Small noise asymptotics of integrated OrnsteinUhlenbeck processes - PowerPoint PPT Presentation

Small noise asymptotics of integrated Ornstein–Uhlenbeck processes driven by α -stable Lévy processes Robert Hintze and Ilya Pavlyukevich Friedrich–Schiller–Universität Jena Fifth Workshop on Random Dynamical Systems Bielefeld, 3–5.10.2012 – Typeset by Foil T EX –

C ONTINUOUS L ÉVY F LIGHTS 1 1. Source of randomness: Lévy process L L is a Lévy process if L 0 ✏ 0 , is stochastically continuous and has independent stationary increments (and right continuous paths with left limits). L ✏ Brownian motion � drift ❧♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♥ � jumps ❧♦♦♠♦♦♥ ① x, y ② ✏ ➦ m Lévy–Khintchine formula for L P R m : i ✏ 1 x i y i e i ① λ,y ② ✁ 1 ✁ i ① λ, y ② ✁ t ➺ ✁ E e i ① L t ,λ ② ✏ exp ✑ ✠ ✙ 2 ① Aλ, λ ② � i t ① λ, µ ② � t ν ♣ dy q 1 � ⑥ y ⑥ 2 ❧♦♦♦♦♦♠♦♦♦♦♦♥ ❧♦♦♠♦♦♥ ❧♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♥ Brownian motion drift jumps – Typeset by Foil T EX – 1

C ONTINUOUS L ÉVY F LIGHTS 2 2. α -stable Lévy–Processes (Lévy Flights) L ✏ ♣ L t q t ➙ 0 is a one-dimensional α -stable Lévy process (symmetric: β ✏ 0 ) 1 ✁ i β sgn ♣ u q tan πα ✦ ✁ tc ⑤ u ⑤ α ✁ ✠✮ E e i uL t ✏ exp , α P ♣ 0 , 1 q ❨ ♣ 1 , 2 q 2 6 1 4 0.5 5 10 15 20 2 -0.5 -1 5 10 15 20 -1.5 -2 α ✏ 0 . 75 α ✏ 1 . 75 Pure jump process with enumerable many (small) jumps on any time interval, jump times are dense. 1 1 Cauchy–process α ✏ 1 1 � x 2 π 1 2 π e ✁ x 2 Brownian motion ❄ α ✏ 2 2 – Typeset by Foil T EX – 2

C ONTINUOUS L ÉVY F LIGHTS 3 3. α -stable Lévy process (Lévy flights) Isometric α -stable LP in R m : Γ ♣✁ α c m,α ✏ π m ④ 2 2 q E e i ① L t ,λ ② ✏ exp ✑ ✁ tc m,α ⑥ λ ⑥ α ✙ α P ♣ 0 , 2 q , , Γ ♣ m � α 2 α 2 q dy Jump measure: ν ♣ dy q ✏ ⑥ y ⑥ α � m , α P ♣ 0 , 2 q 1 Cauchy process: α ✏ 1 , probability density p ♣ x q ✒ 1 � ⑥ x ⑥ 2 2.0 4 1.5 1.0 2 0.5 0.0 0 � 0.5 � 1.0 � 2 � 1.5 � 2.0 � 4 � 2.0 � 1.5 � 1.0 � 0.5 0.0 0.5 1.0 1.5 2.0 � 4 � 2 0 2 4 Brownian motion 1 . 50 -stable Lévy process – Typeset by Foil T EX – 3

C ONTINUOUS L ÉVY F LIGHTS 4 4. Motivation and Setting Chechkin, Gonchar, Szydłowski, Physics of Plasmas 2002. l ✏ ♣ l t q t ➙ 0 is an isometric α -stable Lévy process in R 3 , E e i ① u,l t ② ✏ e ✁ t ⑥ u ⑥ α , u P R 3 , α P ♣ 0 , 2 q . Langevin equation for a particle in a external magnetic field B and Lévy electric field ✾ l : x � ε ✾ x ✏ r ✾ x ✂ B s ✁ ν ✾ ✿ l or x ε ✏ v ε , ✩ ✾ ☎ ☞ ν ✁ B 3 B 2 ✬ ✫ v ε ✏ r v ε ✂ B s ✁ νv ε � ε ✾ ✁ B 1 , A ✏ B 3 ν ✾ l ✆ ✌ ❧♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♥ ✁ B 2 B 1 ν ✬ ✪ ✏ : ✁ Av ε In other words, x ε is an integrated OU process: ➺ t ➺ t x ε v ε v ε Av ε t ✏ x 0 � t ✏ v 0 ✁ s d s � εl t s d s, 0 0 – Typeset by Foil T EX – 4

