Rough Paths Methods 4: Application To FBm

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Rough paths methods 4:Application to fBmSamy TindelPurdue UniversityUniversity of Aarhus 2016Samy T. (Purdue)Rough Paths 4Aarhus 20161 / 67

Outline1Main result2Construction of the Levy area: heuristics3Preliminaries on Malliavin calculus4Levy area by Malliavin calculus methods5Algebraic and analytic properties of the Levy area6Levy area by 2d-var methods7Some projectsSamy T. (Purdue)Rough Paths 4Aarhus 20162 / 67

Outline1Main result2Construction of the Levy area: heuristics3Preliminaries on Malliavin calculus4Levy area by Malliavin calculus methods5Algebraic and analytic properties of the Levy area6Levy area by 2d-var methods7Some projectsSamy T. (Purdue)Rough Paths 4Aarhus 20163 / 67

ObjectiveSummary: We have obtained the following pictureRdx ,RRRdxdxRough paths theorySmooth V0 , . . . , VdVj (x ) dx jdy Vj (y )dx jRemaining question:RRHow to definedxdx when x is a fBm with H 1/2?Samy T. (Purdue)Rough Paths 4Aarhus 20164 / 67

Levy area of fBmProposition 1.Let B be a d-dimensional fBm, with H 1/3, and 1/3 γ H. Almost surely, the paths of B:12Belong to C1γAdmit a Levy area B2 C22γ such thatδB2 δB δB,iji. e. B2,ijsut δBsu δButConclusion:The abstract rough paths theory applies to fBm with H 1/3Proof of item 1: Already seen (Kolmogorov criterion)Samy T. (Purdue)Rough Paths 4Aarhus 20165 / 67

Geometric and weakly geometric Levy areaRemark:The stack B2 as defined in Proposition 1 is calleda weakly geometric second order rough path above X, allows a reasonable differential calculusWhen there exists a family B ε such thatIIIB ε is smoothB2,ε is the iterated Riemann integral of B εB2 limε 0 B2,εthen one has a so-called geometric rough path above B, easier physical interpretationSamy T. (Purdue)Rough Paths 4Aarhus 20166 / 67

Levy area construction for fBm: historySituation 1: H 1/4, 3 possible geometric rough paths constructions for B.Malliavin calculus tools (Ferreiro-Utzet)Regularization or linearization of the fBm path (Coutin-Qian)Regularization and covariance computations (Friz et al)Situation 2: d 1, Then one can take B2st (Bt Bs )22Situation 3: H 1/4, d 1The constructions by approximation divergeExistence result by dyadic approximation (Lyons-Victoir)Recent advances (Unterberger, Nualart-T)for weakly geometric Levy area constructionSamy T. (Purdue)Rough Paths 4Aarhus 20167 / 67

Outline1Main result2Construction of the Levy area: heuristics3Preliminaries on Malliavin calculus4Levy area by Malliavin calculus methods5Algebraic and analytic properties of the Levy area6Levy area by 2d-var methods7Some projectsSamy T. (Purdue)Rough Paths 4Aarhus 20168 / 67

fBm kernelRecall: B is a d-dimensional fBm, withBti ZKt (u) dWui ,t 0,Rwhere W is a d-dimensional Wiener process and1Kt (u) (t u)H 2 1{0 u t}3 t Kt (u) (t u)H 2 1{0 u t} .Samy T. (Purdue)Rough Paths 4Aarhus 20169 / 67

Heuristics: fBm differentialFormal differential:Rwe have Bvj 0v Kv (u) dWuj and thus formally for H 1/2Ḃvj Z v0 v Kv (u) dWujFormal definition of the area:Consider B i . Then formallyZ 10BvidBvj Z 10 Bvi Z v0Z 1 Z 10u v Kv (u) dWuj v Kv (u) Bvi dv dvdWujThis works for H 1/2 since H 3/2 1.Samy T. (Purdue)Rough Paths 4Aarhus 201610 / 67

Heuristics: fBm differential for H 1/2Formal definition of the area for H 1/2:Use the regularity of B i and writeZ 10 Bvi dBvj Z 1 Z 10uZ 1 Z 10u v Kv (u) Bvi dvi v Kv (u) δBuv dWuj dv dWuj Z 10K1 (u) Bui dWuj .iControl of singularity: v Kv (u) δBuv (v u)H 3/2 H, Definition works for 2H 3/2 1, i.e. H 1/4!Hypothesis:R10Samy T. (Purdue)Bvi dBvj well defined as stochastic integralRough Paths 4Aarhus 201611 / 67

Outline1Main result2Construction of the Levy area: heuristics3Preliminaries on Malliavin calculus4Levy area by Malliavin calculus methods5Algebraic and analytic properties of the Levy area6Levy area by 2d-var methods7Some projectsSamy T. (Purdue)Rough Paths 4Aarhus 201612 / 67

