math.PR 2026-02-04 0

On the Mean-Field limit of diffusive games through the master equation: $L^{\infty}$ estimates and extreme value behavior

We consider an $N$-player game where the states of the players evolve with time as Stochastic Differential Equations (SDEs) with interaction only in the drift terms. Each player controls the drift of the SDE satisfied by her state process, aiming to …

Authors: Erhan Bayraktar, Nikolaos Kolliopoulos

On the Mean-Field limit of diffusiv e games through the master equation: L ∞ estimates and extreme v alue b eha vior. Erhan Ba yraktar ∗ † Nik olaos K olliop oulos ‡§ ¶ F ebruary 5, 2026 Abstract W e consider an N -pla yer game where the states of the play ers evolv e with time as Sto c hastic Differen tial Equations (SDEs) with interaction only in the drift terms. Each pla yer controls the drift of the SDE satisfied by her state pro cess, aiming to minimize the exp ected v alue of a cost that dep ends on the paths of the play er’s state and the empirical measure of the states of all the pla yers until a terminal time. When N → ∞ , previous works ha ve established Central Limit Theorems and Large Deviation Principles for the state pro cesses when the game is in Nash Equilibrium (the Nash states), b y using the Master Equation to construct approximations of those processes that ev olve with time as SDEs with classical Mean-Field in teraction. Sta ying in this framework, we improv e an existing L 1 estimate for the total error of approximating all the Nash states to an L ∞ one, and w e also establish the N → ∞ asymptotic behavior of the upp er order statistics of the Nash states. The latter initiates the developmen t of an Extreme V alue Theory for Stochastic Differential Games. 1 In tro duction In this pap er we establish an L ∞ estimate for the error of an efficien t approximation of a large Sto c hastic Differential Game under Nash equilibrium, and also a fundamen tal result of Extreme V alue Theory for the states of the play ers when their n um b er grows tow ards infinity . W e work with a game of N play ers on some filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) , where for each i ∈ { 1 , 2 , . . . , N } , the ev olution in time of the i -th play er’s state is mo deled as a diffusiv e pro cess X i,N ,v N = ( X i,N ,v N t ) t ∈ [0 ,T ] that satisfies X i,N ,v N t = X i 0 + Z t 0 b  X i,N ,v N s , µ N ,v N s , v i,N s  ds + σ W i t , t ≥ 0 , (1.1) where µ N ,v N t denotes the empirical measure of the N play ers: µ N ,v N t = 1 N N X ℓ =1 δ X ℓ,N,v N t , (1.2) ∗ Department of Mathematics, Universit y of Michigan, erhan@umich.edu . † F unded in part by the NSF though DMS-2106556 and by the Susan M. Smith Chair. ‡ Department of Mathematics, Universit y of Michigan, nkolliop@umich.edu . § Department of Mathematics and Statistics, University of Cyprus, Kolliopoulos.Nikolaos@ucy.ac.cy . ¶ F unded by the NSF through DMS-2406232. 1 and v i,N = ( v i,N t ) t ∈ [0 ,T ] is an F t -adapted pro cess which represents the i -th play er’s strategy and tak es v alues in some set V ⊂ R - we write V N for the set from which each v i,N is pic ked. In the ab o ve, ( X i 0 ) i ∈ N and ( W i ) i ∈ N are indep enden t sequences of indep enden t F 0 -measurable random v ariables with law µ 0 ∈ P ( R ) and indep enden t F t -adapted standard Bro wnian motions resp ectiv ely , while b : R × P ( R ) × V 7→ R is a sufficiently regular function and σ is a p ositiv e constant. Here, P ( E ) denotes the set of probability measures on a Banach space equipp ed with the 1-W asserstein metric: W 1 ( m 1 , m 2 ) = inf m 1 , 2 ( · ,E )= m 1 m 1 , 2 ( E , · )= m 2 Z E × E k x − y k E m 1 , 2 ( dx, dy ) In b oth X i,N ,v N t and µ N ,v N t , the sup erscript v N denotes the dep endence of the states and the empirical measure on the vector of strategy pro cesses v N = ( v 1 ,N , v 2 ,N , . . . , v N ,N ) ∈ V N N . In this game, the i -th play er w an ts to pic k v i,N in a wa y that minimizes her exp ected cost: J i,N  v N  = E " Z T 0 f  X i,N ,v N s , µ N ,v N s , v i,N s  ds + g  X i,N ,v N T , µ N ,v N T  # (1.3) where f : R × P ( R ) × V 7→ R and g : R × P ( R ) 7→ R are the running and terminal cost functions resp ectiv ely . Then, tw o problems that naturally o ccur are: (a) the search of Nash equilibria, i.e. c hoices v N , ∗ = ( v 1 ,N , ∗ , v 2 ,N , ∗ , . . . , v N ,N, ∗ ) for v N that satisfy: J i,N  v N , ∗  ≤ J i,N  v 1 ,N , ∗ , v 2 ,N , ∗ , . . . , v i − 1 ,N , ∗ , v i,N , v i +1 ,N , ∗ , . . . , v N ,N, ∗  (1.4) for any i ∈ { 1 , 2 , . . . , N } and any pro cess v i,N ∈ V N ; (b) the analysis of the system ( 1.1 ) under a Nash equilibrium v N , ∗ where the i -th play er picks v i,N = v i,N , ∗ for each i ∈ { 1 , 2 , . . . , N } , including the study of this regime as N → ∞ to understand play er dynamics in large p opulation games. The study of a systemic risk mo del that falls in to the ab o ve framework can be found in [ 10 ], while v ariations of our setup arise in other financial con texts as w ell, see e.g [ 40 , 11 , 31 , 22 , 41 , 50 ]. In this w ork we restrict to the case where v N , ∗ is a closed-lo op Nash equilibrium, in which case each v i,N , ∗ is pick ed from a set V N of pro cesses that are deterministic functions of the states of the play ers, i.e: V N = { v i,N : v i,N t = h i,N t ( X 1 ,N ,v N t , X 2 ,N ,v N t , . . . , X N ,N,v N t ) for h i,N t : R N 7→ V , ∀ t ∈ [0 , T ] } . Sufficien t conditions for the existence of a closed-lo op Nash equilibrium v N , ∗ for the Sto chastic Differen tial Game ( 1.1 ) - ( 1.2 ) can b e found in [ 7 ], in whic h case eac h v i,N , ∗ admits the form v i,N , ∗ t = ˆ v  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , U i,N x i  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  , (1.5) where for any x ∈ R , y ∈ R and µ ∈ P ( R ) w e write ˆ v ( x, µ, y ) = argmin v ∈ V { b ( x, µ, v ) y + f ( x, µ, v ) } , (1.6) and U 1 ,N , U 2 ,N , . . . , U N ,N are the v alue functions of the N pla y ers which satisfy the system of nonlinear PDEs 0 = U i,N t ( t, x ) + min v ∈ V  b  x i , µ N x , v  U i,N x i ( t, x ) + f  x i , µ N x , v  2 + N X j =1 j  = i U i,N x j ( t, x ) · b  x j , µ N x , ˆ v  x j , µ N x , U j,N x j ( t, x )  + σ 2 2 N X j =1 U i,N x j x j ( t, x ) , (1.7) along with terminal conditions U i,N ( T , x ) = g  x i , µ N x  , (1.8) in which µ N x := 1 N P N ℓ =1 δ x ℓ for x = ( x 1 , x 2 , . . . , x N ) . Thus, the evolution of the states of the N pla yers under Nash equilibrium is given b y X i,N ,v N, ∗ t = X i 0 + Z t 0 ˆ b  X i,N ,v N, ∗ s , µ N ,v N, ∗ s , U i,N x i  s, X 1 ,N ,v N, ∗ s , X 2 ,N ,v N, ∗ s , . . . , X N ,N,v N, ∗ s  ds + σ W i t (1.9) for t ≥ 0 , where w e write: ˆ b ( x, m, y ) = b ( x, m, ˆ v ( x, m, y )) . (1.10) It is generally exp ected that the N → ∞ asymptotics of N -play er Sto c hastic Differential Games are go verned by a prop ert y known as Propagation of Chaos, as happ ens with uncontrolled diffu- sions with standard Mean-Field interaction [ 44 , 48 , 24 ]. In the game framework, this prop ert y is summarized as follows: (i) The empirical measure µ N ,v N, ∗ := 1 N N X ℓ =1 δ X ℓ,N,v N, ∗ on C [0 , T ] satisfies µ N ,v N, ∗ → L ( X v ∗ ) (1.11) w eakly as N → ∞ , where X v = ( X v t ) t ∈ [0 ,T ] denotes the state pro cess of a represen tative pla yer in the limiting regime as N → ∞ - which is frequently called "Mean-Field Game" - under an F t -adapted strategy pro cess: v = ( v t ) t ∈ [0 ,T ] ∈ V ∞ := { v : v t = h t ( X v t ) for h t : R 7→ V , ∀ t ∈ [0 , T ] } , L ( X v ) b eing the law of the C [0 , T ] -v alued random path X v = ( X v t ) t ∈ [0 ,T ] and v ∗ = ( v ∗ t ) t ∈ [0 ,T ] b eing the optimal strategy process for the representativ e play er. (ii) F or any fixed k ∈ N it holds that:  X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X k,N ,v N, ∗ t  →  X 1 ,v ∗ t , X 2 ,v ∗ t , . . . , X k,v ∗ t  (1.12) in distribution as N → ∞ , with X 1 ,v ∗ , X 2 ,v ∗ , . . . , X k,v ∗ b eing indep enden t copies of X v ∗ . F or a standard Bro wnian motion ( W t ) t ∈ [0 ,T ] and a random v ariable X 0 with la w µ 0 , the ev olution of ( X v ∗ , v ∗ ) is giv en b y X v ∗ t = X 0 + Z t 0 b  X v ∗ s , L ( X v ∗ s ) , v ∗ s  ds + σ W t 3 v ∗ = argmin v ∈V ∞ E " Z T 0 f  X v s , L ( X v ∗ s ) , v s  ds + g  X v T , L ( X v ∗ T )  # . (1.13) The deriv ation of Propagation of Chaos for a class of Sto c hastic Differential Games can b e found in [ 7 ]. As shown in [ 39 ] through a simple example of uncontrolled diffusions, ( 1.12 ) is generally not true for k = N in any sense and, for that reason, the asymptotic relation H N  X 1 ,N ,v N, ∗ , X 2 ,N ,v N, ∗ , . . . , X k,N ,v N, ∗  ∼ H N  X 1 ,v ∗ , X 2 ,v ∗ , . . . , X k,v ∗  as N → ∞ (1.14) holds in some sense only for a very limited class of functions H N with N argumen ts; ( 1.11 ) is essen tially a Law of Large Numbers for the interacting v ariables X i,N ,v N, ∗ , with the corresponding Cen tral Limit Theorems and Large Deviation Principles b eing also av ailable in the literature [ 17 , 18 ], and all these results can be seen as ( 1.14 ) for appropriate c hoices of H N . F or example, the La w of Large Num b ers ( 1.11 ) is precisely ( 1.14 ) for H N ( x 1 , x 2 , . . . , x N ) = 1 N P N i =1 δ x i , since the left-hand side of the latter is