Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911212465291264 |
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| author | Hambly, Ben Jettkant, Philipp |
| author_facet | Hambly, Ben Jettkant, Philipp |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation Hambly, Ben Jettkant, Philipp Optimization and Control Probability 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. |
| title | Optimal Control of the Nonlinear Stochastic Fokker--Planck Equation |
| topic | Optimization and Control Probability |
| url | https://arxiv.org/abs/2406.16512 |