Some sufficient condition for the ergodicity of the L evy transform - PowerPoint PPT Presentation

Some sufficient condition for the ergodicity of the L´ evy transform Vilmos Prokaj E¨ otv¨ os Lor´ and University, Hungary Probability, Control and Finance A Conference in Honor of Ioannis Karatzas 2012, New York

L´ evy transformation of the path space ◮ β is a Brownian motion � sign ( β s ) d β s = |β| − L 0 ( β ) . T β = ◮ T is a transformation of the path space. ◮ T preserves the Wiener measure. ◮ Is T ergodic? ◮ A deep result of Marc Malric claims that the L´ evy transform is topologically recurrent, i.e., on an almost sure event { T n β : n ≥ 0 } ∩ G � = ∅ , for all nonempty open G ⊂ C [ 0 , ∞ ) . ◮ We use only a weaker form, also due to Marc Malric, the density of zeros of iterated paths, i.e.: ∞ � { t > 0 : ( T n β ) t = 0 } is dense in [ 0 , ∞ ) . n = 0

Ergodicity and Strong mixing (reminder) P ◦ T − 1 = P T : Ω → Ω , ◮ T is ergodic, if N − 1 1 � A ∩ T − n B � � P → P ( A ) P ( B ) , for all A , B . ◮ N n = 0 N − 1 ◮ or, 1 X ◦ T n → E ( X ) , � for each r.v. X ∈ L 1 . N n = 0 ◮ or, the invariant σ –field, is trivial. � A ∩ T − n B � ◮ T is strongly mixing if P → P ( A ) P ( B ) , for all A , B .

Ergodicity and weak convergence In our case Ω = C [ 0 , ∞ ) is a polish space (complete, separable, metric space). Theorem Ω polish, T is a measure preserving transform of (Ω , B (Ω) , P ) . Then n − 1 ◮ T is ergodic iff 1 w � P ◦ ( T 0 , T k ) − 1 − → P ⊗ P as n → ∞. n k = 0 w ◮ T is strongly mixing iff P ◦ ( T 0 , T n ) − 1 − → P ⊗ P as n → ∞.

Ergodicity and weak convergence In our case Ω = C [ 0 , ∞ ) is a polish space (complete, separable, metric space). Theorem Ω polish, T is a measure preserving transform of (Ω , B (Ω) , P ) . Then n − 1 ◮ T is ergodic iff 1 w � P ◦ ( T 0 , T k ) − 1 − → P ⊗ P as n → ∞. n k = 0 w ◮ T is strongly mixing iff P ◦ ( T 0 , T n ) − 1 − → P ⊗ P as n → ∞. Note that both families of measures are tight: � n − 1 � 1 P ◦ ( T 0 , T n ) − 1 : n ≥ 0 P ◦ ( T 0 , T n ) − 1 : n ≥ 0 � � � and n k = 0 If C ⊂ Ω compact, with P (Ω \ C ) < ε then T 0 / T k / � ( T 0 , T k ) / � � � � � ∈ C × C ≤ P ∈ C + P ∈ C < 2 ε . P

f . d . Convergence of finite dim. marginals ( − → ) is enough Some notations: ◮ β is the canonical process on Ω = C [ 0 , ∞ ), ◮ h : [ 0 , ∞ ) × C [ 0 , ∞ ) progressive, | h | = 1 dt ⊗ d P a.e. � . T : Ω → Ω , T β = h ( s , β ) d β s , (e.g. h ( s , β ) = sign ( β s )) . 0 ◮ β ( n ) = T n β is the n -th iterated path. � t ◮ h ( 0 ) = 1 , h ( n ) k = 0 h ( s , β ( k ) ) for n > 0 , so β ( n ) 0 h ( n ) = � n − 1 = s d β s . s t