Space HNotation: LetE be the set of step-functions f : R RB be a 1-d fBmRecall:1RH (s, t) E[Bt Bs ] ( s 2H t 2H t s 2H )2Space H: Closure of E with respect to the inner productD1[s1 ,s2 ] , 1[t1 ,t2 ]EH E [δBs1 s2 δBt1 t2 ](1) RH (s2 , t2 ) RH (s1 , t2 ) RH (s2 , t1 ) RH (s1 , t1 ) [s1 ,s2 ] [t1 ,t2 ] RHSamy T. (Purdue)Rough Paths 4Aarhus 201613 / 67

Isonormal processFirst chaos of B: We setH1 (B) closure in L2 (Ω) of linear combinations of δBstFundamental isometry: The mapping1[t,t 0 ] 7 Bt 0 Btcan be extended to an isometry between H and H1 (B), We denote this isometry by ϕ 7 B(ϕ).Isonormal process: B can be interpreted asA centered Gaussian family {B(ϕ); ϕ H}Covariance function given by E[B(ϕ1 )B(ϕ2 )] hϕ1 , ϕ2 iHSamy T. (Purdue)Rough Paths 4Aarhus 201614 / 67

Underlying Wiener process on compact intervalsVolterra type representation for B:Bt Zt 0Kt (u) dWu ,RwithW Wiener processKt (u) defined byKt (u) cH 1 Hu 2t1(t u)H 21 H2 u1 H2Z tvuH 23(v u)H 12 dv 1{0 u t}Bounds on K : If H 1/211 Kt (u) . (t u)H 2 u H 2 ,Samy T. (Purdue)3and t Kt (u) . (t u)H 2 .Rough Paths 4Aarhus 201615 / 67

Underlying Wiener process on RMandelbrot’s representation for B:Bt ZKt (u) dWu ,t 0RwithW two-sided Wiener processKt (u) defined by Kt (u) cH (t H 1/2u) H 1/2( u) 1{ u t}Bounds on K : If H 1/2 and 0 u t1 Kt (u) . (t u)H 2 ,Samy T. (Purdue)3and t Kt (u) . (t u)H 2 .Rough Paths 4Aarhus 201616 / 67

Fractional derivativesDefinition: For α (0, 1), u R and f smooth enough,Z αfr fu r drΓ(1 α) 0r 1 αfr1 Z αdrI fu Γ(α) u (r u)1 ααD fuInversion property: ααI α D f D I α f fSamy T. (Purdue)Rough Paths 4Aarhus 201617 / 67

Fractional derivatives on intervalsNotation: For f : [a, b] R, extend f by setting f ? f 1[a,b]Definition:Z bfuαfu fr drΓ(1 α)(b u)α Γ(1 α) u (r u)1 α1 ZbfrαI α fu? Ib fu drΓ(α) u (r u)1 αα ?D fuαDb fuA related operator: For H 1/2,1/2 HKf D Samy T. (Purdue)Rough Paths 4fAarhus 201618 / 67

Wiener space and fractional derivativesProposition 2.For H 1/2 we haveK isometry between H and a closed subspace of L2 (R)For φ, ψ H,E [B(φ)B(ψ)] hφ, ψiH hKφ, KψiL2 (R) ,In particular, for φ H,hiE B(φ) 2 kϕkH kKϕkL2 (R)Notation:B(φ) is called Wiener integral of φ w.r.t BSamy T. (Purdue)Rough Paths 4Aarhus 201619 / 67

Cylindrical random variablesDefinition 3.Letf Cb (Rk ; R)ϕi H, for i {1, . . . , k}F a random variable defined byF f (B(ϕ1 ), . . . , B(ϕk ))We say that F is a smooth cylindrical random variableNotation:S Set of smooth cylindrical random variablesSamy T. (Purdue)Rough Paths 4Aarhus 201620 / 67

Malliavin’s derivative operatorDefinition for cylindrical random variables:If F S, DF H defined byDF kXi 1 f(B(ϕ1 ), . . . , B(ϕk ))ϕi . xiProposition 4.D is closable from Lp (Ω) into Lp (Ω; H).Notation: D1,2 closure of S with respect to the normhihikF k21,2 E F 2 E kDF k2H .Samy T. (Purdue)Rough Paths 4Aarhus 201621 / 67

Divergence operatorDefinition 5.Domain of definition:nDom(δ ) φ L2 (Ω; H); E [hDF , φiH ] cφ kF kL2 (Ω)oDefinition by duality: For φ Dom(I) and F D1,2E [F δ (φ)] E [hDF , φiH ]Samy T. (Purdue)Rough Paths 4(2)Aarhus 201622 / 67

