Pathwise integration with respect to paths of finite quadratic variation

Transcription

1 JMPA Journal de Mahémaques Pures e Applquées (216) Pahwse negraon wh respec o pahs of fne quadrac varaon Anna ANANOVA a & Rama CONT a,b a Deparmen of Mahemacs, Imperal College London. b Laboraore de Probablés e Modèles Aléaores, UMR 7599 CNRS-Pars VI. Absrac We sudy a pahwse negral wh respec o pahs of fne quadrac varaon, defned as he lm of non-ancpave Remann sums for graden-ype negrands. We show ha he negral sasfes a pahwse somery propery, analogous o he well-known Io somery for sochasc negrals. Ths propery s hen used o represen he negral as a connuous map on an appropraely defned vecor space of negrands. Fnally, we oban a pahwse sgnal plus nose decomposon for regular funconals of an rregular pah wh non-vanshng quadrac varaon, as a unque sum of a pahwse negral and a componen wh zero quadrac varaon. Résumé Inégraon rajecorelle par rappor à des rajecores de varaon quadraque fne. Nous consrusons une négrale, défne comme lme de sommes de Remann non-ancpaves, par rappor à des rajecores muldmensonnels rrégulères de varaon nfne mas de varaon quadraque fne. Cee consrucon perme d négrer des négrands qu se représenen comme une dérvée dreconelle d une fonconelle régulère de la rajecore. Nous monrons une formula d somére, qu es une verson rajecorelle de la formule d somére d Io. Cee propréé perme d obenr une verson rajecorelle de la décomposon sgnal plus bru : éan donné une rajecore rrégulère de varaon quadraque srcemen crossane, nous monrons que oue fonconelle régulère de cee rajecore adme une unque décomposon comme somme d une négrale rajecorelle e une composane régulère de varaon quadraque zéro. MSC 21 : Prmary 6H5 Secondary 26B15, 26A42 Emal addresses: a.ananova14@mperal.ac.uk (Anna ANANOVA), Rama.Con@mah.cnrs.fr (Rama CONT). URL: hp://rama.con.perso.mah.cnrs.fr (Rama CONT). Arcle publshed n Journal de Mahémaques Pures e Applquées 16 (216) 1 25

2 In hs semnal paper Calcul d Io sans probablés [12], Hans Föllmer proved a change of varable formula for smooh funcons of pahs wh nfne varaon, usng he concep of quadrac varaon along a sequence of parons. A pah ω C ([, T ], R) s sad o have fne quadrac varaon along he sequence of parons π m = ( = m < m 1 < < m k(m) = T ) f for any [, T ], he lm [ω] π () := lm m m (ω( m +1) ω( m )) 2 < (1) exss and defnes a connuous ncreasng funcon [ω] π : [, T ] R + called he quadrac varaon of ω along π. Exendng hs defnon o vecor-valued pahs (see Secon 1) Föllmer [12] showed ha, for negrands of he form f(ω()), f C 2 (R d ) one may defne a pahwse negral f(ω())dω() as a ponwse lm of Remann sums along he sequence of parons π m. Ths consrucon has been exended n varous drecons, o less regular funcons [1,9,23] and pah-dependen funconals [4,7,8]. In parcular, Con & Fourné [4] consruced pahwse negrals of he ype ω F (, ω)d π ω where ω F s a dreconal dervave (Dupre dervave) of a non-ancpave funconal F, and proved a change of varable formulas for such negrals. Ths paper conrbues several new resuls o he sudy of he pahwse approach o sochasc negraon [12] and s exenson o pah-dependen funconals [4]. Our man resul s an somery formula for he pahwse negral: we gve condons on he negrand φ and he pah ω C ([, T ], R) under whch. [ φd π ω] π () = φ 2 d[ω] π. Theorem 2.1 gves a precse saemen of hs propery for muldmensonal pahs. Our condons noably allow for connuous pahs wh Hölder exponen srcly less han 1/2 and apply o ypcal pahs of Brownan moon and dffuson processes. Ths resul may be undersood as a pahwse verson of he well-known Io somery formula [19], showng ha he Io somery formula holds no jus as an equaly n expecaon bu n fac pah by pah. Ths resul has several neresng consequences. Frs, mples ha he pahwse negral may be defned as a a connuous mappng for he quadrac 2

3 varaon merc on an approprae space of negrands (Proposon 2.2). Ths connuy propery, ogeher wh s nerpreaon as a lm of Remann sums, s wha dsngushes our consrucon from oher pahwse consrucons of sochasc negrals [2,21,23,25] whch lack eher a connuy propery or an nerpreaon as a lm of non-ancpave Remann sums. A second consequence of he pahwse somery propery s a deermnsc sgnal plus nose decomposon for funconals of an rregular pah: we show ha any regular funconal of pah ω wh non-vanshng quadrac varaon may be unquely decomposed as he sum of a pahwse negral wh respec o ω and a smooh componen wh zero quadrac varaon (Proposon 4.1). Ths may be seen as a pahwse analogue of he well-known semmarngale decomposon [1] (or, more precsely he decomposon of Drchle processes [13]). A smlar resul was obaned by Harer and Plla [18] n he conex of rough pah heory, usng dfferen echnques. Fnally, we clarfy he pahwse naure of he negral defned n [4]: we show (Theorem 3.2) ha hs negral s ndeed a pahwse lm of non-ancpave Remann sums, whch s mporan for nerpreaon and n applcaons. Oulne Secon 1 recalls some key defnons and resuls on funconal calculus from [4,3] and recalls he defnon of he Föllmer negral [12] and s exenson o pah-dependen negrands by Con & Fourné [4]. The somery formula for hs negral s derved n Secon 2 (Theorems 2.1). In Secon 2.2 we dscuss he somery formula for Lebesgue parons. We use he of Secon 2 n Secon 2.3 o represen he negral as a connuous map (Proposon 2.2). In Secon 3 we clarfy he pahwse naure of hs negral (Theorem 3.2). Secon 4 derves a pahwse sgnal plus nose decomposon for funconals of rregular pahs (Proposon 4.1), whch may be vewed as a deermnsc analogue of he semmarngale decomposon for sochasc processes. 1 Non-ancpave funconal calculus and pahwse negraon of graden funconals Our approach reles on he Non-ancpave Funconal Calculus [4,3], a funconal calculus whch apples o non-ancpave funconals of càdlàg pahs wh fne quadrac varaon, n he sense of Föllmer [12]. We recall here some key conceps and resuls of hs approach, followng [3]. Le X be he canoncal process on Ω = D([, T ], R d ), and (F ) [,T ] be he flraon generaed by X. We are neresed n causal, or non-ancpave 3