f . d . Convergence of finite dim. marginals ( − → ) is enough Some notations: ◮ β is the canonical process on Ω = C [ 0 , ∞ ), ◮ h : [ 0 , ∞ ) × C [ 0 , ∞ ) progressive, | h | = 1 dt ⊗ d P a.e. � . T : Ω → Ω , T β = h ( s , β ) d β s , (e.g. h ( s , β ) = sign ( β s )) . 0 ◮ β ( n ) = T n β is the n -th iterated path. � t ◮ h ( 0 ) = 1 , h ( n ) k = 0 h ( s , β ( k ) ) for n > 0 , so β ( n ) 0 h ( n ) = � n − 1 = s d β s . s t Then ◮ The distribution of ( β, β ( n ) ) is P ◦ ( T 0 , T n ) − 1 ◮ Let κ n is uniform on { 0 , 1 , . . . , n − 1 } and independent of β . � n − 1 k = 0 P ◦ ( T 0 , T k ) − 1 . The law of ( β, β ( κ n ) ) is 1 n f . d . ◮ T is strongly mixing, iff ( β, β ( n ) ) − → BM-2. f . d . ◮ Similarly, T is ergodic, iff ( β, β ( κ n ) ) − → BM-2. f . d . ◮ Reason: Tightness + − → = convergence in law.

Characteristic function ◮ Fix t 1 , . . . , t r ≥ 0 and α = ( a 1 , . . . , a r , b 1 , . . . , b r ) ∈ R 2 r ◮ The characteristic function of ( β t 1 , . . . , β t r , β ( n ) t 1 , . . . , β ( n ) t r ) at α is � � � � � �� g ( s ) d β ( n ) � � f ( s )+ g ( s ) h ( n ) � f ( s ) d β s + � d β s i i s s φ n = E e = E e , where f = � r j = 1 a j 1 [ 0 , t j ] , g = � r j = 1 b j 1 [ 0 , t j ] . ◮ Finite dim. marginals has the right limit, if for all choices r ≥ 1 , α ∈ R 2 r , t 1 , . . . , t r ≥ 0 � � − 1 � f 2 + g 2 φ n → exp for strong mixing , 2 n − 1 � � � 1 − 1 f 2 + g 2 � φ k → exp for ergodicity . n 2 k = 0

Estimate for |φ n − φ| , where φ = e − 1 � f 2 + g 2 2 � t 0 ( f ( s ) + h ( n ) ◮ M t = s g ( s )) d β s . ◮ M is a closed martingale and so is Z = exp � iM + 1 � 2 � M � . ◮ Z 0 = 1 = ⇒ E ( Z ∞ ) = 1 . � ∞ � ∞ 0 h ( n ) ◮ � M � ∞ = 0 f 2 ( s ) + g 2 ( s ) ds + 2 s f ( s ) g ( s ) ds ◮ � ∞ � ∞ � � ( fd β + gd β ( n ) ) + fgh ( n ) φ = φ E ( Z ∞ ) = E exp i 0 0 � t 0 h ( n ) ◮ Recall that fg = � j a j b j 1 [ 0 , t j ] . Then with X n ( t ) = s ds � ∞ r � � � ∞ � 0 fgh ( n ) � � � | fg | E � fgh ( n ) � | φ n − φ | ≤ E � 1 − e � ≤ e � ≤ C E | X n ( t j ) | , � � � � � 0 j = 1 where C = C ( f , g ) = C ( α , t 1 , . . . , t r ) does not depend on n .

� t p 0 h ( n ) X n ( t ) = s ds → 0 for all t ≥ 0 would be enough Theorem p 1. If X n ( t ) → 0 for all t ≥ 0 , then T is strongly mixing. p � n − 1 2. T is ergodic if and only if 1 k = 0 X 2 k ( t ) → 0 for all t ≥ 0 . n Strong mixing: ◮ The only missing part is the convergence of finite dimensional marginals. p ◮ If X n ( t ) → 0 then E | X n ( t ) | → 0 since | X n ( t ) | ≤ t . f . d . ◮ Then | φ n − φ | ≤ C � ⇒ ( β, β ( n ) ) j E | X n ( t j ) | → 0 = − → BM-2. Remember, that: f . d . D ◮ ( β, β ( n ) ) → BM-2 + tightness gives: ( β, β ( n ) ) − → BM-2. D ◮ ( β, β ( n ) ) → BM-2 ⇔ T strong mixing.

