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Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation

Ben Hambly, Philipp Jettkant

TL;DR

The paper develops a comprehensive framework for the optimal control of nonlinear stochastic Fokker–Planck dynamics with common noise, establishing well-posedness and existence of optimal controls. By shifting along the noise, the authors transform the SPFP into a random-coefficient FP equation and derive both necessary and sufficient stochastic maximum principles, with adjoint processes governed by nonlocal BSPDEs and forward-backward SPDE systems. They also prove an equivalence between the FP control problem and a McKean–Vlasov control problem, and provide a detailed application to government interventions in financial systems, including numerical demonstrations. The results extend existing MFC/MFG theory to nonlinear stochastic FP dynamics and deliver a rigorous, implementable optimal-control framework for mean-field systems with common noise.

Abstract

We consider a control problem for the nonlinear stochastic Fokker--Planck equation. This equation describes the evolution of the distribution of nonlocally interacting particles affected by a common source of noise. The system is directed by a controller that acts on the drift term with the goal of minimising a cost functional. We establish the well-posedness of the state equation, prove the existence of optimal controls, and formulate a stochastic maximum principle (SMP) that provides necessary and sufficient optimality conditions for the control problem. The adjoint process arising in the SMP is characterised by a nonlocal (semi)linear backward SPDE for which we study existence and uniqueness. We also rigorously connect the control problem for the nonlinear stochastic Fokker--Planck equation to the control of the corresponding McKean--Vlasov SDE that describes the motion of a representative particle. Our work extends existing results for the control of the Fokker--Planck equation to nonlinear and stochastic dynamics. In particular, the sufficient SMP, which we obtain by exploiting the special structure of the Fokker--Planck equation, seems to be novel even in the linear deterministic setting. We illustrate our results with an application to a model of government interventions in financial systems, supplemented by numerical illustrations.

Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation

TL;DR

The paper develops a comprehensive framework for the optimal control of nonlinear stochastic Fokker–Planck dynamics with common noise, establishing well-posedness and existence of optimal controls. By shifting along the noise, the authors transform the SPFP into a random-coefficient FP equation and derive both necessary and sufficient stochastic maximum principles, with adjoint processes governed by nonlocal BSPDEs and forward-backward SPDE systems. They also prove an equivalence between the FP control problem and a McKean–Vlasov control problem, and provide a detailed application to government interventions in financial systems, including numerical demonstrations. The results extend existing MFC/MFG theory to nonlinear stochastic FP dynamics and deliver a rigorous, implementable optimal-control framework for mean-field systems with common noise.

Abstract

We consider a control problem for the nonlinear stochastic Fokker--Planck equation. This equation describes the evolution of the distribution of nonlocally interacting particles affected by a common source of noise. The system is directed by a controller that acts on the drift term with the goal of minimising a cost functional. We establish the well-posedness of the state equation, prove the existence of optimal controls, and formulate a stochastic maximum principle (SMP) that provides necessary and sufficient optimality conditions for the control problem. The adjoint process arising in the SMP is characterised by a nonlocal (semi)linear backward SPDE for which we study existence and uniqueness. We also rigorously connect the control problem for the nonlinear stochastic Fokker--Planck equation to the control of the corresponding McKean--Vlasov SDE that describes the motion of a representative particle. Our work extends existing results for the control of the Fokker--Planck equation to nonlinear and stochastic dynamics. In particular, the sufficient SMP, which we obtain by exploiting the special structure of the Fokker--Planck equation, seems to be novel even in the linear deterministic setting. We illustrate our results with an application to a model of government interventions in financial systems, supplemented by numerical illustrations.
Paper Structure (24 sections, 26 theorems, 170 equations, 1 figure)

This paper contains 24 sections, 26 theorems, 170 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Main Contributions and Structure of the Paper
  4. Notation
  5. Main Results
  6. The Control Problem
  7. The Necessary Stochastic Maximum Principle
  8. The Sufficient Stochastic Maximum Principle
  9. Practical Implementation of the Criticality Condition
  10. Semilinear BSPDEs with Càdlàg Noise
  11. Càdlàg Martingales with Values in Hilbert Spaces
  12. Itô's Formula for Càdlàg Hilbert Space-Valued Semimartingales
  13. Semilinear BSPDEs
  14. Optimal Control for the Random Fokker--Planck Equation
  15. Well-Posedness of the Fokker--Planck Equation
  16. ...and 9 more sections

Key Result

Proposition 2.3

Let Assumption ass:fpe be satisfied, fix $\gamma \in \mathbb{G}$, and define $\tilde{\gamma} \in \mathbb{G}$ by $\tilde{\gamma}_t(x) = \gamma_t(x + \sigma_0 W_t)$. If $\nu = (\nu_t)_{0 \leq t \leq T}$ is a continuous $\mathcal{M}(\mathbb{R}^d)$-valued process that satisfies SPDE eq:sfpe with control

Figures (1)

  • Figure 1: Plot of numerical approximations of the adjoint process $u_t(x)$ for $\kappa = 0$ (left-hand side) and $\kappa = 1$ (right-hand side). The dotted lines indicate the interval for which $\partial_x u_t(x) \leq - w$.

Theorems & Definitions (55)

  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.9
  • Theorem 2.10
  • Proposition 2.12
  • Corollary 2.13
  • ...and 45 more