Large systems of diffusions interacting through their ranks - PowerPoint PPT Presentation

Large systems of diffusions interacting through their ranks Mykhaylo Shkolnikov INTECH Investment Management/UC Berkeley June 6, 2012

Outline Setting and history 1 Questions 2 Previous results 3 Our results 4 Proof sketch 5 Under the rug 6

Setting Fix a natural number N ∈ N , real numbers b 1 , b 2 , . . . , b N and positive real number σ 1 , σ 2 , . . . , σ N . Consider a system of interacting diffusions (particles) on R : N N � � d X i ( t ) = b j 1 { X i ( t )= X ( j ) ( t ) } d t + σ j 1 { X i ( t )= X ( j ) ( t ) } d W i ( t ) , j =1 j =1 where W 1 , W 2 , . . . , W N are i.i.d. standard Brownian motions and some initial values X 1 (0) , X 2 (0) , . . . , X N (0) are fixed.

Some historic remarks Model appeared ’87 in this form in paper by Bass and Pardoux in the context of filtering theory. They proved existence and uniqueness in law. Model reappeared in stochastic porfolio theory (’02 book by Fernholz , ’06 survey by Fernholz and Karatzas ): diffusions X 1 , X 2 , . . . , X N represent logarithmic capitalizations : log ( stock price × number of stocks ) . (1) Model assumes that dynamics depends only on ranks . True in the long run : explains following picture.

Capital distribution curves 1e � 02 1e � 04 1e � 06 1e � 08 1 5 10 50 100 500 1000 5000 Figure: Capital distribution curves 1929-1989

Concentration results Chatterjee and Pal ’07: for particle system above, vector of market weights e X i ( t ) � � j =1 e X j ( t ) , i = 1 , 2 , . . . , N (2) � N is a Markov process and its invariant distribution concentrates around curves of above type as N → ∞ . Moreover, the limit N → ∞ is given by a Poisson-Dirichlet point process of first kind. Pal , S. ’10 and Ichiba , Pal , S. ’11: strong concentration of paths of market weights on any [0 , t ] as N → ∞ and fast mean-reversion as t → ∞ for any fixed N ∈ N .

Back to particle system All previous results for market weights , which correspond to the spacings process in N N � � d X i ( t ) = b j 1 { X i ( t )= X ( j ) ( t ) } d t + σ j 1 { X i ( t )= X ( j ) ( t ) } d W i ( t ) , j =1 j =1 What about the particle system itself? Can we understand evolution of particle density : N ̺ N ( t ) = 1 � δ X i ( t ) , t ≥ 0 . N i =1

Preliminaries Here: will look at limit N → ∞ , which corresponds to a hydrodynamic limit . First question: how to choose drift and diffusion coefficients for different N to have a meaningful limit? Crucial observation: for fixed N the particle system can be written as d X i ( t ) = b ( F ̺ N ( t ) ( X i ( t ))) d t + σ ( F ̺ N ( t ) ( X i ( t ))) d W i ( t ) for functions b : [0 , 1] → R , σ : [0 , 1] → (0 , ∞ ). ⇒ particle system is of mean-field type.

Aside on mean-field models Systems of the form d X i ( t ) = ˆ b ( ̺ N ( t ) , X i ( t )) d t + ˆ σ ( ̺ N ( t ) , X i ( t )) d W i ( t ) (3) appeared in statistical physics . See: McKean ’69, Funaki ’84, Oelschlager ’84, Nagasawa, Tanaka ’85, ’87, Sznitman ’86, Leonard ’86, Dawson, G¨ artner ’87, G¨ artner ’88. In Gartner ’88, limit lim N →∞ ̺ N ( t ) is obtained under two assumptions .

