Near Optimality of Discrete-Time Approximations for Controlled McKean-Vlasov Diffusions and Interacting Particle Systems
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2025
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| author | Pradhan, Somnath Yuksel, Serdar |
| author_facet | Pradhan, Somnath Yuksel, Serdar |
| contents | We study stochastic optimal control problems for (possibly degenerate) McKean-Vlasov controlled diffusions and obtain discrete-time as well as finite interacting particle approximations. (i) Under mild assumptions, we first prove the existence of optimal relaxed controls by endowing the space of relaxed policies with a compact weak topology. (ii) Establishing continuity of the cost in control policy, we establish near-optimality of piecewise-constant strict policies, show that the discrete-time value functions (finite-horizon and discounted infinite-horizon) converge to their continuous-time counterparts as the timestep converges to zero, and that optimal discrete-time policies are near-optimal for the original continuous-time problem, where rates of convergence are also obtained. (iii) We then extend these approximation and near-optimality results to $N$-particle interacting systems under centralized or decentralized mean-field sharing information structure, proving that the discrete-time McKean-Vlasov policy is asymptotically optimal as $N\to \infty$ and the time step goes to zero. We thus develop an approximation of McKean-Vlasov optimal control problems via discrete-time McKean-Vlasov control problems (and associated numerical methods such as finite model approximation), and also show the near optimality of such approximate policy solutions for the $N$-agent interacting models under centralized and decentralized control. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_21208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Near Optimality of Discrete-Time Approximations for Controlled McKean-Vlasov Diffusions and Interacting Particle Systems Pradhan, Somnath Yuksel, Serdar Optimization and Control 93E20, 49K45, 60K35, 60H30 We study stochastic optimal control problems for (possibly degenerate) McKean-Vlasov controlled diffusions and obtain discrete-time as well as finite interacting particle approximations. (i) Under mild assumptions, we first prove the existence of optimal relaxed controls by endowing the space of relaxed policies with a compact weak topology. (ii) Establishing continuity of the cost in control policy, we establish near-optimality of piecewise-constant strict policies, show that the discrete-time value functions (finite-horizon and discounted infinite-horizon) converge to their continuous-time counterparts as the timestep converges to zero, and that optimal discrete-time policies are near-optimal for the original continuous-time problem, where rates of convergence are also obtained. (iii) We then extend these approximation and near-optimality results to $N$-particle interacting systems under centralized or decentralized mean-field sharing information structure, proving that the discrete-time McKean-Vlasov policy is asymptotically optimal as $N\to \infty$ and the time step goes to zero. We thus develop an approximation of McKean-Vlasov optimal control problems via discrete-time McKean-Vlasov control problems (and associated numerical methods such as finite model approximation), and also show the near optimality of such approximate policy solutions for the $N$-agent interacting models under centralized and decentralized control. |
| title | Near Optimality of Discrete-Time Approximations for Controlled McKean-Vlasov Diffusions and Interacting Particle Systems |
| topic | Optimization and Control 93E20, 49K45, 60K35, 60H30 |
| url | https://arxiv.org/abs/2510.21208 |