Rough paths methods 1: Introduction

Transcription

1 Rough paths methods 1: Introduction Samy Tindel Purdue University University of Aarhus 216 Samy T. (Purdue) Rough Paths 1 Aarhus / 16

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

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

4 Equation under consideration Equation: Standard differential equation driven by fbm, R n -valued with t [, 1]. d Y t = a + V (Y s ) ds + V j (Y s ) dbs, j (1) j=1 Vector fields V,..., 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 / 16

5 Fractional Brownian motion B = (B 1,..., B d ) B j centered Gaussian process, independence of coordinates Variance of the increments: E[ Bt j Bs j 2 ] = t s 2H 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 / 16

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

7 Paths for a linear SDE driven by fbm dyt =.5Yt dt + 2Yt dbt, H =.5 Blue: (Bt )t [,1] Samy T. (Purdue) Y = 1 H =.7 Red: (Yt )t [,1] Rough Paths 1 Aarhus / 16

8 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. 28) Captures long range dependences between transactions Samy T. (Purdue) Rough Paths 1 Aarhus / 16

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

10 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 s < t 1, set x 2,n,i,j st Rough paths assumption 1: s<u<v<t x is a C γ function with γ > 1/3. dx n,i u dx n,j v The process x 2,n converges to a process x 2 as n in a C 2γ space. Rough paths assumption 2: Vector fields V,..., V j in C b. Samy T. (Purdue) Rough Paths 1 Aarhus / 16

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

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

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

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

15 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 in the rough paths sense. dy t = V (Y t ) dt + V j (Y t ) db j t, Remark: Extensions to H 1/4 (Unterberger, Nualart-T). Samy T. (Purdue) Rough Paths 1 Aarhus / 16

16 Study of equations driven by fbm Basic properties: 1 Moments of the solution 2 Continuity w.r.t initial condition, noise More advanced natural problems: 1 Density estimates Hu-Nualart + Lots of people 2 Numerical schemes Neuenkirch-T, Friz-Riedel 3 Invariant measures, ergodicity Hairer-Pillai, Deya-Panloup-T 4 Statistical estimation (H, coeff. V j ) Berzin-León, Hu-Nualart, Neuenkirch-T Samy T. (Purdue) Rough Paths 1 Aarhus / 16

17 Extensions of the rough paths formalism Stochastic PDEs: Equation: t Y t (ξ) = Y t (ξ) + σ(y t (ξ)) ẋ t (ξ) (t, ξ) [, 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 + ẋ t (ξ) (t, ξ) [, 1] R ẋ 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 / 16

18 Aim 1 Definition and properties of fractional Brownian motion 2 Some estimates for Young s integral, case H > 1/2 3 Extension to 1/3 < H 1/2 Samy T. (Purdue) Rough Paths 1 Aarhus / 16

19 General strategy 1 In order to solve our equation, we shall go through the following steps: Young integral for H > 1/2 Case 1/3 < H < 1/2, with a semi-pathwise method 2 For each case, 2 main steps: Definition of a stochastic integral us db s 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 / 16