4. Applications to stochastic analysis. Guide for this section

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1 Guide for thi ection 4. Alication to tochatic analyi So far, I have reented the theory of rough differential equation a a urely determinitic theory of differential equation driven by multi-cale time indexed ignal. Lyon, however, contructed hi theory firt a a determinitic alternative to Itô integration theory, after ome hint by Föllmer in the early 8 that Itô formula can be undertood in a determinitic way, and other work by Bichteler, Karandikhar... on the athwie contruction of tochatic integral. Recall that tochatic integral are obtained a limit in robability of Riemann um, with no hoe for a tronger convergence to hold a a rule. Lyon wa not only looking for a determinitic way of contructing Itô integral, he wa alo looking for a way of obtaining them a continuou function of their integrator! Thi required a notion of integrator different from the claical one... Rough ath were born a uch integrator, with the rough integral of controlled integrand, defined in exercie 3, in the role of Itô integral. What link thee two notion of integral i the following fudamental fact. Brownian motion ha a natural lift into a Hölder -rough ath, for any <<3, called the Brownian rough ath. Thi object i contructed in ection 4. uing Kolmogorov claical regularity criterion, and ued in ection 4. to ee that the tochatic and rough integral coincide whenever they both make ene. Thi fundamental fact i ued in ection 4.3 to ee that tochatic differential equation can be olved in a two te roce. i Purely robabilitic te. Lift Brownian motion into the Brownian rough ath. ii Purely determinitic te. Solve the rough differential equation aociated with the tochatic differential equation. Thi require from the driving vector field to be Cb 3, for the machinery of rough differential equation to make ene, which i more demanding than the Lichitz regularity required in the Itô etting. Thi contraint come with an enormou gain yet: the olution ath to the tochatic differential equation i now a continuou function of the driving Brownian rough ath, thi i Lyon univeral limit theorem, in triking contrat with the meaurable character of thi olution, when een a a function of Brownian motion itelf. The twit i that the econd level of the Brownian rough ath i itelf jut a meaurable function of the Brownian ath. Together with the above olution cheme for olving tochatic differential equation, thi rovide a imle and dee undertanding of ome fundamental reult on diffuion rocee, a ection 4.4 on Freidlin-Wentzell theory of large deviation will demontrate. We follow the excellent forthcoming lecture note [5] in ection 4. and The Brownian rough ath Definition and roertie. Let B t t be an Rl -valued Brownian motion defined on ome robability ace Ω, F, P. There i no difficulty in uing Itô

2 theory of tochatic integral to define the two-index continuou roce 4. B Itô t := u That roce atifie Chen relation B Itô t db r db u = = BItô tu + BItô u + B u B tu B u db u. for any u t. A i B well-known to have almot-urely -Hölder continuou amle ath, for any >, the roce B Itô = B,B Itô will aear a a Hölder -rough ath if one can how that B Itô i almot-urely - Hölder continuou. Thi can be done eaily uing Kolmogorov regularity criterion, which we recall and rove for comletene. Denote for that uroe by D the et of dyadic rational in [, ] and write D n for { k n ; k =.. n}. Theorem Kolmogorov criterion. Let S, d be a metric ace, and q and β>/q > be given. Let alo X t be an S-valued roce defined on ome t D robability ace, uch that one ha 4. d Xt,X L q C t β, for ome finite contant C, for all, t D. Then,forallα [,β q, there exit a random variable C α L q uch that one ha almot-urely d X,X t Cα t α, for all, t D; o the roce X ha an α-hölder modification defined on [, ]. Proof Given, t D with <t,letm be the only integer uch that m+ t < m. The interval [, t contain at mot one interval [ r m+,r m+ + m+ with r m+ D m+. If o, each of the interval [, r m+ and [ r m+ + m+,t contain at mot one interval [ r m+,r m+ + m+ with r m+ D m+. Reeating thi remark u to exhaution of the dyadic interval [, t by uch dyadic ub-interval, we ee, uing the triangle inequality, that d X t,x S n, n m+ where S n =u t Dn d X t,x t+ n.sowehave d X t,x t α S n m+α C α n m+ where C α := n nα S n. But a the aumtion 4. imlie E [ [ ] Sn q ] q E d Xt,X t+ n n C n qβ, t D n we have C L α q nα S n q C α β+ n <, n n o C α i almot-urely finite. The concluion follow in a traightforward way.

