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Weierstrass Institute forApplied Analysis and StochasticsRough paths and rough partial differentialequationsChristian BayerMohrenstrasse 39 · 10117 Berlin · Germany · Tel. 49 30 20372 0 · www.wias-berlin.deMarch 18, 2016

Outline1Motivation and introduction2Rough path spaces3Integration against rough paths4Integration of controlled rough paths5Rough differential equations6Applications of the universal limit theorem7Rough partial differential equationsRough paths and rough partial differential equations · March 18, 2016 · Page 2 (48)

Controlled differential equationsStandard ordinary differential equationẏt V(yt ),y0 ξ Rd ,t [0, 1]V : Rd Rd smoothControlled differential equationdyt V(yt )dxt ,y0 ξ Rd ,IV : Rd Rd e smoothIxt path taking values in ReIxt may contain component t, i.e., includest [0, 1]dyt V0 (yt )dt V(yt )dxtRough paths and rough partial differential equations · March 18, 2016 · Page 3 (48)

Controlled differential equationsStandard ordinary differential equationẏt V(yt ),y0 ξ Rd ,t [0, 1]V : Rd Rd smoothControlled differential equationdyt V(yt )dxt ,y0 ξ Rd ,IV : Rd Rd e smoothIxt path taking values in ReIxt may contain component t, i.e., includest [0, 1]dyt V0 (yt )dt V(yt )dxtRough paths and rough partial differential equations · March 18, 2016 · Page 3 (48)

Controlled differential equationsStandard ordinary differential equationẏt V(yt ),y0 ξ Rd ,t [0, 1]V : Rd Rd smoothControlled differential equationdyt V(yt )dxt ,y0 ξ Rd ,IV : Rd Rd e smoothIxt path taking values in ReIxt may contain component t, i.e., includest [0, 1]dyt V0 (yt )dt V(yt )dxtRough paths and rough partial differential equations · March 18, 2016 · Page 3 (48)

Examples of controlled differential equationsdyt V(yt )dxt ,Iy0 ξ Rd ,t [0, 1]xt smooth:ẏt V(yt ) ẋtIxt Wt (ω) is a path of a Brownian motion, i.e., yt yt (ω) ispathwise solution of the stochastic differential equationdyt (ω) V(yt (ω))dWt (ω)(Ito, Stratonovich or some other sense?)Ixt Zt (ω) for some other stochastic process, such as fractionalBrownian motion, yt yt (ω) is pathwise solution of thecorresponding stochastic differential equationRough paths and rough partial differential equations · March 18, 2016 · Page 4 (48)

Examples of controlled differential equationsdyt V(yt )dxt ,Iy0 ξ Rd ,t [0, 1]xt smooth:ẏt V(yt ) ẋtIxt Wt (ω) is a path of a Brownian motion, i.e., yt yt (ω) ispathwise solution of the stochastic differential equationdyt (ω) V(yt (ω))dWt (ω)(Ito, Stratonovich or some other sense?)Ixt Zt (ω) for some other stochastic process, such as fractionalBrownian motion, yt yt (ω) is pathwise solution of thecorresponding stochastic differential equationRough paths and rough partial differential equations · March 18, 2016 · Page 4 (48)

Examples of controlled differential equationsdyt V(yt )dxt ,Iy0 ξ Rd ,t [0, 1]xt smooth:ẏt V(yt ) ẋtIxt Wt (ω) is a path of a Brownian motion, i.e., yt yt (ω) ispathwise solution of the stochastic differential equationdyt (ω) V(yt (ω))dWt (ω)(Ito, Stratonovich or some other sense?)Ixt Zt (ω) for some other stochastic process, such as fractionalBrownian motion, yt yt (ω) is pathwise solution of thecorresponding stochastic differential equationRough paths and rough partial differential equations · March 18, 2016 · Page 4 (48)

Integral formdyt V(yt )dxt ,y0 ξ Rd ,t [0, 1]Assume that xt is not smooth, sayα(ex C ([0, 1]; R ) BIe x C [0, 1]; R) x s xt supα C kxkα , α 1s,t s t While ẋ does not “easily” make sense, maybe the integral formdoes:Ztyt ξ V(y s )dx s ,t [0, 1]0IINotice: If x C α , then generically y C α (and no better), as well.Need to make sense of expressions of the formZty s dx s ,x, y C α ([0, 1])0Rough paths and rough partial differential equations · March 18, 2016 · Page 5 (48)

Integral formdyt V(yt )dxt ,y0 ξ Rd ,t [0, 1]Assume that xt is not smooth, sayα(ex C ([0, 1]; R ) BIe x C [0, 1]; R) x s xt supα C kxkα , α 1s,t s t While ẋ does not “easily” make sense, maybe the integral formdoes:Ztyt ξ V(y s )dx s ,t [0, 1]0IINotice: If x C α , then generically y C α (and no better), as well.Need to make sense of expressions of the formtZy s dx s ,x, y C α ([0, 1])0Rough paths and rough partial differential equations · March 18, 2016 · Page 5 (48)