precisely µ N ,v N, ∗ and its right-hand side conv erges to L ( X v ∗ ) as N → ∞ by the La w of Large Num bers for indep enden t and iden tically distributed (i.i.d.) random v ariables. It is also shown in [ 7 ] that the N → ∞ limit of our setup is gov erned by the Master equation, whic h is the follo wing nonlinear P D E on [0 , T ] × R × P ( R ) : 0 = U t ( t, x, m ) + min v ∈ V { b ( x, m, v ) U x ( t, x, m ) + f ( x, m, v ) } + σ 2 2 U xx ( t, x, m ) + Z R U m ( t, x, m, z 1 ) × b ( z 1 , m, ˆ v ( z 1 , m, U x ( t, z 1 , m ))) m ( dz 1 ) + σ 2 2 Z R U mz ( t, x, m, z 1 ) m ( dz 1 ) (1.15) under the terminal condition U ( T , x, m ) = g ( x, m ) , with U m b eing our notation for the deriv ative with resp ect to m ∈ P ( R ) whic h is defined in e.g [ 8 , 7 , 18 ] (all notations are introduced in Section 2). In some sense, ( 1.15 ) plays the role of the N → ∞ limit of ( 1.7 ), and the analogue of ( 1.5 ) in the limit that gives the optimal strategy pro cess v ∗ is: v ∗ t = ˆ v  X v ∗ t , L ( X v ∗ t ) , U x  t, X v ∗ t , L ( X v ∗ t )  , (1.16) so the controlled McKean-Vlaso v SDE ( 1.13 ) that describ es the evolution of the representativ e pla yer in the N → ∞ limit of the game b ecomes: X v ∗ t = X 0 + Z t 0 ˆ b  X v ∗ s , L ( X v ∗ s ) , U x  s, X v ∗ s , L ( X v ∗ s )  ds + σ W t . (1.17) T o obtain a Central Limit Theorem and a Large Deviations Principle for the Nash states X i,N ,v N, ∗ t in [ 17 ] and [ 18 ] resp ectively , the authors utilize that when N is large, the system ( 1.9 ) satisfied b y the Nash states X i,N ,v N, ∗ is sufficien tly close to the following simpler system: ¯ X i,N t = X i 0 + Z t 0 ˆ b  ¯ X i,N s , ¯ µ N s , U x  s, ¯ X i,N s , ¯ µ N s  ds + σ W i t , ¯ µ N t = 1 N N X ℓ =1 δ ¯ X ℓ,N t (1.18) 4 for t ∈ [0 , T ] . Sp ecifically , ( 1.18 ) describ es a standard Mean-Field System that is easier to handle compared to ( 1.9 ), and its components ¯ X i,N t constitute a significantly b etter appro ximation of the Nash states X i,N ,v N, ∗ compared to the i.i.d. processes X i,v ∗ . The latter allow ed the authors of [ 17 , 18 ] to obtain the strong L 1 (Ω) -estimate      N X i =1 ( ˆ b  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , U i,N x i  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  − ˆ b  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , U x  t, X i,N ,v N, ∗ t , µ N ,v N, ∗ t  ) 2      L 1 (Ω) ≤ C N , (1.19) whic h is a O ( N − 1 ) b ound for the relative entrop y that arises when we apply Girsanov’s theorem to transform the law of ( X 1 ,N ,v N, ∗ , X 2 ,N ,v N, ∗ , . . . , X N ,N,v N, ∗ ) into that of ( ¯ X 1 ,N t , ¯ X 2 ,N t , . . . , ¯ X N ,N t ) , and as N → ∞ , it is the fast deca y of this entrop y that allows for existing asymptotic results for the standard Mean-Field System of the pro cesses ¯ X i,N to b e extended to the Nash states X i,N ,v N, ∗ , including a Central Limit Theorem and a Large Deviations Principle. More generally , using Pinsker’s inequality and the entrop y b ound ( 1.19 ), the deriv ation of the asymptotic relation ( 1.14 ) reduces to showing that H N  ¯ X 1 ,N t , ¯ X 2 ,N t , . . . , ¯ X N ,N t  ∼ H N  X 1 ,v ∗ t , X 2 ,v ∗ t , . . . , X k,v ∗ t  as N → ∞ (1.20) whenev er H N is simply measurable and b ounded. Then, since ( 1.17 ) is the Mck ean-Vlaso v SDE that captures the N → ∞ behavior of both systems ( 1.9 ) and ( 1.18 ), the latter is actually the reduction of the establishment of an arbitrary propagation of chaos prop ert y for the Nash states to obtaining the same prop ert y for a standard system of uncontrolled diffusions with Mean-Field in teraction in the drifts. Our first contribution to the ab o v e framew ork is the improv ement of the L 1 (Ω) -estimate ( 1.19 ) where the L 1 (Ω) norm is replaced by the L ∞ (Ω) norm. A p oten tial application of this result could b e the numerical simulation of large Sto c hastic Differential Games under Nash equilibrium, since it tells us that if we simulate the classical Mean-Field System ( 1.18 ) instead, the additional error will certainly not exceed a deterministic threshold of order O ( N − 1 ) . It should b e mentioned that it could b e possible to make the total error negligible in that case, as there is a growing literature on the developmen t of new metho ds for the numerical simulation of Mean-Field Systems like ( 1.18 ) (see e.g [ 34 , 29 , 35 , 33 ] for the Random Batc h Metho d and [ 36 , 43 ] for tw o applications). T o improv e ( 1.19 ), we obtain BSDE estimates similar to those in [ 17 , Pro of of Theorem 4.2], using, how ever, the deriv ativ es of the solutions to the PDEs ( 1.7 ) and ( 1.15 ) instead of the solutions themselves. The required assumptions include b oundedness of the deriv ativ es of the functions U and U i,N that are hard to verify , but we provide an example which sho ws that b oundedness of the deriv ativ es of the terminal condition g ( x, m ) is sufficient in some simple settings. Our second result in this work is to extend the list of asymptotic results that ha v e already b een established for the Nash states X i,N ,v N, ∗ as N → ∞ , whic h con tains a Cen tral Limit Theorem and a Large Deviations Principle, b y also obtaining a fundamental result of Extreme V alue Theory . The probabilistic results of classical Extreme V alue Theory (EVT) concern indep endent random observ ations Z 1 , Z 2 , . . . , Z N with some common cumulativ e distribution function F : R 7→ [0 , 1] , and the most fundamental question is whether there exist deterministic sequences ( a N ) N ∈ N and 5 ( b N ) N ∈ N suc h that the normalized maximum max i ≤ N Z i − b N a N = Z i N 1 − b N a N con verges weakly as N → ∞ , with the limiting distribution b eing non-degenerate. When this is the case, the limiting cumulativ e distribution function admits the form G γ ( ax + b ) for a, b, γ ∈ R , where w e define: G γ ( x ) := e − (1+ γ x ) − 1 γ for 1 + γ x > 0 . It is then said that the distribution of the v ariables Z i b elongs to the domain of attraction of the extreme v alue distribution G γ ( x ) , with the parameter γ called "the extreme v alue index", and we t ypically hav e the join t conv ergence of multiple normalized upp er order statistics. The latter means that if we sort our random v ariables as Z i N 1 ≥ Z i N 2 ≥ . . . ≥ Z i N N , for any fixed k ∈ N w e ha v e Z i N 1 − b N a N , Z i N 2 − b N a N t , . . . , Z i N k − b N a N ! →  ( E 1 ) − γ − 1 γ , ( E 1 + E 2 ) − γ − 1 γ , · · · , ( E 1 + E 2 + · · · + E k ) − γ − 1 γ  (1.21) w eakly as N → ∞ , where E 1 , E 2 , . . . , E k are i.i.d. standard exp onential random v ariables. More- o ver, when the distribution of the v ariables Z i b elongs to the domain of attraction of the extreme v alue distribution, the intermediate order statistics Z i N k N with k N → ∞ , k N N → 0 as N → ∞ are t ypically asymptotically normal, meaning that: Z ( k N ) − ˆ b N ˜ a N ∼ N (0 , 1) as N → ∞ , (1.22) for some deterministic normalizing sequences ( ˜ a N ) N ∈ N and ( ˆ b N ) N ∈ N whic h are different from ( a N ) N ∈ N and ( b N ) N ∈ N . W e refer to [ 14 , Chapters 1 and 2] for a deep er study of probabilistic EVT. The main use of the ab ov e asymptotic results is: (a) the construction of consistent statistical estimators for γ , a N , ˜ a N , b N and ˆ b N whic h are functions of m ultiple intermediate order statistics Z i N k N ; (b) the estimation of tail probabilities 1 − F ( z ) for large z after γ , a N , ˜ a N , b N and ˆ b N ha ve b een estimated. Those techniques are particularly useful when z > max i ≤ N Z i , in which case the simple estimation 1 − F ( z ) ≈ 1 N N X i =1 1 { Z i >z } is not effective due to the right-hand side b eing equal to 0 , which can lead to the incorrect conclusion that a rare but highly impactful even t can nev er o ccur. W e refer to [ 14 , Chapters 3 and 4] for an in tro duction to statistical EVT, see also [ 46 , 30 , 49 ] for the earliest works in this area, [ 16 , 20 , 12 , 26 , 28 , 25 , 5 , 27 , 51 , 15 , 6 , 45 ] for relev an t developmen ts ov er the last 40 years, and [ 19 , 4 , 1 , 32 , 2 , 23 ] for some more recent works with more contemporary metho ds. The weak conv ergence ( 1.21 ) has already b een extended to a setup of N uncontrolled diffusions { X 1 ,N , X 2 ,N , . . . , X 1 ,N } with a standard Mean-Field interaction in the drifts - i.e. interaction through the direct dep endence of the drifts on the systemic empirical measure - where for some fixed t ≥ 0 we hav e Z i = X i,N t for all i ∈ { 1 , 2 , . . . , N } , see [ 37 ] for the k = 1 case and [ 38 ] for the extension to arbitrary fixed 6 k ∈ N . Here we further extend ( 1.21 ) to our diffusive game setup under Nash equilibrium, where for some fixed t ≥ 0 we hav e Z i = X i,N ,v N, ∗ t for i ∈ { 1 , 2 , . . . , N } . While this is exp ected to op en the wa y for the prediction of extreme v alues in contin uously evolving diffusiv e p opulations with a game structure, the extension of ( 1.22 ) which concerns intermediate order statistics to our setup and the developmen t of statistical metho ds are left for subsequent works. T o extend ( 1.21 ) to our diffusive game setup, w e sort the Nash states at a time t ∈ [0 , T ] as X i N 1 ( t ) ,N ,v N, ∗ t ≥ X i N 2 ( t ) ,N ,v N, ∗ t ≥ . . . ≥ X i N N ( t ) ,N ,v N, ∗ t , with the