Journal of Multivariate Analysis. Least squares estimators for discretely observed stochastic processes driven by small Lévy noises

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1 Author' peroal copy Joural of Multivariate Aalyi 6 (23) Cotet lit available at SciVere ScieceDirect Joural of Multivariate Aalyi joural homepage: Leat quare etimator for dicretely oberved tochatic procee drive by mall Lévy oie Hogwei Log a,, Yautaka Shimizu b, Wei Su c a Departmet of Mathematical Sciece, Florida Atlatic Uiverity, Boca Rato, FL , USA b Graduate School of Egieerig Sciece, Oaka Uiverity, Toyoaka, Oaka , Japa c Departmet of Mathematic ad Statitic, Cocordia Uiverity, Motreal, Quebec H3G M8, Caada a r t i c l e i f o a b t r a c t Article hitory: Received 2 May 22 Available olie 25 Jauary 23 AMS 2 ubject claificatio: primary 62F2 62M5 ecodary 6G52 6J75 Keyword: Aymptotic ditributio of LSE Coitecy of LSE Dicrete obervatio Leat quare method Stochatic procee Parameter etimatio Small Lévy oie We tudy the problem of parameter etimatio for dicretely oberved tochatic procee drive by additive mall Lévy oie. We do ot impoe ay momet coditio o the drivig Lévy proce. Uder certai regularity coditio o the drift fuctio, we obtai coitecy ad rate of covergece of the leat quare etimator (LSE) of the drift parameter whe a mall diperio coefficiet ε ad imultaeouly. The aymptotic ditributio of the LSE i our geeral ettig i how to be the covolutio of a ormal ditributio ad a ditributio related to the jump part of the Lévy proce. Moreover, we briefly remark that our methodology ca be eaily exteded to the more geeral cae of emi-martigale oie. 23 Elevier Ic. All right reerved.. Itroductio Let (Ω, F, P) be a baic probability pace equipped with a right cotiuou ad icreaig family of σ -algebra (F t, t ). Let (L t, t ) be a R d -valued Lévy proce, which i give by t t L t = at + σ B t + zñ(d, dz) + zn(d, dz), (.) z z > where a = (a,..., a d ) R d, σ = (σ ij ) d r i a d r real-valued matrix, B t = (B,..., t Br t ) i a r-dimeioal tadard Browia motio, N(d, dz) i a idepedet Poio radom meaure o R + (R d \ {}) with characteritic meaure dtν(dz), ad Ñ(d, dz) = N(d, dz) ν(dz)d i a martigale meaure. Here we aume that ν(dz) i a Lévy meaure o d R d \ {} atifyig R d \{} ( z 2 )ν(dz) < with z = i= z2 i. The tochatic proce X = (X t, t ), tartig from x R d, i defied a the uique trog olutio to the followig tochatic differetial equatio (SDE) dx t = b(x t, θ)dt + εdl t, t [, ]; X = x, (.2) where θ Θ = Θ (the cloure of Θ ) with Θ beig a ope bouded covex ubet of R p, ad b = (b,..., b d ) : R d Θ R d i a kow fuctio. Without lo of geerality, we aume that ε (, ]. The regularity coditio o b will Correpodig author. addre: hlog@fau.edu (H. Log) X/$ ee frot matter 23 Elevier Ic. All right reerved. doi:.6/j.jmva.23..2