4 funconals of X,ha s, funconals F : [, T ] D([, T ], R d ) R such ha ω Ω, F (, ω) = F (, ω ), (2) where ω := ω( ) s he sopped pah of ω a me. The process F (, X ) hen only depends on he pah of X up o and s (F )-adaped. I s convenen o defne such funconals on he space of sopped pahs [3]: a sopped pah s an equvalence class n [, T ] D([, T ], R d ) for he followng equvalence relaon: (, ω) (, ω ) ( = and ω(.) = ω (.) ). (3) The space of sopped pahs s defned as he quoen of [, T ] D([, T ], R d ) by he equvalence relaon (3): Λ T = {(, ω( )), (, ω) [, T ] D([, T ], R d )} = ( [, T ] D([, T ], R d ) ) / We denoe W T he subse of Λ T conssng of connuous sopped pahs. We endow he se Λ T wh a merc space srucure by defnng he followng dsance: d ((, ω), (, ω )) = sup ω(u ) ω (u ) + := ω ω + u [,T ] (Λ T, d ) s hen a complee merc space. Any map F : [, T ] D([, T ], R d ) R verfyng he causaly condon (2) can be equvalenly vewed as a funconal on he space Λ T of sopped pahs. We call non-ancpave funconal on D([, T ], R d ) a measurable map F : (Λ T, d ) R and denoe by C, (Λ d T ) he se of such maps whch are connuous for he d ) merc. Some weaker noons of connuy for non-ancpave funconals urn ou o be useful [5]: Defnon 1.1. A non-ancpave funconal F s sad o be: connuous a fxed mes f for any [, T ], F (, ) : D([, T ], ) R n [, T ],.e. ω D([, T ], R d ), ɛ >, η >, ω D([, T ], R d ), ω ω < η = F (, ω) F (, ω ) < ɛ lef-connuous f (, ω) Λ T, ɛ >, η > such ha (, ω ) Λ T, ( < and d ((, ω), (, ω )) < η) = F (, ω) F (, ω ) < ɛ We denoe by C, l (Λ d T ) he se of lef-connuous funconals. Smlarly, we can defne he se C, r (Λ d T ) of rgh-connuous funconals. 4

5 We also nroduce a noon of local boundedness for funconals: Defnon 1.2. A non-ancpave funconal F s sad o be boundednesspreservng f for every compac subse K of R d, [, T ], C(K, ) > such ha: [, ], (, ω) Λ T, ω([, ]) K = F (, ω) < C(K, ). We denoe by B(Λ d T ) he se of boundedness-preservng funconals. We now recall some noons of dfferenably for funconals followng [4,3,11]. For e R d and ω D([, T ], R d ), we defne he vercal perurbaon ω e of (, ω) as he càdlàg pah obaned by shfng he pah by e afer : ω e = ω + e1 [,T ]. Defnon 1.3. A non-ancpave funconal F s sad o be: horzonally dfferenable a (, ω) Λ T f DF (, ω) = lm h + F ( + h, ω ) F (, ω ) h (4) exss. If DF (, ω) exss for all (, ω) Λ T, hen (4) defnes a non-ancpave funconal DF, called he horzonal dervave of F. vercally dfferenable a (, ω) Λ T f he map: g (,ω) : R d R e F (, ω + e1 [,T ] ) s dfferenable a. Is graden a s called he Dupre dervave (or vercal dervave) of F a (, ω): ω F (, ω) = g (,ω) () R d (5).e. ω F (, ω) = ( F (, ω), = 1,, d) wh F (, ω) = lm h F (, ω + he 1 [,T ] ) F (, ω ) h where (e, = 1,, d) s he canoncal bass of R d. If F s vercally dfferenable a all (, ω) Λ d T, ω F : Λ T R d defnes a non-ancpave funconal called he vercal dervave of F. We may repea he same operaon on ω F and defne smlarly 2 ωf, 3 ωf, ec. 5

6 Remark 1. Noe ha DF (, ω) s no he paral dervave n : F ( + h, ω) F (, ω) DF (, ω) F (, ω) = lm. h h F (, ω) corresponds o a Lagrangan dervave whch follows he ncremen of F along he pah from o + h, whereas n (4) he pah s sopped a and he ncremen s aken along he sopped pah. We consder he followng classes of smooh funconals: Defnon 1.4. We defne C 1,k b (Λ d T ) as he se of non-ancpave funconals F : (Λ d T, d ) R whch are horzonally dfferenable wh DF connuous a fxed mes; k mes vercally dfferenable wh j ωf C, l (Λ d T ) for j =,, k; DF, ω F,, k ωf B(Λ d T ). Consder now a sequence π n = ( = n < n 1 < < n k(n) = T ) of parons of [, T ]. π n = sup{ n +1 n, = 1 k(n)} wll denoe he mesh sze of he paron. A càdlàg pah x D([, T ], R) s sad o have fne quadrac varaon along he sequence of parons (π n ) n 1 f for any [, T ] he lm [x] π () := lm n n +1 (x( n +1) x( n )) 2 < (6) exss and he ncreasng funcon [x] has Lebesgue decomposon [x] π () = [x] c π() + <s x(s) 2 where [x] c π s a connuous, ncreasng funcon. We denoe he se of such pahs Q π ([, T ], R). A d-dmensonal pah x = (x 1,..., x d ) D([, T ], R d ) s sad o have fne quadrac varaon along π f x Q π ([, T ], R) and x + x j Q π ([, T ], R) for all, j = 1..d. Then for any, j = 1,, d and [, T ], we have (x ( n k+1) x ( n k)).(x j ( n k+1) x j ( n k)) n [x] j π () = [x + x j ] π () [x ] π () [x j ] π (). n k πn, n k 2 The marx-valued funcon [x] : [, T ] S + d whose elemens are gven by [x] j π () = [x + x j ] π () [x ] π () [x j ] π () 2 s called he quadrac covaraon of he pah x along he sequence of parons π. For furher dscusson of hs concep, we refer o [3,26]. 6

