Mean field games: convergence of a finite difference method

1 Introduction

• The model describes stochastic-game equilibria as the number of players tends to infinity, through coupled equations for agent density and potential.The paper assumes a two-dimensional state space for its finite-difference analysis, noting that computation becomes too demanding in practice for dimensions d ≥ 4.Source: §1 · p. 1
• The paper treats terminal costs that depend directly on x; costs depending on the terminal density are left outside its discussion.The authors state that their methods can handle density-dependent terminal costs, but do not discuss that case here.Source: §1 · p. 2
• The forward Bellman and backward Fokker–Planck equations form a coupled system that requires specific strategies for analysis and numerical approximation.The setup uses a convex Hamiltonian and allows local or nonlocal cost operators.Source: §1 · p. 2
• Earlier finite-difference schemes use monotone Hamiltonian approximations and a weak formulation of the Fokker–Planck equation, with existence, uniqueness, robustness, and grid-uniform bounds among their established properties.The cited prior work includes schemes for infinite- and finite-horizon games.Source: §1 · p. 3
• This paper proves convergence of the schemes proposed in for H(x, p) = H(x) + |p|^β, with β > 1, under suitable monotonicity assumptions on Φ.The stated results include uniqueness and stability estimates under further assumptions.Source: §1 · p. 3
• The analysis distinguishes nonlocal smoothing operators from local operators, with convergence theorems for both cases.The introduction also notes uniqueness and stability for the continuous model under reasonable assumptions, unlike the Nash system for individual players, where uniqueness hardly occurs.Source: §1 · pp. 2–3

2 Finite difference schemes

• The scheme approximates H(x, ∇u) with a numerical Hamiltonian g that is monotone, consistent with H, and differentiable.The construction uses four one-sided finite differences; convexity is an additional condition used in the uniqueness result.Source: §2 · p. 4
• For H(x, p) = H(x) + |p|^β, the paper selects a numerical Hamiltonian built from the four one-sided differences.The form is also stated to apply to a positive constant multiple of the specified function G.Source: §2 · p. 4
• The discrete coupling operator uses F(m_i,j) for local costs, while nonlocal operators are required to satisfy additional properties.For nonlocal costs, the construction includes discrete probability densities and piecewise bilinear interpolation; a bilaplacian example is provided.Source: §2 · pp. 4–5
• The Bellman equation is discretized by a semi-implicit Euler scheme, and the Fokker–Planck equation by a weak formulation with a transport operator.The density update is implicit in m and explicit in u; starting from a discrete probability density, the update preserves membership in the discrete probability set.Source: §2 · pp. 5–6
• For ν > 0 and the stated assumptions, the discrete system has a solution with nonnegative probability densities; adding convexity and coupling assumptions gives uniqueness.The existence result also assumes the specified conditions on g and Φ_h, continuous u0, and mT in K; uniqueness requires (g1)–(g4) and (Φh1)–(Φh2).Source: §2 · pp. 6–7

3 Basic facts for numerical Hamiltonians of the form (10)-(11)

• The numerical Hamiltonians considered have form (10)–(11), with the stated results also applying to g(x, q) = H(x) + cG(q) for positive c.Source: §3 · p. 7

3.1 Basic lemmas

• For β ≥2, the Hessian Gpp(p) − β|p|^(β−2)I4 is positive semidefinite.Source: §3.1 · p. 7
• The numerical Hamiltonian’s increment after subtracting its first-order term is bounded below by the corresponding increment for G.Source: §3.1 · p. 7

3.2 A nonlinear functional G(m, u, ũ)

• Under (g4), the functional G(m, u, ũ) is nonnegative for every nonnegative grid function m.Source: §3.2 · p. 8
• Lemma 5 is stated for a nonnegative grid function m and a function g of form (10)–(11) with β ≥ 2.Source: §3.2 · p. 8

3.3 A fundamental identity

• Under assumptions (g4) and (Φh2), three terms in the fundamental identity are nonnegative, yielding uniqueness and, with further assumptions, a priori or stability estimates.The text cites Lemma 5 as an example of further assumptions.Source: §3.3 · p. 10

4 Study of the convergence in the case when Φ is a nonlocal smoothing operator

• The analysis assumes a unique classical solution with m > 0 for monotone, nonlocal smoothing Φ, alongside smooth data and a numerical cost function satisfying (Φh1)–(Φh4).Source: §4 · p. 11

4.1 The case when β ≥2

• For β ≥2, uh converges uniformly and in Lβ(0, T; W1,β(T2)), while mh converges in C0([0, T]; L2(T2)) ∩ L2(0, T; H1(T2)).The authors note that error estimates would require additional assumptions on the order of the discrete scheme.Source: §4.1 · pp. 11, 13

4.2 The case when 1 < β < 2

• For 1 < β < 2, uh converges uniformly and in L2(0, T; W1,2(T2)), while mh converges in L2((0, T) × T2).The proof obtains relative compactness of mh, then uses monotonicity of Φ to identify the limit and conclude convergence of the whole family.Source: §4.2 · pp. 13–15

5.1 A priori estimates for (21)-(22) with local operators Φh

• The interpolated spatial averages Uh are bounded in W 1,1(0, T) uniformly in h and ∆t.Source: §5.1 · p. 17

5.2 Convergence theorems

• Local operators lack a priori Lipschitz estimates on uh, so the convergence analysis assumes a classical solution of the continuous problem.For example, existence is cited for β = 2, C1 F bounded below, and C2 data u0 and mT.Source: §5.2 · p. 17
• When (F2) holds with γ > 2 and F is nondecreasing, the stationary problem has a classical solution for any β > 1.The passage attributes this existence result to the weak Bernstein method.Source: §5.2 · p. 17
• The discrete evolutive solutions converge in Lβ(0, T; W 1,β(T2)) for u and L2−η2((0, T)×T2) for m, assuming a unique classical solution with m > 0.The estimates also bound the interpolated spatial averages Uh in W 1,1(0, T), independently of h and ∆t.Source: §5.1 · p. 17, §5.2 · p. 18
• For the stationary system, uh converges in W 1,β(T2), mh in L2−η2(T2), and λh to λ, assuming a unique classical solution with m > 0 and zero-mean u.Source: §5.2 · p. 21

• The numerical Hamiltonian derivative satisfies a lower bound involving −Gp(p) · (˜p − p), yielding inequality (59).Source: A Proofs of some technical lemmas · p. 22
• The remainder of g is bounded below by the corresponding remainder of G, supporting inequality (31).Source: A Proofs of some technical lemmas · p. 22
• For 1 < β < 2, the proof bounds the intermediate vector term using the maximum of |p|∞ and |˜p|∞.Source: A Proofs of some technical lemmas · pp. 22–23