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A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems

Matteo Burzoni, Alekos Cecchin, Andrea Cosso

Abstract

We provide a new version of the Tikhonov theorem for both two-scale forward systems and also two-scale forward-backward systems of stochastic differential equations, which also covers the McKean-Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the ''fast'' variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose ''fast'' component converges to the optimal control process, while the ''slow'' component converges to the optimal state process. The interest in such a procedure is that it allows to approximate the solution of the control problem avoiding the usual step of the minimization of the Hamiltonian.

A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems

Abstract

We provide a new version of the Tikhonov theorem for both two-scale forward systems and also two-scale forward-backward systems of stochastic differential equations, which also covers the McKean-Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the ''fast'' variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose ''fast'' component converges to the optimal control process, while the ''slow'' component converges to the optimal state process. The interest in such a procedure is that it allows to approximate the solution of the control problem avoiding the usual step of the minimization of the Hamiltonian.
Paper Structure (9 sections, 12 theorems, 122 equations, 2 figures, 1 table)

This paper contains 9 sections, 12 theorems, 122 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Two-scale stochastic systems: stochastic Tikhonov theorems
  3. Stochastic Tikhonov theorem I
  4. Stochastic Tikhonov theorem II
  5. Approximating control problems with two-scale FBSDEs
  6. A McKean-Vlasov linear quadratic example
  7. The two-scale approximation.
  8. The curse of dimension.
  9. Lions differentiability

Key Result

Lemma 2.4

Let Assumption ass:two_scale hold. Let $\xi\in L^2(\Omega,{\cal G},\mathbb{P};\mathbb{R}^d)$, $\alpha,\alpha'\in\mathbb{H}_k^2$. Suppose that $X^\alpha\in\mathbb{S}_d^2$ is a solution to the following controlled stochastic differential equation on $[0,T]$: Similarly, suppose that $X^{\alpha'}\in\mathbb{S}_d^2$ is a solution to equation SDE_X_alpha_Estimate with $\alpha'$ in place of $\alpha$. The

Figures (2)

  • Figure 1: Approximation errors with varying $\varepsilon$ (Left). An example of trajectory of $X,\hat{\alpha},X^\varepsilon,A^\varepsilon$ for $\varepsilon=5\cdot10^{-5}$ (Right).
  • Figure 2: Execution times with varying dimension.

Theorems & Definitions (32)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • ...and 22 more