P ROBLEM 1 How to show concentration of the shape of market weights? - PowerPoint PPT Presentation

T RANSPORT INEQUALITIES FOR STOCHASTIC PROCESSES Soumik Pal University of Washington, Seattle Jun 6, 2012

I NTRODUCTION Three problems about rank-based processes

T HE A TLAS MODEL ◮ Define ranks: x ( 1 ) ≥ x ( 2 ) . . . ≥ x ( n ) . Fix δ > 0. ◮ SDE in R n : � t 1 { X i ( s ) = X ( n ) ( s ) } ds + W i ( t ) , X i ( t ) = x 0 + δ ∀ i . 0 ◮ The market weight: S i ( t ) = exp ( X i ( t )) , S i µ i ( t ) = ( t ) . S 1 + S 2 + . . . + S n ◮ Banner, Fernholz, Karatzas, P .- (Pitman, Chatterjee), Shkolnikov, Ichiba and several more.

A CURIOUS SHAPE Power law decay of real market weights with rank: 1e � 02 ◮ log µ ( i ) vs. log i . 1e � 04 ◮ Dec 31, 1929 - 1999 . 1e � 06 ◮ Includes all NYSE, AMEX, and NASDAQ. 1e � 08 1 5 10 50 100 500 1000 5000 Figure 1: Capital distribution curves: 1929–1999

P ROBLEM 1 ◮ How to show concentration of the shape of market weights? ◮ Fix J ≪ N . Linear regression through ( log i , log µ ( i ) ( t )) , 1 ≤ i ≤ J . ◮ Slope − α ( t ) . ◮ Estimate fluctuation of the process { α ( s ) , 0 ≤ s ≤ T } .

P ROBLEM 2 ◮ Lipschitz functions F 1 ( T , B ( T )) , . . . , F d ( T , B ( T )) . ◮ Define M i ( t ) := E [ F i ( T , B ( T )) | B ( t )] . ◮ Suppose � � sup M i ( t ) ≤ a ( t ) , 0 ≤ t ≤ T ≥ 1 / 2 . P i ◮ What is √ � � sup t , 1 ≤ t ≤ T | sup M i ( 1 ) > a ( 1 ) P M i ( t ) > a ( t ) + α ? i i

P ROBLEM 3 ◮ Back to rank-based models. ◮ Suppose V π ( t ) wealth ( V π ( 0 ) = 1 ) - portfolio π . ◮ π = µ - market portfolio. ◮ How does V π compare with V µ ? P ( V π ( t ) / V µ ( t ) ≥ a ( t )) ≤ exp ( − r ( t )) , explicit a ( t ) and r ( t ) .

And now the answers ...

P ROBLEM 1: FLUCTUATION OF SLOPE T HEOREM (P.-’10, P.-S HKOLNIKOV ’10) Suppose market weights are running at equilibrium. Take T = N /δ 2 . α = sup 0 ≤ s ≤ T α ( s ) . Let ¯ √ − r 2 � � � � ≤ 2 exp P α > m α + r ¯ N . 2 σ 2 Here m α = median and σ 2 = σ 2 ( δ, J ) .

P ROBLEM 2: BAD PRICES T HEOREM (P. ’12) For some absolute constant C > 0 : √ � � ≈ CT − α 2 / 8 . sup t , 1 ≤ t ≤ T | sup M i ( 1 ) > a ( 1 ) M i ( t ) > a ( t ) + α P i i Compare with square-root boundary crossing.

P ROBLEM 3: P ERFORMANCE OF PORTFOLIOS ◮ Symmetric functionally generated portfolio G . ◮ π depends only on market weights. ◮ Market, diversity-weighted, entropy-weighted portfolios. T HEOREM (I CHIBA -P.-S HKOLNIKOV ’11) Let R ( t ) = V π ( t ) / V µ ( t ) . R ( t ) ≥ c + G ( µ ( t )) / G ( µ ( 0 )) ≤ exp − α + t � � � � P R ( t ) ≤ c − G ( µ ( t )) / G ( µ ( 0 )) ≤ exp − α − t � � � � P . Here c ± , α ± explicit.

Transportation - Entropy - Information Inequalities

T RANSPORTATION I NEQUALITIES TCI (Ω , d ) - metric space. P , Q - prob measures. π [ Ed p ( X , X ′ )] 1 / p . W p ( Q , P ) = inf

T RANSPORTATION I NEQUALITIES TCI (Ω , d ) - metric space. P , Q - prob measures. π [ Ed p ( X , X ′ )] 1 / p . W p ( Q , P ) = inf ◮ P satisfies T p if ∃ C > 0: � 2 CH ( Q | P ) . W p ( Q , P ) ≤ ◮ H ( Q | P ) = E Q log ( dQ / dP ) or ∞ .

