# Rough Paths Methods 1: Introduction

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Rough paths methods 1: IntroductionSamy TindelPurdue UniversityUniversity of Aarhus 2016Samy T. (Purdue)Rough Paths 1Aarhus 20161 / 16

Outline1Motivations for rough paths techniques2Summary of rough paths theorySamy T. (Purdue)Rough Paths 1Aarhus 20162 / 16

Outline1Motivations for rough paths techniques2Summary of rough paths theorySamy T. (Purdue)Rough Paths 1Aarhus 20163 / 16

Equation under considerationEquation:Standard differential equation driven by fBm, Rn -valuedYt a Z t0V0 (Ys ) ds d Z tXj 1 0Vj (Ys ) dBsj ,(1)witht [0, 1].Vector fields V0 , . . . , Vd in Cb .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 1Aarhus 20164 / 16

Fractional Brownian motionB (B 1 , . . . , B d )B j centered Gaussian process, independence of coordinatesVariance of the increments:E[ Btj Bsj 2 ] t s 2HH Hölder-continuity exponent of BIf H 1/2, B Brownian motionIf H 6 1/2 natural generalization of BMRemark: FBm widely used in applicationsSamy T. (Purdue)Rough Paths 1Aarhus 20165 / 16

Examples of fBm pathsH 0.3Samy T. (Purdue)H 0.5Rough Paths 1H 0.7Aarhus 20166 / 16

Paths for a linear SDE driven by fBmdYt 0.5Yt dt 2Yt dBt ,H 0.5Blue: (Bt )t [0,1]Samy T. (Purdue)Y0 1H 0.7Red: (Yt )t [0,1]Rough Paths 1Aarhus 20167 / 16

Some applications of fBm driven systemsBiophysics, fluctuations of a protein:New experiments at molecule scale, Anomalous fluctuations recordedModel: Volterra equation driven by fBm, Samuel KouStatistical estimation neededFinance:Stochastic volatility driven by fBm (Sun et al. 2008)Captures long range dependences between transactionsSamy T. (Purdue)Rough Paths 1Aarhus 20168 / 16

Outline1Motivations for rough paths techniques2Summary of rough paths theorySamy T. (Purdue)Rough Paths 1Aarhus 20169 / 16

Rough paths assumptionsContext: Consider a Hölder path x andFor n 1, x n linearization of x with mesh 1/n, x n piecewise linear.For 0 s t 1, set2,n,i,jxst Zs u v tdxun,i dxvn,jRough paths assumption 1:x is a C γ function with γ 1/3.The process x2,n converges to a process x2 as n , in a C 2γ space.Rough paths assumption 2:Vector fields V0 , . . . , Vj in Cb .Samy T. (Purdue)Rough Paths 1Aarhus 201610 / 16

Brief summary of rough paths theoryMain rough paths theorem (Lyons): Under previous assumptions, Consider y n solution to equationytn a Z t0V0 (yun ) du d Z tXj 1 0Vj (yun ) dxun,j .Theny n converges to a function Y in C γ .Y can be seenR as solution toRP, Yt a 0t V0 (Yu ) du dj 1 0t Vj (Yu ) dxuj .Samy T. (Purdue)Rough Paths 1Aarhus 201611 / 16

Brief summary of rough paths theoryMain rough paths theorem (Lyons): Under previous assumptions, Consider y n solution to equationytn a Z t0V0 (yun ) du d Z tXj 1 0Vj (yun ) dxun,j .Theny n converges to a function Y in C γ .Y can be seenR as solution toRP, Yt a 0t V0 (Yu ) du dj 1 0t Vj (Yu ) dxuj .Rough paths theorySamy T. (Purdue)Rough Paths 1Aarhus 201611 / 16

Brief summary of rough paths theoryMain rough paths theorem (Lyons): Under previous assumptions, Consider y n solution to equationytn a Z t0V0 (yun ) du d Z tXj 1 0Vj (yun ) dxun,j .Theny n converges to a function Y in C γ .Y can be seenR as solution toRP, Yt a 0t V0 (Yu ) du dj 1 0t Vj (Yu ) dxuj .Rdx ,RRdxdxRough paths theorySmooth V0 , . . . , VdSamy T. (Purdue)Rough Paths 1Aarhus 201611 / 16