ranks i N j ( t ) dep ending on t ≥ 0 since the Brownian motions W i are independent, and w e sho w that for fixed k ∈ N , the top k order statistics { X i N j ( t ) ,N ,v N, ∗ t : j ∈ { 1 , 2 , . . . , k }} satisfy: X i N 1 ( t ) ,N ,v N, ∗ t − b N t a N t , X i N 2 ( t ) ,N ,v N, ∗ t − b N t a N t , . . . , X i N k ( t ) ,N ,v N, ∗ t − b N t a N t ! →  ( E 1 ) − γ t − 1 γ t , ( E 1 + E 2 ) − γ t − 1 γ t , · · · , ( E 1 + E 2 + · · · + E k ) − γ t − 1 γ t  (1.23) w eakly as N → ∞ for an appropriate choice of the Extreme V alue Index γ t ≤ 0 and the deterministic normalizing sequences { a N t } ∞ N =1 and { b N t } ∞ N =1 , where E 1 , E 2 , . . . , E k are i.i.d. standard exp onen tial random v ariables. In view of classical Extreme V alue Theory which concerns i.i.d. random v ariables, w e pick { a N t } ∞ N =1 , { b N t } ∞ N =1 and γ t suc h that ( 1.23 ) holds when the Nash states X i,N ,v N, ∗ t are replaced by the i.i.d. random v ariables X i,v ∗ t , and then it suffices to deriv e ( 1.14 ) when H N is an appropriate b ounded function that captures the N → ∞ b ehavior of the normalized upp er order statistics of its arguments. In that case, w e reduce ( 1.14 ) to ( 1.20 ) b y using ( 1.19 ) as describ ed earlier, and the latter is a propagation of chaos prop ert y for the upp er order statistics of the pro cesses ¯ X i,N whose establishment is now sufficient for our result to hold. How ever, while propagation of c haos for the upp er order statistics of diffusions with a certain type of Mean-Field interaction in their drifts has b een obtained in the second author’s previous work [ 37 , 38 ], the drift terms of the diffusiv e processes ¯ X i,N in ( 1.18 ) dep end on the empirical measure ¯ µ N t in a more general wa y that is not cov ered by those pap ers. F or this reason, we use a T aylor expansion in the space P ( R ) to linearize those drift functions with resp ect to their measure argument, which leads to the follo wing appro ximation of the system ( 1.18 ) that is cov ered by the existing propagation of c haos results for upp er order statistics: ˜ X i,N t = X i 0 + Z t 0 ˆ b  ˜ X i,N s , L ( X v ∗ s ) , U x  s, ˜ X i,N s , L ( X v ∗ s )  ds + Z t 0 Z R δ δ m n ˆ b  ˜ X i,N s , m, U x  s, ˜ X i,N s , m o ( z 1 )    m = L ( X v ∗ s ) × ( ˜ µ N s − L ( X v ∗ s ))( dz 1 ) ds + σ W i t , ˜ µ N t = 1 N N X ℓ =1 δ ˜ X ℓ,N t , (1.24) with δ δ m b eing our notation for a F rechet deriv ativ e with respect to m ∈ P ( R ) . After that, we will see that the McKean-Vlasov SDE that gov erns the N → ∞ b eha vior of the Mean-Field System 7 ( 1.24 ) is again ( 1.17 ), i.e the same as for the systems ( 1.18 ) and ( 1.9 ) that ( 1.24 ) appro ximates. Therefore, controlling the relative entrop y that arises by applying Girsanov’s theorem to transform the law of ( ¯ X 1 ,N t , ¯ X 2 ,N t , . . . , ¯ X N ,N t ) into that of ( ˜ X 1 ,N t , ˜ X 2 ,N t , . . . , ˜ X N ,N t ) , we further reduce the establishmen t of our result to ha ving a propagation of chaos prop erty for upp er order statistics that is provided by the existing literature. Finally , it should b e noted that our drift linearization tec hnique was also used in [ 37 , Step 3 of Section 6] and [ 38 , Section 4], but it was p erformed in R instead of P ( R ) to eliminate a nonlinear dep endence on a linear functional of the empirical measure. 2 Assumptions and main results W e begin with the most fundamen tal assumption of this pap er: Assumption 2.1. F or any ( x, y , m ) ∈ R 2 × P ( R ) , the function: V 3 v 7→ b ( x, m, v ) y + f ( x, m, v ) admits a minimizer v = ˆ v ( x, m, y ) . W e can no w in tro duce the notations that will be used in ev erything that follo ws: Notation. The fol lowing notations wil l b e use d thr oughout this p ap er: (i) W e write C for the arbitr ary p ositive c onstant, which is deterministic, dep ends only on the p ar ameters of the system ( 1.1 ) - ( 1.3 ) (i.e the functions b, f , g , the volatility σ and the distri- bution of the initial values X i 0 ), and wil l gener al ly change fr om line to line. (ii) Having assumme d the existenc e of the minimizer ˆ v ( x, m, y ) in A ssumption 2.1 , we denote: ˆ b ( x, m, y ) = b ( x, m, ˆ v ( x, m, y )) and ˆ f ( x, m, y ) = f ( x, m, ˆ v ( x, m, y )) (2.1) (iii) The differ entiation of any function q with r esp e ct to an R -value d ar gument w is denote d by q w , wher e: • F or q b eing ˆ b , ˆ f , ˆ v or any of their derivatives we may have w = x, y or z i , which denote, r esp e ctively, the first, thir d and (3 + i ) -th ar gument (for i ≥ 1 and only for derivatives of q that ar e of or der ≥ i with r esp e ct to the me asur e ar gument m ). • F or q b eing any derivative of U we may have w = t, x or z i , which denote, r esp e ctively, the first, se c ond and (3 + i ) -th ar gument (for i ≥ 1 and only for derivatives of q that ar e of or der ≥ i with r esp e ct to the me asur e ar gument m ). The F r e chet derivative of any function q with r esp e ct to the me asur e ar gument m is denote d by δ q δ m , and we write q m for the c orr esp onding intrinsic derivative. Henc e, we wil l write for example δ q δ m ( x, m, U x ( t, x, m ) , z 1 ) for the function that o c curs when we plug y = U x ( t, x, m ) in δ q δ m ( x, m, y , z 1 ) , and δ δ m { q ( x, m, U x ( t, x, m )) } ( z 1 ) for the total derivative of q ( x, m, U x ( t, x, m )) with r esp e ct to m : δ δ m { q ( x, m, U x ( t, x, m )) } ( z 1 ) = δ q δ m ( x, m, U x ( t, x, m ) , z 1 )+ q y ( x, m, U x ( t, x, m )) δ U x δ m ( t, x, m, z 1 ) , which we c an differ entiate with r esp e ct to z 1 to obtain { q ( x, m, U x ( t, x, m )) } m ( z 1 ) = q m ( x, m, U x ( t, x, m ) , z 1 ) + q y ( x, m, U x ( t, x, m )) U xm ( t, x, m, z 1 ) . 8 W e introduce now the conditions necessary for our results to hold, which highly o verlap with those required for the establishmen t of a Central Limit Theorem and a Large Deviations Principle in [ 17 , 18 ]. The first set of conditions is needed for both results we establish, and it is giv en b elow: Assumption 2.2. The fol lowing c onditions ar e in for c e: (i) The initial values X i 0 ar e i.i.d. r andom variables, and their c ommon law µ 0 satisfies Z R x p ′ µ 0 ( dx ) < ∞ for some p ′ > 4 . (ii) The volatility σ is assume d to b e a p ositive c onstant. (iii) The classic al p artial derivatives ˆ b x , ˆ b m , ˆ b y , ˆ f y and the se c ond or der classic al p artial derivative ˆ b yy of the functions ˆ b ( x, m, y ) and ˆ f ( x, m, y ) exist and ar e b ounde d. (iv) F or e ach N ∈ N , the system of PDEs ( 1.7 ) with the terminal c onditions ( 1.8 ) admits classic al solutions U i,N , in the sense that e ach U i,N ( t, x ) is c ontinuously differ entiable in t > 0 and twic e c ontinuously differ entiable in x = ( x 1 , x 2 , . . . , x N ) ∈ R N . Final ly, for e ach N ∈ N , ther e exists a c onstant K N > 0 such that    U i,N x j ( t, x )    ≤ K N  1 + q x 2 1 + x 2 2 + . . . + x 2 N  and   U i,N ( t, x )   ≤ K N  1 + x 2 1 + x 2 2 + . . . + x 2 N  for al l i, j ∈ { 1 , 2 , . . . , N } , al l t ∈ [0 , T ] and al l x = ( x 1 , x 2 , . . . , x N ) ∈ R N . (v) The Master e quation ( 1.15 ) with the terminal c ondition U ( T , x, m ) = g ( x, m ) admits a solution U = U ( t, x, m ) , whose p artial derivatives U t , U x , U m , U xx , U xm , U mx , U mm , U mz 1 exist and ar e c ontinuous under the pr o duct topolo gies on the sp ac es they ar e define d, e.g [0 , T ] × R × P ( R ) for U xx and [0 , T ] × R × P ( R ) × R 2 for U mm . Final ly, the derivatives U x , U m , U xx , U xm = U mx and U mm ar e b ounde d, and the derivative U xmm exists and is also b ounde d. Remark 2.3. W e mak e the follo wing important notes: (i) Assumption 2.2 is essentially [ 17 , Assumptions A and B] for p ∗ = 1 , which are required for recalling results and computations from that pap er, accompanied by the existence and b oundedness of ˆ b yy and U xmm . There is also a simplification in the sense that we ask for ˆ b and ˆ f to hav e bounded classical deriv ativ es instead of simply b eing Lipsc hitz; for one of our results (Theorem 2.9 b elow), our pro of seems to work when ˆ b is Lipschitz in x and w and when ˆ f is Lipschitz in y , but e.g ˆ b y and ˆ b yy ha ve to b e classical deriv ativ es that exist everywhere. (ii) The conditions in Assumption 2.2 that are hard to verify are the existence of sufficien tly regular solutions U i,N to ( 1.7 ) and a sufficiently regular solution U to the Master equation ( 1.15 ). Assumptions A and B in [ 17 ] are accompanied b y references that verify all these conditions for a wide range of setups, except from the existence and b oundedness of U xmm : for ( 1.7 ), the results of [ 42 ] are applicable to Sto c hastic Differential Games where the Hamiltonian H ( x, m, y ) := ˆ b ( x, m, y ) y + ˆ f ( x, m, y ) is globally Lipsc hitz in y , while [ 3 ] and [ 9 , Section 6.3.1] 9 con tain results for setups where H ( x, m, y ) is quadratic in y ; for ( 1.15 ), results for the existence of a sufficently regular solution can also b e found in [ 9 ]. F or the existence and b oundedness of U xmm and of other deriv ativ es of U in Assumptions 2.4 and 2.5 b elow that are not cov ered b y the ab ov e references, we highly exp ect that having a regular enough terminal cost function g ( x, m ) is sufficient. P ossible wa ys for showing that could include the adaptation to higher order deriv ativ es of the computations p erformed in [ 9 ] for obtaining b ounded second order deriv ativ es (which w e exp ect to b e long and tedious but not requiring any no vel ideas), and the extension of the following classical b ootstraping metho d to PDEs on [0 , T ] × R × P ( R ) and high-dimensional quasilinear parab olic systems: if U is a classical solution to U t + U xx + q ( U x ) U x = 0 with U ( T , · ) and q ( · ) b eing sufficien tly regular, U solves also the linear parab olic PDE U t + U xx + q U x = 0 with q = q ( U x ) seen as a known C 1 , 1 function. F rom there, standard regularity theory of linear parabolic PDEs gives U ∈ C 1 , 3 when U ( T , · ) ∈ C 3 . (iii) F or our setup, the Central Limit Theorem in [ 17 ] and the Large Deviations Princip e in [ 18 ] are also obtained when [ 17 , Assumption B] is replaced by [ 17 , Assumption B’], and a class of games for which this assumption is in force is also provided. In our pap er, this amendment translates to imp osing b oundedness on U and uniform boundedness (in i and N ) on U i,N , but allo wing for ˆ f y to gro w linearly in y . As [ 17 , Assumption B] is only used in the pro of of Theorem 2.9 b elo w when ( 1.19 ) is recalled, and since the latter also holds under [ 17 , Assumption B’] for large N , we deduce that Theorem 2.9 b elo w (which gives the N → ∞ b eha vior of the upp er order statistics of the Nash states) holds under this mo dification as w ell. (iv) F or a p opular class of linear-quadratic games, g ( x, m ) is a quadratic function and thus the first order deriv atives U x and U m are not bounded. How ev er, the authors in [ 17 , 18 ] show that the Cen tral Limit Theorem and the Large Deviations Principle they obtain can still hold in these setups, by studying the systemic risk mo del from [ 10 ] (see e.g [ 18 , Section 6]). This is also the case for our result on the N → ∞ b eha vior of the upp er order statistics of the Nash states which is given in Theorem 2.9 b elo w. The reason is that the pro of uses the b oundedness of U x and U m only for recalling ( 1.19 ) from [ 17 , 18 ], which means that this b oundedness of U x and U m can b e ignored if ( 1.19 ) can b e verified directly . This is exactly what we do in Example 2.12 below, where we also consider the systemic risk mo del from [ 10 ] The next set of conditions is only required for our result on the weak conv ergence of the normal- ized upp er order statistics of the Nash states; it contains the conditions required in addition to Assumption 2.2 for the pro of w e will give for that result. Assumption 2.4. The fol lowing c onditions ar e in for c e: (i) F or some κ 0 > 0 , the law µ 0 of the indep endent initial values X i 0 satisfies Z R e κ 0 x 2 µ 0 ( dx ) < ∞ , which implies that it also has finite moments (satisfying, in p articular, (i) of A ssumption 2.2 ). 10 (ii) The derivatives ˆ b yx , ˆ b ym , ˆ b mm , δ U x δ m , ( δ U x δ m ) x , ( δ ˆ b δ m ) x , ( δ ˆ b δ m ) y and ( δ ˆ b δ m ) yz 1 exist and ar e b ounde d. Finally , we hav e the following conditions that are required in addition to Assumption 2.2 for the impro vemen t of ( 1.19 ) to an L ∞ (Ω) estimate: Assumption 2.5. The fol lowing c onditions ar e in for c e: (i) F or the functions ˆ b ( x, m, y ) and ˆ f ( x, m, y ) , the classic al first or der derivatives f x , f m exist, and the classic al se c ond or der derivatives ˆ b xy , ˆ b my , ˆ f xy , ˆ f my , ˆ f yy exist and ar e b ounde d. (ii) The thir d or der p artial derivatives of e ach U i,N that do not involve t , i.e U i,N x j 1 x j 2 x j 3 for j 1 , j 2 , j 3 ∈ { 1 , 2 , . . . , N } , exist and ar e c ontinuous in ( t, x ) ∈ R N +1 . Mor e over, the se c ond or der p artial derivatives of e ach U i,N that involve t , i.e U i,N tx j and U i,N x j t for j ∈ { 1 , 2 , . . . , N } , exist and ar e c ontinuous in ( t, x ) ∈ R N +1 . Final ly, we assume that the first or der p artial derivatives U i,N x i ar e b ounde d uniformly in N and i ∈ { 1 , 2 , . . . , N } , and the se c ond or der p artial derivatives U i,N x i x j ar e b ounde d (not ne c essarily uniformly in N and i ∈ { 1 , 2 , . . . , N } ). (iii) The thir d or der p artial derivatives of U that do not involve differ entiation in t , i.e U xxx , U xxm , U xmx , U mxx , U mmx , U mxm , U xmm , U mmm , U xmz 1 , U mz 1 x , U mxz 1 , U mz 1 z 1 , U mmz 1 , U mz 1 m and U mmz 2 , exist and ar e c ontinuous under the pr o duct top olo gies on the sp ac es they ar e define d, e.g [0 , T ] × R × P ( R ) for U xxx and [0 , T ] × R × P ( R ) × R 2 for U xmm . Mor e over, the se c ond or der p artial derivatives that involve a single differ entiation in t , i.e U tx , U xt , U tm and U mt , exist and ar e c ontinuous under the pr o duct top olo gies on the sp ac es they ar e define d. Final ly, the derivatives U mz 1 , U xxm , U xmz 1 , U mmm , U mmz 1 and U mmz 2 ar e b ounde d. Remark 2.6. W e ha ve the following imp ortan t observ ations: (i) Under Assumption 2.5 , since U x and U i,N x i are assumed to b e b ounded, the b oundedness in y of ˆ b x , ˆ b m , ˆ b y , ˆ f y and ˆ b yy in Assumption 2.2 can b e reduced to lo cal boundedness. (ii) Assumption 2.5 imp oses roughly that the functions ˆ b, ˆ f , U and U i,N can be differentiated an additional time compared to the setup of [ 17 , 18 ]. The reason is that the replacemen t of L 1 (Ω) with L ∞ (Ω) in ( 1.19 ) requires the reestablishment of an L ∞ (Ω) estimate for the differences U  t, X i,N ,v N, ∗ t , µ N ,v N, ∗ t  − U i,N  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  whic h was obtained in those pap ers ([ 17 , Display (4.18)]), but this time with U and U i,N replaced by U x and U i,N x i resp ectiv ely . That L ∞ (Ω) estimate was obtained in [ 17 , 18 ] via the manipulation of certain equations that inv olv e ˆ b, ˆ f , U, U i,N and their deriv atives, meaning that the ab o ve reestablishment requires computations of a similar nature after differentiating those equations, and the later increases by 1 the orders of the highest order deriv ativ es that app ear in the computations whic h are the least sufficient highest orders of differen tiability . W e can now present the tw o results we obtain in this pap er. Our first result is the improv ement of ( 1.19 ) to an L ∞ (Ω) estimate: 11 Theorem 2.7. Consider the Nash states { X 1 ,N ,v N, ∗ , X 2 ,N ,v N, ∗ , . . . , X N ,N,v N, ∗ } that satisfy the system ( 1.18 ) . If A ssumptions 2.1 , 2.2 and 2.5 hold, we c an find a C > 0 such that P -almost sur ely, for al l t ∈ [0 , T ] and N ∈ N we have: N X i =1 ( ˆ b  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , U i,N x i  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  − ˆ b  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , U x  t, X i,N ,v N, ∗ t , µ N ,v N, ∗ t  ) 2 ≤ C N . (2.2) Remark 2.8. It can be seen from the pro of of the abov e estimate that w e actually hav e an L ∞ ( R N +1 ) bound of order O ( N − 1 ) for both U i,N x i ( t, x 1 , x 2 , . . . , x N ) − U x   t, x i , N − 1 N X j =1 δ x j   and U i,N x j ( t, x 1 , x 2 , . . . , x N ) with i 6 = j , which is uniform in i and N . Then, we hav e our second result, whic h establishes the asymptotic b eha vior of the upp er order statistics of the Nash states: Theorem 2.9. Supp ose that A ssumptions 2.1 , 2.2 and 2.4 hold. F or e ach i ∈ N , let ( X i,v ∗ t ) t ∈ [0 ,T ] b e the unique solution ( X v ∗ t ) t ∈ [0 ,T ] to the McK e an-Vlasov SDE ( 1.17 ) when ( W t ) t ∈ [0 ,T ] is r eplac e d by ( W i t ) t ∈ [0 ,T ] . Supp ose also that for a fixe d t ≥ 0 , L ( X v ∗ t ) b elongs to the domain of attr action of an extr eme value distribution with cumulative distribution function G γ t ( x ) = e − (1+ γ t x ) − 1 γ t , 1 + γ t x ≥ 0 , for some value γ t ∈ R , in which c ase we c an find deterministic se quenc es ( a N t ) N ∈ N and ( b N t ) N ∈ N such that P max i ≤ N X i,v ∗ t − b N t a N t ≤ x ! → G γ t ( x ) (2.3) for any x ∈ R as N → ∞ . Then, sorting the Nash states X i,N ,v N, ∗ t as X i N 1 ( t ) ,N ,v N, ∗ t ≥ X i N 2 ( t ) ,N ,v N, ∗ t ≥ . . . ≥ X i N N ( t ) ,N ,v N, ∗ t , for any k ∈ N we have the fol lowing c onver genc e for the top k or der statistics of the Nash system: X i N 1 ( t ) ,N ,v N, ∗ t − b N t a N t , X i N 2 ( t ) ,N ,v N, ∗ t − b N t a N t , . . . , X i N k ( t ) ,N ,v N, ∗ t − b N t a N t ! →  ( E 1 ) − γ t − 1 γ t , ( E 1 + E 2 ) − γ t − 1 γ t , · · · , ( E 1 + E 2 + · · · + E k ) − γ t − 1 γ t  (2.4) we akly as N → ∞ , wher e E 1 , E 2 , . . . , E k ar e i.i.d. standar d exp onential r andom variables. 