T RANSPORTATION I NEQUALITIES TCI (Ω , d ) - metric space. P , Q - prob measures. π [ Ed p ( X , X ′ )] 1 / p . W p ( Q , P ) = inf ◮ P satisfies T p if ∃ C > 0: � 2 CH ( Q | P ) . W p ( Q , P ) ≤ ◮ H ( Q | P ) = E Q log ( dQ / dP ) or ∞ . ◮ Related: Bobkov and Götze, Bobkov-Gentil-Ledoux, Dembo, Gozlan-Roberto-Samson, Otto and Villani, Talagrand.

M ARTON ’ S ARGUMENT ◮ T p , p ≥ 1 ⇒ Gaussian concentration of Lipschitz functions. ◮ If f : Ω → R - Lipschitz. | f ( x ) − f ( y ) | ≤ σ d ( x , y ) . ◮ Then f has Gaussian tails: P ( | f − m f | > r ) ≤ 2 e − r 2 / 2 C σ 2 . ◮ Fix p = 2 from now on.

Idea of proof for Problem 1

T HE W IENER MEASURE ◮ Consider Ω = C [ 0 , T ] , d ( ω, ω ′ ) = sup 0 ≤ s ≤ T | ω ( s ) − ω ′ ( s ) | . ◮ (Feyel-Üstünel ’04, P . ’10) P = Wiener measure satisfies T 2 with C = T . ◮ Related: Djellout-Guillin-Wu, Fang-Shao, Fang-Wang-Wu, Wu-Zhang.

P ROOF ◮ Proof: If Q ≪ P , then by Girsanov d ω ( t ) = b ( t , ω ) dt + d β ( t ) . Here β ∼ P . � 1 / 2 ≤ E Q d 2 ( ω, β ) ◮ W 2 ( Q , P ) ≤ � � 2 TH ( Q | P ) .

E XAMPLES ◮ How to show local time at zero has Gaussian tail? ◮ L 0 ( T ) is not Lipschitz w.r.t uniform norm. ◮ Lévy representation: L 0 ( T ) = − inf 0 ≤ s ≤ t β ( s ) ∧ 0 . ◮ Lipschitz function of the entire path. Thus P ( | L 0 ( T ) − m T | > r ) ≤ 2 e − r 2 / 2 T .

BM IN R n ◮ Multidimensional Wiener measure satisfies T 2 . ◮ Uniform metric n d 2 ( ω, ω ′ ) = 1 | ω ( s ) − ω ′ ( s ) | 2 . � sup n 0 ≤ s ≤ T i = 1 ◮ Skorokhod map S : BM in R n �→ RBM in polyhedra . ◮ Deterministic map. Rather abstract and complicated. ◮ But Lipschitz.

T HE L IPSCHITZ CONSTANT T HEOREM (P. - S HKOLNIKOV ’10) The Lipschitz constant of S is ≤ 2 n 5 / 2 . ◮ The slope α ( t ) is a linear map. BM on R n → RBM on wedge → slope of regression . ◮ Evaluate Lipschitz constant. Estimate concentration.

Idea of proof for Problem 2

A DIFFERENT METRIC ◮ For ω, ω ′ ∈ C d [ 0 , ∞ ) : σ r = inf { t ≥ 0 : σ r ( ω, ω ′ ) > r } . ◮ Consider ϕ : R + → R + � ∞ � � ϕ 2 ( s ) ds ≤ 1 ϕ ≥ 0 , ϕ ↓ , Φ 1 := . 0 ◮ (P . ’12) A metric on paths: � 1 / 2 � ∞ � ρ ( ω, ω ′ ) := sup ϕ ( σ r ) dr . 0 ϕ ∈ Φ 1

G ENERALIZED TCI T HEOREM (P. ’12) P - multidimension Wiener measure. � 2 H ( Q | P ) . 4 W 2 ( Q , P ) ≤ With respect to ρ .

A N EXAMPLE ◮ P -Wiener measure. Two event processes: 1 ≤ t ≤ T . √ √ s , 1 ≤ s ≤ T β ( s ) ≥ 2 s , 1 ≤ s ≤ T � � � � A T = β ( s ) ≤ B T = .

A N EXAMPLE ◮ P -Wiener measure. Two event processes: 1 ≤ t ≤ T . √ √ s , 1 ≤ s ≤ T β ( s ) ≥ 2 s , 1 ≤ s ≤ T � � � � A T = β ( s ) ≤ B T = . ◮ Let Q 1 = P ( · | A T ) , Q 2 = P ( · | B T ) .

A N EXAMPLE ◮ P -Wiener measure. Two event processes: 1 ≤ t ≤ T . √ √ s , 1 ≤ s ≤ T β ( s ) ≥ 2 s , 1 ≤ s ≤ T � � � � A T = β ( s ) ≤ B T = . ◮ Let Q 1 = P ( · | A T ) , Q 2 = P ( · | B T ) . ◮ Couple ( X , Y ) ∼ ( Q 1 , Q 2 ) . 1 ≤ s ≤ T . σ √ s ( X , Y ) ≤ s ,

A N EXAMPLE ◮ P -Wiener measure. Two event processes: 1 ≤ t ≤ T . √ √ s , 1 ≤ s ≤ T β ( s ) ≥ 2 s , 1 ≤ s ≤ T � � � � A T = β ( s ) ≤ B T = . ◮ Let Q 1 = P ( · | A T ) , Q 2 = P ( · | B T ) . ◮ Couple ( X , Y ) ∼ ( Q 1 , Q 2 ) . 1 ≤ s ≤ T . σ √ s ( X , Y ) ≤ s , ◮ ϕ ↓ and ≥ 0: � √ � T � T T ϕ ( σ √ s ) ds ϕ ( s ) ds ϕ ( σ r ) dr = 2 √ s ≥ 2 √ s . 1 1 1

A N EXAMPLE ◮ P -Wiener measure. Two event processes: 1 ≤ t ≤ T . √ √ s , 1 ≤ s ≤ T β ( s ) ≥ 2 s , 1 ≤ s ≤ T � � � � A T = β ( s ) ≤ B T = . ◮ Let Q 1 = P ( · | A T ) , Q 2 = P ( · | B T ) . ◮ Couple ( X , Y ) ∼ ( Q 1 , Q 2 ) . 1 ≤ s ≤ T . σ √ s ( X , Y ) ≤ s , ◮ ϕ ↓ and ≥ 0: � √ � T � T T ϕ ( σ √ s ) ds ϕ ( s ) ds ϕ ( σ r ) dr = 2 √ s ≥ 2 √ s . 1 1 1 ◮ Take 2 1 2 √ s 1 { 1 ≤ s ≤ T } . ϕ ( s ) = � log T

E XAMPLE CONTD . ◮ Thus 2 ( Q 1 , Q 2 ) ≥ 1 W 2 � log T . 2

E XAMPLE CONTD . ◮ Thus 2 ( Q 1 , Q 2 ) ≥ 1 W 2 � log T . 2 1 log T ≤ W 2 ( Q 1 , Q 2 ) ≤ W 2 ( Q 1 , P ) + W 2 ( Q 2 , P ) � 4 √ ◮ 2

E XAMPLE CONTD . ◮ Thus 2 ( Q 1 , Q 2 ) ≥ 1 W 2 � log T . 2 1 log T ≤ W 2 ( Q 1 , Q 2 ) ≤ W 2 ( Q 1 , P ) + W 2 ( Q 2 , P ) � 4 √ ◮ 2 � 2 H ( Q 1 | P ) + � 2 H ( Q 2 | P ) 4 4 ≤

E XAMPLE CONTD . ◮ Thus 2 ( Q 1 , Q 2 ) ≥ 1 W 2 � log T . 2 1 log T ≤ W 2 ( Q 1 , Q 2 ) ≤ W 2 ( Q 1 , P ) + W 2 ( Q 2 , P ) � 4 √ ◮ 2 � 2 H ( Q 1 | P ) + � 2 H ( Q 2 | P ) 4 4 ≤ � � 1 1 2 log 2 log P ( B T ) . ≤ 4 P ( A T ) + 4

Idea of proof for problem 3.

T RANSPORTATION -I NFORMATION INEQUALITY ◮ E - Dirichlet form. Fisher Information: √ √ if d ν = fd µ. I ( ν | µ ) := E ( f , f ) , ◮ µ satisfies W 1 I ( c ) inequality if W 2 1 ( ν, µ ) ≤ 4 c 2 I ( ν | µ ) , ∀ ν.

T RANSPORTATION -I NFORMATION INEQUALITY ◮ E - Dirichlet form. Fisher Information: √ √ if d ν = fd µ. I ( ν | µ ) := E ( f , f ) , ◮ µ satisfies W 1 I ( c ) inequality if W 2 1 ( ν, µ ) ≤ 4 c 2 I ( ν | µ ) , ∀ ν. ◮ Allows precise control of additive functionals.

P OINCARÉ INEQUALITIES T HEOREM (G UILLIN ET AL .) Consider W 1 ( ν, µ ) = � ν − µ � TV . ( X t , t ≥ 0 ) Markov - invariant distribution µ . Suppose µ - Poincaré ineq. Then W 1 I holds.

P OINCARÉ INEQUALITIES T HEOREM (G UILLIN ET AL .) Consider W 1 ( ν, µ ) = � ν − µ � TV . ( X t , t ≥ 0 ) Markov - invariant distribution µ . Suppose µ - Poincaré ineq. Then W 1 I holds. Gaps of rank-based processes stationary. Poincaré ineq holds.

Thank you Ioannis. Happy birthday.