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Extended mean field control: a global numerical solution via finite-dimensional approximation

Athena Picarelli, Marco Scaratti, Jonathan Tam

Abstract

We investigate the global numerical approximation of a class of extended mean field control problems (MFC), where the dynamics and costs depend on the joint distribution of the state and the control. We propose a framework to approximate the value function globally over the Wasserstein space, moving beyond the restriction of fixed initial conditions. Our approach exploits the propagation of chaos by approximating the infinite-dimensional MFC problem by an $N$-player cooperative game, together with the usage of finite-dimensional solvers. This method avoids the need to parametrise functions on an infinite-dimensional space, offering a balance between probabilistic rigor and computational efficiency.

Extended mean field control: a global numerical solution via finite-dimensional approximation

Abstract

We investigate the global numerical approximation of a class of extended mean field control problems (MFC), where the dynamics and costs depend on the joint distribution of the state and the control. We propose a framework to approximate the value function globally over the Wasserstein space, moving beyond the restriction of fixed initial conditions. Our approach exploits the propagation of chaos by approximating the infinite-dimensional MFC problem by an -player cooperative game, together with the usage of finite-dimensional solvers. This method avoids the need to parametrise functions on an infinite-dimensional space, offering a balance between probabilistic rigor and computational efficiency.

Paper Structure

This paper contains 18 sections, 12 theorems, 84 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Extended mean field control problem
  3. Probabilistic framework
  4. The control problem
  5. Approximation of the value function
  6. Case 1: sigma non-degenerate
  7. Case 2: sigma degenerate
  8. Approximation by a non-degenerate control problem
  9. Approximation by a cooperative stochastic differential game of finitely many players
  10. Approximation result
  11. A computational scheme with neural networks
  12. Obtaining an approximately optimal policy
  13. Computing the finite-dimensional value function
  14. Recovering the mean field value function
  15. Numerical experiments
  16. ...and 3 more sections

Key Result

Proposition 2.3

Given ass:ex_uniq_SDE, for any $t \in [0,T]$, ${\alpha \in \mathcal{A}}$, and $\xi,{\xi}^{\prime}\in L^2(\Omega,\mathcal{F}_t,\mathbb{P};\mathbb{R}^d)$, there exists a unique, indistinguishable solution $(X^{t,\xi,\alpha}_r)_{r\in[t,T]}$ to SDE, satisfying for a positive constant $C_1$ independent of $t,\xi,\alpha$. Also, for all $s \in [t,T]$, for a positive constant $C_2$ independent of $t,s,\

Figures (4)

  • Figure 5.1: (Example 1) Top: empirical convergence of the finite-dimensional value function as the number of particles $N\to\infty$. Bottom: exact mean field solution $v(0,\delta_{x_0})$ versus generated Monte Carlo estimates, for $x_0\in\{0.000, 0.125, 0.250, 0.375, 0.500\}$, number of players $N$ increasing.
  • Figure 5.2: (Example 1) Plot of the distribution quantiles over time, for one simulated player of the process $X^n_t$, starting at $\xi\sim\delta_{0.5}$. The histogram of $X^n_T$ at the terminal time $T=1$ compared with theoretical density $\mathcal{N}(0.5,1)$ is plotted simultaneously on the right vertical axis.
  • Figure 5.3: (Example 2) Left: The absolute relative error of the expected stock price, comparing the explicit solution with the finite-dimensional approximation. Right: empirical convergence of the finite-dimensional value function as the number of particles $N\to\infty$.
  • Figure 5.4: (Example 3) Left column: Tracing $\bar{v}^{*}_{N,\eta}$ along the diagonal of the hypercube $[0,2]^{100}$. Right column: Tracing $\bar{v}^{*}_{N,\eta}$ along a circle on the first two dimensions.

Theorems & Definitions (31)

  • Remark 2.1
  • Proposition 2.3
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Theorem 2.8
  • proof
  • Corollary 2.9
  • ...and 21 more