Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlocal Stochastic Optimal Control for Diffusion Processes: Existence, Maximum Principle and Financial Applications

Stefana-Lucia Anita, Luca Di Persio

TL;DR

The paper addresses optimal control of diffusion processes where the control acts nonlocally on the diffusion coefficient via a spatial convolution. It develops existence results for optimal controls using compactness and weak convergence, and derives a stochastic maximum principle that couples the state, adjoint, and the nonlocal structure. A concrete financial application to mean-variance portfolio optimization demonstrates an integral representation for the optimal strategy when both drift and volatility are affected by the nonlocal control. Overall, the work provides a rigorous framework for nonlocal diffusion control with broad implications for stochastic optimization and financial engineering.

Abstract

This paper investigates the optimal control problem for a class of parabolic equations where the diffusion coefficient is influenced by a control function acting nonlocally. Specifically, we consider the optimization of a cost functional that incorporates a controlled probability density evolving under a Fokker-Planck equation with state-dependent drift and diffusion terms. The control variable is subject to spatial convolution through a kernel, inducing nonlocal interactions in both drift and diffusion terms. We establish the existence of optimal controls under appropriate convexity and regularity conditions, leveraging compactness arguments in function spaces. A maximum principle is derived to characterize the optimal control explicitly, revealing its dependence on the adjoint state and the nonlocal structure of the system. We further provide a rigorous financial application in the context of mean-variance portfolio optimization, where both the asset drift and volatility are controlled nonlocally, leading to an integral representation of the optimal investment strategy. The results offer a mathematically rigorous framework for optimizing diffusion-driven systems with spatially distributed control effects, broadening the applicability of nonlocal control methods to stochastic optimization and financial engineering.

Nonlocal Stochastic Optimal Control for Diffusion Processes: Existence, Maximum Principle and Financial Applications

TL;DR

The paper addresses optimal control of diffusion processes where the control acts nonlocally on the diffusion coefficient via a spatial convolution. It develops existence results for optimal controls using compactness and weak convergence, and derives a stochastic maximum principle that couples the state, adjoint, and the nonlocal structure. A concrete financial application to mean-variance portfolio optimization demonstrates an integral representation for the optimal strategy when both drift and volatility are affected by the nonlocal control. Overall, the work provides a rigorous framework for nonlocal diffusion control with broad implications for stochastic optimization and financial engineering.

Abstract

This paper investigates the optimal control problem for a class of parabolic equations where the diffusion coefficient is influenced by a control function acting nonlocally. Specifically, we consider the optimization of a cost functional that incorporates a controlled probability density evolving under a Fokker-Planck equation with state-dependent drift and diffusion terms. The control variable is subject to spatial convolution through a kernel, inducing nonlocal interactions in both drift and diffusion terms. We establish the existence of optimal controls under appropriate convexity and regularity conditions, leveraging compactness arguments in function spaces. A maximum principle is derived to characterize the optimal control explicitly, revealing its dependence on the adjoint state and the nonlocal structure of the system. We further provide a rigorous financial application in the context of mean-variance portfolio optimization, where both the asset drift and volatility are controlled nonlocally, leading to an integral representation of the optimal investment strategy. The results offer a mathematically rigorous framework for optimizing diffusion-driven systems with spatially distributed control effects, broadening the applicability of nonlocal control methods to stochastic optimization and financial engineering.

Paper Structure

This paper contains 7 sections, 3 theorems, 105 equations.

Table of Contents

  1. Keywords:
  2. MSC:
  3. Introduction
  4. Existence of an optimal control
  5. The maximum principle
  6. Mathematical Finance application
  7. Final comments and further directions

Key Result

Lemma 2.1

If $\{ u_n\}_{n\in \mathbb{N}^*}\subset {\cal U}$ satisfies $u_n\longrightarrow u$ weakly in $L^{\ell ^*}(\omega )$, then for any $i\in \{1,2, ...,d\}$.

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof