TRANSPORT INEQUALITIES FOR STOCHASTIC PROCESSES Soumik Pal University of Washington, Seattle Jun 6, 2012
INTRODUCTION Three problems about rank-based processes
THE ATLAS MODEL Define ranks: x (1) x (2)... x (n). Fix δ > 0. SDE in R n : t X i (t) = x 0 + δ 1{X i (s) = X (n) (s)} ds + W i (t), i. 0 The market weight: S i (t) = exp(x i (t)), µ i (t) = S i S 1 + S 2 +... + S n (t). Banner, Fernholz, Karatzas, P.- (Pitman, Chatterjee), Shkolnikov, Ichiba and several more.
A CURIOUS SHAPE Power law decay of real market weights with rank: 1e!08 1e!06 1e!04 1e!02 log µ (i) vs. log i. Dec 31, 1929-1999. Includes all NYSE, AMEX, and NASDAQ. 1 5 10 50 100 500 1000 5000 Figure 1: Capital distribution curves: 1929 1999 cesses represented by continuous semimartingales (see, e.g., Duffie (1992) or Karatzas and Shreve (1998)). The representation of market weights in terms of continuous semimartingales is straightforward, but in order to represent the ranked market weights, it is necessary to use semimartingale local times to capture the behavior when ranks change. The methodology for this analysis was developed in Fernholz (2001), and is outlined here in an Appendix. By using the representation of ranked market weights given in Fernholz (2001), we are able to determine the asymptotic behavior of the capital distribution. For a market with a stable capital distribution, this asymptotic behavior provides insight into the steady-state structure of the market. We shall assume that we operate in a continuously-traded, frictionless market in which the stock prices vary continuously and the companies pay no dividends. We assume that companies neither enter nor leave the market, nor do they merge or break up, and that the total number of shares of a
PROBLEM 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 }.
PROBLEM 2 Lipschitz functions F 1 (T, B(T )),..., F d (T, B(T )). Define Suppose What is ( P sup i M i (t) := E [F i (T, B(T )) B(t)]. ( ) P sup M i (t) a(t), 0 t T i 1/2. M i (t) > a(t) + α ) t, 1 t T sup M i (1) > a(1)? i
PROBLEM 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...
PROBLEM 1: FLUCTUATION OF SLOPE THEOREM (P.- 10, P.-SHKOLNIKOV 10) Suppose market weights are running at equilibrium. Take T = N/δ 2. Let ᾱ = sup 0 s T α(s). ( P ᾱ > m α + r ) N 2 exp ( r 2 ) 2σ 2. Here m α =median and σ 2 = σ 2 (δ, J).
PROBLEM 2: BAD PRICES THEOREM (P. 12) For some absolute constant C > 0: ( P sup M i (t) > a(t) + α ) t, 1 t T sup M i (1) > a(1) i i Compare with square-root boundary crossing. CT α2 /8.
PROBLEM 3: PERFORMANCE OF PORTFOLIOS Symmetric functionally generated portfolio G. π depends only on market weights. Market, diversity-weighted, entropy-weighted portfolios. THEOREM (ICHIBA-P.-SHKOLNIKOV 11) Let R(t) = V π (t)/v µ (t). Here c ±, α ± explicit. P ( 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 ].
Transportation - Entropy - Information Inequalities
TRANSPORTATION INEQUALITIES TCI (Ω, d) - metric space. P, Q - prob measures. W p (Q, P) = inf π [Ed p (X, X )] 1/p.
TRANSPORTATION INEQUALITIES TCI (Ω, d) - metric space. P, Q - prob measures. P satisfies T p if C > 0: W p (Q, P) = inf π [Ed p (X, X )] 1/p. W p (Q, P) 2CH(Q P). H(Q P) = E Q log(dq/dp) or.
TRANSPORTATION INEQUALITIES TCI (Ω, d) - metric space. P, Q - prob measures. P satisfies T p if C > 0: W p (Q, P) = inf π [Ed p (X, X )] 1/p. W p (Q, P) 2CH(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.
MARTON S ARGUMENT T p, p 1 Gaussian concentration of Lipschitz functions. If f : Ω R - Lipschitz. f (x) f (y) σd(x, y). Then f has Gaussian tails: P ( f m f > r) 2e r 2 /2Cσ 2. Fix p = 2 from now on.
Idea of proof for Problem 1
THE WIENER 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.
PROOF Proof: If Q P, then by Girsanov dω(t) = b(t, ω)dt + dβ(t). Here β P. W 2 (Q, P) [ E Q d 2 (ω, β) ] 1/2 2TH(Q P).
EXAMPLES 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) 2e r 2 /2T.