12 Remark 2.10. It should be noted that: (i) When γ t = 0 , we define G 0 ( x ) := lim γ → 0 + G γ ( x ) = e − e − x (the standard Gumbel distribution), in whic h case the limit in ( 2.4 ) is in terpreted as ( − log ( E 1 ) , − log ( E 1 + E 2 ) , . . . , − log ( E 1 + E 2 + . . . + E k )) . (ii) When the initial law µ 0 is Gaussian and ˆ b ( x, m, U x ( s, x, m )) is linear in x , one can solve ( 1.17 ) explicitly to find that X v ∗ is a Gaussian pro cess, i.e X v ∗ t ∼ N ( m t , σ 2 t ) for all t ≥ 0 . In that case, by [ 14 , Example 1.1.7] w e ha ve that for eac h t ≥ 0 , the law L ( X v ∗ t ) b elongs to the domain of attraction of the standard Gum b el distribution and we hav e ( 2.3 ) with γ t = 0 and b N t = σ t p 2 log ( N ) − log(log ( N )) − log(4 π ) + m t (2.5) and finally a N t = σ t p 2 log ( N ) − log(log ( N )) − log(4 π ) (2.6) (a relev an t setup that arises in systemic risk is analyzed in Example 2.12 b elo w). A standard w ay for verifying ( 2.3 ) is the deriv ation of the V on Mises condition (see [ 14 , Theorem 1.1.8]), and some recent computations that use Malliavin Calculus hav e shown that this is p ossible with γ t = 0 when ˆ b ( x, m, U x ( s, x, m )) admits a limit as x → + ∞ and when the initial la w µ 0 is a dirac measure (so X i 0 = x 0 for all i ∈ { 1 , 2 , . . . , N } , for some x 0 ∈ R ), but the normalizing sequences ( a N t ) N ∈ N and ( b N t ) N ∈ N ha ve not b een computed explicitly . The deriv ation of an as wide as p ossible class of McKean-Vlasov SDEs ( 1.17 ) for which ( 2.3 ) holds will b e a ma jor researc h direction in the near future, as well as the explicit computation of the sequences ( a N t ) N ∈ N and ( b N t ) N ∈ N whenev er it is p ossible; having ( 2.3 ) with γ t = 0 whenever Assumptions 2.2 and 2.4 are satisfied seems lik e a quite p ossible scenario. (iii) As [ 14 , Theorem 1.2.1] suggests, ( 2.3 ) could hold with γ t > 0 only when X v ∗ t has regularly v arying tails, and with γ t < 0 only when X v ∗ t is upp er b ounded by a deterministic constant. While the first is incompatible with Assumptions 2.2 , and 2.4 by ( 2.13 ) in Lemma 2.15 b elo w, the second is feasible in the extended setup where σ dep ends on X v ∗ (see also Remark 2.11 ). F or example, this would b e the case if − X v ∗ could resemble a CIR pro cess, i.e ( Y t ) t ∈ [0 ,T ] with Y t = y + Z t 0 k ( θ − Y s ) ds + ξ Z t 0 p | Y s | dW s , for t ≥ 0 , since one can apply [ 14 , Theorem 1.2.6] on the known distribution function of Y t (see [ 13 ]) to deduce that the law of − Y t is in the domain of attraction of G γ t with γ t = − ξ 2 2 kθ Remark 2.11. Theorem 2.9 can be extended to setups where the constant σ is replaced b y a state-dep enden t volatilit y σ ( X i,N ,v N t ) . A technique that allows this extension is discussed in the final section of the pap er, and our argumen t is solid for the case where σ : R 7→ R + is b ounded and has upper b ounded deriv atives up to order 3 . Example 2.12. Consider the linear-quadratic game introduced in [ 10 ], where: b ( x, m, v ) = ¯ b ( ¯ m − x ) + v , 13 f ( x, m, v ) = 1 2 v 2 − q v ( ¯ m − x ) + ϵ 2 ( ¯ m − x ) 2 , g ( x, m ) = ˜ g 2 ( ¯ m − x ) 2 in which ¯ m = R y · m ( dy ) and ¯ b, q , ϵ, ¯ g are constants. In this setup, the state pro cesses X i,N ,v N describ e the log-monetary reserves of N banks that b orrow from each other at a fixed rate ¯ b that is common for any t wo banks, and at any time t ∈ [0 , T ] , the i -th bank with log-w ealth X i,N ,v N can also borrow from a central bank at a rate v i,N t it c ho oses (where v i,N t < 0 corresp onds to lending). The exp ected cost J i,N that the i -th bank wan ts to minimize (defined in 1.3 ) is the expected v alue of an accumulated (until a future time T ) p enalty for deviating from the av erage p erformance of all the banks, combined with an accum ulated (also until time T ) cost of b orro wing from the central bank. The ab o v e setup is also the main example provided in [ 17 ] and [ 18 ], which establish a Central Limit Theorem and a Large Deviations Principle for our setup, and the authors obtain v i,N , ∗ =  q + ϕ N ( t )  1 − 1 N    1 N N X j =1 X j,N ,v N, ∗ t − X i,N ,v N, ∗ t   where ϕ N is the solution to an ODE of Riccati type, and also: U ( t, x, m ) = ϕ ∞ ( t ) 2 ( x − m ) 2 for ϕ ∞ ( t ) = lim N →∞ ϕ N ( t ) . In that case, the system ( 1.9 ) b ecomes X i,N ,v N, ∗ t = X i 0 + Z t 0  ¯ b + q + ϕ N ( s )  1 − 1 N    1 N N X j =1 X j,N ,v N, ∗ s − X i,N ,v N, ∗ s   ds + σ W i t , while both systems ( 1.18 ) and ( 1.24 ) coincide with ¯ X i,N t = X i 0 + Z t 0  ¯ b + q + ϕ ∞ ( s )    1 N N X j =1 ¯ X j,N s − ¯ X i,N s   ds + σ W i t , with the corresp onding McKean-Vlaso v SDE ( 1.17 ) being X v ∗ t = X 0 + Z t 0  ¯ b + q + ϕ ∞ ( s )   E h X v ∗ t i − ¯ X i,N s  ds + σ W t . It is not difficult to v erify that ( 1.19 ) holds when the initial v alues X i 0 satisfy (i) of Assumption 2.4 (see [ 17 , Page 43, Display (6.6) and b elo w] for concrete argumen ts), so as discussed in (iv) of Remark 2.3 , we can ignore the non-b oundedness of U x and U m and recall Theorem 2.9 . Then, ( 2.3 ) holds with γ t = 0 as discussed in (ii) of Remark 2.10 , with the deterministic normalizing sequences ( a N t ) N ∈ N and ( b N t ) N ∈ N giv en by ( 2.5 ) and ( 2.6 ) when the X i 0 are Gaussian, meaning that we also ha ve the conv ergence ( 2.4 ) with γ t = 0 and the same sequences ( a N t ) N ∈ N and ( b N t ) N ∈ N . 14 Remark 2.13. The authors in [ 10 ] also introduce a default barrier D < 0 , meaning that X i,N ,v N t < D at some t ∈ [0 , T ] ⇔ the i-th bank is in default at time t. In a relatively stable economy , there can b e a large num b er N of banks but no defaults observed un til a time t 0 ∈ [0 , T ] , meaning that at time t 0 , one has no picture for the probability of default. In that case, metho ds of Extreme V alue Theory extended to our game setup can help with the estimation of the likelihoo d of the rare (but p ossibly highly impactful) even t of k banks b eing in default at the future time t 1 > t 0 . Indeed, replacing each X i,N ,v N with − X i,N ,v N for each i , we obtain the same game with different initial conditions X i 0 , and the ev ent of interest has probability P  X i N k ,N ,v N t 1 > − D  = P   X i N k ,N ,v N t 1 − b N t 1 a N t 1 > − D − b N t 1 a N t 1   . T aking D = D N = − a N t 1 M − b N t 1 with M not extremely large compared to N , the large- N b eha vior of the ab ov e probabilit y of interest is gov erned b y ( 2.4 ) given b y Theorem 2.9 . Of course, this is a simplified regime where the the state processes are not stopp ed up on hitting the default barrier; applying the same idea to this more complex setup (whic h could also incorp orate other realistic features lik e common noise) could b e p ossible, but this is b ey ond the scop e of this pap er. Moreov er, the terms a N t 1 and b N t 1 of the normalizing sequences need to b e kno wn, but an extension of statistical metho ds of Extreme V alue Theory to the regime of Mean-Field Systems and Sto chastic Differen tial Games (which is in our research plans for the near future) would allow for the estimation of a N t and b N t for t ≤ t 0 (supp osing that the states are fully observ ed un til the present time t 0 ), and the prediction of a N t 1 and b N t 1 at the future time t 1 > t 0 should then b e p ossible by using to ols lik e regression. Finally , it should b e noted that the ab ov e analysis do es not require knowledge of functions that drive the Nash states (e.g ˆ b ( x, m, y ) ), meaning that it could b e applicable to game mo dels for systemic risk where the co efficients in equilibrium are not computable. T o prov e Theorem 2.7 and Theorem 2.9 , w e present a few lemmata. First, b ecause the proof of Theorem 2.7 is very long and highly computational, we isolate some of its computations into the follo wing lemma: Lemma 2.14. W e define U i,N ( t, x ) = U  t, x i , µ N x  . Then, for j, k ∈ { 1 , 2 , . . . , N } we c an find functions r 1 j ( t, x ) and r 2 j,k ( t, x ) with | r 1 j ( t, x ) | < C N and | r 2 j,k ( t, x ) | < C N for al l t ≥ 0 and x ∈ R N , such that 0 = U i,N x k t ( t, x ) + ˆ b x ( x k , µ N x , U k,N x k ( t, x ) − r 1 k ( t, x )) U i,N x k ( t, x ) + ˆ f x ( x i , µ N x , U i,N x i ( t, x ) − r 1 i ( t, x )) δ ik + 1 N ˆ f m ( x i , µ N x , U i,N x i ( t, x ) − r 1 i ( t, x ) , x k ) + ˆ f y ( x i , µ N x , U i,N x i ( t, x ) − r 1 i ( t, x )) U i,N x k x i ( t, x ) + N X j =1 1 N ˆ b m ( x j , µ N x , U j,N x j ( t, x ) − r 1 j ( t, x ) , x k ) U i,N x j ( t, x ) + N X j =1 ˆ b y ( x j , µ N x , U j,N x j ( t, x ) − r 1 j ( t, x )) U i,N x j ( t, x ) U j,N x k x j ( t, x ) 15 + N X j =1 ˆ b ( x j , µ N x , U x ( t, x j , µ N x )) U i,N x k x j ( t, x ) + σ 2 2 N X j =1 U i,N x k x j x j ( t, x ) − r 2 i,k ( t, x ) , (2.7) for al l i, k ∈ { 1 , 2 , . . . , N } and al l t ≥ 0 and x ∈ R N . Next, for the pro of of Theorem 2.9 , we will make use of the following lemma, which gives efficien t b ounds for the moments of a solution to a McKean-Vlaso v SDE, and for the momen ts of the W asserstein distances of the la w of the solution to that SDE from the empirical measures that appro ximate this law. Some of the b ounds obtained are natural and probably already known, but they are included for the sake of completeness. Lemma 2.15. L et Z 1 0 , Z 2 0 , . . . b e a se quenc e of indep endent r andom variables with a c ommon law µ 0 ∈ P ( R ) , and c onsider the empiric al me asur es µ N ,Z N t = 1 N N X ℓ =1 δ Z ℓ,N t and µ N ,Z t = 1 N N X ℓ =1 δ Z ℓ t (2.8) wher e the dynamics of the p articles Z i,N ar e given by the system Z i,N t = Z i 0 + Z t 0 B  s, Z i,N s , µ N ,Z N s  ds + W i t , i = 1 , 2 , . . . , N (2.9) and Z i = ( Z i t ) t ∈ [0 ,T ] is the solution to the c orr esp onding McK e an-Vlasov SDE: Z t = Z 0 + Z t 0 B ( s, Z s , L ( Z s )) ds + W t L ( Z 0 ) = µ 0 . (2.10) when ( W t ) t ∈ [0 ,T ] and Z 0 ar e r eplac e d by ( W i t ) t ∈ [0 ,T ] and Z i 0 r esp e ctively, for e ach i ∈ { 1 , 2 , . . . , N } . Supp ose that ther e exist κ 0 , L > 0 such that E h e κ 0 Z 2 0 i < ∞ (2.11) and | B ( t, z 1 , m 1 ) − B ( t, z 2 , m 2 ) | ≤ L ( | z 1 − z 2 | + W 1 ( m 1 , m 2 )) (2.12) for any two p airs ( z 1 , m 1 ) , ( z 2 , m 2 ) ∈ R × ( P ( R ) , W 1 ) and any t ∈ [0 , T ] . Then, we have that: E [ Z p t ] ≤ C p p Γ  p 2  (2.13) and E h W p 1  µ N ,Z t , L ( Z t ) i ≤ C p p Γ  p 2  N p 2 . (2.14) and also E h W p 1  µ N ,Z N t , L ( Z t ) i ≤ C p p Γ  p 2  N p 2 . (2.15) for any t ∈ [0 , T ] , any p ≥ 1 , and some C > 0 that do es not dep end on t or p . 