2 Author' peroal copy H. Log et al. / Joural of Multivariate Aalyi 6 (23) be provided i Sectio 2. Aume that thi proce i oberved at regularly paced time poit {t k = k/, k =, 2,..., }. The oly ukow quatity i SDE (.2) i the parameter θ. Let θ Θ be the true value of the parameter θ. The purpoe of thi paper i to tudy the leat quare etimator for the true value θ baed o the amplig data (X tk ) with mall diperio ε ad large ample ize. I the cae of diffuio procee drive by Browia motio, a popular method i the maximum likelihood etimator (MLE) baed o the Giraov deity whe the procee ca be oberved cotiuouly (ee Prakaa Rao [3], Lipter ad Shiryaev [9], Kutoyat [6], ad Bihwal [2]). Whe a diffuio proce i oberved oly at dicrete time, i mot cae the traitio deity ad hece the likelihood fuctio of the obervatio i ot explicitly computable. I order to overcome thi difficulty, ome approximate likelihood method have bee propoed by Lo [2], Pedere [27,28], Poule [29], ad Aït-Sahalia []. For a compreheive review o MLE ad other related method, we refer to Søree [37]. The leat quare etimator (LSE) i aymptotically equivalet to the MLE. For the LSE, the covergece i probability wa proved i Dorogovcev [5] ad Le Breto [8], the trog coitecy wa tudied i Kaoga [2], ad the aymptotic ditributio wa tudied i Prakaa Rao [3]. For a more recet compreheive dicuio, we refer to Prakaa Rao [3], Kutoyat [6], Bihwal [2] ad the referece therei. The parametric etimatio problem for diffuio procee with jump baed o dicrete obervatio have bee tudied by Shimizu ad Yohida [35] ad Shimizu [33] via the quai-maximum likelihood. They etablihed coitecy ad aymptotic ormality for the propoed etimator. Moreover, Ogihara ad Yohida [26] howed ome troger reult tha the oe by Shimizu ad Yohida [35], ad alo ivetigated a adaptive Baye-type etimator with it aymptotic propertie. The drivig jump procee coidered i Shimizu ad Yohida [35], Shimizu [33] ad Ogihara ad Yohida [26] iclude a large cla of Lévy procee uch a compoud Poio procee, gamma, ivere Gauia, variace gamma, ormal ivere Gauia or ome geeralized tempered table procee. Mauda [24] dealt with the coitecy ad aymptotic ormality of the TFE (trajectory-fittig etimator) ad LSE whe the drivig proce i a zero-mea adapted proce (icludig Lévy proce) with fiite momet. The parametric etimatio for Lévy-drive Ortei Uhlebeck procee wa alo tudied by Brockwell et al. [3], Spiliopoulo [39], ad Valdivieo et al. [46]. However, the aforemetioed paper were uable to cover a importat cla of drivig Lévy procee, amely α-table Lévy motio with α (, 2). Recetly, Hu ad Log [9,] have tarted the tudy o parameter etimatio for Ortei Uhlebeck procee drive by α-table Lévy motio. They obtaied ome ew aymptotic reult o the propoed TFE ad LSE uder cotiuou or dicrete obervatio, which are differet from the claical cae where aymptotic ditributio are ormal. Fae [6] exteded the reult of Hu ad Log [] to multivariate Ortei Uhlebeck procee drive by α-table Lévy motio. Mauda [25] propoed a elf-weighted leat abolute deviatio etimator for dicretely oberved ergodic Ortei Uhlebeck procee drive by ymmetric Lévy procee. The aymptotic theory of parametric etimatio for diffuio procee with mall white oie baed o cotiuou-time obervatio ha bee well developed (ee, e.g., Kutoyat [4,5], Yohida [48,5], Uchida ad Yohida [44]). There have bee may applicatio of mall oie aymptotic to mathematical fiace, ee for example Yohida [49], Takahahi [4], Kuitomo ad Takahahi [3], Takahahi ad Yohida [4], Uchida ad Yohida [45]. From a practical poit of view i parametric iferece, it i more realitic ad iteretig to coider aymptotic etimatio for diffuio procee with mall oie baed o dicrete obervatio. Subtatial progre ha bee made i thi directio. Geo-Catalot [7] ad Laredo [7] tudied the efficiet etimatio of drift parameter of mall diffuio from dicrete obervatio whe ε ad. Søree [36] ued martigale etimatig fuctio to etablih coitecy ad aymptotic ormality of the etimator of drift ad diffuio coefficiet parameter whe ε ad i fixed. Søree ad Uchida [38] ad Gloter ad Søree [8] ued a cotrat fuctio to tudy the efficiet etimatio for ukow parameter i both drift ad diffuio coefficiet fuctio. Uchida [42,43] ued the martigale etimatig fuctio approach to tudy etimatio of drift parameter for mall diffuio uder weaker coditio. Thu, i the cae of mall diffuio, the aymptotic ditributio of the etimator are ormal uder uitable coditio o ε ad. Log [2] tudied the parameter etimatio problem for dicretely oberved oe-dimeioal Ortei Uhlebeck procee with mall Lévy oie. I that paper, the drift fuctio i liear i both x ad θ (b(x, θ) = θx), the drivig Lévy proce i L t = ab t + bz t, where a ad b are kow cotat, (B t, t ) i the tadard Browia motio ad Z t i a α-table Lévy motio idepedet of (B t, t ). The coitecy ad rate of covergece of the leat quare etimator are etablihed. The aymptotic ditributio of the LSE i how to be the covolutio of a ormal ditributio ad a table ditributio. I a imilar framework, Log [22] dicued the tatitical etimatio of the drift parameter for a cla of SDE with pecial drift fuctio b(x, θ) = θb(x). Ma [23] exteded the reult of Log [2] to the cae whe the drivig oie i a geeral Lévy proce. However, all the drift fuctio dicued i Log [2,22] ad Ma [23] are liear i θ, which retrict the applicability of their model ad reult. I thi paper, we allow the drift fuctio b(x, θ) to be oliear i both x ad θ, ad the drivig oie to be a geeral Lévy proce. We are itereted i etimatig the drift parameter i SDE (.2) baed o dicrete obervatio {X ti } i= whe ε ad. We hall ue the leat quare method to obtai a aymptotically coitet etimator. Coider the followig cotrat fuctio Ψ (θ) = X tk X tk b(x tk, θ) 2 ε 2,

3 Author' peroal copy 424 H. Log et al. / Joural of Multivariate Aalyi 6 (23) where = t k = /. The the LSE ˆθ i defied a ˆθ := arg mi Ψ (θ). Sice miimizig Ψ (θ) i equivalet to miimizig Φ (θ) := ε 2 (Ψ (θ) Ψ (θ )), we may write the LSE a ˆθ = arg mi Φ (θ). We hall ue thi fact later for coveiece of the proof. I the oliear cae, it i geerally very difficult or impoible to obtai a explicit formula for the leat quare etimator ˆθ. However, we ca ue ome ice criteria i tatitical iferece (ee Chapter 5 of va der Vaart [47] ad Shimizu [34] for a more geeral criterio) to etablih the coitecy of the LSE a well a it aymptotic behavior (aymptotic ditributio ad rate of covergece). I thi paper, we coider the aymptotic of the LSE ˆθ with high frequecy ( ) ad mall diperio (ε ). Our goal i to prove that ˆθ θ i probability ad to etablih it rate of covergece ad aymptotic ditributio. We obtai ome ew aymptotic ditributio for the LSE i our geeral ettig, which are the covolutio of ormal ditributio ad a ditributio related to the jump part of the drivig Lévy proce. Some imilar but more geeral reult are alo etablihed whe the drivig Lévy proce i replaced by a geeral emi-martigale. The paper i orgaized a follow. I Sectio 2, we tate our mai reult with ome remark ad example. We etablih the coitecy of the LSE ˆθ ad give it aymptotic ditributio, which i a atural exteio of the claical malldiffuio cae. All the proof are give i Sectio 3. I Sectio 4, we dicu the exteio of mai reult i Sectio 2 to the geeral cae whe the drivig oie i a emi-martigale. Some imulatio tudie are provided i Sectio Mai reult 2.. Notatio ad aumptio Let X = (X t, t ) be the olutio to the uderlyig ordiary differetial equatio (ODE) uder the true value of the drift parameter: dx t = b(x, θ t )dt, X = x. m k z k, where m i z i For a multi-idex m = (m,..., m k ), we defie a derivative operator i z R k a m z := m z := m m i / z i i. Let C k,l (R d Θ; R q ) be the pace of all fuctio f : R d Θ R q which i k ad l time cotiuouly differetiable with repect to x ad θ, repectively. Moreover C k,l (Rd Θ; R q ) i a cla of f C k,l (R d Θ; R q ) atifyig that up θ α β x f (x, θ) C( + x ) λ for uiveral poitive cotat C ad λ, where α = (α,..., α p ) ad β = (β,..., β d ) are multi-idice with p α i= i l ad d β i= i k, repectively. We itroduce the followig et of aumptio. (A) There exit a cotat K > uch that b(x, θ) b(y, θ) K x y ; b(x, θ) K( + x ) for each x, y R d ad θ Θ. (A2) b(, ) C 2,3 (Rd Θ; R d ). (A3) θ θ b(x, θ) t b(x, θ t ) for at leat oe value of t [, ]. (A4) I(θ ) = (I ij (θ )) i,jp i poitive defiite, where I ij (θ) = ( θi b) T (X, θ) θ j b(x, θ)d. It i well-kow that SDE (.2) ha a uique trog olutio uder (A). For coveiece, we hall ue C to deote a geeric cotat whoe value may vary from place to place. For a matrix A, we defie A 2 = tr(aa T ), where A T i the trapoe of A. I particular, σ 2 = d r σ 2 i= ij Aymptotic behavior of LSE The coitecy of our etimator ˆθ i give a follow. Theorem 2.. Uder coditio (A) (A3), we have ˆθ Pθ θ a ε ad.