BM IN R n Multidimensional Wiener measure satisfies T 2. Uniform metric d 2 (ω, ω ) = 1 n n i=1 sup 0 s T ω(s) ω (s) 2. Skorokhod map S : BM in R n RBM in polyhedra. Deterministic map. Rather abstract and complicated. But Lipschitz.
THE LIPSCHITZ CONSTANT THEOREM (P. - SHKOLNIKOV 10) The Lipschitz constant of S is 2n 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, ): Consider ϕ : R + R + Φ 1 := σ r = inf {t 0 : σ r (ω, ω ) > r}. { ϕ 0, ϕ, (P. 12) A metric on paths: ρ(ω, ω ) := [ sup ϕ Φ 1 0 0 } ϕ 2 (s)ds 1. ϕ (σ r ) dr] 1/2.
GENERALIZED TCI THEOREM (P. 12) P - multidimension Wiener measure. With respect to ρ. W 2 (Q, P) 4 2H(Q P).
AN EXAMPLE P-Wiener measure. Two event processes: 1 t T. A T = { β(s) s, 1 s T } B T = { β(s) 2 s, 1 s T }.
AN EXAMPLE P-Wiener measure. Two event processes: 1 t T. A T = { β(s) s, 1 s T } B T = { β(s) 2 s, 1 s T }. Let Q 1 = P ( A T ), Q 2 = P ( B T ).
AN EXAMPLE P-Wiener measure. Two event processes: 1 t T. A T = { β(s) s, 1 s T } B T = { β(s) 2 s, 1 s T }. Let Couple (X, Y ) (Q 1, Q 2 ). Q 1 = P ( A T ), Q 2 = P ( B T ). σ s (X, Y ) s, 1 s T.
AN EXAMPLE P-Wiener measure. Two event processes: 1 t T. A T = { β(s) s, 1 s T } B T = { β(s) 2 s, 1 s T }. Let Couple (X, Y ) (Q 1, Q 2 ). ϕ and 0: T 1 Q 1 = P ( A T ), Q 2 = P ( B T ). σ s (X, Y ) s, 1 s T. ϕ(σ r )dr = T 1 ϕ(σ s ) ds T 2 s ϕ(s) ds 1 2 s.
AN EXAMPLE P-Wiener measure. Two event processes: 1 t T. A T = { β(s) s, 1 s T } B T = { β(s) 2 s, 1 s T }. Let Couple (X, Y ) (Q 1, Q 2 ). ϕ and 0: Q 1 = P ( A T ), Q 2 = P ( B T ). σ s (X, Y ) s, 1 s T. Take T 1 ϕ(σ r )dr = ϕ(s) = T 1 ϕ(σ s ) ds T 2 s ϕ(s) ds 1 2 s. 2 1 log T 2 1{1 s T }. s
EXAMPLE CONTD. Thus W 2 2 (Q 1, Q 2 ) 1 2 log T.
EXAMPLE CONTD. Thus W 2 2 (Q 1, Q 2 ) 1 2 log T. 1 2 4 log T W 2 (Q 1, Q 2 ) W 2 (Q 1, P) + W 2 (Q 2, P)
EXAMPLE CONTD. Thus W 2 2 (Q 1, Q 2 ) 1 2 log T. 1 2 4 log T W 2 (Q 1, Q 2 ) W 2 (Q 1, P) + W 2 (Q 2, P) 4 2H(Q 1 P) + 4 2H(Q 2 P)
EXAMPLE CONTD. Thus W 2 2 (Q 1, Q 2 ) 1 2 log T. 1 2 4 log T W 2 (Q 1, Q 2 ) W 2 (Q 1, P) + W 2 (Q 2, P) 4 2H(Q 1 P) + 4 2H(Q 2 P) 2 4 log 1 P(A T ) + 4 2 log 1 P(B T ).
Idea of proof for problem 3.
TRANSPORTATION-INFORMATION INEQUALITY E - Dirichlet form. Fisher Information: I(ν µ) := E( f, f ), if dν = fdµ. µ satisfies W 1 I(c) inequality if W 2 1 (ν, µ) 4c 2 I (ν µ), ν.
TRANSPORTATION-INFORMATION INEQUALITY E - Dirichlet form. Fisher Information: I(ν µ) := E( f, f ), if dν = fdµ. µ satisfies W 1 I(c) inequality if W 2 1 (ν, µ) 4c 2 I (ν µ), ν. Allows precise control of additive functionals.
POINCARÉ INEQUALITIES THEOREM (GUILLIN 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.
POINCARÉ INEQUALITIES THEOREM (GUILLIN 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.