An equilibrium mean field games model of transaction volumes Min Shen, Gabriel Turinici CEREMADE, Universit´ e Paris Dauphine Graz, Oct 10th-14th 2011 Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 1 / 59
Outline Motivation and introduction to Mean Field Games (MFG) 1 Mathematical objects: SDEs, Ito, Fokker-Planck 2 Optimal control theory: gradient and adjoint 3 Theoretical results of Lasry-Lions 4 Some numerical approaches 5 General monotonic algorithms (J. Salomon, G.T.) 6 Related applications: bi-linear problems Framework Construction of monotonic algorithms Technology choice modelling (A. Lachapelle, J. Salomon, G.T.) 7 The model Numerical simulations Liquidity source: heterogenous beliefs and analysis costs 8 Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 2 / 59
Mean field games: introduction • MFG = model for interaction among a large number of agent / players ... not particles. An agent can decide, based on a set of preferences and by acting on parameters ( ... control theory). Note: in standard rumor spreading (or opinion making) modeling agent is supposed to be a mechanical black-box, not the case here. This situation is included as particular case. • distinctive properties: the existence of a collective behavior (fashion trends, financial crises, real estates valuation, etc.). One agent by itself cannot influence the collective behavior, it only optimizes its own decisions given the environmental situation. References: Lasry Lions CRAS notes (2006), Lions online course at College de France. Further references latter on. Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 3 / 59
Mean field games: introduction • Nash equilibrium: a game of N players is in a Nash equilibrium if, for any player j supposing other N − 1 remain the same, there is no decision of the player j that can improve its outcome. • MFG = Nash equilibrium equations for N → ∞ . All players are the same. • Agent follows an evolution equation involving some controlling action. Its decision criterion depend on the others, more precisely on the density of other players. • Will consider here stochastic diff. equations, but deterministic case is a particular situation and can be treated. Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 4 / 59
Mathematical framework of MFG What follows is the most simple model that shows the properties of MFG models. Cf. references for more involved modeling. X x t = the characteristics at time t of a player starting in x at time 0. It evolves with SDE: dX x t = α ( t , X x t ) dt + σ dW x t , X x 0 = x (1) • α ( t , X x t ) = control can be changed by the agent/ player. • independent brownians (!) • m ( t , x ) = the density of players at time t and position x ∈ E ; E is the state space. Optimization problem of the agent: fixed T = finite horizon �� T � L ( X x t , α ( t , X x t )) + V ( X x t ; m ( t , · )) dt + V 0 ( X x inf T ; m ( T , · )) (2) α E 0 static case (infinite horizon): � 1 �� T � � L ( X x t , α ( t , X x t )) + V ( X x + V 0 ( X x inf α lim inf t ; m ( t , · )) dt T ; m ( T , · )) T →∞ E T 0 (3) Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 5 / 59
Mathematical framework of MFG: examples Example: choice of a holiday destination. Particular case: deterministic, no dependence on the initial condition, no dependence on the control. Each individual minimizes distance to an ideal destination and a term depending on the presence of others: V 0 ( y ; m ) = F 0 ( y ) + F 1 ( m ). Question: what is the solution ? X x T will be chosen as the minimum of y �→ F 0 ( y ) + F 1 ( m ( y )). Then m is the distribution of such X x T . COUPLING between m and X x T !! Particular case: F 0 ( y ) = y 2 on R . Origin is the most preferred point for all individuals, distance increases slowly in neighborhood, fast outside. Take F 1 ( m ) = cm . Modelization: c > 0 = crowd aversion, c < 0 = propensity to crowd. Remark: all points y in the the support of m have to be minimums of V 0 ! Solution: c > 0: semi-circular distribution m ( y ) = ( λ − y 2 ) + c c < 0: Dirac masses at minimum of F 0 . Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 6 / 59
