[論文レビュー] Stochastic Maximum Principles and Linear-Quadratic Optimal Control Problems for Fractional Backward Stochastic Evolution Equations in Hilbert Spaces

Javad A. Asadzade, Nazım I. Mahmudov|arXiv (Cornell University)|Jan 4, 2026

Stochastic processes and financial applications被引用数 0

ひとこと要約

The paper develops a stochastic maximum principle for fractional backward stochastic evolution equations in Hilbert spaces and applies it to explicitly solve linear-quadratic optimal control problems, yielding closed-form controls via an adjoint system.

ABSTRACT

This paper develops a comprehensive framework for optimal control of systems governed by fractional backward stochastic evolution equations (FBSEEs) in Hilbert spaces. We first establish a stochastic maximum principle (SMP) as a necessary condition for optimality. This is achieved by introducing spike variations, deriving precise estimates for the associated variational equations, and constructing an adjoint process tailored to the fractional dynamics. Subsequently, we apply this general principle to solve the linear-quadratic (LQ) optimal control problem explicitly. The resulting optimal control is characterized in closed form via the adjoint process and is shown to be governed by a system of coupled fractional forward-backward stochastic equations. Our work bridges fractional calculus with stochastic control theory, providing a rigorous foundation for controlling infinite-dimensional systems with memory and long-range dependencies.

研究の動機と目的

• Motivated by memory effects and nonlocal dynamics in infinite-dimensional systems, establish a stochastic maximum principle for FBSEEs in Hilbert spaces.
• Derive first-order variational formulas and adjoint processes tailored to fractional dynamics.
• Apply the SMP to solve linear-quadratic optimal control problems explicitly.
• Provide a rigorous bridge between fractional calculus and stochastic control theory in infinite dimensions.

提案手法

• Introduce spike variations of the control to derive variational equations.
• Develop a fractional adjoint process and a Pontryagin-type maximum principle.
• Prove a first-order variation formula for the cost functional under fractional dynamics.
• Solve the backward linear-quadratic problem by deriving an adjoint equation via fractional integration by parts.
• Obtain a closed-form representation of the optimal control in terms of the adjoint processes.

実験結果

リサーチクエスチョン

• RQ1 How can a stochastic maximum principle be formulated for fractional backward stochastic evolution equations in Hilbert spaces?
• RQ2 What are the first-order variations and adjoint equations corresponding to FBSEEs under spike perturbations?
• RQ3 How can the linear-quadratic optimal control problem be explicitly solved in the fractional infinite-dimensional setting?
• RQ4 What role do fractional resolvent operators play in the SMP and LQ problem formulation?

主な発見

• A stochastic maximum principle is established as a necessary condition for optimality for FBSEEs in Hilbert spaces.
• Spike variations and precise variational estimates yield a first-order variation formula for the cost functional.
• An adjoint process tailored to the fractional dynamics is constructed and used to derive the SMP.
• In the LQ setting, the optimal control is characterized in closed form via the adjoint process.
• The optimal control is governed by a system of coupled fractional forward-backward stochastic equations.
• This work integrates fractional calculus with stochastic control to handle memory effects in infinite-dimensional systems.