4 Author' peroal copy H. Log et al. / Joural of Multivariate Aalyi 6 (23) The ext theorem give the aymptotic ditributio of ˆθ. A i eaily ee, our reult iclude the cae of Søree ad Uchida [38] a a pecial cae. Theorem 2.2. Uder coditio (A) (A4), we have ε (ˆθ θ ) P θ I (θ )S(θ ), a ε, ad ε, where S(θ ) := ( θ b) T (X, θ )dl,..., ( θp b) T (X, θ )dl T. (2.) Remark 2.3. Oe of our mai cotributio i that we o loger require ay high-order momet coditio o X a i, e.g., Søree ad Uchida [38] ad other, which make our reult applicable i may practical model. Remark 2.4. I geeral, the limitig ditributio o the right-had ide of (2.) i a covolutio of a ormal ditributio ad a ditributio related to the jump part of the Lévy proce. I particular, if the drivig Lévy proce L i the liear combiatio of tadard Browia motio ad α-table motio, the limitig ditributio become the covolutio of a ormal ditributio ad a table ditributio. Remark 2.5. Whe d = ad b(x, θ) = θx, i.e., SDE (.2) i liear ad drive by a geeral Lévy proce, Theorem 2.2 reduce to Theorem. of Ma [23]. Whe the drivig Lévy proce i a liear combiatio of tadard Browia motio ad α-table motio, Theorem 2.2 wa dicued i Log [2] ad Ma [23]. Example 2.6. We coider a oe-dimeioal tochatic proce i (.2) with drift fuctio b(x, θ) = θ + θ 2 x. We aume that the true value θ = (θ, θ 2 ) of θ = (θ, θ 2 ) belog to Θ = (c, c 2 ) (c 3, c 4 ) R 2 with c < c 2 ad c 3 < c 4. The, X atifie the followig ODE dx t = (θ + θ 2 X t )dt, X = x. (2.2) The explicit olutio i give by X t = e θ 2 t x + θ (eθ 2 t ) θ 2 ˆθ = (ˆθ,, ˆθ,2 ) T of θ i give by ˆθ, = (X X ) ˆθ,2 X tk, ˆθ,2 = (X tk X tk )X tk (X X ) Xt 2 k X tk 2. X tk whe θ 2 ; X t = x + θ t whe θ 2 =. The LSE Note that θ b(x, θ) = ad θ2 b(x, θ) = x. I thi cae, the limitig radom vector i Theorem 2.2 i I (θ )( dl, X dl ) T, where I(θ ) = d X d. X d (X )2 d Example 2.7. We coider a oe-dimeioal tochatic proce i (.2) with drift fuctio b(x, θ) = θ + x 2. We aume that the true value θ of θ belog to Θ = (c, c 2 ) R with < c < c 2 <. The, X atifie the followig ODE dx t = θ + (Xt ) 2 dt, X = x. The explicit olutio i give by X t = (x + oliear equatio X tk X tk =. θ + Xt 2 k 2(x + θ +x 2 )2 e 2t θ θ +x 2 )et. It i eay to verify that the LSE ˆθ of θ i a olutio to the followig