7 Consder now a pah ω Q π ([, T ], R d ) wh fne quadrac varaon along π. Snce ω has a mos a counable se of jump mes, we may assume ha he paron exhauss he jump mes n he sense ha sup ω() ω( ) n. (7) [,T ] π n Then he pecewse-consan approxmaon ω n () = k(n) 1 = ω( +1 )1 [, +1 [() + ω(t )1 {T } () (8) converges unformly o ω: sup [,T ] ω n () ω() n. Noe ha wh he noaon (8), ω n ( n ) = ω( n ) bu ω n ( n ) = ω( n +1 ). If we defne ω n, ω(n ) n = ω n + ω( n )1 [ n,t ], hen ω n, ω(n ) n ( n ) = ω( n ). Approxmang he varaons of he funconal by vercal and horzonal ncremens along he paron π n, we oban he followng pahwse change of varable formula for C 1,2 (Λ d T ) funconals, derved n [4] under more general assumpons: Theorem 1.5 (Pahwse change of varable formula for C 1,2 funconals [4]). Le ω Q π ([, T ], R d ) verfyng (7). Then for any F C 1,2 b (Λd T ) he lm exss and k(n) 1 ω F (, ω )d π ω := lm ω F ( n n, ω n, ω(n ) ) n.(ω( n +1 ) ω( n )) (9) = 1 F (T, ω T ) F (, ω ) = DF (, ω )d + 2 r ( 2 ωf (, ω )d[ω] c π() ) + ω F (, ω )d π ω + (F (, ω ) F (, ω ) ω F (, ω ). ω()). ],T ] A consequence of hs heorem s he ably o defne he pahwse negral. ωf (, ω )d π ω as a lm of Remann sums compued along he sequence of parons π. For a connuous pah ω C ([, T ], R d ) hs smplfes o: k(n) 1 ω F (, ω)d π ω := lm ω F ( ) n n, ω n n.(ω( n +1 ) ω( n )) (1) = 7

8 Ths negral, frs consruced n [4], exends H. Föllmer s consrucon [12] for negrands of he form f, f C 2 (R d ) o (pah-dependen) negrands of he form ω F, F C 1,2 b (Λ d T ). Ths consrucon apples o ypcal pahs of Brownan moon and semmarngales, whch sasfy he quadrac varaon propery almos-surely, alhough hey have nfne p-varaon for p = 2 so (1) canno be consruced as a Young negral [27]. The goal of hs paper s o explore some properes of hs negral. 2 Isomery propery of he pahwse negral 2.1 Pahwse somery formula Le ω Q π ([, T ], R d ) be a gven pah wh fne quadrac varaon along a nesed sequence of parons π = (π n ) n 1. Our goal s o provde condons under whch he he followng deny holds: [F (, ω )] π () = ω F (s, ω s ) ω F (s, ω s ), d[ω] π (s), (11) where we use he noaon A, B := r(ab), for square marces A, B. Ths relaon was frs noed n [3] for funconals of he form F (ω) = φ.dω wh φ pecewse-consan. To exend hs propery o a more general seng we assume a Lpschz-connuy condon on he funconal F and a Hölderype regulary condon for he pah ω. Assumpon 1 (Unform Lpschz connuy). K >, ω, ω D([, T ], R d ), [, T ], F (, ω) F (, ω ) K ω ω. We denoe by Lp(Λ d T, ) he space of all non-ancpave funconals sasfyng he above propery. Consder a nesed sequence of parons π n = { n, =,..., m(n)} of [, T ]. We defne he oscllaon osc(f, π n ) of a funcon f : [, T ] R d beween pons of he paron π n as osc(f, π n ) = max j=..m(n) 1 sup f() f( n ( n j,n j+1 ] j ) 8

9 We make he followng assumpons on he sequence of parons: Assumpon 2 (Vanshng oscllaon along π). Assumpon 3. osc(ω, π n ) n +. max F {,..., m(n) 1} (n +1, ω) F ( n, ω) n +. Remark 2 (Parons wh vanshng mesh). If π n and F : (Λ d T, d ) R s Lpschz-connuous hen Assumpons 2 and 3 are sasfed for any ω C ([, T ], R d ). To see hs, se ɛ >. By unform connuy of ω on [, T ] here exss δ(ɛ) > such ha ω() ω(s) ɛ whenever s δ. Snce π n here exss N 1 1 such ha π n ɛ δ(ɛ) for n N 1. Then for n N 1 we have osc(ω, π n ) ɛ so Assumpon 2 holds. Furhermore, for n N 1, d (( n +1, ω), ( n, ω)) = n +1 n + ω n +1 ω n π n + sup ω() ω( n [ n,n +1 ] ) 2ɛ so d (( n +1, ω), ( n, ω)) as n. Lpschz-connuy of F hen enals max {,..., m(n) 1} F ( n +1, ω) F ( n, ω) n +. In secon 2.2 we wll also consder sequences of parons whose mesh may no converge o zero. For such parons Assumpon 3 s no redundan. For < ν < 1, denoe by C ν ([, T ]) he space of ν Hölder connuous funcons: C ν ([, T ], R d ) = {f C ([, T ]), f() f(s) sup < + }, (,s) [,T ] 2, s s ν and C ν ([, T ], R d ) = α<ν Cα ([, T ], R d ) he space of funcons whch are α Hölder for every α < ν. For ω C ν ([, T ], R d ), he followng pecewse consan approxmaon ω n := m(n) 1 = sasfes: ω ω n π n ν. Our man resul s he followng: ω( n +1 )1 [ n, n +1 ) + ω(t )1 {T } (12) 9

10 Theorem 2.1 (Pahwse Isomery formula). Le π = (π n ) n 1 be a sequence of parons of [, T ], and ω Q π ([, T ], R d ) C ν ([, T ], R d ) for ν > 3 1, 2 sasfyng Assumpon 2. Le F C 1,2 b (Λ d T ) Lp(Λ d T, ) wh ω F C 1,1 b (Λ d T ) such ha Assumpon 3 holds. Then: [. [F (, ω )] π (T ) = ] T ω F (s, ω s )d π ω (T ) = ω F (s, ω s ) ω F (s, ω s ), d[ω] π. π (13) Remark 3. We noe ha for ypcal pahs of a Brownan dffuson or connuous semmarngale wh non-degenerae local marngale componen, he assumpons of Theorem 2.1 hold almos surely as soon as Assumpon 1 s sasfed. More generally he resul holds for ω C 1 2 ([, T ], R d ), whch corresponds o he Hölder regulary of Brownan pahs [24]. We sar wh an mporan lemma whch s essenal n provng boh he pahwse somery formula and he unqueness of pahwse negral. Ths resul also esablshes a connecon beween he funconal dervaves nroduced n Secon 1 and conrolled rough pahs [17]. Lemma 2.2. Le ω C ν ([, T ], R d ) for some ν (1/3, 1/2] and F C 1,2 b (Λ d T, R n ) wh ω F C 1,1 b (Λ d T, R n d ) and F Lp(Λ d T, ). Defne R F s,(ω) := F (s, ω s ) F (, ω ) ω F (, ω )(ω(s) ω()). (14) Then here exss a consan C F,T, ω ν ncreasng n T and ω ν, such ha R F s,(ω) C F,T, ω ν s 2βν, wh 2β ν = ν(1 + ν). Proof. We wll prove only he case when he values of F are scalar,.e. n = 1; he exenson o he general case s sraghforward. We sar by recallng he followng resul from [3, Proposon 5.26]: Lemma 2.3. Le G C 1,1 b (Λ d T ) and λ : [, s] R a connuous funcon wh fne varaon. Then G(s, λ s ) G(, λ ) = DG(u, λ u )du + Here he negrals are defned n he Remann-Seljes sense. ω G(u, λ u )dλ(u). (*) 1