3 3 Recall the definition of the homogeneou norm on T, l a = a a = a + a introduced in equation??, with it aociated ditance function da, b = a b. To ee that B Itô i a Hölder -rough ath we need to ee that it i almot-urely -Hölder continuou a a T, l, -valued ath. Thi can be obtained from Kolmologorv criterion rovided one ha B Itô t L q C t, for ome contant q with < <,andc. Given the form of the norm on q T, l, thi i equivalent to requiring 4.3 Bt L q C t, B Itô t L q C t. Thee two inequalitie hold a a traightforward conequence of the caling roerty of Brownian motion. The random variable B Itô i in any Lq a a conequence of the BDG inequality for intance. Corollary. The roce B Itô i almot-urely a Hölder -rough ath, for any with < <. It i called the Itô Brownian rough ath. 3 Note that B Itô i not weak geometric a the ymmetric art of B Itô t i equal to B t B t t Id. Note alo that we may a well have ued Stratonovich integral in the definition of the iterated integral B Str t := u db r db u = thi doe not make a big difference a riori ince B Str t = B Itô t + t Id. B u db u ; So one can define another Hölder -rough ath B Str t = B t, B Str t above Brownian motion, called Stratonovich Brownian rough ath. Unlike Itô Brownian rough ath, it i weak geometric. Comute the ymmetric art of B Str t! Whatever choice of Brownian rough ath we do, it definition eem to involve Itô theory of tochatic integral. It will haen to be imortant for alication thee two rough ath can actually be contructed in a athwie way from the Brownian ath itelf. Given n, define on the ambiant robability ace the σ-algebra F n := σ { B k n ; k n}, and let B n tand for the continuou iecewie linear ath that coincide with B at dyadic time in D n and i linear in between. Denote by B n,i the coordinate of B n. There i no difficulty in defining B n t := B n u db n u a a genuine integral a B n i iecewie linear, and one ha acutally, for j k, 4.4 B n t = E [ ] B Fn n,jk t, B t = E [ ] B Str,jk t Fn

4 4 and B n,ii t = B n,i t. Prooition 3. The Hölder -rough ath B n = B n, B n converge almoturely to B Str in the Hölder -rough ath toology. Proof We ue the interolation reult tated in rooition?? to rove the above convergence reult. The almot-ure ointwie convergence follow from the martingale convergence theorem alied to the martingale in 4.4. To get the almot-ure uniform bound 4.5 u B n < n it uffice to notice that the etimate Bt C t, B Str,jk t C t obtained from Kolmogorov regularity criterion with C L q for any q>, give n [ ] B t E C Fn t, B n,jk [ ] t E C Fn t, o the uniform etimate 4.5 follow from Doob maximal inequality, which imlie that almot-ure finite character of the maximum of the martingale E [ ] C or Fn, ince thi maximum i integrable How big i the Brownian rough ath? The uer bound of B Itô rovided by the contant C of Kolmogorov regularity reult ay u that B Itô i in all the L q ace. The ituation i actually much better! A a firt hint, notice that ince B Itô t ha the ame ditribution a δ t B Itô,andthenormofB Itô ha a Gauian tail thi i elementary, we have [ ] B Itô [ t B Itô 4.6 E ex = E ex ] < t The following Beov embedding i ueful in etimating the Hölder norm of a ath from it two-oint moment. Theorem 4 Beov. Given α [, there exit an integer kα and a oitive contant C α with the following roertie. For any metric ace S, d and any S-valued continuou ath x t t we have x α C α dxt,x t k d dt It can be roved a a direct conequence of the famou Garia-Rodemich-Rumey lemma. Alied to the Brownian rough ath B Itô, Beov etimate give [ B Itô E ] Bt k [ k C k E ] d dt = C k E B k. t k.

