Skorohod and Stratonovich in the plane Samy Tindel Université de Lorraine Workshop on Random Dynamical Systems - Tianjin 2013 Ongoing joint work with Khalil Chouk (Paris Dauphine) Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 1 / 30
Sketch Introduction 1 Rough paths integrals Skorohod and Stratonovich in 1-d 2d program 2d Stratonovich and Skorohod formulas 2 Young case Rough case Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 2 / 30
Sketch Introduction 1 Rough paths integrals Skorohod and Stratonovich in 1-d 2d program 2d Stratonovich and Skorohod formulas 2 Young case Rough case Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 3 / 30
Overview of relations between integrals Case of Brownian motion W : � t � t Brackets s z u d ⋄ W u Stratonovich: s z u dW u ← → Itô: Case of fractional Brownian motion B : � t � t Trace terms s z u d ⋄ B u Strato, Rough paths: s z u dB u ← → Skorohod: Case of fractional Brownian sheet x indexed by [ 0 , 1 ] 2 : � s 2 � t 2 ?? z u d ⋄ W u Strato, Rough paths: (??) ← → Skorohod: s 1 t 1 Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 4 / 30
Sketch Introduction 1 Rough paths integrals Skorohod and Stratonovich in 1-d 2d program 2d Stratonovich and Skorohod formulas 2 Young case Rough case Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 5 / 30
Rough paths assumptions Context: Consider a Hölder path x and For n ≥ 1, x n ≡ linearization of x with mesh 1 / n → x n piecewise linear. ֒ For 0 ≤ s < t ≤ 1, set � x 1 , n , i x 2 , n , i , j ≡ x i t − x j s < u < v < t dx n , i u dx n , j s , ≡ st st v Main rough paths assumption: x is a C γ function with γ > 1 / 3. The process x 2 , n converges to a process x 2 as n → ∞ → in a C 2 γ space. ֒ Notation: X ≡ ( x 1 , x 2 ) called rough path above x Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 6 / 30
Guiding example: fractional Brownian motion FBm definition: B = ( B 1 , . . . , B d ) B j centered Gaussian process, independence of coordinates Variance of the increments: s | 2 ] = | t − s | 2 H E [ | B j t − B j H − ≡ Hölder-continuity exponent of B If H = 1 / 2, B = Brownian motion If H � = 1 / 2 natural generalization of BM Remarks: FBm widely used in applications 1 Main rough path assumption verified for fBm 2 Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 7 / 30
� Rough paths definition of ϕ ( x ) dx Proposition 1. Consider: x satisfying the rough paths assumptions ϕ : R d → R d of class C 3 b Then: � t � d s ϕ i ( x n u ) dx i , n lim n →∞ exists i = 1 u � t � d s ϕ i ( x u ) dx i We call the limit i = 1 u Convergence of Riemann sums along partition π st = ( s j ) : � t � � s ϕ i ( x u ) dx i � ϕ i 1 ( x s j ) x 1 , i 1 s j s j + 1 + ∂ i 2 ϕ i 1 ( x s j ) x 2 , i 1 i 2 u = lim s j s j + 1 | π st |→ 0 s j ∈ π st Stratonovich type formula holds true Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 8 / 30
Sketch Introduction 1 Rough paths integrals Skorohod and Stratonovich in 1-d 2d program 2d Stratonovich and Skorohod formulas 2 Young case Rough case Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 9 / 30
Stratonovich and Skorohod in 1-d Notation: π n typical partition whose mesh goes to 0 Typical pathwise (Stratonovich, rough paths) integral: � t � s z u dx u = lim z s i ( x s i + 1 − x s i ) + Higher order terms n →∞ π n Wick products, crash course: H n = n th Hermite polynomial, G 1 , G 2 independent Gaussian G ⋄ , n 1 ⋄ G ⋄ , n 2 ≡ H n 1 ( G 1 ) H n 2 ( G 2 ) 1 2 Fundamental property: E [ G ⋄ , n 1 ⋄ G ⋄ , n 2 ] = 0 1 2 Typical Skorohod integral: for a Gaussian process x � t s z u d ⋄ x u = lim � z s i ⋄ ( x s i + 1 − x s i ) + Higher order terms n →∞ π n Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 10 / 30
Stratonovich and Skorohod in 1-d (2) Other way to look at Skorohod: for test r.v L on Wiener space, ��� t � � s z u d ⋄ x u = E [ � z , D x L � H ] L E ֒ → Divergence on Wiener space Assumptions for integration: For Stratonovich: Regularity and rough paths hypothesis on z , x For Skorohod: Regularity on the Wiener space Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 11 / 30