11 Now, le us fx a Lpschz connuous pah λ: [, T ] R d, usng he above lemma repeaedly for G = F, ω F, we wll oban an negral formula for R F,s(λ) n erms of he dervaves F. For he sake of convenence we denoe by F and λ respecvely he -h coordnaes of ω F and λ. We also use Ensen s convenon of summaon n repeaed ndexes and he followng noaon δλ,s := λ(s) λ(). Usng (*) for G = F, we have R F,s(λ) = DF (u, λ u )du + ( F (u, λ u ) F (, λ )) dλ (u) (15) For he second erm on he rgh-hand sde of he above deny we use he (*) wh G = F and hen Fubn s heorem o ge = ( F (u, λ u ) F (, λ )) dλ (u) = D F (r, λ r )(λ (s) λ (r))dr + + u s u D F (r, λ r )drdλ (u) 2 jf (r, λ r )dλ j (r)dλ (u) 2 jf (r, λ r ) λ j (r)(λ (s) λ (r))dr. (16) Thus R F,s(λ) = (DF (u, λ u ) + D F (u, λ u )δλ u,s)du + 2 jf (r, λ r )δλ r,sdλ j (r) (17) To use he above formula for esmae he error erm R F for Hölder connuous pahs we wll use a pecewse lnear approxmaon. For ω C ν le ω N be he pecewse lnear approxmaon of ω on [, s] defned by ω N (r) = ω(r), r [, ], ω N (τ N k ) = ω(τ N k ), k = {,..., N}, where τ N k = + k s N, ω N s lnear on each nerval [τ N k, τ N k+1]. Then, we have ω N s ν ω C ω ν, ω N N ν ν C ω ν, ω N = δω τ,τ +1 τ +1 τ 1 (τ,τ +1 ) N 1 ν ω ν s ν 1, (18) δω a,b ω ν b a ν. 11

12 Usng he local boundedness propery of DF, D ω F and 2 ωf, and he represenaon (17) for λ = ω N, we oban R F,s(ω N ) C F s + C F ω N ν s 1+ν + C F ω N 2 νn 1 ν s 2ν. (19) On he oher hand snce ω N = ω and ω N (s) = ω(s) R F,s(ω N ) R F,s(ω) = F (s, ω N ) F (s, ω) C F ω N ω C F ω ν N ν s ν. (2) from above esmaes (19), (2) and rangle nequaly R F,s(ω) C F s + C F ω ν s 1+ν + C F ω 2 νn 1 ν s 2ν + N ν s ν. To opmze he above bound, we choose N so ha ω 2 νn 1 ν s 2ν N ν s ν.e. N ω 2 ν s ν. Hence he resul R F,s(ω) C F,T (1 + ω ν ) s + C F,ν ω 2ν ν s ν+ν2. We are now ready o prove he somery propery: Proof of Theorem 2.1. To prove he somery formula we noe ha ( F ( n +1, ω n ) F (n +1, ω n ) ) 2 ω F (, ω n ) ω F (, ω n ), δω n, n δω +1 n, n +1 R F n,n (ω) 2 + C +1 F R F (ω) δω n,n n +1,n +1 From he Assumpon 2 follows ha M n := max R F n, n (ω) n and +1 from par a) of Theorem (3.1) we have R F (ω) n,n +1 C n +1 n ν2 +ν. Snce ν > ( 3 1)/2 s equvalen o ν 2 + ν > 1, we ge 2 R F (ω) n,n +1 CM n ν 2 +ν n +1 n CT M 2 1 ν 2 +ν n. Consequenly, snce δω n, n +1 2 r([ω] π ), we have by Cauchy -Schwarz nequaly R F (ω) δω n,n n +1,n +1 R F n (ω) 2,n +1 δω n, n

13 We conclude ( F ( n +1, ω n ) F (n +1, ω n ) ) 2 ω F (, ω n ) ω F (, ω n ), δω n, n δω +1 n, n +1 R F n,n (ω) 2 + C +1 F R F (ω) δω n,n n +1,n +1 whch ogeher wh ω F (, ω n ) ω F (, ω n ), δω n, n +1 mples [. T δω n, n +1 ω F (s, ω s ) ω F (s, ω s ), d[ω] π. ] ω F (s, ω s )d π ω (T ) = ω F (s, ω s ) ω F (s, ω s ), d[ω] π. π To prove he equaly [F (, ω )] π (T ) = [. ωf (s, ω s )d π ω] π (T ) we noe, usng he change of varable formula (Theorem (1.5)) s enough o prove ha he pahs DF (, ω )d, 2 ωf (, ω ), d[ω] π () have zero quadrac varaon along π n. For he second pah follows from Assumpon 2 as mples ha max [ω]( n +1) [ω]( n ). For he frs pah noe ha snce s obvously a pah of bounded varaon we jus need o prove ha max n +1 n DF (, ω )d. For ha we wre n +1 n DF (, ω )d = = = n +1 n n +1 n n +1 n [DF (, ω ) DF (, ω n )]d + n +1 n DF (, ω n )d [DF (, ω ) DF (, ω n )]d + F ( n +1, ω n ) F ( n, ω n ) [DF (, ω ) DF (, ω n )]d + ( F ( n +1, ω n ) F ( n +1, ω n +1 ) ) + ( F ( n +1, ω n +1 ) F ( n, ω n ) ). By he connuy of F, DF and Assumpon 2 he frs wo summands n he rgh hand sde of above equaly unformly converge o as n, by Assumpon 3 he hrd summand also converges unformly o. Remark 4 (Relaon wh Io somery formula). Le P be he Wener measure on C ([, T ], R). Then he pahwse negral (, ω) ωf (u, ω)d π ω s defned P almos surely and defnes a process whch s a verson of he Io negral. ωf (, W )dw. Inegrang (13) wh respec o P yelds he well-known Io somery formula [19]:. ) ( ) ( T ) T E ([ P ω F (, W )dw ](T ) = E P ω F (, W )dw 2 = E P ω F (, W ) 2 d. 13