5 Author' peroal copy 426 H. Log et al. / Joural of Multivariate Aalyi 6 (23) Sice it i impoible to get the explicit expreio for ˆθ, we olve the above equatio umerically (e.g. by uig Newto method). Note that θ b(x, θ) =. It i clear that the limitig radom variable i Theorem 2.2 i 2 θ+x2 I (θ ) dl, where I(θ ) = d. I particular, we aume that L 2 θ +(X ) 2 4 θ +(X t = ab t + σ Z t, where B t i the ) 2 tadard Browia motio ad Z t i a tadard α-table Lévy motio idepedet of B t. Let u deote by N a radom variable with the tadard ormal ditributio ad U a radom variable with the tadard α-table ditributio S α (, β, ), where α (, 2) i the idex of tability ad β [, ] i the kewe parameter. By uig the elf-imilarity ad time chage, we ca eaily how that the limitig radom variable i Theorem 2.2 ha ditributio give by α /α ai 2 (θ )N + σ I (θ ) d U. 2 θ + (X )2 Example 2.8. We coider a two-dimeioal tochatic proce i (.2) with drift fuctio b(x, θ) = C + Ax, where C = (c, c 2 ) T, A = (A ij ) i,j2 ad x = (x, x 2 ) T. We aume that the eigevalue of A have poitive real part. We wat to etimate θ = (θ,..., θ 6 ) T = (c, A, A 2, c 2, A 2, A 22 ) T Θ R 6, whoe true value i θ = (c, A, A, 2 c, 2 A, 2 A 22 )T. The X t atifie the followig ODE dx t = (C + A X t )dt, X = x. The explicit olutio i give by X t (ˆθ,i ) i6 i give by ˆθ, ˆθ,2 = Λ ˆθ,3 Y () k Y () k X () = e A t x + t ea (t ) C d. After ome baic calculatio, we fid that the LSE ˆθ = Y () k X (2) ad ˆθ,4 ˆθ,5 ˆθ,6 = Λ Y (2) k Y (2) k X () Y (2) k X (2) where X (i) (i =, 2) are the compoet of X tk, Y (i) k (i =, 2) are the compoet of Y k = X tk X tk, ad X () X (2) Λ = X () X () 2 X () X (2). X (2) X () X (2) X (2) 2 Sice it i eay ad traightforward to compute the partial derivative θi b(x, θ), i 6, ad the limitig radom vector i Theorem 2.2, we omit the detail here. 3. Proof 3.. Proof of Theorem 2. We firt etablih ome prelimiary lemma. I the equel, we hall ue the otatio Y t := X [t]/ for the tochatic proce X defied by (.2), where [t] deote the iteger part of t. Lemma 3.. The equece {Y t } coverge to the determiitic proce {X t } uiformly o compact i probability a ε ad. Proof. Note that X t X t = t (b(x, θ ) b(x, θ ))d + εl t.,

6 Author' peroal copy H. Log et al. / Joural of Multivariate Aalyi 6 (23) By the Lipchitz coditio o b( ) i (A) ad the Cauchy Schwarz iequality, we fid that t 2 X t X t 2 2 (b(x, θ ) b(x, θ ))d + 2ε 2 L t 2 2t t 2K 2 t t By Growall iequality, it follow that X t X t 2 2ε 2 e 2K 2 t 2 up L 2 t ad coequetly b(x, θ ) b(x, θ ) 2 d + 2ε 2 X X 2 d + 2ε 2 up L 2. t up X t X t 2εe K 2 T 2 up L t, tt tt up L 2 t which goe to zero i probability a ε for each T >. Sice [t]/ t a, we coclude that the tatemet hold. Lemma 3.2. Let τ m Proof. Note that X t = x + t = if{t : X t b(x, θ )d + εl t. m or Yt m}. The, τ m a.. uiformly i ad ε a m. By the liear growth coditio o b ad the Cauchy Schwarz iequality, we get t 2 X t 2 2( x + ε L t ) b(x, θ )d 2 t 2 x + ε up L + 2t b(x, θ ) 2 d t 2 t 2 x + ε up L + 2K 2 t ( + X ) 2 d t 2 t 2 x + ε up L + 4K 2 t 2 + 4K 2 t X 2 d. t Growall iequality yield that 2 X t 2 2 x + ε up L + 4K 2 t 2 t ad X t 2 x + ε up L t Thu, it follow that Y t = X [t]/ 2 + 2Kt x + up L t e 4K 2 t 2 e 2K 2 t Kt e 2K 2 t 2, which i almot urely fiite. Therefore the proof i complete. We hall ue x f (x, θ) = ( x f (x, θ),..., xd f (x, θ)) T to deote the gradiet operator of f (x, θ) with repect to x. Lemma 3.3. Let f C, (Rd Θ; R). Aume (A) (A2). The, we have f (X tk, θ) P θ f (X, θ)d a ε ad, uiformly i θ Θ. (3.)

7 Author' peroal copy 428 H. Log et al. / Joural of Multivariate Aalyi 6 (23) Proof. By the differetiability of the fuctio f (x, θ) ad Lemma 3., we fid that up f (X tk, θ) f (X, θ)d = up f (Y, θ)d f (X, θ)d up up C f (Y, θ) f (X, θ) d C( + X ( x f ) T (X up x f (X + u(y + u(y X ), θ) (Y X ), θ) du + Y ) λ Y X d Y λ + up X + up X up Y X X )du d X d Pθ a ε ad. Lemma 3.4. Let f C, (Rd Θ; R). Aume (A) (A2). The, we have that for each i d ad each θ Θ, f (X tk, θ)(l i t k L i ) P θ f (X, θ)dli a ε ad, where L i t = a it + r σ ij B j t + i the i-th compoet of L t. t Proof. Note that f (X tk, θ)(l i t k L i ) = Let Li t = Li t t z i Ñ(d, dz) + z f (Y, θ)dl i. t z i N(d, dz) z > z > z in(d, dz). The, we have the followig decompoitio f (Y, θ)dl i f (X, θ)dli = Similar to the proof of Lemma 3.3, we have (f (Y, θ) f (X, θ))z in(d, dz) z > z > (f (Y, θ) f (X, θ))z in(d, dz) + z > C f (Y, θ) f (X, θ) z i N(d, dz) C( + X z > (f (Y, θ) f (X, θ))d Li. + Y ) λ Y X z i N(d, dz) λ + up X + up X up Y X z i N(d, dz), z >