Previous results I artner ’88) Fix T > 0 and suppose ˆ Theorem. (G¨ b , ˆ σ continuous (!) , ˆ σ strictly positive . Let Q N be the law of ̺ N ( t ), t ∈ [0 , T ] on C ([0 , T ] , M 1 ( R )). Then the sequence Q N , N ∈ N is tight. Moreover, under any limit point Q ∞ : � t ∀ f : ( ̺ ( t ) , f ) − ( ̺ (0) , f ) = ( ̺ ( s ) , L ̺ ( s ) f ) d s (4) 0 for all t ∈ [0 , T ] almost surely. Hereby: � ( ̺ ( t ) , f ) = f d ̺ ( t ) R b ( ̺ ( s ) , · ) f ′ + 1 L ̺ ( s ) f = ˆ σ ( ̺ ( s ) , · ) 2 f ′′ . 2 ˆ

Previous results II In particular, if � t ∀ f : ( ̺ ( t ) , f ) − ( ̺ (0) , f ) = ( ̺ ( s ) , L ̺ ( s ) f ) d s (5) 0 has a unique solution ̺ ∞ in C ([0 , T ] , M 1 ( R )), then it must hold ̺ N → ̺ ∞ , N → ∞ in probability . (6) This is not known in general (some conditions in work of Sznitman ’86)!

Diffusions interacting through their mean-field, intuition How can one guess the limiting dynamics ? Suppose we already knew ̺ N → ̺ ∞ with ̺ ∞ deterministic Then for large N the system of diffusions should behave as d X i ( t ) = ˆ b ( ̺ ∞ ( t ) , X i ( t )) d t + ˆ σ ( ̺ ∞ ( t ) , X i ( t )) d W i ( t ) (7) Thus, the empirical measure converges to the law of d X ( t ) = ˆ b ( ̺ ∞ ( t ) , X ( t )) d t + ˆ σ ( ̺ ∞ ( t ) , X ( t )) d W ( t ) (8) Ito’s formula for f ( X ( t )) and L ( X ( t )) = ̺ ∞ ( t ) imply: � t ( ̺ ∞ ( t ) , f ) − ( ̺ ∞ (0) , f ) = ( ̺ ∞ ( s ) , L ̺ ∞ ( s ) f ) d s 0

Previous results III artner ’87) Fix T > 0 and suppose ˆ Theorem. (Dawson, G¨ b continuous , σ ≡ 1 (!) ˆ Then the sequence ( ̺ N ( t ), t ∈ [0 , T ]), N ∈ N satisfies a large deviations principle on C ([0 , T ] , M 1 ( R )) with the good rate function � T t g + 1 � � ( γ ( t ) , R γ 2( g x ) 2 ) d t I ( γ ) = sup ( γ ( T ) , g ) − ( γ (0) , g ) − 0 g ∈S and scale N . Hereby: b ( ̺ ( s ) , · ) g x + 1 R γ t g = g t + ˆ 2 g xx .

Relation to Burger’s equation Remarks: Goodness of rate function + LDP imply: ̺ N will concentrate around the set { γ : I ( γ ) = 0 } . If we apply this result with ˆ b ( ̺ ( s ) , · ) = − F ̺ ( s ) ( · ) ( discontinuity! ), then only point with I ( γ ) = 0 is the one, whose path of cdfs R ( t , · ) = F γ ( t ) ( · ) is the unique weak solution of viscous Burger’s equation: R t = − 1 2 ( R 2 ) x + 1 2 R xx . I.e.: particle system approximation of R . Same result for a particle system with local time interactions in Sznitman ’86.

Our results I Theorem 1. (Dembo, Krylov, S., Varadhan, Zeitouni ’12) Fix T > 0 and 2 σ 2 nice. suppose ˆ σ = σ ( F ̺ ( t ) ( x )); b and A := 1 b ( ̺ ( t ) , x ) = b ( F ̺ ( t ) ( x )), ˆ Then, ( ̺ N ( t ), t ∈ [0 , T ]), N ∈ N satisfies an LDP on C ([0 , T ] , M 1 ( R )) with scale N and good rate function J defined by J ( R ) = 1 � σ ( R ) R t − ( A ( R ) R x ) x − b ( R ) 2 � σ ( R )( R x ) 1 / 2 � � � A ( R )( R x ) 1 / 2 2 2 � L 2 ( R T ) for all R ∈ C b ( R T ) with R t , R x , R xx ∈ L 3 / 2 ( R T ), R x ∈ L 3 ( R T ), R x having finite (1 + ε ) moment, t �→ ( R x ( t , · ) , g ( t , · )) abs. cont.; J = ∞ otherwise. Hereby, R = F γ ( · ) ( · ).

Our results II Consequences: Goodness of rate function and LDP imply that ̺ N concentrates around the set { γ : J ( γ ) = 0 } . The only path γ with J ( γ ) = 0 corresponds to the unique weak solution of the generalized porous medium equation with convection: R t = Σ( R ) xx + Θ( R ) x . Hence, we found a particle system approximation for the solution of the latter, which converges exponentially fast.

Our results III In the course of the proof show the following regularity result in nonlinear PDEs: Theorem 2. Consider a weak solution of the Cauchy problem for the tilted generalized porous medium equation : R t − ( A ( R ) R x ) x = h A ( R ) R x , R (0 , · ) = R 0 . such that R ∈ C b ( R T ) and R x ( t , · ) d x is a probability measure for every t . R T h 2 R x d m < ∞ and R x has finite (1 + ε ) moment, then R t , R x , R xx � If exist as elements of L 3 / 2 ( R T ), R x ∈ L 3 ( R T ) and R 2 R 2 � � xx t d m < ∞ , d m < ∞ . R x R x R T R T

Proof sketch, general principles I Localization: LDP holds, if we can show weak/local LDP : 1 N log P ( ̺ N ∈ C ) ≤ − inf lim sup γ ∈ C J ( γ ) for all compacts C , N →∞ 1 N log P ( ̺ N ∈ U ) ≥ − inf lim inf γ ∈ U J ( γ ) for all open sets U N →∞ and exponential tightness : 1 N log P ( ̺ N / ∀ K > 0 ∃ C K compact : lim sup ∈ C K ) ≤ − K . N →∞

Proof sketch, general principles II Alternative characterization of weak/local LDP : 1 N log P ( ̺ N ∈ B ( γ, δ )) ≤ − J ( γ ) , ∀ γ : lim δ ↓ 0 lim sup N →∞ 1 N log P ( ̺ N ∈ B ( γ, δ )) ≥ − J ( γ ) ∀ γ : lim δ ↓ 0 lim inf N →∞ What we prove: Local upper bound holds with a Dawson-G¨ artner type rate function I Local lower bound holds with the desired rate function J J ≤ I .

Proof sketch, upper bound around a path γ Appropriate variational problem : � T t g + 1 � � ( γ ( t ) , R γ 2 A ( R )( g x ) 2 ) d t I ( γ ) = sup ( γ ( T ) , g ) − ( γ (0) , g ) − , 0 g ∈S where R γ t g = g t + b ( R ) g x + A ( R ) g xx , R = F γ ( · ) ( · ). Why appropriate ? On the event ̺ N ∈ B ( γ, δ ), our particle system is close to solution of d Y i ( t ) = b ( F γ ( t ) ( Y i ( t ))) d t + σ ( F γ ( t ) ( Y i ( t ))) d W i ( t ) , i = 1 , 2 , . . . , N . on exponential scale .

Proof sketch, upper bound around a path γ Pick test function g , apply Itˆ o’s formula : d g ( t , Y i ( t )) = ( g t + b ( R ) g x + A ( R ) g xx )( t , Y i ( t )) d t + g x ( t , Y i ( t )) σ ( F γ ( t ) ( Y i ( t ))) d W i ( t ) . Hence, d ( ̺ N Y ( t ) , g ( t , · )) = ( ̺ N Y ( t ) , g t + b ( R ) g x + A ( R ) g xx ) d t N + 1 � g x ( t , Y i ( t )) σ ( F γ ( t ) ( Y i ( t ))) d W i ( t ) . N i =1 Note: martingale part of order 1 N . Freidlin-Wentzell type problem!