Brief summary of rough paths theoryMain rough paths theorem (Lyons): Under previous assumptions, Consider y n solution to equationytn a Z t0V0 (yun ) du d Z tXj 1 0Vj (yun ) dxun,j .Theny n converges to a function Y in C γ .Y can be seenR as solution toRP, Yt a 0t V0 (Yu ) du dj 1 0t Vj (Yu ) dxuj .Rdx ,RRRdxdxRough paths theorySmooth V0 , . . . , VdSamy T. (Purdue)Rough Paths 1Vj (x ) dx jdy Vj (y )dx jAarhus 201611 / 16

Iterated integrals and fBmNice situation: H 1/4, 2 possible constructions for geometric iterated integrals of B.Malliavin calculus tools, Ferreiro-UtzetRegularization or linearization of the fBm path, Coutin-Qian, Friz-Gess-Gulisashvili-RiedelConclusion: for H 1/4, one can solve equationdYt V0 (Yt ) dt Vj (Yt ) dBtj ,in the rough paths sense.Remark: Extensions to H 1/4 (Unterberger, Nualart-T).Samy T. (Purdue)Rough Paths 1Aarhus 201612 / 16

Study of equations driven by fBmBasic properties:12Moments of the solutionContinuity w.r.t initial condition, noiseMore advanced natural problems:1234Density estimates, Hu-Nualart Lots of peopleNumerical schemes, Neuenkirch-T, Friz-RiedelInvariant measures, ergodicity, Hairer-Pillai, Deya-Panloup-TStatistical estimation (H, coeff. Vj ), Berzin-León, Hu-Nualart, Neuenkirch-TSamy T. (Purdue)Rough Paths 10.160.140.120.10.080.060.040.020 10 50Aarhus 201651013 / 16

Extensions of the rough paths formalismStochastic PDEs:Equation: t Yt (ξ) Yt (ξ) σ(Yt (ξ)) ẋt (ξ)(t, ξ) [0, 1] RdEasiest case: x finite-dimensional noiseMethods:, viscosity solutions or adaptation of rough paths methodsKPZ equation:Equation: t Yt (ξ) Yt (ξ) ( ξ Yt (ξ))2 ẋt (ξ) (t, ξ) [0, 1] Rẋ space-time white noiseMethods:IIExtension of rough paths to define ( x Yt (ξ))2Renormalization techniques to remove Samy T. (Purdue)Rough Paths 1Aarhus 201614 / 16

Aim123Definition and properties of fractional Brownian motionSome estimates for Young’s integral, case H 1/2Extension to 1/3 H 1/2Samy T. (Purdue)Rough Paths 1Aarhus 201615 / 16

General strategy1In order to solve our equation, we shall go through the followingsteps:II2Young integral for H 1/2Case 1/3 H 1/2, with a semi-pathwise methodFor each case, 2 main steps:IIRDefinition of a stochastic integral us dBsfor a reasonable class of processes uResolution of the equation by means of a fixed point methodSamy T. (Purdue)Rough Paths 1Aarhus 201616 / 16

Samy T. (Purdue) Rough Paths 1 Aarhus 2016 12 / 16. Study of equations driven by fBm Basicproperties: 1 Momentsofthesolution 2 Continuityw.r.tinitialcondition,noise Moreadvancednaturalproblems: 1 Densityestimates, Hu-Nualart Lotsofpeople 2 Numericalschemes, Neuenkirch-T,Friz-Riedel 3 Invariantmeasures,ergodicity, Hairer-Pillai,Deya-Panloup-T 4 Statisticalestimation(H,coeﬀ. V j .

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main idea of the rough paths theory is to introduce a much stronger topology than the convergence in p-variation. This topology, that we now explain, is related to the continuity of lifts of paths in free nilpotent Lie groups. Let G N(Rd) be the free N-step nilpotent Lie group with dgenerators X 1; ;X d. If x: [0;1] !Rd is continuous with bounded variation, the solution x of the equation x(t .