula and Lebesgue differentiation type theorem. Besides, a compendious example is given to illustrate the motivation and effectiveness of the results.

Robust pointwise second order necessary conditions for singular stochastic optimal control with model uncertainty

TL;DR

This work develops robust second-order necessary conditions for singular stochastic optimal control under model uncertainty. By combining convex variational methods with Malliavin calculus (including the Clark-Ocone formula) and a minimax framework, it derives an integral-type second-order inequality and establishes a pointwise version under additional regularity and monotonicity assumptions. A common reference measure for uncertainty is obtained, enabling tractable, verifiable conditions that excise nonoptimal singular controls. An illustrative example demonstrates the practical discrimination power of the results, highlighting their value in robust controller design under ambiguity.

Abstract

We study the singular stochastic optimal control problem with model uncertainty, where the necessary conditions determined by the corresponding maximum principle are trivial. Robust integral form and pointwise second order necessary optimality conditions under certain compactness conditions are derived. Both the drift and diffusion terms are control dependent but the control region are assumed to be convex. The convex variational method is employed, because linear structure is essential in deriving the weak limit of uncertainty measures. Other main technical ingredients in obtaining the integral type conditions are compact analysis and minimax theorem, while for the pointwise ones it is Clark-Ocone formula and Lebesgue differentiation type theorem. Besides, a compendious example is given to illustrate the motivation and effectiveness of the results.
Paper Structure (10 sections, 13 theorems, 166 equations)

This paper contains 10 sections, 13 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries in Malliavin calculus
  3. Formulation and main results
  4. Motivating example
  5. The proof of Theorem \ref{['thmfirornedes']}
  6. The proof of Theorem \ref{['thm-point-nece-cla']}
  7. Technical lemmas
  8. The proof of Theorem \ref{['thm-point-nece-cla']}
  9. Technical lemmas
  10. A comment on (H6) (ii)

Key Result

Proposition 3.1

Under Assumptions (H1) and (H3), for any $\beta\in[1,\infty)$, $u(\cdot), \bar{u}(\cdot)\in \mathcal{U}^\beta$, it yields Besides, as $\varepsilon \to 0$,

Theorems & Definitions (30)

  • Definition 1.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 20 more