Lévy area of fractional Ornstein-Uhlenbeck process and parameter estimation

Transcription

1 Lévy area of fractional Orntein-Uhlenbeck proce and parameter etimation Zhongmin Qian and Xingcheng Xu April 4, 8 arxiv:83.39v [math.pr] 3 Apr 8 Abtract In thi paper, we tudy the etimation problem of an unknown drift parameter matrix for fractional Orntein-Uhlenbeck proce in multi-dimenional etting. By uing rough path theory, we propoe pathwie rough path etimator baed on both continuou and dicrete obervation of a ingle path. The approach i applicable to the high-frequency data. To formulate the parameter etimator, we define a theory of pathwie Itô integral with repect to fractional Brownian motion. By howing the regularity of fractional Orntein-Uhlenbeck procee and the long time aymptotic behaviour of the aociated Lévy area procee, we prove that the etimator are trong conitent and pathwie table. Numerical tudie and imulation are alo given in thi paper. Keyword. Rough path etimator, Pathwie tability, Fractional Brownian motion, FOU procee, Itô integration, Lévy area, Long time aymptotic, High-frequency data. MSC[]. 6H5, 6F, 6M9, 9G3. Introduction The tatitical analyi of time erie and random procee, parameter etimation, nonparameter etimation and tatitical inference, ha been mainly concentrated on model decribed in term of diffuion procee and emi-martingale, ee e.g. ome tandard reference uch a [3,, 35, 36] and etc. Model which are not emi-martingale have received ome attention in application where long-time memory effect have to be taken into conideration, ee e.g. [, 38, 9] for example. In thi article, we tudy multi-dimenional Orntein-Uhlenbeck procee OU procee for hort driven by fractional Brownian motion fbm, known in recent literature a fractional Orntein-Uhlenbeck procee fou procee for implicity, defined to be the olution of the tochatic differential equation SDE dx t = ΓX t dt + ΣdB H t, X = x,. Mathematical Intitute, Univerity of Oxford, OX 6GG, United Kingdom. qianz@math.ox.ac.uk School of Mathematical Science, Peking Univerity, Beijing 87, China; Current addre: Mathematical Intitute, Univerity of Oxford, OX 6GG, United Kingdom. xuxingcheng@pku.edu.cn. Xingcheng Xu i upported by China Scholarhip Council Grant No. 769.

2 where B H i a d-dimenional fbm with Hurt parameter H,, Γ R d d i the drift matrix and Σ R d d i the volatility matrix which i non-degenerate. The previou SDE ha to be interpreted a the tochatic integral equation X t = x which ha an unique olution given by X t = e Γt x + ΓX d + ΣB H t, e Γt ΣdB H.. The integral on the right hand ide i undertood a an Young integral. Therefore, like ordinary OU procee, X t i a Gauian proce. Equation. can be ued to decribe ytem with linear interaction perturbed by Gauian noie. An example of application i inter-banking lending, ee e.g. [8, 4]. In application, an important quetion i to etimate the interaction tructure Γ from an obervation of a ingle path of the proce, auming that Σ i known and a ingle path Xω can be oberved continuouly or at dicrete time. For one dimenional cae, the maximum likelihood etimator MLE and the leat quare etimator LSE and their propertie have been tudied in literature, ee e.g. [, 38, 9, ]. Kleptyna and Le Breton [] and Tudor and Vien [38] have tudied the maximum likelihood etimator baed on continuou obervation and obtained the trong conitency of the MLE a T goe to infinity. Hu and Nualart [9] tudied the leat quare etimator for the cae where the Hurt parameter H >. The trong conitency a T wa proved for H > and a central limit theorem wa alo etablihed if < H < 3. Hu, Nualart and Zhou [] extended 4 their reult for all H,. There are however few work on parameter etimation for multi-dimenional fou procee. The aim of thi paper i therefore to fill thi gap. Firt, for continuou obervation of a ingle path, we give an etimator baed on the rough path theory ee e.g. Lyon and Qian [7]. In order to formulate the parameter etimator, we hould define an Itô type integration theory for multi-dimenional fbm. Coutin and Qian [] propoed a theory of Stratonovich integration for multi-dimenional fbm with H > by uing the rough path analyi. It remain an open quetion to build a 4 rough path theory for fbm with Hurt parameter H. Therefore, we conider fbm with 4 Hurt parameter H where < H. For thi cae, both fbm and fou procee are of finite 3 p-variation with p < 3, and can be enhanced canonically to be geometric rough path. We may define Itô type integral with repect to fbm and fou procee by correcting their enhanced Lévy area procee, and apply it to the tudy of the parameter etimation problem for fou procee baed on continuou obervation. In order to how that the parameter etimator i trongly conitent, we tudy the regularity of fou procee and long time aymptotic behaviour of their Lévy area procee, which we believe are of interet by their own. We alo tudy the etimation problem baed on dicrete obervation. In application, obervation time are dicrete rather than continuou, though the ampling frequency can be made to tend to infinity, which i the cae for high-frequency financial data. For the tatitical inference in thi direction, we recommend [,, 8, 3, 5,, 4] and reference therein. In thi paper, we contruct a parameter etimator baed on high-frequency dicrete obervation by uing rough path theory, and etablih the trong conitence of thi etimator. We would like mention that Diehl, Friz and Mai [] tudied the maximum likelihood etimator for diffuion procee via

3 the rough path analyi, and initiated a tudy of etimator for the fractional cae, but only for the cae that H = ε, for mall ε. The approach we preent in thi paper ha everal advantage over other method in the exiting literature. Firt, our etimator are for multi-dimenional fou procee where the no trivial role played by Lévy area procee may be revealed, which differ fundamentally from the one dimenional cae. Second, the parameter etimator are pathwie defined, and can be calculated baed on obervation of a ingle path. Third, the parameter etimator are pathwie table and robut, in the ene that, if two obervation are very cloe according to the o called p-variation ditance ee the main text below, then their correponding etimator are cloe too. Fourth, our etimator can be contructed by both continuou obervation and dicrete obervation, in particular for high-frequency financial data. Numerical tudie and imulation are given in thi paper, to demontrate that the parameter etimator we propoe are very good. Let u mention that the approach in thi paper can be extended to Orntein-Uhlenbeck proce X t driven by a general Gauian noie G t, o that X t = e Γt x + e Γt ΣdG.3 where the integral on the right-hand ide i well defined a long a t G t i α-hölder continuou for ome α >. Such ingular OU procee may be ueful in application. The paper i organized a follow. In ection, we recall ome preliminarie of the rough path theory and outline a theory of pathwie Itô integral for both fbm and fou procee. In ection 3, we prove the regularity of fou proce and tudy long time aymptotic of the aociated Lévy area procee. Then we contruct a continuou rough path etimator in Section 4. We give a complete proof for almot ure convergence and pathwie tability of thi etimator. Then in ection 5, the dicrete rough path etimator baed on high-frequency data i preented. In ection 6, we give two concrete experimental example baed on imulated ample path, and the numerical reult are hown there. Rough path and Itô integration In thi ection, we introduce everal notation from the rough path theory, following the tandard reference [5, 6, 7, 6, 7]. We give a definition of Itô integrale for fbm and fou procee.. Preliminary of rough path Define the truncated tenor algebra T R d by T R d := n=r d n, with the convention that R d = R, and ue to denote the implex {, t : < t T }. Let X t be a continuou path on interval [, T ] and X,t =, X,t, X,t be an element of pace T R d. Actually, when X t i of finite p-variation with < p < 3, we may lift it to the pace T R d a a multiple function. The initial motivation i to define integral with repect to X by increaing information to X. We recall Chen identity algebraic information and the definition of finite p-variation analyi information. We call that X,t =, X,t, X,t atifie Chen identity if X,t = X t X,. X,t X,u X u,t = X,u X u,t,. 3