� t p 0 h ( n ) X n ( t ) = s ds → 0 for all t ≥ 0 would be enough Theorem p 1. If X n ( t ) → 0 for all t ≥ 0 , then T is strongly mixing. p � n − 1 2. T is ergodic if and only if 1 k = 0 X 2 k ( t ) → 0 for all t ≥ 0 . n Ergodicity. ⇐ . ◮ By Cauchy-Schwarz and | X k ( t ) | ≤ t � � ≤ E 1 / 2 � � � n − 1 � n − 1 1 1 k = 0 | X k ( t ) | k = 0 X 2 → 0 . E k ( t ) n n � � � n − 1 � n − 1 ◮ Then � 1 j E 1 k = 0 φ k − φ � ≤ C � k = 0 | X k ( t j ) | → 0 = ⇒ � � n n f . d . ( β, β ( κ n ) ) − → BM-2. Remeber that ◮ κ n is uniform on { 0 , . . . , n − 1 } and independent of β f . d . D ◮ ( β, β ( κ n ) ) → BM-2 + tightness gives: ( β, β ( κ n ) ) − → BM-2. D ◮ ( β, β ( κ n ) ) → BM-2 ⇔ T ergodic.

� t p 0 h ( n ) X n ( t ) = s ds → 0 for all t ≥ 0 would be enough Theorem p 1. If X n ( t ) → 0 for all t ≥ 0 , then T is strongly mixing. p � n − 1 2. T is ergodic if and only if 1 k = 0 X 2 k ( t ) → 0 for all t ≥ 0 . n Ergodicity. ⇒ (outline of the proof) ◮ Fix 0 < s < t . Then the following limits exist a.s and in L 2 : n − 1 n − 1 1 1 � � s ( β ( k ) h ( k ) s h ( k ) h ( k ) t − β ( k ) Z u = lim for s ≤ u ≤ t , Z = lim s ) u n n n →∞ n →∞ k = 0 k = 0 moreover | Z | and | Z u | are invariant for T , hence they are non-random. � t ◮ Then h ( k ) s ( β ( k ) − β ( k ) s h ( k ) s h ([ k ) s ) = u d β u and t � t � t � . | Z u | d ˜ ˜ Z = Z u d β u = β u , where β = sign ( Z u ) d β u . s s s ◮ Z ∼ N ( 0 , σ 2 ) since | Z u | is non-random. But | Z | is also non-random. � n − 1 1 = ⇒ Z = 0 . = ⇒ Z u = 0 . = ⇒ k = 0 X 2 k ( t ) → 0 . n

A variant of the mean ergodic theorem ◮ T is a measure preserving transformation of Ω, ◮ ε 0 is r.v. taking values in {− 1 , + 1 } , ε k = ε 0 ◦ T k . ◮ For ξ ∈ L 2 (Ω), U ξ = ξ ◦ T ε 0 is an isometry. ◮ von Neumann’s mean errgodic theorem says, that n − 1 1 � U k ξ → P ξ, ∈ L 2 n k = 0 where P is the projection onto X ∈ L 2 : X ◦ T ε 0 = UX = X � � ◮ | P ξ | is invariant under T . ◮ what is U k ξ ? k − 1 � U 2 ξ = ξ ◦ T 2 ε 1 ε 0 , U k ξ = ξ ◦ T k U ξ = ξ ◦ T ε 0 , . . . ε j , j = 0 ◮ Almost sure convergence also holds by the subadditive ergodic theorem.

L´ evy transformation ◮ The L´ evy transformation T is scaling invariant, that is, if for x > 0 Θ x : C [ 0 , ∞ ) → C [ 0 , ∞ ) denotes Θ x ( w )( t ) = xw ( t / x 2 ) then Θ x T = T Θ x � t ◮ As before β ( n ) = β ◦ T n , h ( n ) k = 0 sign ( β ( k ) 0 h ( n ) = � n − 1 t ), X n ( t ) = s ds . t By scaling we get: Theorem p 1. If X n ( 1 ) → 0 as n → ∞, then T is strongly mixing. p � n − 1 2. T is ergodic, if and only if 1 k = 0 X 2 k ( 1 ) → 0 as n → ∞. n