Rough paths methods 1: Introduction Samy Tindel Purdue University - PowerPoint PPT Presentation

Rough paths methods 1: Introduction Samy Tindel Purdue University University of Aarhus 2016 Samy T. (Purdue) Rough Paths 1 Aarhus 2016 1 / 16

Outline Motivations for rough paths techniques 1 Summary of rough paths theory 2 Samy T. (Purdue) Rough Paths 1 Aarhus 2016 2 / 16

Outline Motivations for rough paths techniques 1 Summary of rough paths theory 2 Samy T. (Purdue) Rough Paths 1 Aarhus 2016 3 / 16

Equation under consideration Equation: Standard differential equation driven by fBm, R n -valued � t � t d 0 V j ( Y s ) dB j � Y t = a + 0 V 0 ( Y s ) ds + s , (1) j =1 with t ∈ [0 , 1]. Vector fields V 0 , . . . , V d in C ∞ b . A d -dimensional fBm B with 1 / 3 < H < 1. Note: some results will be extended to H > 1 / 4. Samy T. (Purdue) Rough Paths 1 Aarhus 2016 4 / 16

Fractional Brownian motion B = ( B 1 , . . . , B d ) B j centered Gaussian process, independence of coordinates Variance of the increments: E [ | B j t − B j s | 2 ] = | t − s | 2 H H − ≡ Hölder-continuity exponent of B If H = 1 / 2, B = Brownian motion If H � = 1 / 2 natural generalization of BM Remark: FBm widely used in applications Samy T. (Purdue) Rough Paths 1 Aarhus 2016 5 / 16

Examples of fBm paths H = 0 . 3 H = 0 . 5 H = 0 . 7 Samy T. (Purdue) Rough Paths 1 Aarhus 2016 6 / 16

Paths for a linear SDE driven by fBm dY t = − 0 . 5 Y t dt + 2 Y t dB t , Y 0 = 1 H = 0 . 5 H = 0 . 7 Blue: ( B t ) t ∈ [0 , 1] Red: ( Y t ) t ∈ [0 , 1] Samy T. (Purdue) Rough Paths 1 Aarhus 2016 7 / 16

Some applications of fBm driven systems Biophysics, fluctuations of a protein: New experiments at molecule scale ֒ → Anomalous fluctuations recorded Model: Volterra equation driven by fBm ֒ → Samuel Kou Statistical estimation needed Finance: Stochastic volatility driven by fBm (Sun et al. 2008) Captures long range dependences between transactions Samy T. (Purdue) Rough Paths 1 Aarhus 2016 8 / 16

Outline Motivations for rough paths techniques 1 Summary of rough paths theory 2 Samy T. (Purdue) Rough Paths 1 Aarhus 2016 9 / 16

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 2 , n , i , j s < u < v < t dx n , i u dx n , j ≡ st v Rough paths assumption 1: 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. ֒ Rough paths assumption 2: Vector fields V 0 , . . . , V j in C ∞ b . Samy T. (Purdue) Rough Paths 1 Aarhus 2016 10 / 16

Brief summary of rough paths theory Main rough paths theorem (Lyons): Under previous assumptions → Consider y n solution to equation ֒ � t � t d y n 0 V 0 ( y n 0 V j ( y n u ) dx n , j � t = a + u ) du + u . j =1 Then y n converges to a function Y in C γ . Y can be seen as solution to � t � t 0 V 0 ( Y u ) du + � d 0 V j ( Y u ) dx j ֒ → Y t = a + u . j =1 Samy T. (Purdue) Rough Paths 1 Aarhus 2016 11 / 16

Brief summary of rough paths theory Main rough paths theorem (Lyons): Under previous assumptions → Consider y n solution to equation ֒ � t � t d y n 0 V 0 ( y n 0 V j ( y n u ) dx n , j � t = a + u ) du + u . j =1 Then y n converges to a function Y in C γ . Y can be seen as solution to � t � t 0 V 0 ( Y u ) du + � d 0 V j ( Y u ) dx j ֒ → Y t = a + u . j =1 Rough paths theory Samy T. (Purdue) Rough Paths 1 Aarhus 2016 11 / 16