4 for all, u, u, t. X =, X,t, X,t ha finite p-variation if up P [,t] P X,t p <, up P [,t] P X,t p/ <, where P i a partition of [, T ]. It i equivalent to that there exit a control ω, t uch that X,t ω, t /p, X,t ω, t /p,, t. A control ω i a non-negative, continuou, uper-additive function on and atifie that ωt, t =. Let < p < 3 be a contant. A function X =, X, X from to T R d i called a p-rough path if it ha finite p-variation, and atifie Chen identity. Denote the pace of p-rough path a Ω p R d. According to Lyon and Qian [7], the integration operator i defined a a linear map from Ω p R d to Ω p R e, i.e. F : Ω p R d Ω p R e, and denote the integral by Y = F Xd R X, where Y u,v v and the econd level Y by Y u,v v u u F Xd R X := lim F Xd R X := lim P [,t] P P [,t] P where the limit take over all finite partition P of interval [u, v].. FBM a rough path F X X,t + DF X X,t,.3 Y u, Y,t + F X F X X,t,.4 Almot all ample path of a d-dimenional fbm with Hurt parameter H, ] have 3 finite p-variation with < < p < 3, which can be enhanced canonically to be geometric H rough path. In fact Coutin and Qian [] contructed the canonical rough path enhancement B H,Str =, B H, B H,Str in the Stratonovich ene by uing dyadic approximation of fbm and their iterated integral. But for the parameter etimation problem dicued in thi paper, if the tochatic integral in the etimator ee ection 4 i undertood in the Stratonovich ene, it will almot urely converge to. Such etimator i not reaonable and therefore it ha no ue. So we need a theory of Itô type integration for fbm and fou proce. In [34], the preent author have contructed an Itô type rough path enhancement B H,Itô =, B H, B H,Itô aociated with an fbm by etting B H,Itô,t = B H,Str,t It, which can be ued to define Itô type pathwie integral with repect to B H. For the firt level of Itô integral, F B H d R B H ω = lim F B H ωb,tω H + DF B H ωb H,tω, P [,t] P for every ω N c, where N i a null et. The econd level are defined imilarly a.4. 4

5 However, thi theory of Itô rough path enhancement and aociated Itô integration work well only for one form, i.e. only work well for function of B H t, and therefore thi theory i not uitable in dealing with fou procee which are not one form of the fbm B H t. An fou proce X t depend on the whole path of {B H, t}. In the preent paper, we will reveal that a different integration theory i.e. with different rough path aociated with fbm i required. To define an Itô rough path enhancement aociated with fbm which i uitable for the tudy of fou procee. Take with U γ t := HΓ ϕ γ t := It U γ t,.5 e Γ u u dud..6 Then ϕ γ t ha finite q-variation with q =, and therefore we can define the Itô type fractional Brownian rough path lift for B H to be the following rough path B H,γ,t =, B H,t, B H,γ,t :=, B H,t, B H,Str,t ϕ γ,t,.7 where ϕ γ,t = ϕ γ t ϕ γ. Remark.. One can verify that, if H =, thi Itô rough path enhancement i conitent with Itô theory for the tandard Brownian motion. When Γ =, thi enhancement i the ame with the one form cae defined in [34]. In the following, we will illutrate why we call it a Itô rough path/itô rough integration..3 FOU a rough path For the fou proce X t defined by tochatic differential equation., it can alo be enhanced a a rough path according to the theory of rough path, which i the eence of the theory of rough differential equation. Although for the exitence and uniquene of the olution to., for thi imple cae, the theory of rough path i not needed. However, when we ak if X t can be enhanced to a rough path, or when we want to integrate F X with repect to X, the rough path analyi i a natural tool to deal with thee problem. We emphaize that the meaning of the olution X to a rough differential equation enhanced by. depend on the rough path B H we ue. Here B H can be either B H,Str in Stratonovich ene or B H,γ in Itô ene. Let Z t = Bt H, X t, t and Z = B H, X, t to be it aociated rough path enhancement. Then equation. i enhanced to d R Z = fzd R Z,.8 where fx, y, tξ, η, τ := ξ, Γyτ + Σξ, τ. According to Theorem 6.. and Corollary 6.. in [7], a unique olution Z, which i a rough path, exit. Formally, Z =, Z, Z ha the following expreion: Z,t = B,t, H X,t, t,.9 B H,t BH,udX u Z,t = X,udBu H t X,t t u dbh u u dx u 5 BH,udu X,udu t..

6 Each component of the econd level Z,t i well-defined a part of the olution to.8. More exactly, we denote Stratonovich olution of RDE.8 a Z Str =, Z, Z Str, where Z Str = Z Str,ij i,j=,,3, and we denote Itô olution of RDE.8 a Z Itô =, Z, Z Itô, where Z Itô = Z Itô,ij i,j=,,3. We therefore may define Stratonovich integral firt level of fou proce with repect to fbm a X d R B H,Str = Z Str,,t + X B H,t,. and Itô integral firt level of fou proce with repect to fbm a X d R B H,γ = Z Itô,,t + X B H,t.. Now we can define tochatic integral with repect to fou rough path enhancement X by equation.3,.4. Note that thee integral are pathwie defined and continuou with repect to the ample path Xω in p-variation metric. In what follow, we denote Stratonovich rough integral a and Itô rough integral a F X d R X = F X d R X =,, F X d R X, F X d R X, F X d R X, F X d R X. A an application we will ue the Itô rough integral to contruct the etimator for parametric matrix Γ and prove the aymptotic propertie and pathwie tability in the following ection..4 Zero expectation Let u illutrate the reaon for naming Itô rough path and Itô rough integral. In tochatic analyi, Itô integral can be defined in term of the martingale property, which i uitable for emi-martingale. While for procee which are not emi-martingale uch a fbm, attempt of making integral with repect to fbm being martingale are of coure hopele. We intead demand that the expectation of integral with repect to fbm are contant e.g., to be zero. We call thi kind of integral a Itô type integral, which i in fact an extenion of claical Itô integration theory. Now let u verify that expectation of the Itô integral of fou proce with repect to fbm t X d R B H,γ or write a X d R B H,γ vanihe. According to the theory of differential equation driven by rough path and the definition of integral above, and auming that coefficient matrice Γ and Σ are commutative for implicity, we have Since X d R B H,γ = X t = e Γt X + X d R B H,Str Σϕ γ t..3 e Γt ΣdB H,.4 6