14 So, Theorem 2.1 reveals ha underlyng he Io somery formula s he pahwse somery (13) whch does no rely on he Wener measure. 2.2 Parons of hng mes An alernave approach o he defnon of he pahwse negral s o consder quadrac varaon and Remann sums compued along Lebesgue parons.e. parons of hng mes of unformly spaced levels [2,9,2,22]: τ n (ω) =, τ n k+1(ω) = nf{ > τ n k, ω() ω(τ n k ) 2 n } T. (21) Alhough hs s dfferen n spr from he orgnal approach of Föllmer, here are some argumens for usng such nrnsc sequences of parons for a gven pah, one of hem beng ha quadrac varaon compued along (21) s nvaran under a me change. Thus a naural queson s wheher he resuls above apply o such parons. We wll now presen a verson of Theorem 2.1 for he sequence of parons defned by (21). Defne m(n) = nf{k 1, τ n k (ω) = T } and denoe τ n (ω) = (τ n k (ω), k =,..., m(n)). I s easy o see ha he vanshng oscllaon condon (Assumpon 2) auomacally holds for he sequence (τ n (ω), n 1) snce by consrucon osc(ω, τ n (ω)) 2 n. However, as he example of a consan pah shows, for a general pah ω C ([, T ], R d ), he parons τ n (ω) may fal o have a vanshng mesh sze. We wll show now ha, f a pah ω has srcly ncreasng quadrac varaon along s Lebesgue paron τ n (ω) hen τ n. Lemma 2.4. Assume ω C ([, T ], R d ) has fne quadrac varaon along he Lebesgue paron τ n (ω) defned by (21): (, T ], n lm ω(τk+1) n ω(τk n ) 2 = [ω] τ(ω) () >. (22) τk n If [ω] τ(ω) () s a srcly ncreasng funcon hen τ n (ω). Proof. Le h >, [, T h]. Denoe by m(n,, +h) he number of paron pons of τ n (ω) n he nerval [, + h]. Then τ n k +h ω(τ n k+1) ω(τ n k ) 2 = 2 n (m(n,, + h) 1). 14

15 Snce [ω] τ(ω) () s srcly ncreasng, τ n k +h ω(τ n k+1) = 2 n (m(n,, + h) 1) n [ω] τ(ω) ( + h) [ω] τ(ω) () > so n parcular m(n,, + h) 2 n. Ths holds for any nerval [, T h] whch mples ha he τ n < h for n large enough. Snce hs s rue for any h >, hs mples τ n. The condon ha [ω] τ(ω) s srcly ncreasng s an rregulary condon on ω: means ha he quadrac varaon over any nerval s non-zero. For pahs verfyng hs condon, Theorem 2.1 may be exended o Lebesgue parons: Theorem 2.5 (Pahwse Isomery formula along Lebesgue parons). Le ω Q τ(ω) ([, T ], R d ) C ν ([, T ], R d ) for some ν ( ) 3 1, and F C 1,1 b (Λ d T ) Lp(Λ d T, ) such ha ω F C 1,1 b (Λ d T ). If ω has srcly ncreasng quadrac varaon [ω] τ(ω) wh respec o τ(ω) hen he somery formula holds along he paron of hng mes defned by (21): [. [F (, ω )] τ(ω) () = ] ω F (s, ω s )d π ω () = ω F (s, ω s ) ω F (s, ω s ), d[ω] τ(ω). Proof. Frs noe ha he sequence of parons (τ n (ω), n 1) auomacally sasfes Assumpon 2 snce osc(ω, τ n (ω)) 2 n. Snce [ω] τ(ω) s srcly ncreasng, Lemma 2.4 mples ha τ n. Snce ω C ν ([, T ], R d ) whch ogeher wh he connuy of F mples ha max F {,..., m(n) 1} (n +1, ω) F ( n, ω) n +. so he sequence of parons τ n (ω) sasfes Assumpon 3. The resul hen follows from he Theorem The pahwse negral as a connuous somery Theorem 2.1 suggess ha he exsence of an somerc mappng underlyng he pahwse negral. We wll now proceed o make hs srucure explc. 15

16 A consequence of Theorem 2.1 s ha, f ω s an rregular pah wh srcly ncreasng quadrac varaon, any regular funconal F (., ω) has zero quadrac varaon along π f and only f ω F vanshes along ω: Proposon 2.1 (Preservaon of rregulary). Le ω Q π ([, T ], R d ) C ν ([, T ], R d ) for some ν ( ] 3 1, such ha d[ω] π := a() > s a rghconnuous, posve defne funcon, and F C 1,2 b (Λ d T ) Lp(Λ d T, ) be d a non-ancpave funconal wh ω F C 1,1 b (Λ d T ). If Assumpons 2 and 3 hold, hen he pah F (, ω) has zero quadrac varaon along he paron π f and only f ω F (, ω) =, [, T ]. Proof. Indeed, from Theorem 2.1 [F (, ω )] (T ) = ω F (s, ω)a(s) ω F (s, ω) ds. Snce a() s posve defne he negrand on he rgh hand sde s nonnegave and srcly posve unless ω F (s, ω s ) =. So by rgh-connuy of he negrand he negral s zero f and only f ω F (, ω ). Le F C 1,2 b (Λ T ) be a funconal sasfyng assumpons of Theorem 2.1. We denoe R(Λ T ) he se of such regular funconals. Proposon 2.1 may be undersood n he followng way: f a pah s rregular n he sense of havng srcly ncreasng quadrac varaon, hs propery s locally preserved by any regular funconal ransformaon F as long as ω F does no vansh. We noe ha for ω C 1 2 ([, T ], R d ), we have F (, ω) C 1 2 ([, T ], R). Le a: [, T ] S d + be a connuous funcon akng values n posve-defne symmerc marces. Defnon 2.6 (Harmonc funconals). F R(Λ T ) s called a harmonc f (, ω) Λ T, DF (, ω ) ωf (, ω ), a() =. We denoe H a (Λ T ) he space of a( )-harmonc funconals. Noe ha for any F R(Λ T ) he funconal defned by G(, ω) = F (, ω) DF (s, ω)ds ωf (s, ω), a(s) ds 16