Outline Motivation and introduction to Mean Field Games (MFG) 1 Mathematical objects: SDEs, Ito, Fokker-Planck 2 Optimal control theory: gradient and adjoint 3 Theoretical results of Lasry-Lions 4 Some numerical approaches 5 General monotonic algorithms (J. Salomon, G.T.) 6 Related applications: bi-linear problems Framework Construction of monotonic algorithms Technology choice modelling (A. Lachapelle, J. Salomon, G.T.) 7 The model Numerical simulations Liquidity source: heterogenous beliefs and analysis costs 8 Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 7 / 59
Mathematical objects: SDEs Brownian motion models a very irregular motion (but continuous). Mathematically it is a set of random variables indexed by time t , denoted W t , with: • W 0 = 0 with probability 1 • a.e. t �→ W t ( ω ) is continuous on [0 , T ] • for 0 ≤ s ≤ t ≤ T the increment W ( t ) − W ( s ) is a random normal variable of mean 0 and variance t − s : W ( t ) − W ( s ) ≈ √ t − s N (0 , 1) ( N (0 , 1) is the standard normal variable) • for 0 ≤ s < t < u < v ≤ T the increments W ( t ) − W ( s ) W ( v ) − W ( u ) are independent. 2 πλ e − x 2 1 2 λ ; W t + dt − W t has as law Recall normal density N (0 , λ ) is √ √ dt N (0 , 1) (of order dt 1 / 2 , cf. Ito formula). Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 8 / 59
Martingales (Ω , A , P ) = probability space, ( A t ) t ≥ 0 filtration. An adapted family ( M t ) t ≥ 0 of integrable r.v. (i.e. E | M t | < ∞ ) is martingale if for all s ≤ t : E ( M t |A s ) = M s . Thus E ( M t ) = E ( M 0 ). Theorem t − t, e σ W t − σ 2 2 t are also Let ( W t ) t ≥ 0 be a Brownian motion, then W t , W 2 martingales. Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 9 / 59
Ito integral � T We want to define 0 f ( t , ω ) dW t . � T For 0 h ( t ) dt Riemann sums � j h ( t j )( t j +1 − t j ) converge to the Riemann integral when the division t 0 = 0 < t 1 < t 2 < ... < t N = T of [0 , T ] becomes finer. For the Riemann-Stiltjes integral we can replace dt by increments of a � bounded variation function g ( t ) and obtain f ( t ) dg ( t ) Similarly one can work with Ito sums � N − 1 j =0 h ( t j )( W t j +1 − W t j ) or j =0 h ( t j + t j +1 Stratonovich � N − 1 )( W t j +1 − W t j ) both are the same for 2 deterministic function h . Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 10 / 59
Ito integral Example: h = W , t j = j · dt . Ito: N − 1 N − 1 � � h ( t j )( W t j +1 − W t j ) = W t j ( W t j +1 − W t j ) (4) j =0 j =0 N − 1 = 1 � W 2 t j +1 − W 2 t j − ( W t j +1 − W t j ) 2 (5) 2 j =0 N − 1 = 1 − 1 � � W 2 T − W 2 � ( W t j +1 − W t j ) 2 . (6) 0 2 2 j =0 j =0 ( W t j +1 − W t j ) 2 has average Ndt = T and variance of � N − 1 The term 1 2 � � order dt so the limit will be 1 W 2 T − T . 2 � T � � 0 W t dW t = 1 W 2 Thus T − T ; in particular the non-martingale 2 (previsible) part of W 2 t will be t . Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 11 / 59
Ito integral Stratonovich: N − 1 N − 1 h ( t j + t j +1 � � )( W t j +1 − W t j ) = W tj + tj +1 ( W t j +1 − W t j ) (7) 2 2 j =0 j =0 N − 1 � W t j + W t j +1 � � + ∆ Z j ( W t j +1 − W t j ) (8) 2 j =0 Here ∆ Z j is a r.v. independent of W t j , of null average and variance dt / 4. Sum will be 1 2 W 2 T . Stratonovich is also limit of N − 1 h ( t j ) + h ( t j +1 ) � ( W t j +1 − W t j ) . (9) 2 j =0 Min Shen, Gabriel Turinici (CEREMADE, Universit´ An equilibrium mean field games model of transaction volumes e Paris Dauphine ) Graz, Oct 10th-14th 2011 12 / 59
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