Rome - May 12th Université Paris-Diderot - Laboratoire Jacques-Louis Lions. Mean field games equations with quadratic

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1 Université Paris-Diderot - Laboratoire Jacques-Louis Lions Rome - May 12th 2011

2 Hamiltonian MFG Hamiltonian on the domain [0, T ] Ω, Ω standing for (0, 1) d : (HJB) (K) t u + σ2 2 u u 2 = f (x, m) t m + (m u) = σ2 2 m Boundary conditions: u n = m n = 0 on (0, T ) Ω Terminal condition: u(t, ) = u T ( ) a given payoff. Initial condition: m(0, ) = m 0 ( ) 0 a positive function in L 1 (Ω), typically a probability distribution function.

3 Change of variable Proposition (u = σ 2 ln(φ), m = φψ) Let s consider a smooth solution (φ, ψ) of the following system with φ > 0: with: t φ + σ2 2 φ = 1 σ 2 f (x, φψ)φ (E φ) t ψ σ2 2 ψ = 1 σ 2 f (x, φψ)ψ (E ψ) Boundary conditions: φ n = ψ n = 0 on (0, T ) Ω ( Terminal condition: φ(t, ) = exp ut ( ) ). σ 2 Initial condition: ψ(0, ) = m 0( ) φ(0, ) Then (u, m) = (σ 2 ln(φ), φψ) is a solution of (MFG).

4 Hypotheses x, ξ f (x, ξ) is a continuous and decreasing function. Similar to the hypothesis in the usual proof of uniqueness. f L f 0 This is not a restriction since f is bounded... f f f u u f t. u T L (Ω) m 0 L 2 (Ω)

5 Notations We define P C([0, T ], L 2 (Ω)) with: g P g L 2 (0, T, H 1 (Ω)) and t g L 2 (0, T, H 1 (Ω)) We also define: P ɛ = {g P, g ɛ}

6 Equation (E φ ) Proposition (Well-posedness) ψ P 0, there is a unique weak solution φ to the following equation (E φ ): t φ + σ2 2 φ = 1 σ 2 f (x, φψ)φ (E φ) with φ n = 0 on (0, T ) Ω and φ(t, ) = exp ( ut ( ) σ 2 ). Hence Φ : ψ P 0 φ P is well defined.

7 Equation (E φ ) (continued) Two results: Proposition (Uniform lower bound) ψ P 0, φ = Φ(ψ) P ɛ for ɛ = exp ( 1 σ 2 ( u T + f T ) ) This uniform bound will allow to define ψ(0, ). Proposition (Monotonicity) ψ 1 ψ 2 P 0, Φ(ψ 1 ) Φ(ψ 2 ) This monotonicity result will be central in the constructive.

8 Equation (E ψ ) Proposition (Well-posedness) Let s fix ɛ > 0 as above. φ P ɛ, there is a unique weak solution ψ to the following equation (E ψ ): t ψ σ2 2 ψ = 1 σ 2 f (x, φψ)ψ (E ψ) with ψ n = 0 on (0, T ) Ω and ψ(0, ) = m 0( ) Hence Ψ : φ P ɛ ψ P is well defined. φ(0, ).

9 Equation (E ψ ) (continued) Two results: Proposition (Positiveness) φ P ɛ, ψ = Ψ(φ) P 0 Proposition (Monotonicity) φ 1 φ 2 P ɛ, Ψ(φ 1 ) Ψ(φ 2 ) This monotonicity result will be central in the constructive.

10 Constructive - Definition The we consider involves two sequences (φ n+ 1 2 ) n and (ψ n ) n that are built using the following recursive equations: with: t φ n σ 2 ψ 0 = 0 2 φn+ 1 2 = 1 σ 2 f (x, φn+ 1 2 ψ n )φ n+ 1 2 t ψ n+1 σ2 2 ψn+1 = 1 σ 2 f (x, φn+ 1 2 ψ n+1 )ψ n+1 1 Boundary conditions: φn+ 2 n = ψn+1 n = 0 on (0, T ) Ω ( Terminal condition: φ n+ 1 2 (T, ) = exp ut ( ) ). σ 2 Initial condition: ψ n+1 (0, ) = m 0( ) φ n+ 1 2 (0, )