7 where X i a contant vector and the integral on the right hand ide i Young integral and equal e Γt Σ d R B H,Str. Therefore E X d R B H,Str = E e Γ X d R B H,Str + E e Γ u Σ d R B H,Str u d R B H,Str The firt term on the right hand ide i zero, and the econd term E e Γ u Σ d R B H,Str u d R B H,Str = e Γ u ΣdR H u, t = Σ It HΓ e Γ u u dud = Σϕ γ t, where R H u, = EBu H B H = u + u i the covariance function of fbm and the integral againt R H u, i defined a an Young integral in D ene ee e.g. [6]. Thu, combining equation above, we have proved the zero expectation property, i.e. E X d R B H,γ =..5 3 Long time aymptotic of Lévy area of fou procee In thi ection, we tudy propertie of fou procee. We how the α-hölder continuity of fou procee, and prove a long time aymptotic property of Lévy area of fou procee. 3. Regularity of fou procee 3.. The covariance of fou procee The covariance function of a general fou proce can be worked out explicitly. For implicity, we firt tudy a tationary verion of fou proce in thi ection. Conider X t = σ e λt db H, which i tationary and ergodic ee e.g. [9], and B H i fbm with Hurt parameter H <. It i well known that the covariance R H, of B H i of finite -variation. The covariance function of {X t = σ e λt db H, t } i given by ee, e.g. [33] rt = CovX, X +t = CovX, X t = σ G + inπh λ π coλtx x + x dx = σ σ G + cohλt λ t F ; H +, H + ; 4 λ t, where G i the Gamma function, coh the hyperbolic coine function, F ;, ; the generalized hypergeometric function, i.e. a n a p n x n pf q a,, a p ; b,, b q ; x = b n b q n n!, 7 n=

8 and a =, a n = aa + a + n, for n. One can ee the figure of thi covariance function r and it firt and econd derivative below, where we take H =., σ = λ = a an example. Figure 3.: Graph of the covariance function r of tationary fou and it firt two derivative, e.g. H =., σ = λ =. Lemma 3.. For the covariance function r of tationary fou proce X with H <, we have the following propertie: i rt i -Hölder continuou on R +, that i, rt r C H t, for any, t R + and C H depend on H, σ, λ only we may ignore σ, λ. ii There exit contant < T < T uch that r T =, and r t > on interval, T, r t < on interval T,. That i, r i convex on [, T ] and concave on T,. Proof. For covariance function r, near t =, rt = σ λ λ HG G + t + ot, and for t large enough ee Theorem.3, [9], rt = N n σ λ n k t n + Ot N. n= k= Since rt i continuou on [, and one can alo ee that rt ha polynomial decay to zero a t large from above equality. For i, we have max t rt = C <, for any, t R + and t, then rt r C C t. 8

9 For any, t R + and t <, we how the tatement in three cae:, t [, ],, t [, and < < t<. For the firt and third term, we actually need to how that for any, t [, ] and t <, there exit a contant C uch that rt r C t. Since rt = ct + ϕt, where ϕ i mooth on R +, then rt r c t + max u ϕ u t C t. For the econd cae, i.e. for any, t [, and t <, we have rt r max u r u t C t. Thu we proved the tatement i. For ii, one can ee that there exit a mall number ε > uch that r ε > and a large number T > uch that, for all t T, r t <. By the continuity of r on,, there exit a T ε, T atifying r T = and r t > for any t, T. Following are important propertie of fou procee when H <. It i well-known that, increment of fbm are negatively correlated when H <, and poitively correlated when H >, while for H = increment over different time period are independent. We found that for fou proce with H <, the dijoint increment are locally negative correlated. If the ditance of the interval correponding the dijoint increment i large, then they are poitively correlated, we call it long-range poitive correlation. See the theorem below. Heuritically, fou proce i locally like fbm o that it ha the locally negative correlation property a fbm when H <. For long ditance the drift become the dominated force, o the fou behave poitively correlated. In the cae where H =, the fou i the tandard OU proce driven by tandard Bownian motion. The propertie of it are well known. Our main concern here i for the true fou proce cae with H <. Theorem 3.. Conider the tationary fou proce X with H <. T, T are given in the previou lemma. i Locally negative correlation For any and t i < t i+ t j < t j+ + T, then EX ti+ X ti X tj+ X tj. 3. ii Long-range poitive correlation For any t i < t i+ < t j < t j+, and if t j t i+ > T, then EX ti+ X ti X tj+ X tj. 3. Proof. i Since EX ti+ X ti X tj+ X tj = rt j+ t i+ rt j+ t i rt j t i+ rt j t i = rx 3 rx 4 rx rx = [rx 4 rx 3 rx rx ], where x := t j t i+, x := t j t i, x 3 := t j+ t i+, x 4 := t j+ t i, then we have x < x x 3 < x 4 T or x < x 3 x < x 4 T, and rx 4 rx 3 x 4 x 3 rx rx x x, by convexity of r. Thi prove 3.. ii The proof of 3. i almot the ame a i. 9

10 Now we have the following propoition. Propoition 3.3. For the tationary fou proce X, it atifie that E X t X C H t, 3.3 for any, t R +, and C H depend on H, σ, λ only we ignore σ, λ here. Proof. Since X i a tationary Gauian proce, then E X t X = EX t + X X t X = r r t C H t. The lat inequality hold by -Hölder continuity of the covariance function rt, i.e. Lemma 3..i. Propoition 3.4. Let X be the tationary fou proce with H,. Then it covariance R X, t = EX X t i of finite -variation on [, +T ] in D ene for any. Moreover, there exit contant C = CH and T > uch that, for all < t in [, + T ], where R X ρ ρ var;[,t] := up i,j R X var;[,t] CH t, 3.4 [ E X ti+ X ti X t j+ X t ρ j ], 3.5 and P = {t i }, P = {t j} are any two partition of interval [, t]. Proof. By Lemma 5.54 of [6], we jut need to how the finite -variation by the ame partition P = {t i } of interval [, t] [, + T ]. Let u conider [ E Xti+ X ti X tj+ X tj ]. 3.6 i,j For a fixed i, and i j, E [ X ti+ X ti X tj+ X tj ] for H < [ ] E Xti,t i+ X tj,t j+ j = j i [ ] E Xti,t i+ X tj,t j+ + E Xti,ti+ by Theorem 3., hence, E j i E + X ti,t i+ X tj,t j+ E Xti,ti+ CH E [ Xti,t i+ X,t ] j + X ti,t i+ X tj,t j+ E Xti,ti+ + E Xti,t i+ + CH E Xti,ti+.