Comparison of 1-d Stratonovich and Skorohod Summary of a paper by Hu-Jolis-T (AOP 2013): Consider a rather general multidimensional Gaussian process x Including fractional Brownian motion for H ∈ ( 0 , 1 ) Assume that x gives raise to a rough path Then a Stratonovich type change of variable holds for x A Malliavin-Skorohod type change of variable is available for x Corrections between both integrals are computed Conclusion: Existence of rough path = ⇒ Skorohod and Strato changes of variables Comparison between Skorohod and Strato integrals Remark: Similar ideas in preprint by Kruk-Russo Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 12 / 30
From Stratonovich to Skorohod Stratonovich/Skorohod corrections: for a sequence of partitions π n • Start from some convergent Riemann-Stratonovich sums involving: ∂ k i 1 ... i k f ( x t q ) x k , i 1 ,..., i k t q , t q + 1 • By Wick calculus, compute the corrections between i k ... i 1 f ( x t q ) ⋄ x k , ⋄ , i 1 ,..., i k ∂ k i k ... i 1 f ( x t q ) x k , i 1 ,..., i k ∂ k and t q , t q + 1 t q , t q + 1 ֒ → Nice combinatorial formula • Analyze limits as | π n | → 0 Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 13 / 30
Sketch Introduction 1 Rough paths integrals Skorohod and Stratonovich in 1-d 2d program 2d Stratonovich and Skorohod formulas 2 Young case Rough case Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 14 / 30
2d-Itô type formulas: history Martingale case (70-80’s): Wong-Zakai, Cairoli-Walsh, Nualart Itô type formulas Stochastic, mixed bracket/stochastic and pure bracket terms Fractional Brownian sheet case (03-06): Tudor-Viens Itô-Skorohod formulas from Malliavin calculus Mixed and bracket terms involve covariance of the fBs Rough sheet case (ongoing): Chouk-Gubinelli Definition of a planar rough path Related Stratonovich differential calculus Study of RRDS: widely open! Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 15 / 30
Aim of the talk For a process x indexed by the plane: Get some handy Strato-Skorohod formulas Make a link between both Strato and Skorohod worlds Additional difficulties with respect to 1d-case: Clumsy formulas for 2d-indexed processes Terrible boundary terms Lack of Riemann sums representations ֒ → They were at the heart of strategy in 1-d Restricted framework: Rather general Gaussian process in the Young case Fractional Brownian sheet with H 1 , H 2 > 1 / 3 Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 16 / 30
Sketch Introduction 1 Rough paths integrals Skorohod and Stratonovich in 1-d 2d program 2d Stratonovich and Skorohod formulas 2 Young case Rough case Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 17 / 30
Sketch Introduction 1 Rough paths integrals Skorohod and Stratonovich in 1-d 2d program 2d Stratonovich and Skorohod formulas 2 Young case Rough case Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 18 / 30
Notations Notations for integration in the plane: x : generic plane-indexed function, here real valued Directions of integration: 1 and 2 y ≡ ϕ ( x ) , y j ≡ ϕ ( j ) ( x ) Planar differentials: d 12 x ≡ ∂ 2 st x and d ˆ 2 x ≡ ∂ s x ∂ t x 1 ˆ Planar increment: δ x s 1 s 2 ; t 1 t 2 ≡ x s 2 ; t 2 − x s 2 ; t 1 − x s 1 ; t 2 + x s 1 ; t 1 Change of variables, smooth x : � [ s 1 , s 2 ] × [ t 1 , t 2 ] ϕ ( 1 ) ( x u ; v ) d uv x u ; v [ δϕ ( x )] s 1 s 2 ; t 1 t 2 = � [ s 1 , s 2 ] × [ t 1 , t 2 ] ϕ ( 2 ) ( x u ; v ) d u x u ; v d v x u ; v , + Change of variables, shorthand: � � � � 2 y 1 d 12 x + 2 y 2 d ˆ 2 x := z 1 + z 2 . δ y = 1 ˆ 1 1 Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 19 / 30
Notations 2 Index conventions: 1 ; 2 ≡ integration of type d 12 , ˆ 1 ; ˆ 2 ≡ integration of type d ˆ 2 = d 1 × d 2 1 ;ˆ 1 ; 0 ≡ integration of type d 1 (1-dimensional) First terms of the rough path: � � � � � � x 1 ; 2 = 2 = ˆ 1 ;ˆ 2 d 12 x = δ x , and x 2 d 1 x d 2 x = 2 d ˆ 2 x 1 ˆ 1 1 1 Increments z 1 and z 2 : z 1 = � � 2 y 1 d 12 x , z 2 = � � 2 y 2 d ˆ and 2 x 1 ˆ 1 1 Young type assumptions: x is ( γ 1 , γ 2 ) -Hölder, with γ 1 , γ 2 > 1 / 2 Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 20 / 30
Young type change of variable Theorem 2. Under Young assumptions → z 1 and z 2 are well defined in the 2d-Young sense. ֒ Moreover: (i) Both z 1 and z 2 can be decomposed as: z 1 = y 1 x 1 ; 2 + ρ 1 , z 2 = y 2 x 2 + ρ 2 , ˆ 1 ;ˆ and where ρ 1 , ρ 2 are increments with double regularity ( 2 γ 1 , 2 γ 2 ) (ii) Nice continuity properties when x n → x and x n smooth Samy T. (Lorraine) Skorohod and Stratonovich in the plane Tianjin 2013 21 / 30
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