11 Constructive - Definition In other words, the constructive is defined as: ψ 0 = 0 n N, φ n+ 1 2 = Φ(ψ n ) n N, ψ n+1 = Ψ(φ n+ 1 2 )

12 Constructive Theorem The above has the following properties: (φ n+ 1 2 ) n is a decreasing sequence of P ɛ. (ψ n ) n is an increasing sequence of P 0, bounded from above in P by Ψ(ɛ) (φ n+ 1 2, ψ n ) n converges for almost every (t, x) (0, T ) Ω, and in L 2 (0, T, L 2 (Ω)) towards a couple (φ, ψ). (φ, ψ) P ɛ P 0 is a weak solution of. It s noteworthy that there is nothing like mass conservation, except asymptotically.

13 Uniform subdivision (t 0,..., t I ) of (0, T ) where t i = i t Uniform subdivision (x 0,..., x J ) of (0, 1) where x j = j x Finite difference : ˆψn i,j and ˆφ n+ 1 2 i,j Neumann conditions: ˆψ n i, 1 = ˆψ n i,1 and ˆψ n i,j+1 = ˆψ n i,j 1 Neumann conditions: ˆφn+ 1 2 i, 1 = ˆφ n+ 1 2 i,1 and ˆφ n+ 1 2 i,j+1 = ˆφ n+ 1 2 i,j 1 M = M I +1,J+1 (R) M ɛ = {(m i,j ) i,j M, i, j, m i,j ɛ}

14 ˆψ 0 i,j =0 Completely implicit for ˆφ n+ 1 2 : ˆφ n+ 1 2 i+1,j ˆφ n+ 1 2 i,j + σ2 ˆφ n+ 1 2 i,j+1 2 ˆφ n ˆφ n+ 1 2 i,j i,j 1 t 2 ( x) 2 = 1 σ 2 f (x j, ˆφ n+ 1 2 i,j ˆφ n+ 2 1 ( ) ut (x I,j =exp j ) σ 2 Completely implicit for ˆψ n+1 : ˆψ n i,j ) ˆφ n+ 1 2 i,j ˆψ i+1,j n+1 ˆψ i,j n+1 σ2 t 2 ˆψ i+1,j+1 n+1 2 ˆψ i+1,j n+1 + ˆψ i+1,j 1 n+1 ( x) 2 = 1 σ 2 f (x j, ˆφ n+ 1 2 n+1 i+1,j ˆψ i+1,j ) ˆψ n+1 i+1,j ˆψ n+1 0,j = m 0 (x j ) ˆφ n ,j

15 Well-posedness I Proposition (Well-posedness) ˆψ M 0, there is a unique solution ˆφ M to the following equation: ˆφ i+1,j ˆφ i,j t + σ2 2 ˆφ i,j+1 2 ˆφ i,j + ˆφ i,j 1 ( x) 2 = 1 σ 2 f (x j, ˆφ i,j ˆψ i,j ) ˆφ i,j with ˆφ ( ) ut (x I,j = exp j ) and the conventions ˆφ σ 2 i, 1 = ˆφ i,1, ˆφ i,j+1 = ˆφ i,j 1. Hence Φ d : ˆψ M 0 ˆφ M is well defined.

16 Uniform lower bound and Monotonicity Proposition (Uniform lower bound) ˆψ M 0, ˆφ = Φ d ( ˆψ) M ɛ for the same ɛ as in the continuous case. Proposition (Monotonicity) ˆψ 1 ˆψ 2 M 0, Φ d ( ˆψ 1 ) Φ d ( ˆψ 2 )

17 Well-posedness II Proposition (Well-posedness) Let s fix ɛ > 0 as above. ˆφ M ɛ, there is a unique solution ˆψ M to the following equation: ˆψ i+1,j ˆψ i,j t σ2 2 ˆψ i+1,j+1 2 ˆψ i+1,j + ˆψ i+1,j 1 ( x) 2 = 1 σ 2 f (x j, ˆφ i+1,j ˆψi+1,j ) ˆψ i+1,j with ˆψ 0,j = m 0(x j ) ˆφ 0,j and the conventions ˆψ i, 1 = ˆψ i,1, ˆψ i,j+1 = ˆψ i,j 1. Hence Ψ d : ˆφ M ɛ ˆψ M is well defined.