11 Therefore, we have [ ] E Xti,t i+ X tj,t j+ i,j CH i E [ Xti,t i+ X,t ] + CH i E Xti,ti+. The econd term on the right hand ide i controlled by CH t by Propoition 3.3. Now we how that [ ] E Xti,t i+ X,t CH t. Since i E [ Xti,t i+ X,t ] = E Xti+ X t X ti X t + X ti X X ti+ X = rt t i+ rt t i + rt i rt i+ rt t i+ rt t i + rt i rt i+ C H t i+ t i + C H t i+ t i C H t i+ t i, thu E [ ] X ti,t i+ X,t Now we have completed the proof. i i CH t i+ t i CH t. Corollary 3.5. Let X be the tationary fou proce with H,. Then it covariance R X, t = EX X t i of finite -variation on [, T ] in D ene. Moreover, there exit a contant C = CH uch that, for all < t in [, T ], R X CH t. 3.7 var;[,t] [ T Proof. We divide the interval [, T ] into m+ = T ]+ piece, denote them a [, T ], [T, T ],, [m T, mt ], [mt, T ]. For any ubinterval [, t] [, T ], there exit q, q N uch that [q T, q T ] and t [q T, q +T ], by the ubadditivity of R X, then var;[, ] we have R X R X + R var;[,t] var;[,q T ] X + + R var;[q T,q +T ] X CH q T + q T q T + + t q T CH t. Thi complete the proof of the corollary. 3.. Regularity of fou procee var;[q T,t] In the following, we tudy the α-hölder continuity of one dimenional, tationary fou proce X t = σ e λt db H. Before howing the regularity, we recall the uual Garia- Rodemich-Rumey inequality ee e.g., page 6, Stroock and Varadhan [37]. Lemma 3.6. Garia-Rodemich-Rumey inequality Let p and Ψ be continuou, trictly increaing function on [, uch that p = Ψ =, and lim t Ψt =.

12 Given T > and φ C[, T ], R d, if there i a contant B uch that T T φt φ Ψ ddt B, 3.8 p t then for all t T, φt φ 8 t A an application of thi lemma above, we have 4B Ψ pdu. 3.9 u Propoition 3.7. Let X be a one dimenional, tationary fou proce with H, on [, T ]. Then there exit a contant < β < and an almot urely finite random variable C independent of T uch that X t X CT β t α, a.. 3. for any α, H, any, t T. Proof. By Propoition 3.3, we know that E X t X C H t. 3. Since X t i Gauian proce, all the norm are equivalent, we get E X t X p C p E X t X p Cp,H t ph, 3. for any p >. Next, we apply the Garia-Rodemich-Rumey inequality. Take Ψx = x p and px = x H. Then inequality 3. implie that T T Xt X E Ψ ddt C p,h T. p t Define B T := T T Ψ Then for any q > 3, we get E n= Xt X T T ddt = p t B n n q = n= EB n n q n= X t X p ddt. t ph Cn n q <. Thu there exit an almot urely finite random variable R independent of n uch that n= B n n q R, a.. So we have B n Rn q, a.. n, q > 3. Take n = [T ], then B T B n+ Rn + q CRT q, a.. T >, q > 3.

13 Then the Garia-Rodemich-Rumey inequality give that X t X 8 t Ψ 4BT u pdu C4B T p t H /p CR p T q p t α, for any α, H, p > 3 [ H α] and 3 < q < p. Thi conclude the lemma. Remark 3.8. When X t = σ e λt db H, it till atifie inequality 3.. Beide, we prove a propoition for a function of fou procee, which will be applied in ection 5. Propoition 3.9. Let B H = B H,, B H,,, B H,d be a d-dimenional fbm with H,, X = X, X,, X d a d-dimenional fou proce, where Xt i = σ e λit db H,i, λ i >, σ R. Define F X t := X t X t = XtX i j t i,j=,,,d, and the norm of matrix A a A = d i,j= a ij. Then there exit a contant < β <, an almot urely finite random variable C independent of T and a random variable R T tend to zero almot urely a T uch that F X t F X CR t α T T β, 3.3 up t for any, t T, any α, H. Proof. Firt, we preent a fact about upremum of one dimenional, tationary fou proce X i t = up t X i below. Since we know that X i and X i have the ame ditribution and their covariance function i r i t = CovX i +t, X i = C λ i G + t + ot, 3.4 for t mall, where C = σ λ i HG and G i Gamma function. So by Theorem 3. of Pickand [3], we know that for t tending to infinity t up X i δ, a.., t t up X i, a.., δ t for any δ >. Since X i t = up t X i up t X i, then X i t t δ, a Since Xt i = X i t e λt X i, o we alo have Xi t, a., where X i t δ t = up t X. i X Now define R t = up i t i=,,d, then R t δ t, a.. a t. For any i, j =,,, d, and, t T, X i tx j t X i X j = X i t X i X j t + X i X j t X j X j t X i t X i + X i X j t X j X j T X i t X i + X i T X j t X j CR T T δ T β t α + CR T T δ T β t α CR T T δ+β t α, where the lat econd inequality i followed from Propoition 3.7. One can chooe δ, β uch that < δ + β =: β <. Thu, we have. Thi complete the proof of the tatement. 3

14 3..3 Lévy area of multi-dimenional fou procee In thi ubection, let B H = B H,, B H,,, B H,d be a d-dimenional fbm with H,, 3 X = X, X,, X d a d-dimenional fou proce, where Xt i = σ e λit db H,i, λ i >, σ R. Then X = X, X,, X d i tationary ee [9]. It covariance function i given by R X, t = diagr, t,, R d, t, where R i, t = EXX i t. i In thi ubection, we will how one etimate for off-diagonal element of Lévy area Xi u d R X j of the multi-dimenional fou proce X. We denote Stratonovich Lévy area of fou proce X a At := X u d R X = Xu i d R X j, i,j=,,,d and A ij t a it component. Before howing the etimate of off-diagonal element, we recall a lemma baed on Wiener chao. We denote H n P a homogeneou Wiener chao of order n and C n P := n j=h j P the Wiener chao or non-homogeneou chao of order n. The lemma below give the hypercontractivity of Wiener chao. Lemma 3.. Refer to, e.g., Lemma 5., [6] Let q N and Z C q P. Then, for p >, E Z E Z p p q + p q E Z. 3.6 Now we illutrate one etimate for off-diagonal element, i.e. following propoition. when i j, we have the Propoition 3.. Let X = X,, X d be a d-dimenional, tationary fou proce with H 3,, and A ijt = Xi u d R X j, i j, be the off-diagonal element of Stratonovich Lévy area of X. Then there exit < β < and an almot urely finite random variable C uch that A ij t A ij Cn β, a for any, t [n, n] and any integer n. Proof. Firt, we rewrite R i, t a and denote R i t = EXX i t, i, t, t R i = EX i u, v Xu,v, i R i = EX i u,txu, i R i = EX i u, v,txu,v. i For the econd moment of the Lévy area, E Xu i d R X j = E = = 4 X i ux i v d R X j d R X j EXuX i vdex i ux j v j u u R i dr j, v v