8 Author' peroal copy H. Log et al. / Joural of Multivariate Aalyi 6 (23) which coverge to zero i probability a ε ad by Lemma 3.. By uig the toppig time τ m, Lemma 3., Markov iequality ad domiated covergece, we fid that for ay give η > ad ome fixed m P + + η (f (Y r η σ 2 ij, θ) f (X, θ)) {τ m E E f (Y f (Y } d Li > η a i η, θ) f (X, θ) 2 {τ m }, θ) f (X, θ) 2 {τ m } E /2 d f (Y which goe to zero a ε ad. The, we have P (f (Y, θ) f (X, θ))d Li > η P(τ < ) + m P, θ) f (X, θ) {τ m } d /2 d z i ν(dz) 2, (3.2) z (f (Y, θ) f (X, θ)) {τ m } d Li > η, which coverge to zero a ε ad by Lemma 3.2 ad (3.2). Thi complete the proof. Lemma 3.5. Let f C, (Rd Θ; R). Aume (A) (A2). The, we have that for i d, f (X tk, θ)(x i t k X i b i (X tk, θ ) ) P θ a ε ad, uiformly i θ Θ, where X i t ad b i are the i-th compoet of X t ad b, repectively. Proof. Note that X i t k = X i + It i eay to ee that tk b i (X, θ )d + ε(l i t k L i ). f (X tk, θ)(x i t k X i b i (X tk, θ ) ) = = + ε tk f (X tk, θ)(b i (X, θ ) b i (X tk, θ ))d f (X tk, θ)(l i t k L i ) f (Y By the give coditio o f ad the Lipchitz coditio o b, we have up f (Y, θ)(b i (X, θ ) b i (Y, θ ))d up f (Y KC KC, θ)(b i (X, θ ) b i (Y, θ ))d + ε, θ) K X Y d ( + Y ) λ ( X X + Y X )d λ + up X t t f (Y, θ)dl i. up X X + up Y X, which coverge to zero i probability a ε ad by Lemma 3.. Next uig the decompoitio of L t, we have up ε f (Y, θ)dl i ε up a i f (Y, θ)d + ε up r f (Y, θ) σ ij db j + ε up f (Y, θ)z i Ñ(d, dz) + ε up f (Y, θ)z i N(d, dz). z z >

9 Author' peroal copy 43 H. Log et al. / Joural of Multivariate Aalyi 6 (23) It i clear that ε up a i f (Y, θ)d ε a i C ε a i C ( + Y ) λ d λ + up X, which coverge to zero i probability a ε ad, ad ε up f (Y, θ)z i N(d, dz) ε up f (Y, θ) z i N(d, dz) z > z > ε C( + Y ) λ z i N(d, dz) εc z > λ + up X which coverge to zero i probability. Note that r P ε up f (Y, θ) σ ij db j > η P(τ < ) + m ε P up z i N(d, dz), z > f (Y, θ) {τ m } r σ ij db j > η. (3.3) Let u i (θ) = ε f (Y, θ) {τ m } r σ ij db j, i d. We wat to prove that u i (θ) i probability a ε ad, uiformly i θ Θ. It uffice to how the poitwie covergece ad the tighte of the equece {u i ( )}. For the poitwie covergece, by the Chebyhev iequality ad Ito iometry, we have P( u i (θ) > η) ε2 η 2 E r 2 f (Y, θ) {τ m } σ ij db j r σ 2 ij ε 2 η 2 E f (Y, θ) 2 {τ m } d r r σ 2 ij σ 2 ij ε 2 η 2 E C 2 ( + Y ) 2λ {τ m } d ε 2 η 2 C 2 ( + m) 2λ, (3.4) which coverge to zero a ε ad with fixed m. For the tighte of {u i ( )}, by uig Theorem 2 i Appedix I of Ibragimov ad Ha mikii [], it i eough to prove the followig two iequalitie E[ u i (θ) 2q ] C, (3.5) E[ u i (θ 2) u i (θ ) 2q ] C θ 2 θ 2q (3.6) for θ, θ, θ 2 Θ, where 2q > p. The proof of (3.5) i very imilar to momet etimate i (3.4) by replacig Ito iometry with the Burkholder Davi Gudy iequality. So we omit the detail here. For (3.6), by uig Taylor formula ad the Burkholder Davi Gudy iequality, we have E[ u i (θ 2) u i (θ ) 2q ] ε 2q C q r ε 2q C q r σ 2 ij q E σ 2 ij q E (f (Y, θ 2 ) f (Y θ 2 θ 2 θ f (Y q, θ )) 2 {τ m } d q, θ + v(θ 2 θ )) 2 {τ m } dvd

10 Author' peroal copy H. Log et al. / Joural of Multivariate Aalyi 6 (23) ε 2q C q r ε 2q C q r σ 2 ij q C 2q θ 2 θ 2q E σ 2 ij q C 2q ( + m) 2λq θ 2 θ 2q. ( + Y q ) 2λ {τ m } d Combiig (3.3) ad the above argumet, we have that ε up f (Y, θ) r σ ijdb j coverge to zero i probability a ε ad. Similarly, we ca prove that ε up f (Y z, θ)z i Ñ(d, dz) coverge to zero i probability a ε ad. Therefore, the proof i complete. Now we are i a poitio to prove Theorem 2.. Proof of Theorem 2.. Note that Φ (θ) = 2 tk, θ) b(x tk, θ )) (b(x T (X tk X tk b(x tk, θ )) + := Φ () (θ) + Φ(2) (θ). b(x tk, θ) b(x tk, θ ) 2 By Lemma 3.5 ad let f (x, θ) = b i (x, θ) b i (x, θ ) ( i d), we have up Φ () (θ) P θ a ε ad. By uig Lemma 3.3 with f (x, θ) = b(x, θ) b(x, θ ) 2, we fid up Φ (θ) (2) F(θ) P θ a ε ad, where F(θ) = b(x, θ) t b(x, θ t ) 2 dt. Thu combiig the previou argumet, we have up Φ (θ) F(θ) P θ a ε ad, ad that (A3) ad the cotiuity of X yield that if F(θ) > F(θ ) =, θ θ >δ Pθ for each δ >. Therefore, by Theorem 5.9 of va der Vaart [47], we have the deired coitecy, i.e., ˆθ θ a ε ad. Thi complete the proof Proof of Theorem 2.2 Note that θ Φ (θ) = 2 ( θ b) T (X tk, θ)(x tk X tk b(x tk, θ) ). Let G (θ) = (G,..., Gp ) T with G i (θ) = ( θi b) T (X tk, θ)(x tk X tk b(x tk, θ) ), i =,..., p, ad let K (θ) = θ G (θ), which i a p p matrix coitig of elemet K(θ) ij = θj G i (θ), i, j p. Moreover, we itroduce the followig fuctio K ij (θ) = ( θj θi b) T (X, θ)(b(x, θ ) b(x, θ))d Iij (θ), i, j p. The we defie the matrix fuctio K(θ) = (K ij (θ)) i,jp. Before provig Theorem 2.2, we prepare ome prelimiary reult. Lemma 3.6. Aume (A) (A2). The, we have that for each i =,..., p ε G i (θ ) P θ ( θi b) T (X, θ )dl a ε, ad ε.