C ONTINUOUS L ÉVY F LIGHTS 5 5. ε -dependent timescale Interesting events should occur on the time intervals of the order O ♣ 1 ε α q , ε Ñ 0 . Time transformation: t ÞÑ ε α . t Self-similarity of an α -stable process: Law ♣ εl t εα , t ➙ 0 q ✏ Law ♣ l q ✏ Law ♣ L q ➺ t ➺ t t ➺ εα εα ✏ ✁ 1 ✏ ✁ 1 Law V t : ✏ v t εα ✏ ✁ Av s d s � εl t εα d s � εl t AV s d s � L t , Av s ε α ε α εα 0 0 0 ➺ t ➺ t t ➺ εα v s d s ✏ 1 εα d s ✏ 1 X t : ✏ x t εα ✏ v s V s d s ε α ε α 0 0 0 From now on: on some probability space consider an α -stable Lévy process L and a family of processes t V ε , X ε ✉ (with big friction parameter 1 ε α Ñ ✽ ) ➺ t ✩ t ✏ ✁ 1 V ε AV ε s d s � L t , ✬ ✬ ε α ✫ Law ♣ V ε t , X ε t , t ➙ 0 q ✏ Law ♣ v ε εα , x ε 0 εα , t ➙ 0 q ➺ t t t 1 X ε V ε t ✏ ✬ s d s ✬ ε α ✪ 0 – Typeset by Foil T EX – 5

C ONTINUOUS L ÉVY F LIGHTS 6 6. Explicit solution Ornstein–Uhlenbeck process: ➺ t ➺ t t ✏ ✁ 1 e ✁ t ✁ s εα A d L s V ε AV ε V ε s d s � L t ñ t ✏ ε α 0 0 Integrated Ornstein–Uhlenbeck process (Fubini): ➺ t ➺ t ✑ ➺ s t ✏ 1 s d s ✏ 1 ✙ εα A d L u A e ✁ s ✁ u AX ε AV ε d s ε α ε α 0 0 0 ➺ t ✑ ➺ t ✏ 1 ✙ A e ✁ s ✁ u εα A d s d L u ε α u 0 ➺ t ✁ εα A ✠ 1 ✁ e ✁ t ✁ u ✏ A ✁ 1 A d L u 0 The process X ε is absolutely continuous, non-Markovian, semimartingale. – Typeset by Foil T EX – 6

C ONTINUOUS L ÉVY F LIGHTS 7 7. Convergence of f.d.d. Theorem 1. For any n ➙ 1 , 0 ↕ t 1 ➔ ☎ ☎ ☎ ➔ t n ➔ ✽ t n q P ♣ AX ε t 1 , . . . , AX ε Ñ ♣ L t 1 , . . . , L t n q , ε Ñ 0 . Assume: E i ① u,L t ② ✏ e ✁⑥ u ⑥ α , u P R d . α P ♣ 0 , 1 q , Show: P AX ε Ñ L t , ε Ñ 0 , t ➙ 0 . t – Typeset by Foil T EX – 7

C ONTINUOUS L ÉVY F LIGHTS 8 8. Proof (convergence of one-dimensional distributions) ➺ t εα A d L s e ✁ t ✁ s AX ε t ✁ L t ✏ ✁ 0 n e ✁ t ✁ sk t ✁ L t q ✏ E exp ✦ ✮ E e i u ♣ AX ε ➳ εα A ∆ L s k ✁ i u lim n k ✏ 1 n E e ✁ i u e ✁ t ✁ sk εα A ∆ L sk ➵ ✏ lim n k ✏ 1 n ✎ ✁ u e ✁ t ✁ sk ✎ α ✎ εα A ✎ ➵ e ∆ s k ✏ lim n k ✏ 1 n α ✮ ✎ u e ✁ t ✁ sk ✎ εα A ✎ ✦ ➳ ✏ exp lim ∆ s k ✎ ✎ ✎ n k ✏ 1 ➺ t α ✎ εα A ✎ ✦ ✮ ✎ e ✁ t ✁ s ⑥ u ⑥ α ✏ exp Ñ 1 , ε Ñ 0 d s ✎ ✎ ✎ 0 ❧♦♦♦♦♠♦♦♦♦♥ Ñ 0 , s ✘ t, ε Ñ 0 – Typeset by Foil T EX – 8

C ONTINUOUS L ÉVY F LIGHTS 9 9. Functional limit theorem? Convergence of f.d.d. does not imply convergence of the first passage times. s ↕ t X ε → a q P ♣ τ a ♣ X ε q ↕ t q ✏ P ♣ sup Need convergence in a path space D ♣r 0 , ✽q , R q with an appropriate metric. Problem: the limit α -stable Lévy process L is (in general) càdlàg the processes t AX ε ✉ ε → 0 are absolutely continuous. 1 2 3 4 5 � 0.1 � 0.2 � 0.3 – Typeset by Foil T EX – 9