15 where the integral which appear on the right hand ide above can be viewed a a -dimenional D Young integral ee e.g. Section 6.4 of Friz and Victoir [6]. Then we have u u t, u u t, u u R i dr j = R i dr j + R i dr j v v, v v v u t u + R i dr j + R i dr j, v v v =: I + II + III + IV. For the firt term I, by Young-Lóeve-Towghi inequality ee e.g. Theorem 6.8 of [6], we have Then by Corollary 3.5, we have that where I C R i var;[,t] R j var;[,t] C max{ R i, R j var;[,t] }. var;[,t] I = R i, u, v u dr j C t 4H. 3.8 v For the econd term II, by Young D etimate ee e.g. Theorem 6.8 of [6], we have, u u II = R i dr j, t C R i R j, var;[,t], t var;[,t] R i = up var;[,t] P = up P up P R i l l tl+ R i tl r i t l+ r i t l C H t l+ t l C H t, l and R j, t = up var;[,t] P = up P R j l l tl+, t EX j t l,t l+ X j,t tl R j, t R j. var;[,t] In above etimate, function r i i the covariance r i t = EX i X i +t. Thu, we have II =, u u R i dr j v C t 4H

16 For the third term III, it i the ame with the econd term II line by line. So u III = R i dr j C t 4H. 3., v v For the lat term IV, IV = R i t R j R j t t = r i r j r j t C t. t R j + R j 3. Now combing inequalitie 3.8,3.9,3. and 3., we get E Let < t and, t [n, n], o we have X i u d R X j C t 4H + C t. 3. E X i u d R X j C t. Now we turn to prove the etimate, for arbitrary p, by the hypercontractivity of Wiener chao ee Lemma 3., we further have t p E[ A ij t A ij p ] = E Xu i d R X j 3 p p E p p Xu i d R X j C t ph. Take Ψx = x p and px = x H, the above inequality implie that Define B n := E n+ n+ n n n+ n+ n n Aij t A ij Ψ p t Aij t A ij Ψ p t ddt = ddt C. n+ n+ n n A ij t A ij p t ph ddt. Then for any q >, we get E n= B n n q = n= EB n n q n= C n q <. Thu there exit an almot urely finite random variable R independent of n uch that n= B n n q R, a.. 6

17 So we have B n Rn q, a.. n, q >. 3.3 Apply the Garia-Rodemich-Rumey inequality, and for any n < t n, we get A ij t A ij 8 t Ψ 4Bn pdu 8H u = 8H H /p 4B n p CR p n q p, a.. 4Bn u p u H du for any p > H and < q < p. Thu we complete thi proof. 3. Long time aymptotic of Lévy area Now in thi ubection, we conider the multi-dimenional fou proce which i the olution to tochatic differential equation dx t = ΓX t dt + σdb H t, X =, 3.4 where Γ i a ymmetric, poitive-definite matrix, σ i a contant, and B H = B H,, B H,,, B H,d i a d-dimenional fbm. Our aim in thi ection i to how a long time aymptotic property of Lévy area At = X d R X of fou procee X. That i to how t At = t X d R X, a.. a t goe to infinity. The component of olution proce X are not independent ince the interaction between each other. We firt make an orthogonal tranformation for thi dynamical ytem. Since the drift matrix Γ i ymmetric and poitive-definite, there exit an orthogonal matrix Σ uch that ΣΓΣ T = Λ, 3.5 where Λ = diag{λ,, λ d } and < λ λ d. Define X t := ΣX t, and B H t := ΣB H t, ince Σ i an orthogonal matrix, BH t i till a d- dimenional fbm with Hurt parameter H. Then tochatic differential equation 3.4 become d X t = Λ X t dt + σd B H t. 3.6 Now the fou proce X t ha independent component. We alo have that X d R X = Σ T X d R X Σ. What we hould prove now i that t X d R X, a.. t a t goe to infinity. 7

18 We may ignore the ymbol tilde and ue X, B H to denote X and B H, repectively, for implicity. Now the d-dimenional fou proce X = X, X,, X d ha independent component and atifie Define X i t = σ e λ it db H,i, i =,,, d. 3.7 X i t = σ e λit db H,i, i =,,, d. 3.8 Then {X i t, t } are tationary, ergodic, Gauian procee, ee [9]. 3.. On-diagonal cae Lemma 3.. For the on-diagonal component of Lévy area At = X d R X, we have a t tend to infinity. t A iit = t Proof. Firt, we how that for any α >, X i d R X i, a.., i =,,, d. 3.9 lim t X i t =, a tα A we know that the covariance of the proce X i t i r i t = CovX i +t, X i λ i = C G + t + ot, 3.3 for t mall, where C = σ λ i HG and G i Gamma function. Then the limit 3.3 follow from Theorem 3. of Pickand [3]. Since X i t = X i t e λ it X i and then from 3.3, it follow that lim t t X i d R X i = Xi t, Thu, we conclude thi lemma for on-diagonal cae. 3.. Off-diagonal cae X i d R X i =, a.. Let X = X, X,, X d be the d-dimenional, tationary Gauian proce given by 3.8. It covariance function i given by where R i, t := EX i X i t. R X, t = diagr, t,, R d, t, 8

19 When i j, we have a proof of equation 3. in Propoition 3. that E X i d R X j Ct 4H + Ct. 3.3 When t, we have E X i d R X j Ct 4H Now we define A ij t = Xi d R X j a in ubection 3..3, and Zn ij := n A ij n we firt how that when t = n N dicrete equence, we have lim n n A ijn =, a Propoition 3.3. For the dicrete equence { A n ijn, n } and H,, we have 3 a n goe to infinity. Proof. By the inequality 3.33, we have n A ijn, a E A ij n Cn 4H. Then up E Zn ij C. n According Propoition 5. of [6], we know that Zn ij C P. By Lemma 3., we have belong to the econd Wiener chao up E Zn ij p 3 p p p upe Zn ij p <. n For any ɛ >, by Chebyhev inequality, we have P A ij n > nɛ = P Z ij n > n ɛ n n p ɛ up p n where p >. Then, P A ij n > nɛ C n p ɛ <. p n n The almot ure convergence follow from the Borel-Cantelli lemma. E Z ij n p Now we can conclude thi ubection, that i, to how the limit for arbitrary t rather than at dicrete time N +. Theorem 3.4. Suppoe tochatic proce X t i fou proce which i the olution to tochatic differential equation 3.4 and Γ i ymmetric and poitive-definite. Then t At = t X d R X, a.., a t, where the above integral i in Stratonovich ene. 9