11 Author' peroal copy 432 H. Log et al. / Joural of Multivariate Aalyi 6 (23) Proof. Note that for i p ε G i (θ ) = ε ( θi b) T (X tk, θ )(X tk X tk b(x tk, θ ) ) = ε ( θi b) T (X tk, θ ) := H () (θ ) + H (2) (θ ). tk (b(x, θ ) b(x tk, θ ))d + By uig Lemma 3.4 ad lettig f (x, θ) = θi b j (x, θ) ( i p, j d) with θ = θ, we have H (2) (θ ) = ( θi b) T (Y Pθ, θ )dl ( θi b) T (X, θ )dl ( θi b) T (X tk, θ )(L tk L tk ) a ε ad. It uffice to prove that H (θ () ) coverge to zero i probability. For H (θ () ), we eed ome delicate etimate for the proce X t. For [, t k ], we have X X tk = (b(x u, θ ) b(x tk, θ ))du + b(x tk, θ )( ) + ε(l L tk ). By the Lipchitz coditio o b ad the Cauchy Schwarz iequality, we fid that 2 X X tk 2 2 (b(x u, θ ) b(x tk, θ ))du + 2 b(x tk, θ ) ( ) + ε L L tk 2 2 2K 2 X u X tk 2 du + 2 b(x tk, θ ) + ε up L L tk. t k By Growall iequality, we get 2 X X tk 2 2 b(x tk, θ ) + ε up L L tk e 2K 2 ( ). t k It further follow that up t k X X tk 2 b(x tk, θ ) + ε up L L tk t k e K 2 / 2. (3.7) Thu, by the Lipchitz coditio o b ad (3.7), we get tk H () (θ ) ε θi b(x tk, θ ) (b(x, θ ) b(x tk, θ ))d tk ε θi b(x tk, θ ) b(x, θ ) b(x tk, θ ) d tk ε θi b(x tk, θ ) K X X tk d (ε) K 2Ke K 2 / 2 ε θi b(x tk, θ ) up t k X X tk 2Ke K 2 / 2 θi b(x tk, θ ) b(x tk, θ ) + := H (,) (θ ) + H (,2) (θ ). It i eay to ee that H (,) θi b(x tk, θ ) b(x tk, θ ) CK (θ ) coverge to zero i probability a ε ice λ+ + up X < a.. θi b(x tk, θ ) up t k L L tk

12 Author' peroal copy H. Log et al. / Joural of Multivariate Aalyi 6 (23) (cf. (3.)). By uig the baic fact that up L L tk = o P (), t k we fid that H (,2) (θ ) 2Ke K 2 / 2 C λ + up X up L L tk, t k which coverge to zero i probability a ε ad. Therefore the proof i complete. Lemma 3.7. Aume (A) (A4). The, we have up K (θ) K(θ) P θ a ε ad. Proof. It uffice to prove that for i, j p up K ij (θ) K ij (θ) P θ a ε ad. Note that K ij (θ) = θ j G i (θ) = ( θj θi b) T (X tk, θ)(x tk X tk b(x tk, θ ) ) + := K ij,() ( θj θi b) T (X tk, θ)(b(x tk, θ ) b(x tk, θ)) ( θi b) T (X tk, θ) θj b(x tk, θ) i= (θ) + K ij,(2) (θ). By uig Lemma 3.5 ad lettig f (x, θ) = θj θi b l (x, θ) ( i, j p, l d), we have that up K ij,() (θ) coverge to zero i probability a ε ad. By uig Lemma 3.3 ad lettig f (x, θ) = ( θj θi b) T (x, θ)(b(x, θ ) b(x, θ)) ( θi b) T (x, θ) θj b(x, θ), it follow that up K ij,(2) (θ) K ij (θ) coverge to zero i probability a ε ad. Thu, the proof i complete. Fially we are ready to prove Theorem 2.2. Proof of Theorem 2.2. The proof idea maily follow Uchida [42]. Let B(θ ; ρ) = {θ : θ θ ρ} for ρ >. The, by the coitecy of ˆθ, there exit a equece η a ε ad uch that B(θ ; η ) Θ, ad that P θ [ˆθ B(θ ; η )]. Whe ˆθ B(θ ; η ), it follow by Taylor formula that D S = ε G (ˆθ ) ε G (θ ), where D = K (θ + u(ˆθ θ ))du ad S = ε (ˆθ θ ) ice B(θ ; η ) i a covex ubet of Θ. We have D K (θ ) {ˆθ B(θ ;η )} up K (θ) K (θ ) θ B(θ ;η ) Coequetly, it follow from Lemma 3.7 that Pθ D K(θ ), ε,. up K (θ) K(θ) + up K(θ) K(θ ) + K (θ ) K(θ ). θ B(θ ;η ) θ B(θ ;η ) Note that K(θ) i cotiuou with repect to θ. Sice K(θ ) = I(θ ) i poitive defiite, there exit a poitive cotat δ > uch that if w = K(θ )w > 2δ. For uch a δ >, there exit ε(δ) > ad N(δ) N uch that for ay ε (, ε(δ)), > N(δ), B(θ ; η ) Θ ad K(θ) K(θ ) < δ/2 for θ B(θ ; η ). For uch δ >, let Γ = up K (θ) K(θ ) < δ θ θ <η,ϵ 2, ˆθ B(θ ; η ).