5 5 So it follow from 4.6 that we have for any oitive contant c c k B Itô k [ E E ex cc B Itô ] k! k k o ex B Itô will be integrable rovided c i mall enough, by 4.6. Corollary 5. The -rough ath norm of the Brownian rough ath ha a Gauian tail. 4.. Rough and tochatic integral. Let X be any R l -valued Hölder -rough ath, with <<3. Recall we defined in exercie the integral of an LR l, R d - valued ath F controlled by X =X, X a the well-defined limit F dx = lim F ti X ti+ t i + F t i X ti+ t i, where the um i over the time t i of finite artition π of [, ] whoe meh tend to. Thi make ene in articular for X = B Itô. At the ame time, if F i adated to the Brownian filtration, the Riemann um F ti B ti+ t i converge in robability to the tochatic integral F db, a the meh of the artition π tend to. Taking ubequence if neceary, one define imultaneouly the tochatic and the rough integral on an event of robability. They actually coincide almot-urely if F i adated to the Brownian filtration! To ee thi, it uffice to ee that F t i X ti+ t i converge in L to along the ubequence of artition ued to define the tochatic integral F db. Aume firt that F i bounded, by M ay. Then, ince it i adated and F ti i indeendent of B ti+ t i, an elementary conditioning give F t i B Itô L t i+ t i F t i B Itô L t i+ t i M Bti+ L t i M π, which rove the reult in that cae. If F i not bounded we ue a localization argument and to the roce at the toing time τ M := inf { u [, ] ; F >M }. The above reaoning how in that cae that we have the almot-ure equality τm F db = F τ M db, from which the reult follow ince τ M tend to a M increae indefinitely. Prooition 6. Let F be an LRl, R d -valued ath controlled by B Itô = B,B, adated to the Brownian filtration, with a derivative roce F alo adated to that filtration. Then we have almot-urely F db Itô = F db.

6 6 If one ue B Str intead of B Itô in the above rough integral, an additional welldefined term := lim F t ti+ π i t i Id aear in the left hand ide, and we have almot-urely F db Str = F db Itô + = F db +. To identify that additional term, denote by SymA the ymmetric art of a matrix A and recall that ti+ t i Id = Sym B Str t i+ t i Sym B Itô t i+ t i = B t i+ t i Sym B Itô t i+ t i ; note alo that the above reaoning howing that F t i B Itô t i+ t i converge to in L alo how that F t i Sym B Itô t i+ t i converge to in L. So i almot-urely equal to the limit a π of the um F t i B t i+ t i. Since F t i B ti+ t i = F ti+ t i + R ti+ t i for ome -Hölder remainder R, the above um equal Fti+ti B ti+ti + o π. We recognize in the right hand ide um a quantity which converge in robability to the bracket of F and B. Corollary 7. Under the aumtion of rooition 6, we have almot-urely F db Str = F db Rough and tochatic differential equation. Equied with the receeding two reult, it i eay to ee that the olution atht to a rough differential equation driven by B Str or B Itô coincide almot-urely with the olution of the correonding Stratonovich or Itô tochatic differential equation. Theorem 8. Let F = V,...,V l be C 3 b vector field on R d. The olution to the rough differential equation 4.7 dx t = Fx t B Str dt coincide almot-urely with the olution to the Stratonovich differential equation dx t = V i x t dbt. i A imilar tatement hold for the Itô Brownian rough ath and olution to Itô equation.

7 Proof Recall we have een in exercie 6 that a ath i a olution to the rough differential equation 4.7 if and only if it i a olution ath to the integral equation x t = x + Fx db Str. Given the reult of corollary 7, the theorem will follow if we can ee that x i adated to the Brownian filtration; for if one et F := Fx then it derivative F = F x Fx, with F the differential of F with reect to x, will alo be adated. But the adatedne of the olution x to equation 4.7 i clear from it contruction in the roof of theorem??. We obtain a a corollary of theorem 8, Lyon univeral limit theorem and the convergence reult roved in rooition 3 for the rough ath aociated with the iecewie linear interolation B n of B the following fundamental reult, firt roved by Wong and Zakai in the mid 6. Corollary 9 Wong-Zakai theorem. The olution ath to the ordinary differential equation 4.8 dx n t = F x n t db n t converge almot-urely to the olution ath to the Stratonovich differential equation dx t = Fx t db t. Proof It uffice to notice that olving the rough differential equation = F B n dt dz n t z n t 7 i equivalent to olving equation Freidlin-Wentzell large deviation theory. We hall cloe thi coure with a ectacular alication of the continuity roerty of the olution ma to a rough differential equation, by howing how one can recover the baic of Freidlin-Wentzell theory of large deviation for diffuion rocee from a unique large deviation rincile for the Stratonovich Brownian rough ath. Exercie 8 alo ue thi continuity roerty to deduce Stroock-Varadhan celebrated uort theorem for diffuion law from the correonding tatement for the Brownian rough ath A large deviation rincile for the Stratonovich Brownian rough ath. a Schilder theorem. Let tart thi ection by recalling Schilder large deviation rincile for Brownian motion. Define for that uroe the real-valued function I on C [, ], R d equal to h H = ḣ d on H,and elewhere. We agree to write IA for inf{i h ; h A}, for any Borel ubet A of C [, ], R d, endowed with the C toology.