20 The on- Proof. Firt, aume that X i the tationary fou proce a in equation 3.8. diagonal cae i proved in Lemma 3.. For the off-diagonal cae, ince t A ijt t A ijt A ij n + n t n A ijn, 3.36 and etting n = [t], by Propoition 3., we have that the firt term on the right hand ide i controlled by Ct n β Cn β, a.. And the econd term alo tend to zero by Propoition 3.3. Thu we have completed the proof of Theorem 3.4 when fou proce X i tationary. If X i not tationary verion but tart at point at t =, we can alo prove thi aymptotic for their Stratonovich integral. Now let X = X,, X d be the tationary verion a above. Then fou proce Xt i = X i t e λit X i, i =,,, d. So t X i u d R X j = t t X i u d R X j + t e λ iu dx j ux i t λ j e λ ju X i udux j λ j e λ i+λ j u dux i X j, where the lat three integral are Young integral. The firt term on the right hand ide tend to zero almot urely, which ha been proved above. The lat term alo goe to zero almot urely, which can be proved eaily. For the econd e λiu dx j u are two Gauian procee. and third term, we can ee that λ je λju X i udu and By almot the ame argument a the proof of the limit t Xi u d R X j, a., we can alo prove that the econd and third term both converge to zero almot urely. Here we jut give a ketch of proof for the econd term. Define Z t = λ je λju X i udu, and ξ = X j. Firt, we how that ξz n n, a. for integer ubequence. Since n E Z n = E λ j e λju Xudu i = max t r it n n n n where C, C independent of n. Then P n ξz n > ε E ξz n Eξ4 n ε r i u ve λ ju+v dudv e λju+v dudv C e λjn λ C, j + EZ 4 n n ε C n ε, by Borel-Cantelli lemma, we proved that ξz n n, a.. Now we how for any n and any, t [n, n + ], there exit a contant β, and an almot urely finite random variable R uch that Z t Z Rn β, a... Since E Z t Z = E λ j e λju Xudu i = r i u ve λju+v dudv C t, where C i a univeral contant, applying Garia-Rodemich-Rumey inequality a Propoition 3., we get Z t Z Rn β, a.. Then, chooe n = [t], t ξz t t ξ Z t Z n + n t n ξz n R ξ n β + n ξz n, a.. Thu we proved the limit of the econd term. The third term follow likely a above. By taking an orthogonal tranformation for X independent component, we get the ame limit for Stratonovich integral of olution to equation 3.4. Therefore, we conclude thi theorem.

21 4 Pathwie Stable etimator 4. Continuou Rough Path Etimator In thi ection, let X be fou proce, i.e. the olution to the following tochatic differential equation dx t = ΓX t dt + ΣdB H t. 4. We contruct an etimator baed on continuou obervation via rough path theory. We uppoe that the rough path enhancement X,t ω, X,t ω of fou proce X t ω could be continuouly oberved in Itô ene defined in ection. It may leave the uer with the quetion of how to undertand data a a rough path in practice. For thi direction, there are in fact work on how to invere data to rough path. We recommend thoe who may be intereted in thee quetion to look at the literature on rough path analyi, in particular [6]. For the contruction of etimator, we adapt the idea of leat quare etimator of Hu and Nualart [9] who derived thi etimator in one dimenional cae, which i formally taken a the minimizer T γ T := arg min Ẋt γx t dt, 4. γ Θ where Θ i the parameter pace. In multi-dimenional cae, we take formally the etimator a the minimizer Γ t := arg min Γ Θ which lead to the olution where L t = S t = Σ Ẋ ΓΣ X d, t [, T ], 4.3 Γ t = L t S t, 4.4 I X T Q I X d LV, V, 4.5 I X T Q d R X V, 4.6 and pace V = R d d, L t i the invere of L t, Q = ΣΣ T, I X = δjx i k i,j,k=,,d, and M T denote tranpoe of matrix M. The integral S t i taken a Itô rough integral of X defined in ection. We call thi etimator a rough path etimator. When Σ = σi I i identity matrix, σ i a contant, the etimator become Γ T t = X X d X d R X. 4.7 Acctually, we can make a rotation to dynamical ytem 4., i.e. act Σ to X t, then we get the above diagonal cae. So without lo of generality, we can uppoe that Σ = σi. Now we give two example for cae d =,. For one dimenional cae, the rough path etimator i γ t = X d R X X d = X,t + X X,t. 4.8 X d

22 For d =, the tranpoe of the rough path etimator i Γ T t t = X d X X d detl t X X X t d X d X d R X t X d R X where X d R X X d R X. 4.9 detl t X = X d X d X X d, 4. X i d R X j = X ij,t + X i X j,t, i, j =,. 4. A a remark, we mention that here in our paper Xω, Xω, and Γω are pathwie-defined almot urely. 4. Strong Conitency Now we conider the aymptotic behavior of thi rough path etimator Γ t. The olution X to 4. i given by X t = e Γt X + e Γt ΣdB H. 4. Without lo of generality, we uppoe that X =. In the following, we will prove chain rule for our rough integral, and then how the almot ure convergence of our rough path etimator. 4.. Chain Rule Firt, we have the following lemma. Lemma 4.. For H, ], we have 3 X d R X = X X d Γ T + σ Here, the integral can be either Stratonovich or Itô rough integral. X d R B H. 4.3 Proof. We ue the relationhip between almot rough path and rough path, ee Theorem 3.. in [7], to prove thi lemma. To implify notation, we how d = cae, i.e. to prove X d R X = γ X d + σ Firt, by the theory of rough differential equation and.8, we know that X d R B H. 4.4 Z,t fz Z,t + DfZ Z,t, 4.5 Z,t fz fz Z,t, 4.6

23 where the right hand ide are actually almot rough path aociated Z,t, and mean the difference i controlled by ω, t θ with θ >, for all, t. So by 4.5, we have Z,t B,t, H γx t + σb,t, H t +, γ X,u du,. Since o we have Thi implie t t X,u du = X u du X t = o t, Z,t B H,t, γx t + σb H,t, t. X,t γx t + σb H,t. 4.7 Actually, thi above formula could be een from tochatic differential equation 4. directly. Now by 4.6, we have where Hence, Z,t X,u db H u X,t σ B H,t u dbh u BH,udX u BH,udu M M M 3, t u dx u t M = σb H,t γx u dbu H, M = σ M 3 = σ X,u du σ B,udX H u γx u dx u, B H,udu γx t. σb H,t γx u dbu H σb H,t, 4.8 B,udX H u γx u dx u σ B H,udX u, 4.9 B H,udu γx t = o t. 4. Combine 4.7 and 4.9, we further have X,t σ B H,t. Now uing the reult above, we can how the equation 4.4 ince we have LHS X X,t + X,t γx t + σx B H,t + σ B H,t RHS. Thu we have completed the proof of thi lemma. A a corollary, we have Corollary 4.. For H, ], the rough path etimator Γ 3 t ha the following expreion: = Γ T X X d X d R B H,γ. 4. Γ T t 3

24 4.. Almot ure convergence In order to etablih the trong conitency of the rough path etimator Γ t, i.e. our aim now i to prove that t t Γ t Γ, a.. a t, 4. Then by Slutky Theorem and Corollary 4., we can get 4.. X X d C H, a X d R B H,γ, a.., 4.4 Propoition 4.3. Suppoe tochatic proce X t i the fou proce to tochatic differential equation 4. and Γ i ymmetric and poitive-definite, then t X X d C H, a.., where the above integral on the left hand ide i Lebegue integral and the contant matrix C H = σ H x e Γx dx. Proof. Define the proce X t := σ e Γt db H, 4.5 then the proce X i tationary Gauian proce and it i ergodic ee ubection 3.. By the ergodic theorem ee [9], we have Since lim t t X X d = EX X, a X t = X t e Γt X, o that t lim X X d = EX X, a t t For the right hand ide, applying integration by part, we have and EX X = σ E = σ Γ E = σ Γ = σ Γ Γ Thu we have proved thi lemma. e Γ db H e Γ+u B H e Γ db H Bu H dud e Γ+u I + u u dud x e Γx dx, x e Γx dx = H 4 x e Γx dx.