Integral formdyt V(yt )dxt ,y0 ξ Rd ,t [0, 1]Assume that xt is not smooth, sayα(ex C ([0, 1]; R ) BIe x C [0, 1]; R) x s xt supα C kxkα , α 1s,t s t While ẋ does not “easily” make sense, maybe the integral formdoes:Ztyt ξ V(y s )dx s ,t [0, 1]0IINotice: If x C α , then generically y C α (and no better), as well.Need to make sense of expressions of the formtZy s dx s ,x, y C α ([0, 1])0Rough paths and rough partial differential equations · March 18, 2016 · Page 5 (48)

Integral formdyt V(yt )dxt ,y0 ξ Rd ,t [0, 1]Assume that xt is not smooth, sayα(ex C ([0, 1]; R ) BIe x C [0, 1]; R) x s xt supα C kxkα , α 1s,t s t While ẋ does not “easily” make sense, maybe the integral formdoes:Ztyt ξ V(y s )dx s ,t [0, 1]0IINotice: If x C α , then generically y C α (and no better), as well.Need to make sense of expressions of the formtZy s dx s ,x, y C α ([0, 1])0Rough paths and rough partial differential equations · March 18, 2016 · Page 5 (48)

Young integraltZy s dx s ,x, y C α ([0, 1])0Recall the Riemann-Stieltjes integral:1Zy s dx s B lim0 P 0X[s,t] Py s (xt x s ) {z }( )Cx s,tP a finite partition of [0, 1]Theorem (Young 1936)(a) Let y C β ([0, 1]; R), x C α ([0, 1]; R) with 0 α, β 1 andα β 1R. Then ( ) converges and theR 1resulting bi-linear map1(x, y) 7 0 y s dx s is continuous, i.e., 0 y s dx s Cα β ( y0 ) kykβ kxkα .(b) Let α β 1. Then there are y C β ([0, 1]; R), x C α ([0, 1]; R)such that ( ) does not converge, i.e., such that different sequences ofpartitions yield different limits (or none at all).Rough paths and rough partial differential equations · March 18, 2016 · Page 6 (48)

Young integraltZy s dx s ,x, y C α ([0, 1])0Recall the Riemann-Stieltjes integral:1Zy s dx s B lim0 P 0X[s,t] Py s (xt x s ) {z }( )Cx s,tP a finite partition of [0, 1]Theorem (Young 1936)(a) Let y C β ([0, 1]; R), x C α ([0, 1]; R) with 0 α, β 1 andα β 1R. Then ( ) converges and theR 1resulting bi-linear map1(x, y) 7 0 y s dx s is continuous, i.e., 0 y s dx s Cα β ( y0 ) kykβ kxkα .(b) Let α β 1. Then there are y C β ([0, 1]; R), x C α ([0, 1]; R)such that ( ) does not converge, i.e., such that different sequences ofpartitions yield different limits (or none at all).Rough paths and rough partial differential equations · March 18, 2016 · Page 6 (48)

Young integraltZy s dx s ,x, y C α ([0, 1])0Recall the Riemann-Stieltjes integral:1Zy s dx s B lim0 P 0X[s,t] Py s (xt x s ) {z }( )Cx s,tP a finite partition of [0, 1]Theorem (Young 1936)(a) Let y C β ([0, 1]; R), x C α ([0, 1]; R) with 0 α, β 1 andα β 1R. Then ( ) converges and theR 1resulting bi-linear map1(x, y) 7 0 y s dx s is continuous, i.e., 0 y s dx s Cα β ( y0 ) kykβ kxkα .(b) Let α β 1. Then there are y C β ([0, 1]; R), x C α ([0, 1]; R)such that ( ) does not converge, i.e., such that different sequences ofpartitions yield different limits (or none at all).Rough paths and rough partial differential equations · March 18, 2016 · Page 6 (48)

Young integral IIdyt V(yt )dxt ,y0 ξ Rd ,t [0, 1]Let x C α ([0, 1]; Re ), α 21 and V Cb2 (Rd ; Rd e ). Then the usualPicard iteration scheme converges and the controlled differentialequation has a unique solution.ExampleLet 0 H 1. The fractional Brownian motion withindex H ish HurstiHHthe Gaussian process (on [0, 1]) with W0 0, E Wt 0 andhi 1 E WtH W sH t2H s2H t s 2H .2IIfBm with H Paths ofWH12is standard Brownian motion;are a.s. α-Hölder for any α H (but no α H ).Hence, we can solve fractional SDEs for H 12 .Rough paths and rough partial differential equations · March 18, 2016 · Page 7 (48)

Young integral IIdyt V(yt )dxt ,y0 ξ Rd ,t [0, 1]Let x C α ([0, 1]; Re ), α 21 and V Cb2 (Rd ; Rd e ). Then the usualPicard iteration scheme converges and the controlled differentialequation has a unique solution.ExampleLet 0 H 1. The fractional Brownian motion withindex H ish HurstiHHthe Gaussian process (on [0, 1]) with W0 0, E Wt 0 andhi 1 E WtH W sH t2H s2H t s 2H .2IIfBm with H Paths ofWH12is standard Brownian motion;are a.s. α-Hölder for any α H (but no α H ).Hence, we can solve fractional SDEs for H 12 .Rough paths and rough partial differential equations · March 18, 2016 · Page 7 (48)