Divergence and integralsCase of a simple process: Considern 10 t1 · · · tnai R constantsThenδ n 1X!ai 1[ti ,ti 1 ) i 0n 1Xai δBti ti 1i 0Case of a deterministic process: if φ H is deterministic,δ (φ) B(φ),hence divergence is an extension of Wiener’s integralSamy T. (Purdue)Rough Paths 4Aarhus 201623 / 67

Divergence and integrals (2)Proposition 6.LetB a fBm with Hurst parameter 1/4 H 1/2f a C 3 function with exponential growth{Πnst ; n 1} set of dyadic partitions of [s, t]DefinenS̃ n, 2X 1f (Btk ) δBtk tk 1 .k 0Then S̃ n, converges in L2 (Ω) to δ (f (B))Remark: In the Brownian case, δ coincides with Itô’s integralSamy T. (Purdue)Rough Paths 4Aarhus 201624 / 67

Criterion for the definition of divergenceProposition 7.Leta b, and E [a,b] step functions in [a, b]H0 ([a, b]) closure of E [a,b] with respect tokϕk2H0 ([a,b]) Z ba!2Z b Z bϕ2u ϕr ϕu du dr du.(b u)1 2Hau (r u)3/2 HThenH0 ([a, b]) is continuously included in HIf φ D1,2 (H0 ([a, b])), then φ Dom(δ )Samy T. (Purdue)Rough Paths 4Aarhus 201625 / 67

Bound on the divergenceCorollary 8.Under the assumptions of Proposition 7,hihE δ (φ) 2 . E kφk2D1,2 (H0 ([a,b]))Samy T. (Purdue)Rough Paths 4iAarhus 201626 / 67

Multidimensional extensionsAim:Define a Malliavin calculus for (B 1 , . . . , B d )First point of view: Rely oniPartial derivatives D B with respect to each componentiDivergences δ ,B , defined by duality, Related to integrals with respect to each B iSecond point of view:Change the underlying Hilbert space and considerĤ H {1, . . . , d}Samy T. (Purdue)Rough Paths 4Aarhus 201627 / 67

Russo-Vallois’ symmetric integralDefinition 9.Letφ be a random pathThenZ baφw d Bwi 1 Zb L limφw Bwi ε Bwi ε dw ,ε 0 2ε a2provided the limit exists.Extension of classical integrals: Russo-Vallois’ integral coincides withYoung’s integral if H 1/2 and φ C 1 H εStratonovich’s integral in the Brownian caseSamy T. (Purdue)Rough Paths 4Aarhus 201628 / 67

Russo-Vallois and MalliavinProposition 10.Let φ be a stochastic process such that1φ1[a,b] D1,2 (H0 ([a, b])), for all a b 2The following is an almost surely finite random variable:1 ZbhDφu , 1[u ε,u ε] iH duTr[a,b] Dφ : L limε 0 2ε a2ThenRbaφu d Bui exists, and verifiesZ baSamy T. (Purdue)φu d Bui δ (φ 1[a,b] ) Tr[a,b] Dφ.Rough Paths 4Aarhus 201629 / 67

Outline1Main result2Construction of the Levy area: heuristics3Preliminaries on Malliavin calculus4Levy area by Malliavin calculus methods5Algebraic and analytic properties of the Levy area6Levy area by 2d-var methods7Some projectsSamy T. (Purdue)Rough Paths 4Aarhus 201630 / 67

Levy area: definition of the divergenceLemma 11.LetH 14B a d-dimensional fBm(H)0 s t Then for any i, j {1, . . . , d} (either i j or i 6 j) we have12ijφju δBsu1[s,t] (u) lies in Dom(δ ,B )The following estimate holds true:ESamy T. (Purdue) i δ ,B φj 2 Rough Paths 4 cH t s 4HAarhus 201631 / 67

ProofCase i j, strategy:We invoke Corollary 8iWe have to prove φi 1[s,t] D1,2,B (H0 ([s, t]))iAbbreviation: we write D1,2,B (H0 ([s, t])) D1,2 (H0 )Norm of φi in H0 : We havehE kφi k2H0iA1st A1st A2st Z tsA2st ESamy T. (Purdue)i 2E [ δBsu ]du1 2H(t u) Z tZ t sui δBur dr3/2 H(r u)Rough Paths 4!2du Aarhus 201632 / 67

Proof (2)Analysis of A1st : u s 2Hdus (t u)1 2H cH (t s)4HA1st Z tu: s (t s)v (t s)4HZ 10v 2Hdv(1 v )1 2HAnalysis of A2st :A2st Z thdus[u,t]2Z tZ tduus cHSamy T. (Purdue)ZZ tsdr1 dr2iiE δBurδBur12i(r1 u)3/2 H (r2 u)3/2 Hdr(r u)3/2 2H!2(t u)4H 1 du cH (t s)4HRough Paths 4Aarhus 201633 / 67