16 Finally , we present the lemma that is used for all the reductions in the pro of of Theorem 2.9 , the pro of of which uses Pinsker’s inequality . Lemma 2.16. F or e ach N ∈ N , we c onsider the N -dimensional pr o c esses Z N ,k = ( Z 1 ,N ,k , Z 2 ,N ,k , . . . , Z N ,N,k ) for k ∈ { 1 , 2 } , with Z i,N ,k = ( Z i,N ,k t ) t ∈ [0 ,T ] , which ar e taken to b e the p athwise unique str ong solutions to the N -dimensional SDEs: Z N , 1 t = Z N 0 + Z t 0 A N  s, Z N , 1 s   B N , 1  s, Z N , 1 s  ds + d W N s  + Z t 0 C N  s, Z N , 1 s  ds (2.16) and Z N , 2 t = Z N 0 + Z t 0 A N  s, Z N , 2 s   B N , 2  s, Z N , 2 s  ds + d W N s  + Z t 0 C N  s, Z N , 2 s  ds. (2.17) In the ab ove, for k ∈ { 1 , 2 } , B N ,k ( · ) and C N ( · ) ar e me asur able R N -value d functions with B N ,k ( · ) b eing lo c al ly b ounde d, A ( · ) is a me asur able function with values in the sp ac e of N × N matric es, and W N := ( W 1 , W 2 , . . . , W N ) . Supp ose that for al l N ∈ N and al l s ∈ [0 , T ] we have: Z T 0 E h   B N , 2  s, Z N , 2 s  − B N , 1  s, Z N , 2 s    2 2 i ds ≤ C N . (2.18) for some C > 0 that do es not dep end on N . Then, it holds that    E h H N  Z N , 1 t i − E h H N  Z N , 2 t i    → 0 (2.19) as N → ∞ , for any me asur able function H N : R N 7→ [0 , 1] and al l t ∈ [0 , T ] . 3 Pro ofs In this section we place all the pro ofs of this pap er. Betw een the pro ofs of our tw o main results, w e will first presen t the pro of of Theorem 2.7 , but for that w e m ust first pro ve Lemma 2.14 . Pr o of of L emma 2.14 . F or a fixed x = ( x 1 , x 2 , . . . , x N ) ∈ R N and for each i ∈ { 1 , 2 , . . . , N } , plug- ging µ → µ N x and x → x i in ( 1.15 ) and recalling the computations in [ 17 , Pro of of Prop osition 4.1] w e get: 0 = U i,N t ( t, x ) + N X j =1 ˆ b ( x j , µ N x , U x ( t, x j , µ N x )) U i,N x j ( t, x ) + ˆ f ( x i , µ N x , U x ( t, x i , µ N x )) + σ 2 2  U i,N x i x i ( t, x ) − 1 N U xm ( t, x i , µ N x , x i ) − 1 N 2 U mm ( t, x i , µ N x , x i , x i )  + σ 2 2 N X j =1 j  = i  U i,N x j x j ( t, x ) − 1 N 2 U mm ( t, x i , µ N x , x j , x j )  . (3.1) 17 Differen tiating the abov e with respect to x k w e get: 0 = U i,N x k t ( t, x ) + ˆ b x ( x k , µ N x , U x ( t, x k , µ N x )) U i,N x k ( t, x ) + ˆ f x ( x i , µ N x , U x ( t, x i , µ N x )) δ ik + 1 N ˆ f m ( x i , µ N x , U x ( t, x i , µ N x ) , x k ) + ˆ f y ( x i , µ N x , U x ( t, x i , µ N x )) ∂ ∂ x k U x ( t, x i , µ N x ) + N X j =1 1 N ˆ b m ( x j , µ N x , U x ( t, x j , µ N x ) , x k ) U i,N x j ( t, x ) + N X j =1 ˆ b y ( x j , µ N x , U x ( t, x j , µ N x )) U i,N x j ( t, x ) ∂ ∂ x k U x ( t, x j , µ N x ) + N X j =1 ˆ b ( x j , µ N x , U x ( t, x j , µ N x )) U i,N x k x j ( t, x ) + σ 2 2 N X j =1 U i,N x k x j x j ( t, x ) − σ 2 2 ∂ ∂ x k  1 N U xm ( t, x i , µ N x , x i ) + 1 N 2 U mm ( t, x i , µ N x , x i , x i )  − σ 2 2 N X j =1 j  = i 1 N 2 ∂ ∂ x k U mm ( t, x i , µ N x , x j , x j ) (3.2) where all deriv atives exist in a classical sense and w e can in terchange the deriv ativ es with each other and with integrals b y the regularity pro vided by Assumptions 2.2 and 2.5 . In the ab ov e, w e ha ve also used that for any differen tiable function F ( µ ) of µ ∈ P we hav e ∂ ∂ x j F ( µ N x ) = 1 N F m ( µ N x , x j ) . (3.3) Using no w ( 3.3 ) again w e can obtain the following relations: ∂ ∂ x k U mm ( t, x i , µ N x , x j , x j ) = U xmm ( t, x i , µ N x , x j , x j ) δ ik + 1 N U mmm ( t, x i , µ N x , x j , x j , x k ) + U mmz 1 ( t, x i , µ N x , x j , x j ) δ j k + U mmz 2 ( t, x i , µ N x , x j , x j ) δ j k (3.4) for an y i, j, k ∈ { 1 , 2 , . . . , N } , U x ( t, x j , µ N x ) = U j,N x j ( t, x ) − 1 N U m ( t, x j , µ N x , x j ) (3.5) whic h implies also that ∂ ∂ x k U x ( t, x j , µ N x ) = U j,N x k x j ( t, x ) − 1 N U mx ( t, x j , µ N x , x j ) δ j k − 1 N U mz 1 ( t, x j , µ N x , x j ) δ j k − 1 N 2 U mm ( t, x j , µ N x , x j , x k ) (3.6) 18 for an y j, k ∈ { 1 , 2 , . . . , N } , and finally ∂ ∂ x k U xm ( t, x i , µ N x , x i ) = U xxm ( t, x i , µ N x , x i ) δ ik + 1 N U xmm ( t, x i , µ N x , x i , x k ) + U xmz 1 ( t, x i , µ N x , x i ) δ ik (3.7) for an y i, k ∈ { 1 , 2 , . . . , N } . Substituting all the abov e in ( 3.2 ) we obtain the desired form with r 1 j ( t, x ) = 1 N U m ( t, x j , µ N x , x j ) (3.8) for all j ∈ { 1 , 2 , . . . , N } and r 2 i,k ( t, x ) = − 1 N ˆ f y ( x i , µ N x , U i,N x i ( t, x ) − r 1 i ( t, x )) ×  U mx ( t, x i , µ N x , x i ) δ ik + U mz 1 ( t, x i , µ N x , x i ) δ ik  + 1 N 2 ˆ f y ( x i , µ N x , U i,N x i ( t, x ) − r 1 i ( t, x )) U mm ( t, x i , µ N x , x i , x k ) − 1 N ˆ b y ( x k , µ N x , U k,N x k ( t, x ) − r 1 k ( t, x )) ×  U mx ( t, x k , µ N x , x k ) + U mz 1 ( t, x k , µ N x , x k )  + 1 N 2 N X j =1 ˆ b y ( x j , µ N x , U j,N x j ( t, x ) − r 1 j ( t, x )) U mm ( t, x j , µ N x , x j , x k ) − σ 2 2 1 N  U xxm ( t, x i , µ N x , x i ) δ ik + 1 N U xmm ( t, x i , µ N x , x i , x k )  − σ 2 2 1 N U xmz 1 ( t, x i , µ N x , x i ) δ ik − σ 2 2 N X j =1 1 N 2  U xmm ( t, x i , µ N x , x j , x j ) δ ik + 1 N U mmm ( t, x i , µ N x , x j , x j , x k )  − σ 2 2 1 N 2  U mmz 1 ( t, x i , µ N x , x k , x k ) + U mmz 2 ( t, x i , µ N x , x k , x k )  (3.9) for all i, k ∈ { 1 , 2 , . . . , N } , which are both b ounded by C N b y Assumptions 2.2 and 2.5 so our proof is complete. W e can no w pro v e the first of our tw o main results. Pr o of of The or em 2.7 . First, under the notation of Lemma 2.14 , we define Y i,k,N t = U i,N x k  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  Z i,j,k,N t = U i,N x k x j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  Y i,k,N t = U i,N x k  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  Z i,j,k,N t = U i,N x k x j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  . (3.10) 19 By Assumptions 2.2 and 2.5 , w e can differentiate the i -th equation of ( 1.7 ) with respect to x k , and use again ( 3.3 ) to obtain 0 = U i,N x k t ( t, x ) + ˆ b x  x k , µ N x , U k,N x k ( t, x )  U i,N x k ( t, x ) + ˆ f x  x i , µ N x , U i,N x i ( t, x )  δ ik + 1 N ˆ f m  x i , µ N x , U i,N x i ( t, x ) , x k  + ˆ f y  x i , µ N x , U i,N x i ( t, x )  U i,N x k x i ( t, x ) + N X j =1 ˆ b  x j , µ N x , U j,N x j ( t, x )  U i,N x k x j ( t, x ) + N X j =1 1 N ˆ b m  x j , µ N x , U j,N x j ( t, x ) , x k  U i,N x j ( t, x ) + N X j =1 ˆ b y  x j , µ N x , U j,N x j ( t, x )  U i,N x j ( t, x ) U j,N x k x j ( t, x ) + σ 2 2 N X j =1 U i,N x k x j x j ( t, x ) , (3.11) along with terminal conditions U i,N x k ( T , x ) = ∂ ∂ x k g  x i , µ N x  . (3.12) Therefore, recalling Assumptions 2.2 and 2.5 again, we can apply Itô’s formula on the pro cess U i,N x k ( t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t ) and use ( 1.9 ) and then ( 3.11 ) to find that d Y i,k,N t = U i,N x k t  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  dt + N X j =1 U i,N x k x j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  × ˆ b  X j,N ,v N, ∗ t , µ N ,v N, ∗ t , U j,N x j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  dt + σ 2 2 N X j =1 U i,N x k x j x j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  dt + σ N X j =1 Z i,j,k,N t dW j t = − ˆ b x  X k,N ,v N, ∗ t , µ N ,v N, ∗ t , Y k,k ,N t  Y i,k,N t dt − ˆ f x  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , Y i,i,N t  δ ik dt − 1 N ˆ f m  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , Y i,i,N t , X k,N ,v N, ∗ t  dt − ˆ f y  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , Y i,i,N t  Z i,i,k,N t dt − 1 N N X j =1 ˆ b m  X j,N ,v N, ∗ t , µ N ,v N, ∗ t , Y j,j,N t , X k,N ,v N, ∗ t  Y i,j,N t dt − N X j =1 ˆ b y  X j,N ,v N, ∗ t , µ N ,v N, ∗ t , Y j,j,N t  Y i,j,N t Z j,j,k,N t dt + σ N X j =1 Z i,j,k,N t dW j t . (3.13) On the other hand, applying Itô’s formula on U i,N x k ( t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t ) and using 20 ( 1.9 ) and then Lemma 2.14 we obtain d Y i,k,N t = U i,N x k t  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  dt + N X j =1 U i,N x k x j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  × ˆ b  X j,N ,v N, ∗ t , µ N ,v N, ∗ t , U j,N x j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  dt + σ 2 2 N X j =1 U i,N x k x j x j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  dt + σ N X j =1 Z i,j,k,N t dW j t = − ˆ b x ( X k,N ,v N, ∗ t , µ N ,v N, ∗ t , Y k,k ,N t − R k, 1 t ) Y i,k,N t dt − ˆ f x ( X i,N ,v N, ∗ t , µ N ,v N, ∗ t , Y i,i,N t − R i, 1 t ) δ ik dt − 1 N ˆ f m ( X i,N ,v N, ∗ t , µ N ,v N, ∗ t , Y i,i,N t − R i, 1 t , X k,N ,v N, ∗ t ) dt − ˆ f y ( X i,N ,v N, ∗ t , µ N ,v N, ∗ t , Y i,i,N t − R i, 1 t ) Z i,i,k,N t dt − N X j =1 1 N ˆ b m ( X j,N ,v N, ∗ t , µ N ,v N, ∗ t , Y j,j,N t − R j, 1 t , X k,N ,v N, ∗ t ) Y i,j,N t dt − N X j =1 ˆ b y ( X j,N ,v N, ∗ t , µ N ,v N, ∗ t , Y j,j,N t − R j, 1 t ) Y i,j,N t Z j,j,k,N t dt + R i,k, 2 t dt + σ N X j =1 Z i,j,k,N t dW j t (3.14) where for j, k ∈ { 1 , 2 , . . . , N } we define R j, 1 t = r 1 j  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  and R j,k, 2 t = r 2 j,k  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  whic h are both b ounded b y C N . As we hav e Y i,k,N T = Y i,k,N T = ∂ ∂ x k g  x i , µ N x  , subtracting ( 3.13 ) from ( 3.14 ) and applying Itô’s formula on ( Y i,k,N t − Y i,k,N t ) 2 w e obtain:  Y i,k,N t − Y i,k,N t  2 = 2 Z T t  Y i,k,N s − Y i,k,N s  ( ˆ b x ( X k,N ,v N, ∗ s , µ N ,v N, ∗ s , Y k,k ,N s − R k, 1 s ) Y i,k,N s 21 − ˆ b x  X k,N ,v N, ∗ s , µ N ,v N, ∗ s , Y k,k ,N s  Y i,k,N s ) ds + 2 Z T t  Y i,k,N s − Y i,k,N s  ( ˆ f x ( X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s − R i, 1 s ) δ ik − ˆ f x  X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s  δ ik ) ds + 2 N Z T t  Y i,k,N s − Y i,k,N s  × ( ˆ f m ( X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s − R i, 1 s , X k,N ,v N, ∗ s ) − ˆ f m  X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s , X k,N ,v N, ∗ s  ) ds + 2 Z T t  Y i,k,N s − Y i,k,N s  ( ˆ f y ( X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s − R i, 1 s ) Z i,i,k,N s − ˆ f y  X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s  Z i,i,k,N s ) ds + 2 N Z T t  Y i,k,N s − Y i,k,N s  × N X j =1 ( ˆ b m ( X j,N ,v N, ∗ s , µ N ,v N, ∗ s , Y j,j,N s − R j, 1 s , X k,N ,v N, ∗ s ) Y i,j,N s − ˆ b m  X j,N ,v N, ∗ s , µ N ,v N, ∗ s , Y j,j,N s , X k,N ,v N, ∗ s  Y i,j,N s ) ds + 2 Z T t  Y i,k,N s − Y i,k,N s  × N X j =1 ( ˆ b y ( X j,N ,v N, ∗ s , µ N ,v N, ∗ s , Y j,j,N s − R j, 1 s ) Y i,j,N s Z j,j,k,N s − ˆ b y  X j,N ,v N, ∗ s , µ N ,v N, ∗ s , Y j,j,N s  Y i,j,N s Z j,j,k,N s ) ds − 2 σ Z T t  Y i,k,N s − Y i,k,N s  N X j =1  Z i,j,k,N s − Z i,j,k,N s  dW j s − σ 2 Z T t N X j =1  Z i,j,k,N s − Z i,j,k,N s  2 ds. (3.15) 22 Next, b y ( 3.3 ) w e ha ve U i,N x j ( t, x ) = U x ( t, x i , µ N x ) δ ij + 1 N U m ( t, x i , µ N x , x j ) (3.16) for an y i, j ∈ { 1 , 2 , . . . , N } , which also implies that U i,N x k x j ( t, x ) = U xx ( t, x i , µ N x ) δ ij δ ik + 1 N U mx ( t, x i , µ N x , x k ) δ ij + 1 N U mx ( t, x i , µ N x , x j ) δ ik − 1 N 2 U mm ( t, x i , µ N x , x j , x k ) + 1 N U mz 1 ( t, x i , µ N x , x j ) δ j k (3.17) for any i, j, k ∈ { 1 , 2 , . . . , N } , so the b oundedness of all the second order deriv atives except U mz 1 in Assumption 2.2 and the b oundedness of U mz 1 in Assumption 2.5 imply that the pro cesses Y i,j,N s and Z i,j,k,N s defined in ( 3.10 ) satisfy   Y i,j,N s   ≤ C  1 N + δ ij  ,   Z i,j,k,N s   ≤ C  1 N 2 + 1 N δ ij + 1 N δ ik + 1 N δ j k + δ ij δ ik  . (3.18) Hence, w e can write    ˆ b x ( X k,N ,v N, ∗ s , µ N ,v N, ∗ s , Y k,k ,N s − R k, 1 s ) Y i,k,N s − ˆ b x  X k,N ,v N, ∗ s , µ N ,v N, ∗ s , Y k,k ,N s  Y i,k,N s    ≤    ˆ b x    ∞   Y i,k,N s − Y i,k,N s   +    ˆ b xy    ∞   Y k,k ,N s − R k, 1 s − Y k,k ,N s     Y i,k,N s   ≤ C    Y i,k,N s − Y i,k,N s   +    Y k,k ,N s − Y k,k ,N s   + 1 N   1 N + δ ik  (3.19) where the first inequality is obtained by b ounding the distance of the t w o terms in the first line b y the sum of the distances of those terms from ˆ b x ( X k,N ,v N, ∗ s , µ N ,v N, ∗ s , Y k,k ,N s ) Y i,k,N s , and by using then the mean v alue theorem, Assumption 2.2 , Assumption 2.5 and the O ( 1 N ) b ounds of the pro cesses | R i, 1 t | and | R i,k, 2 t | . Similarly , w e can obtain:    ˆ f x ( X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s − R i, 1 s ) δ ik − ˆ f x  X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s  δ ik    ≤    ˆ f xy    ∞   Y i,i,N s − R i, 1 s − Y i,i,N s   δ ik ≤ C    Y i,i,N s − Y i,i,N s   + 1 N  δ ik , (3.20)    ˆ f m ( X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s − R i, 1 s , X k,N ,v N, ∗ s ) − ˆ f m  X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s , X k,N ,v N, ∗ s     ≤    ˆ f my    ∞   Y i,i,N s − R i, 1 s − Y i,i,N s   ≤ C    Y i,i,N s − Y i,i,N s   + 1 N  , (3.21) 23    ˆ f y ( X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s − R i, 1 s ) Z i,i,k,N s − ˆ f y  X i,N ,v N, ∗ s , µ N ,v N, ∗ s , Y i,i,N s  Z i,i,k,N s    ≤    ˆ f y    ∞   Z i,i,k,N s − Z i,i,k,N s   +    ˆ f yy    ∞   Y i,i,N s − R i, 1 s − Y i,i,N s     Z i,i,k,N s   ≤ C    Z i,i,k,N s − Z i,i,k,N s   +    Y i,i,N s − Y i,i,N s   + 1 N   1 N + δ ik  , (3.22)    ˆ b m ( X j,N ,v N, ∗ s , µ N ,v N, ∗ s , Y j,j,N s − R j, 1 s , X k,N ,v N, ∗ s ) Y i,j,N s − ˆ b m  X j,N ,v N, ∗ s , µ N ,v N, ∗ s , Y j,j,N s , X k,N ,v N, ∗ s  Y i,j,N s    ≤    ˆ b m    ∞   Y i,j,N s − Y i,j,N s   +    ˆ b my    ∞   Y j,j,N s − R j, 1 s − Y j,j,N s     Y i,j,N s   ≤ C    Y i,j,N s − Y i,j,N s   +    Y j,j,N s − Y j,j,N s   + 1 N   1 N + δ ij  , (3.23) and finally    ˆ b y ( X j,N ,v N, ∗ s , µ N ,v N, ∗ s , Y j,j,N s − R j, 1 s ) Y i,j,N s Z j,j,k,N s − ˆ b y  X j,N ,v N, ∗ s , µ N ,v N, ∗ s , Y j,j,N s  Y i,j,N s Z j,j,k,N s    ≤    ˆ b y    ∞   Y i,j,N s Z j,j,k,N s − Y i,j,N s Z j,j,k,N s   +    ˆ b yy    ∞   Y j,j,N s − R j, 1 s − Y j,j,N s     Y i,j,N s Z j,j,k,N s   =    ˆ b y    ∞      Y i,j,N s − Y i,j,N s  Z j,j,k,N s +  Z j,j,k,N s − Z j,j,k,N s  Y i,j,N s +  Z j,j,k,N s − Z j,j,k,N s   Y i,j,N s − Y i,j,N s      +    ˆ b yy    ∞   Y j,j,N s − R j, 1 s − Y j,j,N s     Y i,j,N s Z j,j,k,N s   ≤    ˆ b y    ∞    Y i,j,N s − Y i,j,N s     Z j,j,k,N s   +   Z j,j,k,N s − Z j,j,k,N s     Y i,j,N s   +   Z j,j,k,N s − Z j,j,k,N s     Y i,j,N s − Y i,j,N s    +    ˆ b yy    ∞   Y j,j,N s − R j, 1 s − Y j,j,N s     Y i,j,N s Z j,j,k,N s   ≤ C    Y i,j,N s − Y i,j,N s    1 N + δ j k  +   Z j,j,k,N s − Z j,j,k,N s    1 N + δ ij  +   Z j,j,k,N s − Z j,j,k,N s     Y i,j,N s − Y i,j,N s   +    Y j,j,N s − Y j,j,N s   + 1 N   1 N 2 + δ ij + δ j k N + δ ij δ j k   . (3.24) 24 Plugging ( 3.19 ) - ( 3.24 ) in ( 3.15 ) we obtain  Y i,k,N t − Y i,k,N t  2 + σ 2 Z T t N X j =1  Z i,j,k,N s − Z i,j,k,N s  2 ds ≤ C ( Z T t  Y i,k,N s − Y i,k,N s  2 ds + Z T t   Y i,k,N s − Y i,k,N s     Y k,k ,N s − Y k,k ,N s   ds + Z T t   Y i,k,N s − Y i,k,N s   1 N  1 N + δ ik  ds + Z T t   Y i,k,N s − Y i,k,N s     Y i,i,N s − Y i,i,N s    1 N + δ ik  ds + Z T t   Y i,k,N s − Y i,k,N s     Z i,i,k,N s − Z i,i,k,N s   ds + Z T t   Y i,k,N s − Y i,k,N s   N X j =1 j  = i   Y i,j,N s − Y i,j,N s    1 N + δ j k  ds + Z T t   Y i,k,N s − Y i,k,N s   N X j =1 j  = i   Y j,j,N s − Y j,j,N s    1 N 2 + δ j k N  ds + Z T t   Y i,k,N s − Y i,k,N s   1 N N X j =1 j  = i   Z j,j,k,N s − Z j,j,k,N s   ds + Z T t   Y i,k,N s − Y i,k,N s   N X j =1   Z j,j,k,N s − Z j,j,k,N s     Y i,j,N s − Y i,j,N s   ds ) − 2 σ Z T t  Y i,k,N s − Y i,k,N s  N X j =1  Z i,j,k,N s − Z i,j,k,N s  dW j s (3.25) Next, we use the inequality 2 ab ≤ a 2 + b 2 , the Cauch y-Sc h w artz inequality and some trivial b ounds to con trol the terms that appear in the ab o ve if we set k = i . Sp ecifically , for every i ∈ { 1 , 2 , . . . , N } and an y ϵ > 0 w e obtain Z T t   Y i,i,N s − Y i,i,N s   1 N ds ≤ 1 2 Z T t  Y i,i,N s − Y i,i,N s  2 ds + T − t 2 N 2 , (3.26) Z T t   Y i,i,N s − Y i,i,N s     Z i,i,i,N s − Z i,i,i,N s   ds ≤ 1 2 ϵ Z T t  Y i,i,N s − Y i,i,N s  2 ds + ϵ 2 Z T t  Z i,i,i,N s − Z i,i,i,N s  2 ds, (3.27) 25 Z T t   Y i,i,N s − Y i,i,N s   N X j =1 j  = i   Y i,j,N s − Y i,j,N s    1 N + δ j i  ds = Z T t   Y i,i,N s − Y i,i,N s   1 N N X j =1 j  = i   Y i,j,N s − Y i,j,N s   ds ≤ 1 2 Z T t  Y i,i,N s − Y i,i,N s  2 ds + 1 2 Z T t 1 N N X j =1 j  = i  Y i,j,N s − Y i,j,N s  2 ds, (3.28) Z T t   Y i,i,N s − Y i,i,N s   N X j =1 j  = i   Y j,j,N s − Y j,j,N s    1 N 2 + δ j i N  ds = Z T t   Y i,i,N s − Y i,i,N s   1 N 2 N X j =1 j  = i   Y j,j,N s − Y j,j,N s   ds ≤ 1 2 Z T t  Y i,i,N s − Y i,i,N s  2 ds + 1 2 Z T t 1 N 3 N X j =1 j  = i  Y j,j,N s − Y j,j,N s  2 ds (3.29) and Z T t   Y i,i,N s − Y i,i,N s   1 N N X j =1 j  = i   Z j,j,i,N s − Z j,j,i,N s   ds ≤ 1 2 ϵ Z T t  Y i,i,N s − Y i,i,N s  2 ds + ϵ 2 Z T t 1 N N X j =1 j  = i  Z j,j,i,N s − Z j,j,i,N s  2 ds, (3.30) and since Y i,i,N s and Y i,i,N s are b ounded uniformly in N and i ∈ { 1 , 2 , . . . , N } due to ( 3.18 ) and the uniform b oundedness of U i,N x i pro vided b y Assumption 2.5 , w e ha v e also Z T t   Y i,i,N s − Y i,i,N s   N X j =1   Z j,j,i,N s − Z j,j,i,N s     Y i,j,N s − Y i,j,N s   ds = Z T t N X j =1   Y i,i,N s − Y i,i,N s     Y i,j,N s − Y i,j,N s     Z j,j,i,N s − Z j,j,i,N s   ds ≤ Z T t N X j =1 1 2 ϵ  Y i,i,N s − Y i,i,N s  2  Y i,j,N s − Y i,j,N s  2 ds + Z T t N X j =1 ϵ 2  Z j,j,i,N s − Z j,j,i,N s  2 ds 26 ≤ C 2 ϵ Z T t N X j =1  Y i,j,N s − Y i,j,N s  2 ds + ϵ 2 Z T t N X j =1  Z j,j,i,N s − Z j,j,i,N s  2 ds. (3.31) T aking k = i in ( 3.25 ), using ( 3.26 ) - ( 3.31 ), summing for all i ∈ { 1 , 2 , . . . , N } , and finally observing that Z i,j,i,N s = Z i,i,j,N s and Z i,j,i,N s = Z i,i,j,N s for all i, j ∈ { 1 , 2 , . . . , N } since the deriv ativ es of U i,N and U i,N in x i and x j can b e interc hanged due to the regularit y provided by Assumption 2.2 , w e find that N X i =1  Y i,i,N t − Y i,i,N t  2 +  σ 2 − C ϵ  Z T t N X i,j =1  Z i,i,j,N s − Z i,i,j,N s  2 ds ≤ C (  1 + 1 ϵ  Z T t N X i =1  Y i,i,N s − Y i,i,N s  2 ds + Z T t  1 N + 1 ϵ  N X i,j =1  Y i,j,N s − Y i,j,N s  2 ds + 1 N ) + 2 σ Z T t  Y i,i,N s − Y i,i,N s  N X i,j =1  Z i,i,j,N s − Z i,i,j,N s  dW j s . (3.32) The sto c hastic integrals in the ab ov e expression are martingales due to the b oundedness of Z i,i,j,N s and Z i,i,j,N s defined in ( 3.10 ), whic h follows from ( 3.18 ) and the b oundedness of the functions U i,N x i x j pro vided by Assumption 2.5 . Observing also that Z i,j s and Z i,j s defined in [ 17 ] are precisely Y i,j,N s and Y i,j,N s defined here resp ectively , by [ 17 , Equation (4.19)] w e ha ve that Z T t E   N X i,j =1  Y i,j,N s − Y i,j,N s  2 ds      F t   ≤ C N P - almost surely . Hence, taking exp ectation giv en F t on ( 3.32 ), picking ϵ to b e sufficiently small and performing the transformation s → T − s , we find that the function g N ( t ) :=      N X i =1  Y i,i,N T − t − Y i,i,N T − t  2      L ∞ (Ω) (3.33) satisfies g N ( t ) ≤ C Z t 0 g N ( s ) ds + C N (3.34) for all t ∈ [0 , T ] . Then, a simple application of Grönw all’s inequality gives g N ( t ) ≤ C N e C t for all t ∈ [0 , T ] , so that P - almost surely w e ha ve N X i =1  Y i,i,N t − Y i,i,N t  2 ≤ C N (3.35) for all t ∈ [0 , T ] and for some deterministic constant C > 0 . 