25 Propoition 4.4. Suppoe tochatic proce X t i the fou proce to tochatic differential equation 4. and Γ i ymmetric and poitive-definite, then t X d R B H,γ, a.., where B H,γ i Itô type rough path enhancement of fbm B H t a in ection with H 3, ]. Proof. Applying integration by part, we have X t = σ e Γt db H = σ B H t Γ e Γt B H d. 4.8 By definition of Itô integration and Stratonovich integration with repect to fbm for fou proce ee rough differential equation.8, we have where X d R B H,γ = X d R B H,Str σϕ γ t, 4.9 ϕ γ t = It U γ t, and U γ t = HΓ e Γ u u dud. For the firt term on the right hand ide, it i defined a Stratonovich integral, and ha the following expreion by Lemma 4. σ Now we repreent U γ t a and we have X d R B H,Str = U γ t = It H + HΓ ϕ γ t = H HΓ X d R X + Γ e Γ d HΓt e Γ d, e Γ d + HΓt e Γ d, Since e Γ α d e Γ α d C a t, then lim t t ϕγ t = HΓ X X d. 4.3 e Γ d e Γ d e Γ d, a Then Combing equation 4.9, 4.3 and Theorem 3.4, and Propoition 4.3, for σ X d R B H,γ = X d R X + Γ 5 X X d σ ϕ γ t,

26 we have σ lim t t Thu we conclude thi propoition. X d R B H,γ = ΓC H σ HΓ e Γ d =, a.. A a corollary of Theorem 3.4, now we have the following tatement, in which the integral i in Itô ene. Corollary 4.5. Suppoe tochatic proce X t i the fou proce to tochatic differential equation 4. and Γ i ymmetric and poitive-definite, then t where the above integral i in Itô ene. X d R X C H, a.., 4.3 Proof. A we can ee from the definition of rough integral and Lemma 4. that for any, t. Thu we have X d R X X X,t + X,t X X,t + σ B H,γ,t X X,t + σ B H,Str X d R X =,t ϕ γ,t X d R X σ ϕ γ,t, By Theorem 3.4 and the limit in equation 4.3, we get lim t t X d R X = σ HΓ X d R X σ ϕ γ t e Γ d =: C H, a Beide, by the definition of C H, we alo have the relation between C H and C H a C H = ΓC H. Now we have trong conitency of the rough path etimator Γ t a t tend to infinity. Theorem 4.6. Suppoe Γ i a parametric matrix and it i ymmetric and poitive-definite. Let Γ t be the rough path etimator a 4.7 of Γ for the tochatic differential equation 4.. Then Γ t Γ, a.., a t Proof. Applying Propoition 4.3 and Propoition 4.4, and by Slutky Theorem, we have t t X X d X d R B H, a.. t t a t goe to infinity. Hence, by Corollary 4., the rough path etimator Γ t almot urely converge to Γ. 6

27 Remark 4.7. Suppoe we take the tochatic integral X d R X in the rough path etimator Γ t equation 4.7 a Stratonovich rough integral rather than Itô rough integral a above, we can ee that Γ t, a.., a t, 4.36 by Theorem 3.4 and Propoition 4.3. That i to ay, we cannot ue Stratonovich rough integral to do thi etimation problem. 4.3 Pathwie Stability In thi ubection, we will how that our rough path etimator i pathwie table and robut. Note that Γ T i a functional on the path pace C[, T ], R d, or exactly on the rough path pace Ω p [, T ], R d. For every obervation ample path Xω or rough path enhancement Xω = Xω, Xω, one ha a correponding etimator Γ T Xω or Γ T Xω = Γ T Xω, Xω. In the following, we will ue the rough path notation rather than ample path, ince our continuou rough path etimator depend on Xω = Xω, Xω rather than jut the firt level ample path Xω. A natural quetion about robutne of etimator arie: if two obervation X and X are very cloe in ome ene, e.g. uniform ditance or p-variation ditance etc, doe it give arie to cloe etimation Γ T X Γ T X? In other word, i the etimator Γ T continuou in ome ditance? Actually, the rough path idea give u a good olution to thi problem. A well-known, in rough path pace, rough integration i continuou with repect to p-variation ditance. Now we firt recall the p-variation rough path ditance d p : d p X, Y = max i=, up P X i t l,t l Yt i l,t l p i l i p, 4.37 where X = X, X and Y = Y, Y are two rough path in rough path pace Ω p [, T ], R d, and P i any partition of interval [, T ]. Now we give the continuity of etimator Γ T under p-variation ditance d p. Theorem 4.8. Let X be a fou proce driven by fbm with Hurt parameter H, ] and 3 X, X be the Itô rough path enhancement. Then rough path etimator Γ T : Xω, Xω Γ T Xω, Xω i continuou with repect to p-variation ditance d p for < p < 3. H Proof. The tatement i a corollary of Theorem 5.3. of Lyon and Qian [7]. 5 Rough Path Etimator Baed on High-Frequency Data In previou ection, the etimator we have conidered up to now i baed on continuou obervation. However, in the real world the proce can be only oberved at dicrete time. Thu deriving an etimator baed on dicrete obervation i neceary. Baed on our continuou rough path etimator, we can contruct a dicrete rough path etimator and it till ha very good propertie. We aume that the fou proce X can be enhanced to an Itô rough path X =, X, X a ection and can be oberved at dicrete time {t l = lh, l =,,,, n}, or equivalently we can get the dicrete data {X t,t, X t,t, X t,t, X t,t,, X tn,t n, X tn,t n } in Itô ene a in ection. Here, n i ample ize, h = h n i the obervation frequency, and 7