17 s a( )-harmonc. The class of harmonc funconals plays a key role n probablsc nerpreaon of he Funconal Io calculus [14,6,3]. We wll now see ha hs class also plays a role n he exenson of he pahwse negral. Le ω Q π ([, T ], R d ) C ν ([, T ], R d ) such ha d[ ω] π /d = a. The funconal change of varable formula (Theorem 1.5 ) hen mples ha F H a (Λ T ), [, T ], F (, ω) = F (, ω) + By Theorem 2.1, we have [F (., ω)] π () = Consder he vecor spaces ω F (u, ω)d π ω. ω F (u, ω)a(u) ω F (u, ω)du = ω F (., ω) 2 L 2 ([,T ],a) <. H a ( ω) := { F (, ω ) F Ha (Λ T ) } Q π ([, T ], R), V a ( ω) := { ω F (, ω ) F Ha (Λ T ) } L 2 ([, T ], a). whch are he mages of ω under all harmonc funconals and her vercal dervaves. Proposon 2.1 hen mples ha he map ω H a ( ω) ω π = [ω] π (T ) defnes a norm on H a ( ω). The pahwse negral can hen be lfed o a connuous map I ω : ( V a ( ω), L 2 ([,T ],a)) (Ha ( ω), π ) whch s n fac an somery: φ = ω F (., ω) I ω (φ) := φd π ω. (23) Proposon 2.2 (Isomery propery). I ω : ( V a ( ω), L 2 ([,T ],a)) (Ha ( ω), π ) s an njecve somery. 3 Pahwse naure of he negral The defnon (1) of he negral ω F (., ω)d π ω nvolves values of he negrand ω F along pecewse consan approxmaons of ω. To be able o 17

18 nepre ω F (., ω)d π ω as a pahwse negral, we mus show ha only depends on he values ω F (, ω), [, T ] of he negrand along he pah ω. Our goal s o provde condons under whch he followng propery holds: Propery 1 (Pahwse naure of he negral). Le F 1, F 2 C 1,2 b (Λ d T ), and ω Q π ([, T ], R) C ν ([, T ]) for some < ν < 1/2. If ω F 1 (, ω ) = ω F 2 (, ω ), [, T ] hen ω F 1 d π ω = ω F 2 d π ω. To gve a precse saemen, le us nroduce he followng assumpon: Assumpon 4 (Horzonal local Lpschz propery). A non-ancpave funconal F : Λ d T V wh values n a fne dmensonal real vecor space V, sasfes he horzonal locally Lpschz propery f ω D([, T ], R d ), C >, η >, h, T h, ω D([, T ], R d ), ω ω < η, F ( + h, ω ) F (, ω ) Ch. The above assumpon assers he Lpschz regulary of F n me as he value of he pah ω s frozen. Noe ha hs propery s weaker han horzonal dfferenably. We sar wh a useful lemma: Lemma 3.1 (Expanson formula for regular funconals). Le ω C ν ([, T ], R d ) for some ν (1/3, 1/2] and F C 1,1 b (Λ d T, R n ) be a non-ancpave funconal such ha ω F C 1,1 b (Λ d T, R n d ), 2 ωf C 1,1 b (Λ d T, R n d d ), F, DF, 3 ωf Lp(Λ d T, ) and 3 ωf horzonally locally Lpschz. Then F (s, ω s ) F (, ω ) = ω F (, ω )(ω(s) ω()) + DF (u, ω u )du ωf (, ω ), (ω(s) ω()) (ω(s) ω()) + O( s 3ν2 +ν ), as s, unformly n, s [, T ]. Proof. For he resul we wll need o furher expand he represenaon (17). 18

19 Frs, noe ha he second erm n he hrd lne of (16) can be wren as u u jf 2 (r, λ r )dλ j (r)dλ (u) = jf 2 (, λ ) dλ j (r)dλ (u) s u ( + 2 j F (r, λ r ) jf 2 (, λ ) ) dλ j (r)dλ (u) (24) Usng he Lemma (*) for G = 2 jf and hen Fubn s heorem, we ge u ( 2 j F (r, λ r ) 2 jf (, λ ) ) dλ j (r)dλ (u) = + = u r s u r D 2 jf (τ, λ τ )Λ j τ,sdτ + D 2 jf (τ, λ τ )dτdλ j (r)dλ (u) 3 jkf (τ, λ τ )dλ k (τ)dλ j (r)dλ (u) 3 jkf (τ, λ τ )Λ j τ,s λ k (τ)dτ, (25) where Λ j a,b = b a u a dλ j (r)dλ (u) = b Combnng (15), (16), (24) and (25) we oban a ( λ j (u) λ j (a) ) λ (u)du. R F,s(λ) = DF (u, λ u )du + + D 2 jf (τ, λ τ )Λ j τ,sdτ + D F (r, λ r )δλ r,sdr + 2 jf (, λ )Λ j,s 3 jkf (τ, λ τ )Λ j τ,s λ k (τ)dτ. (26) Fnally explong he symmery of second and hrd order dervaves: 2 jf = 2 jf, 3 jkf = 3 jkf, from (26) we oban: R F,s(λ) = DF (u, λ u )du + + D 2 jf (τ, λ τ ) Λ j τ,sdτ + D F (r, λ r )δλ r,sdr + 2 jf (, λ ) Λ j,s 3 jkf (τ, λ τ ) Λ j τ,s λ k (τ)dτ, (27) where Λ s he symmerc par of marx Λ. By negraon by pars s easy o compue Λ j a,b = Λj a,b + Λj a,b 2 = 1 2 (λj (b) λ j (a))(λ (b) λ (a)). 19