27 By using now the mean v alue theorem, ( 3.5 ), ( 3.10 ), the triangle inequality , the elementary inequalit y ( a + b ) 2 ≤ 2( a 2 + b 2 ) and ( 3.35 ), we can b ound N X i =1 ( ˆ b  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , U i,N x i  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  − ˆ b  X i,N ,v N, ∗ t , µ N ,v N, ∗ t , U x  t, X i,N ,v N, ∗ t , µ N ,v N, ∗ t  ) 2 ≤    ˆ b y    2 ∞ N X i =1 ( U i,N x i  t, X 1 ,N ,v N, ∗ t , X 2 ,N ,v N, ∗ t , . . . , X N ,N,v N, ∗ t  − U x  t, X i,N ,v N, ∗ t , µ N ,v N, ∗ t  ) 2 ≤    ˆ b y    2 ∞ 2 k U m k 2 ∞ N + 2 N X i =1  Y i,i,N t − Y i,i,N t  2 ! ≤ C N (3.36) P -almost surely , so our pro of is complete. W e pro ceed now to proving the lemmata necessary for our second main result, i.e for Theorem 2.9 . Pr o of of L emma 2.15 . Borrowing the notation from [ 21 ], for γ , t > 0 we consider E 2 ,γ ( L ( Z t )) = E h e γ Z 2 t i (3.37) and we will first show that this is a finite and b ounded in t ∈ [0 , T ] quantit y for sufficiently small γ > 0 . Using ( 2.12 ) and the triangle inequality we get that | Z t | ≤ | Z 0 | + Z t 0 {| B ( s, 0 , L ( Z s )) | + L | Z s |} ds + sup r ∈ [0 ,T ] | W r | (3.38) for t ∈ [0 , T ] , so an application of Grönw all’s inequality gives | Z t | ≤ ( | Z 0 | + Z t 0 | B ( s, 0 , L ( Z s )) | ds + sup r ∈ [0 ,T ] | W r | ) e Lt (3.39) whic h can be used along with sup r ∈ [0 ,T ] | W r | ≤ sup r ∈ [0 ,T ] W r + sup r ∈ [0 ,T ] ( − W r ) and the Cauch y-Sch w artz inequalit y to give E 2 ,γ ( L ( Z t )) ≤ e 4 γ e 2 LT T  T 0 B 2 ( s, 0 , L ( Z s )) ds 28 × r E  e 12 γ e 2 LT Z 2 0  E h e 12 γ e 2 LT ( sup r ∈ [0 ,T ] W r ) 2 i E h e 12 γ e 2 LT ( sup r ∈ [0 ,T ] ( − W r ) ) 2 i . (3.40) W e know now that sup r ∈ [0 ,T ] W r and sup r ∈ [0 ,T ] ( − W r ) hav e the same distribution as the absolute v alue of a Gaussian random v ariable, so using also ( 2.11 ) we can deduce that the right-hand side of ( 3.40 ) is finite for sufficiently small γ > 0 . This gives the desired finiteness and b oundedness of E 2 ,γ ( L ( Z t )) , so b y Mark o v’s inequalit y w e hav e that P ( Z t > x ) = P  e γ Z 2 t > e γ x 2  ≤ E 2 ,γ ( L ( Z t )) e − γ x 2 ≤ C e − γ x 2 (3.41) for any t ∈ [0 , T ] , with C b eing e.g the right-hand side of ( 3.40 ) (which do es not dep end on t ). Moreo ver, the finiteness and b oundedness of E 2 ,γ ( L ( Z t )) allo ws us to recall [ 21 , Theorem 1.2] and deduce that there exist c, C > 0 such that P  W 1  µ N ,Z t , L ( Z t )  > x  ≤ C e − cN x 2 (3.42) for any t ∈ [0 , T ] . Then, ( 2.13 ) and ( 2.14 ) are obtained by applying the argumen t used in the pro of of [ 47 , Lemma 1.4] on ( 3.41 ) and ( 3.42 ) resp ectiv ely . W e will now b ound the momen ts of W 1 ( µ N ,Z N t , L ( Z t )) by using a coupling argument. F or p ≥ 1 , we hav e for each i ∈ { 1 , 2 , . . . , N } and all t ∈ [0 , T ] :     Z i,N t − Z i t  p    =     p Z t 0  Z i,N s − Z i s  p − 1  B  s, Z i,N s , µ N ,Z N s  − B  s, Z i s , L ( Z s )   ds     ≤ pL Z t 0   Z i,N s − Z i s   p − 1    Z i,N s − Z i s   + W 1  µ N ,Z N s , L ( Z s )  ds ≤ pL Z t 0   Z i,N s − Z i s   p ds + pL Z t 0   Z i,N s − Z i s   p − 1 W 1  µ N ,Z N s , µ N ,Z s  ds + pL Z t 0   Z i,N s − Z i s   p − 1 W 1  µ N ,Z s , L ( Z s )  ds ≤ (3 p − 2) L Z t 0   Z i,N s − Z i s   p ds + L Z t 0 W p 1  µ N ,Z N s , µ N ,Z s  ds + L Z t 0 W p 1  µ N ,Z s , L ( Z s )  ds (3.43) where we used ( 2.12 ), the triangle inequality for the W asserstein distance and the elementary in- equalit y pa p − 1 b ≤ ( p − 1) a p + b p . Denoting no w by Π the set of all p erm utations σ : { 1 , 2 , . . . , N } 7→ { 1 , 2 , . . . , N } , it is known that W p 1  µ N ,Z N s , µ N ,Z s  ≤ W p p  µ N ,Z N s , µ N ,Z s  = inf σ ∈ Π 1 N N X i =1    Z i,N t − Z σ ( i ) t    p ≤ 1 N N X i =1    Z i,N t − Z i t    p (3.44) 29 so taking the av erage of ( 3.43 ) ov er all i ∈ { 1 , 2 , . . . , N } we can obtain 1 N N X i =1    Z i,N t − Z i t    p ≤ (3 p − 1) L Z t 0 1 N N X i =1   Z i,N s − Z i s   p ds + L Z t 0 W p 1  µ N ,Z s , L ( Z s )  ds. (3.45) Applying Grön w all’s inequality on the abov e and using ( 3.44 ) again, we find that W p 1  µ N ,Z N s , µ N ,Z s  ≤ 1 N N X i =1    Z i,N t − Z i t    p ≤ e (3 p − 1) LT Z t 0 W p 1  µ N ,Z s , L ( Z s )  ds. (3.46) so using again the triangle inequality along with the inequalit y ( a + b ) p ≤ 2 p ( a p + b p ) w e obtain W p 1  µ N ,Z N s , L ( Z s )  ≤ 2 p W p 1  µ N ,Z t , L ( Z t )  + 2 p W p 1  µ N ,Z t , µ N ,Z N t  ≤ 2 p W p 1  µ N ,Z t , L ( Z t )  + 2 p e (3 p − 1) LT Z t 0 W p 1  µ N ,Z s , L ( Z s )  ds. (3.47) The desired result follows by taking exp ectations on the ab ov e and by recalling ( 2.14 ) Pr o of of L emma 2.16 . The result will b e immediate if we sho w that for any fixed N and any choice of the function H N : R N 7→ [0 , 1] we hav e:    E h H N  Z N , 1 t i − E h H N  Z N , 2 t i    ≤ 1 2 s Z T 0 E     B N , 2  s, Z N , 2 s  − B N , 1  s, Z N , 2 s     2 2  ds. (3.48) W e will first obtain the ab o v e when the co efficien t functions B N ,k ( · ) are b ounded, in which case the right-hand side is simply the relativ e entrop y of a measure c hange via Girsano v’s theorem that transforms the law of Z N , 1 in to that of Z N , 2 . T o do that, we define a Radon-Nikodym density pro cess E N = ( E N t ) t ∈ [0 ,T ] as E N t = exp  M N t − 1 2 h M N i t  (3.49) for t ∈ [0 , T ] , where M N t = Z t 0  B N , 2  s, Z N , 1 s  − B N , 1  s, Z N , 1 s  · d W N s . (3.50) By the b oundedness of B N ,k ( · ) and Novik o v’s condition, the sto c hastic exp onen tial ( E N t ) t ∈ [0 ,T ] is a martingale. By Girsanov’s theorem, the pro cess ˜ W N = ( ˜ W N t ) t ∈ [0 ,T ] with ˜ W N t = W N t − Z t 0  B N , 2  s, Z N , 1 s  − B N , 1  s, Z N , 1 s  ds (3.51) is an N -dimensional standard Brownian motion under the probability measure Q N that satisfies d Q N d P   F t = E N t , and we can write ( 2.16 ) as Z N , 1 t = Z N 0 + Z t 0 A N  s, Z N , 1 s   B N , 2  s, Z N , 1 s  ds + d ˜ W N s  + Z t 0 C N  s, Z N , 1 s  ds (3.52) 30 so that Z N , 1 s is the solution to ( 2.17 ) driven by ˜ W N instead of W . By pathwise uniqueness and th us uniqueness in la w of the solution to the SDE ( 2.17 ), the law of Z N , 1 t under Q N coincides with that of Z N , 2 t under P . Then, the relative entrop y of the measures Q N and P is given by D K L  Q N k P  = E Q N  log  d Q N d P  = E Q N  M N T − 1 2 h M N i T  = E Q N " Z T 0  B N , 2  s, Z N , 1 s  − B N , 1  s, Z N , 1 s  · d W N s − 1 2 Z T 0   ( B N , 2  s, Z N , 1 s  − B N , 1  s, Z N , 1 s    2 2 ds # = E Q N " Z T 0  B N , 2  s, Z N , 1 s  − B N , 1  s, Z N , 1 s  · d ˜ W N s + 1 2 Z T 0   B N , 2  s, Z N , 1 s  − B N , 1  s, Z N , 1 s    2 2 ds # = 1 2 E Q N " Z T 0   B N , 2  s, Z N , 1 s  − B N , 1  s, Z N , 1 s    2 2 ds # = 1 2 E " Z T 0   B N , 2  s, Z N , 2 s  − B N , 1  s, Z N , 2 s    2 2 ds # (3.53) Hence, writing k P − Q N k T V for the total v ariation distance of the probabilit y measures P and Q N , since H N tak es v alues in [0 , 1] , by Pinsker’s inequalit y w e ha v e that    E h H N  Z N , 1 t i − E h H N  Z N , 2 t i    =    E h H N  Z N , 1 t i − E Q N h H N  Z N , 1 t i    =     Z Ω H N  Z N , 1 t  · ( d P − d Q N )     ≤ k P − Q N k T V ≤ r 1 2 D K L ( Q N k P ) (3.54) so when the functions B N ,k ( · ) are b ounded, ( 3.48 ) follows by plugging ( 3.53 ) in to ( 3.54 ). When the functions B N ,k are not b ounded, one can consider the sequences of stopping times ( τ n,k ) n ∈ N with τ n,k := inf { t ≤ T : k Z N ,k t k 2 ≥ n } for k ∈ { 1 , 2 } , and work with the stopp ed pro cesses Z N ,k,n t := Z N ,k t ∧ τ n,k . Then, for every n ∈ N and k ∈ { 1 , 2 } , ( Z N ,k,n t ) t ∈ [0 ,T ] satisfies: Z N ,k,n t = Z N 0 + Z t 0 A N ,n  s, Z N ,k,n s   B N ,k,n  s, Z N ,k,n s  ds + d W N s  + Z t 0 C N ,n  s, Z N ,k,n s  ds, (3.55) 31 with the co efficien ts A N ,n ( s, z ) := A N ( s, z ) 1 {∥ z ∥ 2

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