Young integral IIdyt V(yt )dxt ,y0 ξ Rd ,t [0, 1]Let x C α ([0, 1]; Re ), α 21 and V Cb2 (Rd ; Rd e ). Then the usualPicard iteration scheme converges and the controlled differentialequation has a unique solution.ExampleLet 0 H 1. The fractional Brownian motion withindex H ish HurstiHHthe Gaussian process (on [0, 1]) with W0 0, E Wt 0 andhi 1 E WtH W sH t2H s2H t s 2H .2IIfBm with H Paths ofWH12is standard Brownian motion;are a.s. α-Hölder for any α H (but no α H ).Hence, we can solve fractional SDEs for H 12 .Rough paths and rough partial differential equations · March 18, 2016 · Page 7 (48)

Rough drivers as limits of smooth driversdyt V(yt )dxt ,Iy0 ξ Rd ,t [0, 1]Classical theory works for smooth x, say x C ([0, 1]; Re )IdeaIChoose sequence xn of smooth paths converging to xIAssumesolutions yn converge to some path that corresponding y C α [0, 1]; RdICall y solution of the controlled equationRough paths and rough partial differential equations · March 18, 2016 · Page 8 (48)

Rough drivers as limits of smooth driversdyt V(yt )dxt ,Iy0 ξ Rd ,t [0, 1]Classical theory works for smooth x, say x C ([0, 1]; Re )IdeaIChoose sequence xn of smooth paths converging to xIAssumesolutions yn converge to some path that corresponding y C α [0, 1]; RdICall y solution of the controlled equationRough paths and rough partial differential equations · March 18, 2016 · Page 8 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionzntEven though xn 0 in k·k , we have znt 21 t.Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionznt B12tZ0n,2xn,1s dx s 12tZ0n,1xn,2s dx sEven though xn 0 in k·k , we have znt 21 t.Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionznt 12tZ0Z t 2 ! 2cos n2 s dssin n2 s ds 0Even though xn 0 in k·k , we have znt 21 t.Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionznt 12Z0t11ds t.2Even though xn 0 in k·k , we have znt 21 t.Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionznt B12tZ0n,2xn,1s dx s 12tZ0n,1xn,2s dx sEven though xn 0 in k·k , we have znt 21 t.Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionznt B12tZ0n,2xn,1s dx s 12tZ0n,1xn,2s dx sEven though xn 0 in k·k , we have znt 21 t.Relevance for controlled differential equations: choose 10 V(y) 01 , 1 1 2 y2 2 y1 Then ynt B xtn,1 , xtn,2 , znt solvesdynt V(ynt )dxtn ,y R3y0 (0, 1/n, 0) .Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionznt1B2tZ0n,2xn,1s dx s1 2tZ0n,1xn,2s dx sEven though xn 0 in k·k , we have znt 21 t.RemarkIThe example is not just an instance of “poor choice of norm”:replacing k·k by any other reasonable norm is vulnerable to thesame type of example.I“Curing this example will cure all other counter-examples.”IDoes not work in dimension e 1 (Doss–Sussmanntransformation.)Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionznt1B2tZ0n,2xn,1s dx s1 2tZ0n,1xn,2s dx sEven though xn 0 in k·k , we have znt 21 t.RemarkIThe example is not just an instance of “poor choice of norm”:replacing k·k by any other reasonable norm is vulnerable to thesame type of example.I“Curing this example will cure all other counter-examples.”IDoes not work in dimension e 1 (Doss–Sussmanntransformation.)Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

A counter-example xtn sin(n2 t)/n , cos(n2 t)/n ,t [0, 2π]Consider the area functionznt1B2tZ0n,2xn,1s dx s1 2tZ0n,1xn,2s dx sEven though xn 0 in k·k , we have znt 21 t.RemarkIThe example is not just an instance of “poor choice of norm”:replacing k·k by any other reasonable norm is vulnerable to thesame type of example.I“Curing this example will cure all other counter-examples.”IDoes not work in dimension e 1 (Doss–Sussmanntransformation.)Rough paths and rough partial differential equations · March 18, 2016 · Page 9 (48)

Ito stochastic integrationSuppose you want to cover the case xt Wt (ω), a standard Brownianmotion.Brownian motion is a martingale: i.e., the increments are orthogonal(in L2 (Ω)) to the past: for bounded f , we have Z f ((Wu )0 u s ) E ZW s,t 0 for 0 s t.This

Young integral Z t 0 y sdx s; x;y 2C ([0;1]) Recall theRiemann-Stieltjes integral: Z 1 0 y sdx s B lim jPj!0 X [s;t]2P y s ( x t{z x s}) Cx s;t () Pa ﬁnite partition of [0;1] Th

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