13 Author' peroal copy 434 H. Log et al. / Joural of Multivariate Aalyi 6 (23) The, for ay ε (, ε(δ)) ad > N(δ), we have, o Γ, up (D K(θ ))w up D K(θ + u(ˆθ θ ))du w w = w = + up K(θ + u(ˆθ θ ))du K(θ ) w Thu, o Γ, w = up K (θ) K(θ) + δ θ θ η 2 < δ. if D w if K(θ )w up (D K(θ ))w > 2δ δ = δ >. w = w = w = Hece, lettig D = {D i ivertible, ˆθ B(θ ; η )}, we ee that P θ [D ] P θ [Γ ] a ε ad by Lemma 3.7. Now et U = D D + I p p D c, where I p p i the idetity matrix. The it i eay to ee that Pθ U K(θ ) D K(θ ) D + I p p K(θ ) D c, ice P θ [D ]. Thu, by Lemma 3.6, we obtai that S = U D S D + S D c = U ( ε G (θ )) D + S D c Pθ (I(θ )) ( θ b) T (X, θ )dl,..., a ε, ad ε. Thi complete the proof. ( θp b) T (X, θ )dl T 4. Geeralizatio to emi-martigale oie I thi ectio, we dicu the exteio of our mai reult i Sectio 2 to the geeral cae whe the drivig oie i a emi-martigale. Let Q t = Q + M t + A t be a emi-martigale, where M t i a local martigale ad A t i a fiite variatio proce. The, we ca replace the drivig Lévy proce L t i the SDE (.2) by the emi-martigale Q t to get dx t = b(x t, θ)dt + εdq t, t [, ]; X = x. (4.) All the related iformatio about the LSE of θ dicued i the Itroductio ad Sectio 2 hall be the ame. We are itereted i the coitecy ad aymptotic behavior of the LSE ˆθ uder the geeral model (4.). We tate the ew reult a follow. Theorem 4.. Uder coditio (A) (A3), we have ˆθ Pθ θ a ε ad. Theorem 4.2. Uder coditio (A) (A4), we have ε (ˆθ θ ) P θ I (θ ) S(θ ), a ε, ad ε, where S(θ ) := ( θ b) T (X, θ )dq,..., ( θp b) T (X, θ )dq T.

14 Author' peroal copy H. Log et al. / Joural of Multivariate Aalyi 6 (23) Remark 4.3. Sice the proof of Theorem 4. ad 4.2 are imilar to thoe of Theorem 2. ad 2.2 give i Sectio 3, we metio oly the eceary modificatio. Note that M t i a local martigale, we ca ue the tadard localizatio procedure to make M ad [M, M] beig bouded up to the toppig time {T m }, m =, 2,..., with lim m T m = almot urely. For example, we ca defie T m = if{t > : [M, M] > t m or da > m}. The, we ca modify the defiitio of τ m by τ m = if{t : X t m or Y t m} T m. Lemma 3.2 till hold, i.e. τ m a.. uiformly i ad ε a m. Whe the proof are baed o pathwie argumet, they ca be carried over to the emi-martigale oie cae eaily. Whe the proof are baed o the Markov iequality (or the Chebyhev iequality), Ito iometry ad the Burkholder Davi Gudy iequality (c.f. Theorem 54 ad remark o page 75 i Chapter IV of Protter [32]), we ca apply the modified toppig time τ m to the tochatic itegral with repect to the local martigale M t. Thu all the proof will be till valid i term of the modificatio decribed a above. We omit the detail here. 5. Simulatio Coider a 2-dimeioal model for (.2) with T b(x, θ) = θ + x 2 + x2, θ 2 x 2 2, L t = + x 2 + x2 2 V δ,γ t S α t + B t, (5.) where B i the tadard Browia motio, S α i a tadard ymmetric α-table proce S α (,, ), ad V δ,γ i a variace gamma proce with Lévy deity p V (z) = δ z e γ z, z R, δ, γ >. I thi example, we fid that our LSE of θ, ay ˆθ = (ˆθ,, ˆθ,2 ) atifie X () t k X () ˆθ, + (X () ) 2 + (X (2) ) 2 = ; I the equel, we et value of parameter a ˆθ,2 = X (2) t X (2) k t X (2) k t k +(X () t ) 2 +(X (2) k t ) 2 k (X (), X (2) ) = (, ), (θ, θ 2 ) = (2, ), (δ, γ, α) = (5, 3, 3/2). (X (2) ) 2 +(X () ) 2 +(X (2) ) 2. (5.2) The both X () ad X (2) are ifiite activity jump-procee, but the jump activity of X () i bouded variatio, ad the oe of X (2) i ubouded variatio. A ample path of X = (X () t, X (2) t ) t [,] with ε =.3 i give i Fig.. I each experimet, we geerate a dicrete ample (X tk ) k=,,..., ad compute ˆθ from the ample. Thi procedure i iterated, time, ad the mea ad the tadard deviatio of, ampled etimator are computed i each cae of (ε, ). To optimize the oliear equatio (5.2), we ued lm fuctio i R. O geeratig dicrete ample of Lévy procee, ee e.g., Cot ad Takov [4], Sectio II.6 ad referece therei, or oe ca fid ome radom umber geerator i yuima package of R, which i a package for imulatig SDE with jump; ee For example, we ue rtable to geerate radom ample from α-table ditributio. The reult are how i Table 4. From thoe table, we ca oberve the coitecy reult hold true whe ε. We alo ote that the ize of i ofte le importat i practice for etimatig the drift parameter tha the ize of ε although i eceary i theory. It i ituitively clear becaue the accuracy of etimatig drift highly deped o the termial time T of obervatio i geeral, that i, the larger T become, the more accurately θ i etimated. However, the termial T = i ow fixed. Note that, i the mall oie model, lettig ε correpod to obervig a proce from a macro poit of view, which correpod to the cae T i ome ee. Therefore, icreaig uder fixed ε doe ot improve a bia of etimator, which i improved oly if ε. I geeral, a large ca decreae the tadard error (or tadard deviatio) of etimatio, but the effect eem mall i thi example. Comparig tadard deviatio betwee ˆθ, ad ˆθ,2, the former eem to be etimated more tably tha the latter. Thi i becaue big jump of X () are le frequet tha thoe of X (2). If ε i mall eough, the path of X () i almot imilar to the determiitic curve of (X ) () ice big jump do ot occur o frequetly. However, X (2) ca have more big jump that are ot igorable eve if ε i mall, which make the etimator fluctuatig. To oberve the aymptotic ditributio of ˆθ, we hall compare the above example, ay Model A (o-gauia oie), with a 2-dimeioal proce with the ame drift b a i (5.), but the drivig oie L i 2-dimeioal Browia motio, ay Model

