Mean Field Games: Numerical Methods Y Achdou

Yves Achdou

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Niveau: Secondaire
Mean Field Games: Numerical Methods Y. Achdou October 24, 2011 0-0

• nash point


• theorem there exist


• sup ??rd


• ?∆xui ?h


• solution via convex programming


• numerical results

Sujets

• Nash Point

Informations

• Publié par: profil-shien-2012
• Nombre de lectures: 13
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Mean Field Games: Numerical Methods
Y. Achdou
October 24, 2011
0-0Content
A partial review on the theory of Lasry and Lions
Numerical schemes for the mean field games system
– Description of the scheme
– Existence, bounds, uniqueness
Numerical tests
The optimal planning problem
– Description of the scheme
– Existence of the solution via convex programming
– Numerical results
1I. Mean field games: some aspects of the theory of Lasry and Lions
ConsiderN identical players whose dynamics are
p
i i i
dX = 2dW dt;
t t
i i d
X =x 2 R :
0
1 N d
(W ;:::;W ) independent Brownian motions in R ,
t t
> 0,
i
The control of the playeri, i.e. is a bounded process assumed to be
1 N
adapted to (W ;:::;W ).
t t
For simplicity, all the functions used below are periodic with period 1 in
d
every direction. Let T be the unit torus of R .
2The cost of the playeri at timet is
i
J (t) =
0 0 2 3 1 2 3 1
Z
T
X X
1 1
i i i i
@ @ 4 5 A 4 5 A
E L(X ; ) + V (X ) ds + V (X )
j j
0
s s s T
X X
s
T
N 1 N 1
t
j=i j=i
V andV are operators which continuously map the set of probability
0
measures on T (endowed with the weak * topology) to a bounded
subset ofLip(T).
L is Lipschitz inx uniformly in bounded, and
L(x;)
lim inf = +1.
x
jj!1 jj
Introduce the Hamiltonian
d
H(x;p) = sup (p L(x;)); x2 T;p2 R :
d
2R
1
Assume thatH isC .
3
661 N
Nash equilibria: ( ;:::; ) is a Nash point, if 8i = 1;:::;N,
i 1 i 1 i i+1 N i 1 i 1 i i+1 N
J (t; ;:::; ; ; ;:::; )J (t; ;:::; ; ; ;:::; )
Theorem There existN functions (u (t;x ;:::;x )) such that
j 1 N j=1;:::;N
2 3
8
X X
>
@u @H 1
i
>
4 5 + u H(x ;r u ) (x ;r u )r u = V (x )
X i i x i j x j x i x i
> i j j j
@t @p N 1
<
j=i j=i
2 3
>
X
>
1
> 4 5
u (T; x ; : : : ; x ) = V (x ) i 1 N 0 x i
j
:
N 1
j=i
The feedbacks
@H
i
= (x ;r u )
i x i
i
@p
yield a Nash point.
In general, no uniqueness
4
666Assuming that the players have the same distributionm att = 0, and
0
passing to the limit asN !1, Lasry and Lions get the system of 2 PDEs
8
@u
>
<
+u H(x;ru) = V [m]; in (0;T) T;
@t

@m @H
>
:
m div m (x;ru) = 0; in (0;T) T;
@t @p
with the terminal and initial conditions
u(t =T) =V [m(t =T)]; and m(0;x) =m (x); in T;
0 0
wherem(t;) is the density of players at timet:
Z
m 0; m(t;x)dx = 1:
T
Remark The full justification of the passage to the limit is done in special
cases only.
5Some results on the MFG system
8
@u
>
> u +H(x;ru) =V [m]; in (0;T] T;
>
>
@t
>
>
>
>
@m @H
>
<
+m + div m (x;ru) = 0; in [0;T) T;
@t @p
()
Z
>
>
>
>
mdx = 1; m> 0 in T;
>
>
>
> T
>
:
u(t = 0) =V [m(t = 0)]; m(t =T) =m ;
0
Remark Note the special structure of the system:
1. forward/backward w.r.t. time.
2. the operator in the Fokker-Planck equation is the adjoint of the
linearized version of the operator in the HJB equation.
3. coupling: viaV [m] in the HJB equation and@ H(t;x;ru) in the
p
Fokker Planck equation, and possibly via the initial condition onu.
6Theorem (Lasry-Lions) : Existence for (*)
1) If > 0 and
V andV are operators which continuously map the set of probability
0
measures on T (endowed with the weak * topology) to a bounded
subset ofLip(T), i.e. nonlocal smoothing operators,
d
H is smooth on T R and