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A Fully-discrete Semi-Lagrangian scheme for a price formation MFG model

Yuri Ashrafyan, Diogo Gomes

TL;DR

This work tackles price formation in first-order mean-field games by introducing a fully discrete Semi-Lagrangian scheme that couples a Hamilton-Jacobi equation with a forward transport equation under a deterministic supply $Q(t)$. The method discretizes the HJ equation, the FP transport, and the price balance, and solves the coupled system via an iterative, monotone, multivalued framework that admits a fixed point. The authors prove existence, monotonicity, and convergence of the discretization to a weak solution of the continuous problem, using Kakutani's fixed point theorem and monotonicity arguments. Numerical experiments demonstrate that the Semi-Lagrangian scheme achieves higher accuracy and significantly faster computation than competing variational and neural-network approaches for deterministic price formation MFGs.

Abstract

Here, we examine a fully-discrete Semi-Lagrangian scheme for a mean-field game price formation model. We show the existence of the solution of the discretized problem and that it is monotone as a multivalued operator. Moreover, we show that the limit of the discretization converges to the weak solution of the continuous price formation mean-field game using monotonicity methods. Numerical simulations demonstrate that this scheme can provide results efficiently, comparing favorably with other methods in the examples we tested.

A Fully-discrete Semi-Lagrangian scheme for a price formation MFG model

TL;DR

This work tackles price formation in first-order mean-field games by introducing a fully discrete Semi-Lagrangian scheme that couples a Hamilton-Jacobi equation with a forward transport equation under a deterministic supply . The method discretizes the HJ equation, the FP transport, and the price balance, and solves the coupled system via an iterative, monotone, multivalued framework that admits a fixed point. The authors prove existence, monotonicity, and convergence of the discretization to a weak solution of the continuous problem, using Kakutani's fixed point theorem and monotonicity arguments. Numerical experiments demonstrate that the Semi-Lagrangian scheme achieves higher accuracy and significantly faster computation than competing variational and neural-network approaches for deterministic price formation MFGs.

Abstract

Here, we examine a fully-discrete Semi-Lagrangian scheme for a mean-field game price formation model. We show the existence of the solution of the discretized problem and that it is monotone as a multivalued operator. Moreover, we show that the limit of the discretization converges to the weak solution of the continuous price formation mean-field game using monotonicity methods. Numerical simulations demonstrate that this scheme can provide results efficiently, comparing favorably with other methods in the examples we tested.
Paper Structure (31 sections, 10 theorems, 126 equations, 11 figures, 2 tables)

This paper contains 31 sections, 10 theorems, 126 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Main assumptions
  4. Discretization
  5. Hamilton-Jacobi equation
  6. Continuity equation
  7. Balance condition
  8. Algorithm
  9. Some properties and estimates
  10. Existence of solutions
  11. Step 1.
  12. Step 2.
  13. Step 3.
  14. Step 4.
  15. Existence of a fixed point
  16. ...and 16 more sections

Key Result

Lemma 5.1

Under Assumptions hyp:H_l_0 and hyp:V_uT_Lipschitz_convex, $S_{\rho, h}(f,i,k)$ enjoys the following properties:

Figures (11)

  • Figure 1: Numerical solution of $u$ (left), and the relation of price $\varpi$ and supply functions, $Q$ (right).
  • Figure 2: Numerical solution of $m$ (left) and its density plot (right).
  • Figure 3: Approximated and exact solutions for $\varpi$ (left), and their absolute difference (right).
  • Figure 4: Approximated and exact solutions for $m$ (left), and their difference (right). Note that they only have a major disagreement on the boundary of the support.
  • Figure 5: Approximated and exact solutions for $u$ (left), and their difference (right).
  • ...and 6 more figures

Theorems & Definitions (23)

  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • proof
  • Lemma 5.5
  • proof
  • ...and 13 more