20 Therefore, R F,s(λ) = DF (u, λ u )du + D 2 jf (τ, λ τ )δλ τ,sδλ j τ,sdτ D F (r, λ r )δλ r,sdr jf (, λ )δλ,sδλ j,s we decompose he las erm n he above as follows 3 jkf (τ, λ τ )δλ τ,sδλ j τ,s λ k (τ)dτ, jkf 3 (τ, λ τ )δλ τ,sδλ j λ τ,s k (τ)dτ = jkf 3 (, λ )δλ τ,sδλ j λ τ,s k (τ)dτ s ( + 3 jk F (τ, λ τ ) jkf 3 (, λ ) ) δλ τ,sδλ j λ τ,s k (τ)dτ Explong he cyclcal symmery of ensor 3 jkf n he ndces, j, k: hence 3 jkf (, λ )δλ τ,sδλ j τ,s λ k (τ) = 3 jkf (, λ )δλ τ,sδλ j τ,s d dτ λk τ,s = jkf (, λ )[δλ τ,sδλ j d τ,s dτ λk τ,s + δλ j τ,sδλ k d τ,s dτ λ τ,s + δλ k τ,sδλ d τ,s dτ λj τ,s] = jkf (, λ ) d ( δλ dτ τ,s δλ j τ,sλτ,s) k 3 jkf (, λ )δλ τ,sδλ j τ,s λ k (τ)dτ = jkf (, λ )δλ,sδλ j,sλ k,s. Thanks o hs (28) can be wren as R F,s(λ) = DF (u, λ u )du + D F (r, λ r )δλ r,sdr jf (, λ )δλ,sλ j,s (28) D 2 jf (τ, λ τ )δλ τ,sδλ j τ,sdτ jkf (, λ )δλ,sδλ j,sλ k,s ( 3 jk F (τ, λ τ ) 3 jkf (, λ ) ) δλ τ,sδλ j τ,s λ k (τ)dτ. (29) Now, as n he proof of par a), we ake λ = ω N and esmae R F,s(ω N ). To esmae he las erm n (29) we use ha 3 F s n Lp([, T ], ) and s locally horzonally Lpschz: 3 jkf (τ, ω N τ ) 3 jkf (, ω N ) 3 jkf (τ, ω N τ ) 3 jkf (τ, ω N,τ ) + 3 jkf (τ, ω N,τ ) 3 jkf (, ω N ) C F ω N,τ ω N τ + C F τ C F, ω ν τ ν + C F τ C F, ω ν,t s ν. Pluggng hs esmae no he formula (29) and usng he local boundedness of 2

21 he dervaves DF, D F, 2 jf, D 2 jf, 3 jkf and he esmaes (18) we oban R,s(ω F N ) DF (u, ωu N )du jf (, ω )δω,sδω j,s C F, ω ν s 1+ν + C F, ω ν s 1+2ν +C F, ω ν s 3ν + C F, ω ν N 1 ν s 4ν C F, ω ν,t s 3ν + C F, ω ν N 1 ν s 4ν, (3) where we have used also ha ω N = ω, ω N (s) = ω(s). Recallng (2) we conclude by rangle nequaly RF,s(ω) DF (u, ω u )du jf (, ω )δω,sδω j,s DF (u, ωu N ) DF (u, ω u ) du + CF N ν s ν (31) +C F, ω ν,t s 3ν + C F, ω ν N 1 ν s 4ν, The above nequaly holds for any N > 1,hus we can ake N s 3ν o ge RF,s(ω) DF (u, ω u )du jf (, ω )δω,sδω j,s DF (u, ωu N ) DF (u, ω u ) du + C F, ω ν s ν+3ν2 C F ω ν s 1+ν + C F, ω ν s ν+3ν2, where n he las nequaly we used ha DF s Lpschz connuous. Hence he resul. We are now ready o prove he man resul of hs secon, whch gves suffcen condons on he funconal F under whch he pahwse negral ω F (, ω)d π ω s a (pahwse) lm of Remann sums compued along ω, raher han approxmaons of ω: Theorem 3.2. Le ω Q π ([, T ], R d ) C ν ([, T ], R d ) wh ν > 13 1 sasfyng Assumpon 2. Assume F C 1,2 b (Λ d T ) s such ha ω F, 2 ωf, C 1,1 b (Λ d T ), 6 F, DF, 3 F Lp(Λ d T, ) and 3 ωf s horzonally locally Lpschz. Then he pahwse negral (1) s a lm of non-ancpave Remann sums: ω F (u, ω)d π ω(u) = lm n + In parcular, s value only depends on F (., ω) and [,s] π n ω F (, ω)(ω(s) ω(). ω F (, ω) = = ω F (u, ω)d π ω(u) =. 21

22 Proof. Le us denoe A n := F ( n +1, ω n +1 ) F ( n, ω n ) ω F (, ω n )(ω( n +1) ω( n )) n +1 n DF (u, ω)du ωf (, ω n ), ω n, n +1 ω n, n +1 From Assumpon (2) max A n n and by par b) of Theorem (3.1) A n C n +1 n 3ν2 +ν. From he condon on ν, 3ν 2 + ν > 1, herefore A n C(max A n ) 1 1 3ν 2 +ν n +1 n CT (max A n ) 1 3ν 2 +ν, 1 whch mples A n o. Hence he lm of Remann sums exs and s equal lm n ω F (, ω n )(ω( n +1) ω( n )) = F (T, ω) F (, ω) ωf (, ω), d[ω] π () = DF (u, ω)du ω F (, ω)d π ω(). 4 Rough-smooh decomposon for funconals of rregular pahs As an applcaon of he prevous resuls, we now derve a decomposon heorem for funconals of rregular pahs. Le π = { π n } be a sequence of parons wh π n. Consder an rregular pah wh srcly ncreasng quadrac varaon along π: ω Q π ([, T ], R d ) C 1/2 ([, T ], R d ) wh d[ ω] π d > d a.e. (32) We defne he vecor space of pahs obaned as he mage of ω under regular non-ancpave funconals: U( ω) := { F (, ω) F sasfes he assumpons of Theorem 3.2 } Qπ ([, T ], R). Denoe V a ( ω) := { ω F (, ω) F sasfes he assumpons of Theorem 3.2 } Qπ ([, T ], R d ). The followng resul gves a sgnal plus nose decomposon for such pahs, n he sense of Drchle processes [13]: 22