8 8 Theorem. Let P tand for Wiener meaure on C [, ], R d and B tand for the coordinate roce. Given any Borel ubet A of C [, ], R d, we have A I lim ε log P εb A I A. Proof The traditional roof of the lower bound i a imle alication of the Cameron-Martin theorem. Indeed, if A i the ball of centre h H with radiu δ, and if we define the robability Q by it denity dq dp =ex ε h db ε Ih with reect to P, the roce B := B ε h i a Brownian motion under Q, and we have P εb h δ = P B ε δ [ = E Q ex B ε ε δ B e ε Q Ih ε δ = e ε Ih o ε. ] h db ε Ih One claically ue three fact to rove the uer bound. The iecewie linear aroximation B n of B introduced above obviouly atifie the uer bound, a B n live a a random variable in a finite dimenional ace where it define a Gauian random variable. The equence εb n rovide an exonentially good aroximation of εb, in the ene that lim u ε εb ε log P n εb δ. m 3 The ma I enjoy the following continuity roerty. With A δ := {x ; /, dx, A δ}, wehave I A = lim I A δ. δ The reult follow from the combination of thee three fact. The firt and third oint are eay to ee. A for the econd, jut note that B n B i actually made u of n indeendent coie of a caled Brownian bridge n+ B k,witheach B k defined on the dyadic interval [ k n, k + n]. A it uffice to look at what haen in each coordinate, claical and eay etimate on the real-valued Brownian bridge rovide the reult. b Schilder theorem for Stratonovich Brownian rough ath. The extenion of Schilder theorem to the Brownian rough ath require the introduction of the function J, defined on the et of G l -valued continuou ath e = e,e by the formula J e = I e.

9 Recall the definition of the dilation δ λ on Tl, given in??. Given any <, one can ee the ditribution P ε of δ ε B Str a a robability meaure on the ace of -Hölder G l-valued function, with the correonding norm. Theorem. The family P ε of robability meaure on C [, ], G l atifie a large deviation rincile with good rate function J. It hould be clear to the reader that it i ufficent to rove the claim for the Brownian rough ath above a -dimenional Brownian motion B = B,B,defined on C [, ], R a the coordinate roce. We hall rove thi theorem a a conequence of Schilder theorem; thi would be traightforward if the econd level roce B or rather jut it anti-ymmetric art were a continuou function of the Brownian ath, in uniform toology, which doe not hold true of coure. However, rooition 3 on the aroximation of the Brownian rough ath by it iecewie linear counterart make it clear that it i almot-urely equal to a limit of continuou functional of the Brownian ath. So it i temting to try and ue the following general contraction rincile for large deviation. See the book [?] by Kallenberg for an account of the baic of the theory, and a roof of thi theorem. We tate it here in our etting to avoid unneceary generality, and define the aroximated Lévy area A m t a a real-valued function on the C [, ], R etting A m t := B [m] m db db B [m] m It i a continuou function of B in the uniform tology. The ma A m converge almot-urely uniformly to the Lévy area roce A of B. We ee the roce A a a ma defined on the ace C [, ], R, equal to Lévy area roce on a et of robability and defined in a genuine way on H uing Young integral. Note that element of H are -Hölder continuou. Theorem. Extended contraction rincile If Exonentially good aroximation roerty. 9 lim u ε ε log P ε A m A >δ m, Uniform convergence on I-level et for each r> we have A m A, {I r} m then the ditribution of A under P ε atifie a large deviation rincile C [, ], g with good rate function inf{iω; a = Aω}. Proof of theorem The roof amount to roving oint and in theorem. The econd oint i elementary if one note that for h H [, ], R,