15 Author' peroal copy 436 H. Log et al. / Joural of Multivariate Aalyi 6 (23) X2 X Time Fig.. A ample path of Model (5.) with (θ, θ 2, δ, γ, α) = (2,, 5, 3, 3/2) ad ε =.3. Table Mea (upper) ad tadard deviatio (parethee) of etimate through, experimet i the cae ε =.3 ad (δ, γ, α) = (5, 3, 3/2). ε =.3 = 5 = = 3 True ˆθ, (.877) (.8248) (.7926) ˆθ, (2.8493) (2.8685) (2.7667) Table 2 Mea (upper) ad tadard deviatio (parethee) of etimate through, experimet i the cae ε =. ad (δ, γ, α) = (5, 3, 3/2). ε =. = 5 = = 3 True ˆθ, (.5829) (.5836) (.5833) ˆθ, (.224) (.22) (.97) Table 3 Mea (upper) ad tadard deviatio (parethee) of etimate through, experimet i the cae ε =.5 ad (δ, γ, α) = (5, 3, 3/2). ε =.5 = 5 = = 3 True ˆθ, (.296) (.295) (.293) ˆθ, (.4364) (.326) (.6773) B (Gauia oie). Fig. 2 ad 3 repectively how (ormal) QQ-plot for, iterated ample of ε (ˆθ,i θ i ) (i =, 2) i Model A with (ε, ) = (., 3), ad Fig. 4 ad 5 are thoe for Model B with (ε, ) = (., 3). I Model B, (margial) aymptotic ditributio of ε (ˆθ,i θ i ) (i =, 2) mut theoretically be ormal, which are upported by Fig. 4 ad 5. O the other had, tail of the correpodig ditributio i Model A hould be heavier tha ormal ditributio due

16 Author' peroal copy H. Log et al. / Joural of Multivariate Aalyi 6 (23) Table 4 Mea (upper) ad tadard deviatio (parethee) of etimate through, experimet i the cae ε =. ad (δ, γ, α) = (5, 3, 3/2). ε =. = 5 = = 3 True θˆ, 2.5 (.583).38 (.2626) 2.6 (.583).572 (.454) 2.8 (.578) (.37) 2. θˆ,2. Fig. 2. Normal QQ-plot for, iterated ample of ε (θˆ, θ ) i Model A (o-gauia); (ε, ) = (., 3). Fig. 3. Normal QQ-plot for, iterated ample of ε (θˆ,2 θ2 ) i Model A (o-gauia); (ε, ) = (., 3). to jump activitie, ad we ca oberve thoe fact from Fig. 2 ad 3. We ca alo oberve that the aymptotic ditributio of ε (θˆ,2 θ2 ) i much heavier tha the oe of ε (θˆ, θ ) becaue of the high frequecy of jump i X (2). Thee fact are coitet with the theory.

17 Author' peroal copy 438 H. Log et al. / Joural of Multivariate Aalyi 6 (23) Fig. 4. Normal QQ-plot for, iterated ample of ε (θˆ, θ ) i Model B (Gauia); (ε, ) = (., 3). Fig. 5. Normal QQ-plot for, iterated ample of ε (θˆ,2 θ2 ) i Model B (Gauia); (ε, ) = (., 3). Ackowledgmet The author are grateful to aoymou referee for uggetig to add ectio for emi-martigale oie ad imulatio (Sectio 4 ad 5). Thi reearch wa upported by JSPS KAKENHI Grat Number 24746, Japa Sciece ad Techology Agecy, CREST (the 2d author), ad NSERC Grat Number (the 3rd author). Referece [] Y. Aït-Sahalia, Maximum likelihood etimatio of dicretely ampled diffuio: a cloed-form approximatio approach, Ecoometrica 7 (22) [2] J.P.N. Bihwal, Parameter Etimatio i Stochatic Differetial Equatio, i: Lecture Note i Mathematic, Vol. 923, Spriger-Verlag, Berli, Heidelberg, New York, 28. [3] P.J. Brockwell, R.A. Davi, Y. Yag, Etimatio for o-egative Lévy-drive Ortei Uhlebeck procee, J. Appl. Probab. 44 (27) [4] R. Cot, P. Takov, Fiacial Modellig with Jump Procee, Chapma & Hall/CRC, Boca Rato, FL, 24. [5] A.Ja. Dorogovcev, The coitecy of a etimate of a parameter of a tochatic differetial equatio, Theory Probab. Math. Stat. (976)

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