The Master Equation in a Bounded Domain with Neumann Conditions The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi Università di Roma “Tor Vergata” Université Paris-Dauphine Two Days Online Workshop on Mean Field Games 18/06/2020 The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Mean Field Games Theory is a branch of mathematics introduced by J.-M. Lasry and P.-L. Lions in 2006, in order to describe Nash equilibria in differential games with infinitely many agents. The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Mean Field Games Theory is a branch of mathematics introduced by J.-M. Lasry and P.-L. Lions in 2006, in order to describe Nash equilibria in differential games with infinitely many agents. In non-cooperative differential games with N players, each agent chooses his own strategy in order to minimize a certain cost functional. The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Mean Field Games Theory is a branch of mathematics introduced by J.-M. Lasry and P.-L. Lions in 2006, in order to describe Nash equilibria in differential games with infinitely many agents. In non-cooperative differential games with N players, each agent chooses his own strategy in order to minimize a certain cost functional. Dynamic of the player i , 1 ≤ i ≤ N : √ � dX i t = b ( X i t , α i 2 σ ( X i t ) dB i t ) dt + t , X i t 0 = x i 0 . The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Mean Field Games Theory is a branch of mathematics introduced by J.-M. Lasry and P.-L. Lions in 2006, in order to describe Nash equilibria in differential games with infinitely many agents. In non-cooperative differential games with N players, each agent chooses his own strategy in order to minimize a certain cost functional. Dynamic of the player i , 1 ≤ i ≤ N : √ � dX i t = b ( X i t , α i 2 σ ( X i t ) dB i t ) dt + t , X i t 0 = x i 0 . Here, x i 0 ∈ R d , α i t is the control, b and σ are the drift term and the diffusion matrix and ( B t ) i are independent d -dimensional Brownian motions. The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Cost for the player i : �� T � J N � L ( s , X i s , α i s ) + F N i ( s , X s ) � ds + G N i ( t 0 , x 0 , α · ) = E i ( X T ) , t 0 where F N and G N are the cost functions of the player i and L is the i i Langrangian cost for the control. The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Cost for the player i : �� T � J N � L ( s , X i s , α i s ) + F N i ( s , X s ) � ds + G N i ( t 0 , x 0 , α · ) = E i ( X T ) , t 0 where F N and G N are the cost functions of the player i and L is the i i Langrangian cost for the control. We say that a control α ∗ · provides a Nash equilibrium if, for all controls α · and for all i we have J N i ( t 0 , x 0 , α ∗ · ) ≤ J N i ( t 0 , x 0 , α i , ( α ∗ j ) j � = i ) , i.e., each player chooses his optimal strategy, if we “freeze" the other players’ strategies. The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Cost for the player i : �� T � J N � L ( s , X i s , α i s ) + F N i ( s , X s ) � ds + G N i ( t 0 , x 0 , α · ) = E i ( X T ) , t 0 where F N and G N are the cost functions of the player i and L is the i i Langrangian cost for the control. We say that a control α ∗ · provides a Nash equilibrium if, for all controls α · and for all i we have J N i ( t 0 , x 0 , α ∗ · ) ≤ J N i ( t 0 , x 0 , α i , ( α ∗ j ) j � = i ) , i.e., each player chooses his optimal strategy, if we “freeze" the other players’ strategies. Value function : i ( t 0 , x 0 , α ∗ ) . v N i ( t 0 , x 0 ) = J N The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Using Ito’s formula and the dynamic programming principle, one can prove that v N solves the so-called Nash system: i � − ∂ t v N tr ( a ( x j ) D 2 x j x j v N + H ( x i , D x i v N i ( t , x ) − i ( t , x )) i ( t , x )) j � H p ( x j , D x j v N j ( x )) · D x j v N = F N (1) + i ( t , x ) i ( x ) , j � = i v N i ( T , x ) = G N i ( x ) , for ( t , x ) ∈ [0 , T ] × R Nd . Here H is the Hamiltonian of the system and a = σσ ∗ . The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction Using Ito’s formula and the dynamic programming principle, one can prove that v N solves the so-called Nash system: i � − ∂ t v N tr ( a ( x j ) D 2 x j x j v N + H ( x i , D x i v N i ( t , x ) − i ( t , x )) i ( t , x )) j � H p ( x j , D x j v N j ( x )) · D x j v N = F N (1) + i ( t , x ) i ( x ) , j � = i v N i ( T , x ) = G N i ( x ) , for ( t , x ) ∈ [0 , T ] × R Nd . Here H is the Hamiltonian of the system and a = σσ ∗ . The idea of Lasry and Lions is to simplify the Nash system, with suitable symmetry conditions for the agents and their dynamics, for N ≫ 1. This leads us to the study of the so-called Mean Field Games System . The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction We suppose that F N and G N are of this form: i i F N i ( x ) = F ( x i , m N , i G N i ( t , x ) = G ( x i , m N , i x ) , x ) , The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction We suppose that F N and G N are of this form: i i F N i ( x ) = F ( x i , m N , i G N i ( t , x ) = G ( x i , m N , i x ) , x ) , where m N , i 1 � = δ x j , with δ x the Dirac function at x . x N − 1 j � = i The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction We suppose that F N and G N are of this form: i i F N i ( x ) = F ( x i , m N , i G N i ( t , x ) = G ( x i , m N , i x ) , x ) , where m N , i 1 � = δ x j , with δ x the Dirac function at x . x N − 1 j � = i Heuristically, when N → + ∞ , the Mean Field Games system takes the following form: − ∂ t u − tr ( a ( x ) D 2 u ) + H ( x , Du ) = F ( x , m ) , ∂ 2 ∂ t m − � ij ( a ij ( x ) m ) − div ( mH p ( x , Du )) = 0 , (2) i , j m (0) = m 0 , u ( T ) = G ( x , m ( T )) , with a Hamilton-Jacobi-Bellman equation for u coupled with a Fokker-Planck equation for the law of the population m . The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction We suppose that F N and G N are of this form: i i F N i ( x ) = F ( x i , m N , i G N i ( t , x ) = G ( x i , m N , i x ) , x ) , where m N , i 1 � = δ x j , with δ x the Dirac function at x . x N − 1 j � = i Heuristically, when N → + ∞ , the Mean Field Games system takes the following form: − ∂ t u − tr ( a ( x ) D 2 u ) + H ( x , Du ) = F ( x , m ) , ∂ 2 ∂ t m − � ij ( a ij ( x ) m ) − div ( mH p ( x , Du )) = 0 , (2) i , j m (0) = m 0 , u ( T ) = G ( x , m ( T )) , with a Hamilton-Jacobi-Bellman equation for u coupled with a Fokker-Planck equation for the law of the population m . In order to describe this limit problem, Lasry and Lions introduced the Master Equation , which summarizes the MFG system in a unique equation. The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
The Master Equation in a Bounded Domain with Neumann Conditions Introduction We consider the solution ( u , m ) of (2) with m ( t 0 ) = m 0 ∈ P ( R d ), where P ( R d ) is the set of Borel probability measures, and we define U : [0 , T ] × R d × P ( R d ) → R , U ( t 0 , x , m 0 ) = u ( t 0 , x ) , (3) provided MFG system has a unique solution. The Master Equation in a Bounded Domain with Neumann Conditions Michele Ricciardi
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