28 t := nh i the time horizon. We further aume that a the ample ize n tend to infinity, the obervation frequency h = h n and time horizon t = nh. In other word, the data i high-frequency. Beide, we alo hould give more aumption to balance the rate of ample ize n and the frequency h in order to get good etimator below. Now we give the theorem of almot ure convergence for our high-frequency rough path etimator. Theorem 5.. Suppoe the fou proce X which i the olution to tochatic differential equation 3.4 with H 3, ] can be oberved at dicrete time {t l = lh, l =,,,, n} and a ample ize n, n and h atify l= nh, h = h n, nh p, 5. for ome p, +H+β, and < β <. Let +β n n Γ T n = X lh X lh h X lh X lh,l+h + X lh,l+h, 5. where Γ T denote tranpoe of matrix Γ. Then a n. Proof. Let and L n = L nh = nh l= Γ n Γ, a X u X u du, A nh = nh X u d R X, n n X lh X lh h, Ã n = X lh X lh,l+h + X lh,l+h. l= By Propoition 4.3, we know that nh L nh = nh From Corollary 4.5, we have nh A nh = nh In the following, we how that nh nh l= X u X u du C H, a.. a n. 5.4 X u d R X C H, a.. a n. 5.5 L nh nh L n, a.., 5.6 and A nh nh Ãn, a If o, combining 5.4, 5.5, 5.6 and 5.7, we can conclude thi theorem, that i, L n Ãn = Ln L nh + nh nh L nh Ãn A nh + nh nh A nh C H C H = Γ, a.. 8

29 Now we firt how the limit 5.7, we have A nh nh Ãn = nh n X nh u d R X X lh X lh,l+h + X lh,l+h l= = n l+h X u d R X X lh X nh lh,l+h + X lh,l+h l= lh n l+h X u d R X X lh X nh lh,l+h + X lh,l+h. Since X lh,l+h = l+h lh l= lh X lh,u d R X = l+h lh X u d R X X lh X lh,l+h, o we have got A nh nh Ãn =. For the limit 5.6, L nh nh L nh n n = X nh u X u du X lh X lh h l= = n l+h X u X u du X lh X lh h nh l= lh n l+h X u X u du X lh X lh h nh. l= Let F X t = X t X t, and any < t T. Then by Propoition 3.9, we get F X u du F X t CR T T β t +α, α, H. Take = lh, t = l + h, and T = nh, we have L nh L n n CR nh nh β h +α = CR nh n +β h +α+β = CR nh l= lh nh +α+β +β +β. By aumption, there exit a number p, +H+β +β uch that nhp and nh a n. And R nh, a.. So we get L nh L n, a.. We may aume that the component of fou proce X are independent, we hould make an orthogonal tranformation for X. Thu, we have completed the proof of Theorem Numerical Study In thi ection, we give ome example baed on imulation to demontrate our theoretical reult. We imulate ample from one and two dimenional fou procee X to tochatic differential equation. by Euler cheme. In one dimenional cae, there i no need to 9

30 imulate the Lévy area. For two dimenional example, we exploit the trapezoidal cheme For thoe who may be intereted in thi apect, ee e.g. [9] to get ome idea to dicretize the fractional Lévy area in order to get the econd level X of fou rough path X, X. Thu we can get the imulation of ample path firt level procee X by Euler cheme and Lévy area econd level procee X by trapezoidal cheme. Then we ue our theoretical reult to do etimation for the drift parameter. We get one etimation for each ample path and imulate path of fou proce by Monte Carlo iteration. We demontrate that our rough path etimator perform very good. 6. One-dimenional example In thi ubection, we demontrate an example of one dimenional fou proce with Hurt parameter H, ] to tochatic differential equation 3 dx t = X t dt + db H t, X =, t [, T ]. 6. We ue Euler cheme to draw n equiditant ample on a time horizon T with obervation frequency h = T for each ample path Xω. The ample are n X h ω, X h ω,, X nh ω, and through Monte Carlo iteration, we get ample path {Xω j, j =,,, }. See Figure 6. below, they are imulated ample path of fbm and repective fou procee for varying Hurt parameter H. One could ee from Figure 6. that the ample path of fbm and fou procee become rougher and rougher a Hurt parameter H become maller, and locally ample path of fou proce look like fbm who generate it. By our theory, the dicretized rough path etimator γ n in thi model i given by γ n ω = n l= X lhωx lh,l+h ω + X lh,l+h ω n l= X, 6. lhω h where X lh,l+h ω = X l+h ω X lh ω are increment of ample path Xω and the econd level/lévy area X lh,l+h ω = X lh,l+hω ϕ lh,l+h = l+h lh X lh,u ω d R Xω ϕ lh,l+h = X l+hω X lh ω ϕ lh,l+h, where ϕ lh,l+h = ϕl + h ϕlh, ϕt = t Ut, and Ut = e u u dud. In ummary, X lh,l+h ω, X lh,l+h ω l=,,,n are our dicrete obervation for etimation. Note that ince the dimenion d = in thi model, there i no Lévy area to be dicretized. In the real world application uch a the Vaicek interet rate model, the problem left for etimation i how to enhance high-frequency data to Itô rough path. We may refer to [6] a an inpiration for anwering thi problem. 3

31 Figure 6.: Sample Path of fbm and fou. From left to right and from top to bottom, H =.5,.45,.4,.35, repectively. Blue path are fbm while red path are fou. 3

32 We illutrate our imulation reult in Table, where, one can ee the mean and tandard deviation of our dicretized rough path etimator γ n ω for varying Hurt parameter H baed on Monte Carlo iteration of ample path Xω. We take the time horizon T = {, 3, 4}, ample ize n = {,, }, and Hurt parameter H = {.5,.45,.4,.35}. From Table, we can ee that the etimated value are very good. The mean i very cloe to the true parameter value γ =, and the tandard deviation i mall. That mean the etimator are table for each ample path. Table : Mean and tandard deviation of rough path etimator baed on Monte Carlo iteration in dimenion d =, n ample ize, T time horizon, H Hurt parameter, and true parameter value γ =. H =.5 H =.45 H =.4 H =.35 T n Mean Std dev Mean Std dev Mean Std dev Mean Std dev In addition, we can ee even more information about our rough path etimator from Table. Actually, the ample ize n, time horizon T and ampling frequency h = T affect the value of n thee etimator for different Hurt parameter H. One can ee that when ample ize n fixed, a time horizon T = nh become larger, the mean and tandard deviation become maller. When time horizon T fixed, a ample ize n become large, the etimated value change regularly according to T. One may notice that it doe not become better even if T and n become larger and larger. The reaon behind that i T and n are not the only two variable which effect the etimator. Actually, the ampling frequency h alo work. The aumption in Theorem 5. T = nh, h, nh p, a n, 6.3 for ome p, +H+β, i very important when one compute the value of etimator. The +β three limit above mean that the convergence rate of meh ize/ampling frequency hould be not too low or too large. One hould give a proper meh ize/ampling frequency h in order to obtain better value of etimator. Beide, the proper ampling frequency h alo depend on Hurt parameter H. In Table, when T, n, h are fixed, the mean value and tandard deviation both become maller. It i better to ue different ampling frequency h according to Hurt parameter H. Following i a Box-and-Whiker Plot, in which the central red mark of each blue box indicate the median of rough path etimator γ n ω baed on Monte Carlo iteration of ample path Xω, and the bottom and top edge of the box indicate the 5th and 75th percentile, repectively. Beide, the outlier are plotted individually uing the red + ymbol. 3