23 Proposon 4.1 (Rough-smooh decomposon of pahs). Any pah ω U( ω) has a unque decomposon ω() = ω() + φd π ω + s() where φ V a ( ω), [s] π =. (33) The erm φ.dπ ω s he rough componen of ω whch nhers he rregulary of ω whle s(.) represens a smooh componen wh zero quadrac varaon. Remark 5 (Pahwse Doob-Meyer decomposon). Ths resul may be vewed as a pahwse analogue of he well-known decomposon of a connuous semmarngale no a local marngale (a process wh srcly ncreasng quadrac varaon A smlar pahwse decomposon resul was obaned by Harer and Plla [18] usng rough pah echnques (see also [16, Theorem 6.5] and [15]). Unlke he semmarngale decomposon, he resul of Harer and Plla [18] nvolves Hölder-ype regulary assumpons on he componens, as well as a unform Hölder roughness condon on he pah. Our seng s closer o he orgnal semmarngale decomposon n ha he componens are dsngushed based on quadrac varaon and he rregulary condon (32) on ω s also expressed n erms of quadrac varaon. Proof. Le ω U( ω). Then here exss F R(Λ d T ) verfyng he assumpons of Theorem 3.2, wh ω() = F (, ω). The funconal change of varable formula appled o F hen yelds he decomposon wh φ = ω F (., ω) V a ( ω) and s() = F (, ω ) + du(df (u, ω) + 1 < 2 2 ωf (u, ω), d[ω] >). The connuy of hs negrand mples ha s has fne varaon so [s] π =. Consder now wo dfferen decomposons ω() ω() = φ 1 d π ω + s 1 () = φ 2 d π ω + s 2 (). so s 1 s 2 = (φ 1 φ 2 )d π ω. Snce φ 1 φ 2 V a ( ω) hs represenaon shows ha s 1 s 2 Q π ([, T ], R) so he pahwse quadrac covaraon [s 1, s 2 ] π s well defned and we can apply he polarzaon formula o conclude ha [s 1 s 2 ] π =. Applyng Proposon 2.1 o s 1 s 2 = (φ 1 φ 2 )d π ω. hen mples ha ha φ 1 = φ 2. Fnally, applyng Theorem 3.2 o (φ 1 φ 2 )d π ω hen shows ha s 1 = s 2 whch yelds unqueness of he decomposon (33). Remark 6. As noed by Sched [26], Q π ([, T ], R) s no a vecor space and, gven wo pahs (ω 1, ω 2 ) Q π ([, T ], R) he quadrac covaraon along π canno be defned n general. 23

24 By conras, he space U( ω) nroduced above s a vecor space of pahs wh fne quadrac varaon along π. Moreover, for any par of elemens (ω 1, ω 2 ) U( ω) 2, he quadrac covaraon along π s well defned; f ω = φ.d π ω + s s he rough-smooh decomposon of ω he quadrac covaraon s gven by [ω 1, ω 2 ] π () = < φ 1φ 2, d[ω] >. (34) Acknowledgemens We hank Ncolas Perkowsk for helpful dscussons. Anna Ananova s research s suppored by an Imperal College PhD Fellowshp. References [1] J. Beron, Temps locaux e négraon sochasque pour les processus de Drchle, n Sémnare de Probablés, XXI, vol of Lecure Noes n Mah., Sprnger, Berln, 1987, pp [2] R. V. Chacon, Y. Le Jan, E. Perkns, and S. J. Taylor, Generalsed arc lengh for Brownan moon and Lévy processes, Z. Wahrsch. Verw. Gebee, 57 (1981), pp [3] R. Con, Funconal Io Calculus and funconal Kolmogorov equaons, n Sochasc Inegraon by Pars and Funconal Io Calculus (Lecure Noes of he Barcelona Summer School n Sochasc Analyss, July 212), Advanced Courses n Mahemacs, Brkhauser Basel, 216, pp [4] R. Con and D.-A. Fourné, Change of varable formulas for nonancpave funconals on pah space, J. Func. Anal., 259 (21), pp [5], A funconal exenson of he Io formula, C. R. Mah. Acad. Sc. Pars, 348 (21), pp [6] R. Con and D.-A. Fourné, Funconal Io calculus and sochasc negral represenaon of marngales, Annals of Probably, 41 (213), pp [7] R. Con and Y. Lu, Weak approxmaon of marngale represenaons, Sochasc Process. Appl., 126 (216), pp [8] R. Con and C. Rga, Pahwse analyss and robusness of hedgng sraeges, workng paper, Laboraore de Probablés e Modèles Aléaores, 215. [9] M. Davs, J. Ob lój, and P. Sorpaes, Pahwse sochasc calculus wh local mes, workng paper, Imperal College London,

25 [1] C. Dellachere and P.-A. Meyer, Probables and poenal, vol. 29 of Norh-Holland Mahemacs Sudes, Norh-Holland Publshng Co., Amserdam, [11] B. Dupre, Funconal Iô calculus, Bloomberg Porfolo Research paper, (29). [12] H. Föllmer, Calcul d Iô sans probablés, n Semnar on Probably, XV (Unv. Srasbourg, Srasbourg, 1979/198) (French), vol. 85 of Lecure Noes n Mah., Sprnger, Berln, 1981, pp [13] H. Föllmer, Drchle processes, n Sochasc Inegrals: Proceedngs of he LMS Durham Symposum, July 7 17, 198, D. Wllams, ed., Sprnger, Berln, 1981, pp [14] D.-A. Fourne, Funconal Io calculus and applcaons, ProQues LLC, 21. PhD Thess, Columba Unversy. [15] P. Frz and A. Shekhar, Doob-Meyer for rough pahs, Bullen of he Insue of Mahemacs. Academa Snca., 8 (213), pp [16] P. K. Frz and M. Harer, A course on rough pahs, Unversex, Sprnger, 214. [17] M. Gubnell, Conrollng rough pahs, J. Func. Anal., 216 (24), pp [18] M. Harer and N. S. Plla, Regulary of laws and ergodcy of hypoellpc SDEs drven by rough pahs, Ann. Probab., 41 (213), pp [19] N. Ikeda and S. Waanabe, Sochasc dfferenal equaons and dffuson processes, vol. 24 of Norh-Holland Mahemacal Lbrary, Norh-Holland Publshng Co., Amserdam; Kodansha, Ld., Tokyo, second ed., [2] R. L. Karandkar, On pahwse sochasc negraon, Sochasc Process. Appl., 57 (1995), pp [21] M. Nuz, Pahwse consrucon of sochasc negrals, Elecron. Commun. Probab., 17 (212), pp. no. 24, 7. [22] N. Perkowsk and D. J. Prömel, Local mes for ypcal prce pahs and pahwse Tanaka formulas, Elecronc Journal of Probably, 2 (215), pp. no. 46, 15. [23], Pahwse sochasc negrals for model free fnance, (215). [24] D. Revuz and M. Yor, Connuous marngales and Brownan moon, vol. 293 of Grundlehren der Mahemaschen Wssenschafen, Sprnger-Verlag, Berln, hrd ed., [25] F. Russo and P. Vallos, Elemens of sochasc calculus va regularzaon, n Sémnare de Probablés, XL, vol of Lecure Noes n Mah., Sprnger, Berln, 27, pp [26] A. Sched, On a class of generalzed Takag funcons wh lnear pahwse quadrac varaon, J. Mah. Anal. Appl., 433 (216), pp [27] L. C. Young, An nequaly of he Hölder ype, conneced wh Seljes negraon, Aca Mah., 67 (1936), pp