10 we have h [m] h dh h h m m h H m. A for the firt oint, it uffice to rove that lim u ε log P B B [m] ε m u t [,] db ε δ, m which we can do uing elementary martingale inequalitie. Indeed, denoting by M t the martingale defined by the above tochatic integral, with bracket B B [m] d, the claical exonential inequality give m P M δε, M ε m ex δ ε m, while we alo have P M ε m B P m ε m So the concluion follow from the fact that the -Hölder norm of B ha a Gauian tail Freidlin-Wentzell large deviation theory for diffuion rocee. All together, theorem on the rough ath interretation of Stratonovich differential equation, Lyon univeral limit theorem and the large deviation rincile atified by the Brownian rough ath rove the following baic reult of Freidlin-Wentzell theory of large deviation for diffuion rocee. Given ome Cb 3 vector field V,...,V l on R d,andh H,denotebyy h the olution to the well-defined controlled ordinary differential equation dyt h = εv i y h t dh i t. Theorem 3 Freidlin-Wentzell. Denote by P ε the ditribution of the olution to the Stratonovich differential equation dx t = εv i x t db i t, tarted from ome initial condition x. Given any <, one can conider P ε a a robability meaure on C [, ], R d. Then the family P ε atifie a large deviation rincile with good rate function Jz = inf { Ih; y h = z }.

11 4.5. Exercie on rough and tochatic analyi. Exercie 7 rovide another illutration of the ower of Lyon univeral limit theorem and the continuity of the olution ma to a rough differential equation, called the Itô ma. It how how to obtain a groundbreaking reult of Stroock and Varadhan on the uort of diffuion law by identifying the uort of the ditribution of the Brownian rough ath. Exercie 8 give an intereting examle of a rough ath obtained a the limit of a -dimenional ignal made u of a Brownian ath and a delayed verion of it. While the firt level concentrate on a degenerate ignal with identical coordinate and null area roce a a conequence, the econd level converge to a non-trivial function. Lat, exercie 9 i a continuation of exercie on the airing of two rough ath. 7. Suort theorem for the Brownian rough ath and diffuion law. We how in thi exercie how the continuity of the Itô ma lead to a dee reult of Stroock and Varadhan on the uort of diffuion law. The reader unacquainted with thi reult may have a look at the olohed roof given in the book by Ikeda and Watanabe [5] to ee the benefit of the rough ath aroach. a Tranlating a rough ath. Given a Lichitz continuou ath h and a - rough ath a = a a,with <<3, check that we define another -rough ath etting τ h a t := a t + h t t a t + a u dh u + h u da u + h u dh u, where the integral h u da u i defined a a Young integral by the integration be art formula h u da u := h t a t dh u a u. b Given any R l -valued coninuou ath x, denote a in ection 4. by x n the iecewie linear coninuou interolationof x on dyadic time of order n, andlet X n tand for it aociated rough ath, for <<3. We define a ma X : C [, ], R l G, l π Xx := x, [,] etting π Xx jk t := lim u x n,j u dx n,k u. n So the random variable Xx i almot-urely equal to Stratonovich Browian rough ath under Wiener meaure P. i Show that one ha P-almot-urely Xx + h =τ h Xx for any Lichitz continuou ath h. ii Prove that th law of the random variable τ h X i equivalent to the law of X under P. Recall that the uort of a robability meaure on a toological ace if the mallet cloed etof full meaure. We conider X, under P, aac [, ], G l -valued random variable.