18 Positiveness and monotonicity Proposition (Positiveness) ˆφ M ɛ, ˆψ = Ψ d ( ˆφ) M 0 Proposition (Monotonicity) ˆφ 1 ˆφ 2 M ɛ, Ψ d ( ˆφ 1 ) Ψ d ( ˆφ 2 )

19 Monotonicity of the and limit behavior Theorem Assume that m 0 is bounded. The numerical verifies the following properties: ( ˆφ n+ 1 2 ) n is a decreasing sequence of M ɛ. ( ˆψ n ) n is an increasing sequence of M 0, bounded from above, independently of the subdivision. ( ˆφ n+ 1 2, ˆψ n ) n converges towards a couple ( ˆφ, ˆψ) M ɛ M 0.

20 Convergence of the I Definition of the norm Hypothesis m = (m i,j ) i,j M, m 2 = sup 0 i I 1 J + 1 We suppose that f, u T and m 0 are bounded. J j=0 m 2 i,j We also suppose that f is Lipschitz with respect to ξ (Lipschitz constant: K)

21 Convergence of the II Consistency errors η n+ 1 φ n i+1,j φn+ 2 1 i,j = i,j t η i,j n+1 ψ n+1 = t i+1,j ψn+1 i,j σ2 2 φ n+ 1 2 i,j = φ n+ 1 2 (ti, x j ) ψ n+1 i,j = ψ n+1 (t i, x j ) + σ2 2 φ n i,j+1 2φn+ 2 +φ n+ 1 2 i,j i,j 1 ( x) σ 2 f (x j,φn+ 2 i,j ψ n i,j )φn+ 1 2 i,j ψ i+1,j+1 n+1 2ψn+1 i+1,j +ψn+1 i+1,j 1 ( x) σ 2 f (x j,φn+ 2 i+1,j ψn+1 i+1,j )ψn+1 i+1,j

22 Convergence of the III Theorem (Stability bounds) ( ) 1 If t > 1 + K max e 2 u T σ 2 σ 2, ψ 2, then n N, C n+ 1 2, C n+1, D n+ 1, D n+1 such that: 2 ˆφ n+ 1 2 φ n+ 1 2 Cn+ 1 ˆψ n ψ n + D 2 n+ 1 η n ˆψ n+1 ψ n+1 C n+1 ˆφ n+ 1 2 φ n Dn+1 η n+1

23 Convergence of the IV Theorem (Convergence) Let s suppose that u T, m 0 and f are so that n N, φ n+ 1 2, ψ n C 1,2 ([0, T ] [0, 1]) and φ, ψ C 1,2 ([0, T ] [0, 1]) and still f a Lipschitz function with respect to ξ. Then: lim lim ˆφ n+ 1 2 φ = 0 t, x 0 n lim t, x 0 lim n ˆψ n+1 ψ = 0

24 A first framework People willing to live at the center, but not together. Ω = (0, 1), T = 2, σ = 1 f (x, ξ) = 16(x 1/2) min(5, max(0, ξ)) ( ( m 0 (x) = cos π 2x 2)) 3 2 u T (x) = 0 51 points in time and 51 points in space. Convergence after 7 iterations for n.

25 Solution φ Phi x t

26 Solution ψ Psi t x

27 Solution u u t x 0.4

28 Solution m m x t

29 Solution optimal control α = u alpha x t

30 A first framework People willing to live at x = 1 4 or x = 3 4 during the game and at the center at the end, but never together. Ω = (0, 1), T = 2, σ = 1 ( ( f (x, ξ) = 2 cos π 2x 2)) min(5, max(0, ξ)) m 0 (x) = 1, u T = 1 x(1 x) 2 51 points in time and 51 points in space. Convergence after 28 iterations for n.

31 Solution φ Phi x t

32 Solution ψ Psi t x

33 Solution u u t x

34 Solution m 1.00 m x t

35 Solution optimal control α = u alpha t x