Proof (3)Conclusion for kφi kH0 : We have foundhiE kφi k2H0 cH (t s)4HiDerivative term, strategy: setting D D B we haveWe have Dv φiu 1[s,u] (v )We have to evaluate Dφi H0u HvComputation of the H-norm: According to (1),hi2 2kDφi k2H E δBsu u s 2HSamy T. (Purdue)Rough Paths 4Aarhus 201634 / 67

Proof (4)Computation for Dφi : We gethE kDφi k2H0 Hi Bst1 Bst2Bst1 hZ tE (u s)2Hs(t u)1 2H Z tBst2 E sZ tuidu r s H u s Hdr(r u)3/2 H!2 du Moreover:0 r s H u s H r u HHence, as for the terms A1st , A2st , we gethiE kDφi k2H0 H cH (t s)4HSamy T. (Purdue)Rough Paths 4Aarhus 201635 / 67

Proof (5)Summary: We have foundihihE kφi k2H0 E kDφi k2H0 H cH (t s)4HConclusion for B i : According to Proposition 7 and Corollary 8iδBs·i 1[s,t] Dom(δ ,B )We haveESamy T. (Purdue) δ ,B i δBs·i 1[s,t] 2 Rough Paths 4 cH t s 4HAarhus 201636 / 67

Proof (6)Case i 6 j, strategy: Conditioned on F BB j and φj δBs·j are deterministiciδ ,B (φj ) is a Wiener integraljComputation: For i 6 j we haveE δ ,B i φj 2 E Ehδ ,B i E kφj k2H jφ 2FBj ih cH E kφj k2H0(3)i cH t s 4H ,where computations for the last step are the same as for i j.Samy T. (Purdue)Rough Paths 4Aarhus 201637 / 67

Definition of the Levy areaProposition 12.LetH 14B a d-dimensional fBm(H)0 s t Then for any i, j {1, . . . , d} (either i j or i 6 j) we have122,jijBst st δBsud Bui defined in the Russo-Vallois senseThe following estimate holds true:R ESamy T. (Purdue)2B2,jist cH t s 4HRough Paths 4Aarhus 201638 / 67

ProofStrategy:We apply Proposition 10, and check the assumptionsProposition 10, item 1: proved in Lemma 11Proposition 10, item 2: need to compute trace termTrace term, case i j: We haveiDvB φiu 1[s,u] (v )HencehDφiu , 1[u ε,u ε] iH [s,u] [u ε,u ε] RHSamy T. (Purdue)Rough Paths 4Aarhus 201639 / 67

Proof (2)Computation of the rectangular increment: We have [s,u] [u ε,u ε] RH RH (u, u ε) RH (s, u ε) RH (u, u ε) RH (s, u ε)i1 h 2H ε (u s ε)2H ε2H (u s ε)2H 2i1h (u s ε)2H (u s ε)2H2Computation of the integral: Thanks to an elementary integration,Z ts [s,u] [u ε,u ε] RH duhi1(t s ε)2H 1 ε2H 1 (t s ε)2H 12(2H 1)Samy T. (Purdue)Rough Paths 4Aarhus 201640 / 67

Proof (3)Computation of the trace term: Differentiating we getTr[s,t] Dφi(t s ε)2H 1 ε2H 1 (t s ε)2H 11lim2(2H 1) ε 02ε2H(t s) 2 Expression for the Stratonovich integral: According to Proposition 10B2,iistSamy T. (Purdue) Z tsiiδsud Bui δ ,B (φi 1[s,t] ) Rough Paths 4(t s)2H2Aarhus 2016(4)41 / 67

Proof (4)Moment estimate: Thanks to relation (4) we have E2B2,iist cH t s 4HCase i 6 j: We haveTrace term is 0 ,B iB2,ji(φj 1[s,t] )st δMoment estimate follows from Lemma 11Samy T. (Purdue)Rough Paths 4Aarhus 201642 / 67

RemarkAnother expression for Bii :Rules of Stratonovich calculus for B show that2Biist (δBsti )2Much simpler expression!Samy T. (Purdue)Rough

Samy T. (Purdue) Rough Paths 4 Aarhus 2016 27 / 67. Russo-Vallois’symmetricintegral Let φbearandompath Then Z b a φ w d Bi w L 2 lim ε 0 1 2ε Z b a φ w Bi w ε B i w ε dw, providedthelimitexists. Definition9. gralcoincideswith Young’sintegralifH 1/2andφ C1 H ε Stratonovich’sintegralintheBrowniancase Samy T .

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