12 iii Prove that if a i an element of the uort of the law of X under P, then τ h a a well. c i Ue the ame kind of argument a in rooition3 to how that one can ind an element a in the uort of the law of X under P, and ome Lichitz coninuou ath x n uch that τ x na tend to a n. ii Prove that the uort of the law of B Str in C [, ], G l i the cloure Hölder toology of the et of of Lichitz continuou ath. in d Stroock-Varadhan uort theorem. Let P tand for the ditribution of the olution to the rough differential equation in R d dx t = V i x t dbt i, driven by Brownian motion and ome Cb 3 vector field V i. Jutify that one can ee P a a robability on C [, ], R d.letalowritey h for the olution to the ordinary differential equation dyt h = V i y h t dh i t driven by a Lichitz R l -valued ath h. Prove that the uort of P i the cloure in C [, ], R d of the et of all y h,forh ranging in the et of Lichitz R l -valued ath. 8. Delayed Brownian motion. Let B t t be a real-valued Brownian motion. Given ɛ>, we define a -dimenional roce etting x t = B t ɛ,b t ; it area roce A ɛ t := B u ɛ, ɛ db u B u db u ɛ i well-defined for t <ɛ,ab ɛ and B are indeendent on [, t] in that cae. Show that we define a rough ath X ɛ etting X ɛ t := ex x t + At ɛ T. Recall that d tand for the ambiant metric in T. Prove that one can find a oitive contant a uch that the nequality [ E ex a d X ɛ t, ] Xɛ C< t hold fo a oitive contant C indeendent of <ɛ and t. A in ection 4.., it follow from Beov embedding theorem that, for ny <<3, the weak geometric Hölder -rough ath X ɛ haa Gauian tail, with [ u E ex a X ɛ ] < <ɛ for ome oitive contant a.

13 3 Define a the vector of R with coordinate and in the canonical bai, and et Y t := ex B t t Id. Write d for the ditance on the etof Hölder -rough ath defined in definition??. Prove that d X ɛ, Y converge to in L q, for any q<. 9. Joint lit of a random and a determinitic rough ath. Let << 3 and X =X, X be an R d -valued Hölder -rough ath. Denote by B the Itô Brownian rough ath over R l.givenj,d and k,l, the integral Z jk t := X k udb j u a a genuine Itô integral, and define the integral Bj u dxk u by integration by art, etting Z kj t := Bu j dxk u := Bj txt k Xu k dbj u. Prove that one define a Hölder -rough ath Z over X, B R d+l defining the jk-comonent of it econd order level, a equal to X jk if j, k d, equalto B jk if d + j, k d + l, and by the above fomula otherwie. 3

14 4 Reference [] Lyon, T.J. and Caruana, M. and Lévy, Th. Differential equation driven by rough ath. Lecture Note in Mathematic, 98, Sringer 7. [] Lyon, T. and Qian, Z. Sytem control and rough ath. Oxford Mathematical Monograh, Oxford Univerity Pre. [3] Friz, P. and Victoir, N. Multidimenional tochatic rocee a rough ath. CUP, Cambridge Studie in Advanced Mathematic,,. [4] Baudoin, F., Rough ath theory. Lecture note, htt://fabricebaudoin.wordre.com/category/roughath-theory/, 3. [5] Friz, P. and Hairer, M., A hort coure on rough ath. Lect. Note Math., 4. [6] Bailleul, I., Flow driven by rough ath. arxiv:3.888, 3. [7] Feyel, D. and de La Pradelle, A. Curvilinear integral along enriched ath. Electron. J. Probab., :86 89, 6. [8] Feyel, D. and de La Pradelle, A. and Mokobodzki, G. A non-commutative ewing lemma. Electron. Commun. Probab., 3:4 34, 8. [9] Gubinelli, M., Controlling rough ath. J. Funct. Anal., 6:86 4, 4. [] Lyon, T.. Differential equation driven by rough ignal. Rev. Mat. Iberoamericana, 4 :5 3, 998. [] Montgomery, R., A tour of ubriemannian geometrie, their geodeic and alication. Mathematical Survey and Monograh, 9,. [] Lejay, A., Yet another introduction to rough ath. Séminaire de Probabilité, LNM 979:, 9. [3] Chen, K.T. Iterated ath integral. Bull. Amer. Math. Soc., 835:83 879, 977. [4] Friz, P. and Victoir, N. A note on the notion of geometric rough ath. Probab. Theory Related Field, 36 3:395 46, 6. [5] Ikeda, N. and Watanabe, S. Stochatic differential equation and diffuion rocee. North- Holland Publihing Comany, 988. IRMAR, 63 Avenue du General Leclerc, 354 RENNES, France addre: imael.bailleul@univ-renne.fr