C ONTINUOUS L ÉVY F LIGHTS 10 10. Uniform convergence does not hold Consider the space D ♣r 0 , ✽q , R q with a (local) uniform topology associated with the metric d U,T ♣ x, x ✶ q : ✏ sup ⑤ x t ✁ x ✶ t ⑤ , T → 0 , t Pr 0 ,T s ➺ ✽ d U ♣ x, x ✶ q : ✏ e ✁ T ♣ 1 ❫ d U,T ♣ x, x ✶ qq d T 0 No U -convergence unless L is continuous (Brownian motion with drift): t ✁ L t ⑤ P d U,T ♣ AX ε , L q : ✏ sup ⑤ AX ε Û 0 , ε Ñ 0 . t Pr 0 ,T s – Typeset by Foil T EX – 10

C ONTINUOUS L ÉVY F LIGHTS 11 11. Skorohod J 1 -convergence does not hold Skorohod (1956): J 1 -topology (as well as J 2 , M 1 , M 2 topologies) Consider continuous time changes ✦ ✮ λ : R � Ñ R � , strictly increasing and continuous , λ ♣ 0 q ✏ 0 , λ ♣�✽q ✏ �✽q Λ ✏ x n Ñ x there exists a sequence t λ n ✉ ⑨ Λ such that ô ⑤ λ n ♣ t q ✁ t ⑤ Ñ 0 , sup t ➙ 0 ⑤ x n ♣ λ n ♣ t qq ✁ x ♣ t q⑤ Ñ 0 for all T → 0 . sup t Pr 0 ,T s This topology in metrizable and the space D is Polish. No J 1 -convergence unless L is continuous (Brownian motion with drift): d J 1 ,T ♣ AX ε , L q P Û 0 , ε Ñ 0 . We need a weaker metric, such that the sup -functional is still continuous. – Typeset by Foil T EX – 11

C ONTINUOUS L ÉVY F LIGHTS 12 12. Skorohod M 1 -covergence I For x P D ♣r 0 , T s , R q define a completed graph Γ x : Γ x : ✏ t♣ x 0 , 0 q✉ ❨ t♣ z, t q P R ✂ ♣ 0 , T s : z ✏ cx t ✁ � ♣ 1 ✁ c q x t for some c, c P r 0 , 1 s✉ , Γ x ⑨ R 2 . x Γ x 0 T 0 T Natural order on Γ x : t ➔ t ✶ or t ✏ t ✶ and ⑤ x t ✁ ✁ z ⑤ ↕ ⑤ x t ✁ ✁ z ✶ ⑤ . ♣ z, t q ↕ ♣ z ✶ , t ✶ q if – Typeset by Foil T EX – 12

C ONTINUOUS L ÉVY F LIGHTS 13 13. Skorohod M 1 -convergence II Parametric representation of Γ x : continuous nondecreasing w.r.t. order mapping ♣ z u , t u q : r 0 , 1 s Ñ Γ x . Denote Π x the set of all parametric representations of Γ x . Skorohod M 1 -convergence on D ♣r 0 , T s , R q : x n Ñ x for any ♣ z, t q P Π x there is ♣ z n , t n q ⑨ Π x n such that ô ✦ ✮ ⑤ z n ⑤ t n u ✁ z u ⑤ , sup u ✁ t u ⑤ Ñ 0 , n Ñ ✽ . max sup u Pr 0 , 1 s u Pr 0 , 1 s This topology in metrizable and the space D ♣ R � , R ; M 1 q is Polish (see Whitt, Chapter 12.8). The sup -functional is continuous. Goal: Prove convergence AX ε Ñ L in D ♣r 0 , ✽q , R ; M 1 q in probability i.e. convergence of f.d.d. (done) and tightness . – Typeset by Foil T EX – 13

C ONTINUOUS L ÉVY F LIGHTS 14 14. M 1 -oscillation function For x, y P R denote the segment ✈ x, y ✇ : ✏ t z P R : z ✏ x � c ♣ y ✁ x q , c P r 0 , 1 s✉ . M 1 -oscillation function M : R 3 Ñ r 0 , ✽q , ★ if x ❘ ✈ x 1 , x 2 ✇ , min t⑤ x ✁ x 1 ⑤ , ⑤ x 2 ✁ x ⑤✉ , M ♣ x 1 , x, x 2 q : ✏ x P ✈ x 1 , x 2 ✇ . 0 , M ♣ x 1 , x, x 2 q ✏ euclidean distance between the point x and the segment ✈ x 1 , x 2 ✇ . M(x ,x,x ) 1 2 x x 1 2 x – Typeset by Foil T EX – 14