Brief summary of rough paths theory Main rough paths theorem (Lyons): Under previous assumptions → Consider y n solution to equation ֒ � t � t d y n 0 V 0 ( y n 0 V j ( y n u ) dx n , j � t = a + u ) du + u . j =1 Then y n converges to a function Y in C γ . Y can be seen as solution to � t � t 0 V 0 ( Y u ) du + � d 0 V j ( Y u ) dx j ֒ → Y t = a + u . j =1 � dx , � � dxdx Rough paths theory Smooth V 0 , . . . , V d Samy T. (Purdue) Rough Paths 1 Aarhus 2016 11 / 16

Brief summary of rough paths theory Main rough paths theorem (Lyons): Under previous assumptions → Consider y n solution to equation ֒ � t � t d y n 0 V 0 ( y n 0 V j ( y n u ) dx n , j � t = a + u ) du + u . j =1 Then y n converges to a function Y in C γ . Y can be seen as solution to � t � t 0 V 0 ( Y u ) du + � d 0 V j ( Y u ) dx j ֒ → Y t = a + u . j =1 � dx , � � dxdx � V j ( x ) dx j Rough paths theory dy = V j ( y ) dx j Smooth V 0 , . . . , V d Samy T. (Purdue) Rough Paths 1 Aarhus 2016 11 / 16

Iterated integrals and fBm Nice situation: H > 1 / 4 ֒ → 2 possible constructions for geometric iterated integrals of B . Malliavin calculus tools ֒ → Ferreiro-Utzet Regularization or linearization of the fBm path ֒ → Coutin-Qian, Friz-Gess-Gulisashvili-Riedel Conclusion: for H > 1 / 4, one can solve equation dY t = V 0 ( Y t ) dt + V j ( Y t ) dB j t , in the rough paths sense. Remark: Extensions to H ≤ 1 / 4 (Unterberger, Nualart-T). Samy T. (Purdue) Rough Paths 1 Aarhus 2016 12 / 16

Study of equations driven by fBm Basic properties: Moments of the solution 1 Continuity w.r.t initial condition, noise 2 More advanced natural problems: Density estimates 1 ֒ → Hu-Nualart + Lots of people 0.16 0.14 Numerical schemes 2 0.12 ֒ → Neuenkirch-T, Friz-Riedel 0.1 0.08 Invariant measures, ergodicity 0.06 3 0.04 ֒ → Hairer-Pillai, Deya-Panloup-T 0.02 0 −10 −5 0 5 10 Statistical estimation ( H , coeff. V j ) 4 ֒ → Berzin-León, Hu-Nualart, Neuenkirch-T Samy T. (Purdue) Rough Paths 1 Aarhus 2016 13 / 16

Extensions of the rough paths formalism Stochastic PDEs: Equation: ∂ t Y t ( ξ ) = ∆ Y t ( ξ ) + σ ( Y t ( ξ )) ˙ x t ( ξ ) ( t , ξ ) ∈ [0 , 1] × R d Easiest case: x finite-dimensional noise Methods: ֒ → viscosity solutions or adaptation of rough paths methods KPZ equation: Equation: ∂ t Y t ( ξ ) = ∆ Y t ( ξ ) + ( ∂ ξ Y t ( ξ )) 2 + ˙ x t ( ξ ) − ∞ ( t , ξ ) ∈ [0 , 1] × R x ≡ space-time white noise ˙ Methods: ◮ Extension of rough paths to define ( ∂ x Y t ( ξ )) 2 ◮ Renormalization techniques to remove ∞ Samy T. (Purdue) Rough Paths 1 Aarhus 2016 14 / 16

Aim Definition and properties of fractional Brownian motion 1 Some estimates for Young’s integral, case H > 1 / 2 2 Extension to 1 / 3 < H ≤ 1 / 2 3 Samy T. (Purdue) Rough Paths 1 Aarhus 2016 15 / 16

General strategy In order to solve our equation, we shall go through the following 1 steps: ◮ Young integral for H > 1 / 2 ◮ Case 1 / 3 < H < 1 / 2, with a semi-pathwise method For each case, 2 main steps: 2 � u s dB s ◮ Definition of a stochastic integral for a reasonable class of processes u ◮ Resolution of the equation by means of a fixed point method Samy T. (Purdue) Rough Paths 1 Aarhus 2016 16 / 16