PROPERTIES OF MAXIMUM LIKELIHOOD ESTIMATES IN DIFFUSION AND FRACTIONAL-BROWNIAN MODELS

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1 eor Imov r. a Maem. Sais. heor. Probabiliy and Mah. Sais. Vip. 68, 3 S 94-9(4)6-3 Aricle elecronically published on May 4, 4 PROPERIES OF MAXIMUM LIKELIHOOD ESIMAES IN DIFFUSION AND FRACIONAL-BROWNIAN MODELS UDC NADIYA RUDOMINO-DUSYAS KA Absrac. A mixed Brownian-fracional-Brownian model is considered. wo esimaes for he shif parameer are consruced and compared. he local asympoic normaliy and asympoic efficiency of he esimaes are esablished for he pure linear Brownian and fracional-brownian models. 1. Esimaes of he shif parameer in a mixed Brownian-fracional-Brownian diffusion model where W and B H are independen Recall ha (B H, F, (F ), P) is he fracional Brownian moion wih Hurs index H ( 1 ;1) if: (A1) B H has saionary incremens; (A) B H =andebh =forall>; (A3) E(B H) = H for all >; (A4) B H is a Gaussian process; (A5) B H has coninuous rajecories. Assume ha B H is defined on a probabiliy space (Ω, F, P) and denoe by (F ) he filraion generaed by B H. Le he diffusion equaion conain sochasic differenials wih respec o he fracional Brownian moion and he Wiener process, dx = θx d + σ 1 X dw + σ X db H, X = = X R,, >, {θ, σ 1,σ R\{. he consrucion of he sochasic differenial wih respec o he fracional Brownian moion is given in [1]. Assume ha he processes W and B H are independen (he case of a special dependence of W and B H is considered in []). Le θ be he parameer o be esimaed. We inroduce wo probabiliy measures Q() and Q() as follows. he probabiliy measure Q() is deermined by he following condiion: d Q() { dp () ψ s dŵs 1 ψs ds for a nonrandom funcion ψ s such ha E ψ s ds < and [ { E exp ψ s dŵs 1 ] ψs ds =1. Mahemaics Subjec Classificaion. Primary 6H1, 6F1. Key words and phrases. Fracional Brownian moion, Wiener process, diffusion model, Girsanov heorem, likelihood raio. 139 c 4 American Mahemaical Sociey License or copyrigh resricions may apply o redisribuion; see hp://

2 14 NADIYA RUDOMINO-DUSYAS KA According o he Girsanov heorem, (1) Ŵ = W ψ s ds is a sandard Wiener process wih respec o he probabiliy measure Q(). Le Q() be anoher probabiliy measure saisfying he relaion d Q() { dp () s H 1/ δ s d W s 1 s H 1 δs ds, where δ s is such ha K(, s) δ s ds <, [,], for a kernel K(, s) =C s 1/ H ( s) 1/ H χ{s (; ). Assume ha δ s admis he following inegral represenaion: () K(, s)ϕ s ds = δ s ds, and le he Wiener process W be given by he equaion he process K(, s) db H s (3) BH := B H = s 1/ H d W s. ϕ s ds is a fracional Brownian moion on [,] wih respec o he measure Q() by an analogue of he Girsanov heorem for fracional Brownian moions ([3, heorem.1]; see also [4]). Excluding he shif θx d, he oal shif is σ 1 ψ s ds + σ ϕ s ds = θ, or (4) σ 1 ψ + σ ϕ = θ. Since W and B H are independen, he final probabiliy measure Q() is he produc of he measures Q() and Q(). hus he final likelihood raio is [{ dq() dp () ψ s dŵs 1 ψs ds { (5) s H 1/ δ s d W s 1 ] s h 1 δs ds { ψ s dŵs + s H 1/ δ s d W s 1 [ψs + sh 1 δs ] ds. Solving equaions (4) and () wih respec o he funcions ψ and δ, respecively, we obain (6) (7) ψ = 1 (θ + σ ϕ ), σ 1 ( δ = K(, s)ϕ s ds. License or copyrigh resricions may apply o redisribuion; see hp://

3 PROPERIES OF MAXIMUM LIKELIHOOD ESIMAES 141 Subsiuing equaliies (6) and (7) ino likelihood raio (5), we ge ha a he poin = { dq( ) dp ( ) 1 ( s (θ + σ ϕ s ) dŵs + s H 1/ K(s, u)ϕ u du d W s σ 1 s (8) 1 [ ( ( 1 s ) ] (θ + σ ϕ s ) + s H 1 K(s, u)ϕ u du ds. σ 1 If follows from (8) ha he maximum likelihood esimae θ 1 of he parameer θ saisfies he equaliy 1 dŵs 1 σ 1 σ1 (σ ϕ s + θ) ds = ha can be rewrien as follows: σ 1 Ŵ + σ ϕ s ds + θ =. his gives us he following esimae of he parameer θ: (9) θ1 = σ 1Ŵ σ ϕ s ds. Now we solve equaion (4) wih respec o he funcion ϕ and subsiue i ino equaion (9): θ 1 = σ 1Ŵ whence ( (1) θ1 = θ + σ 1 Subsiuing (1) ino (1) yields ( θ σ 1ψ s ) ds, ) ψ s ds Ŵ. W (11) θ1 = θ σ 1. I is eviden ha he esimae (11) of he parameer θ 1 is srongly consisen. here is anoher esimae of he parameer θ. Solving (4) we deermine he funcion ϕ. he funcion δ is expressed via ϕ by equaliy (): s (1) ϕ = 1 σ (θ + σ 1 ψ ), (13) ( δ = K(, s)ϕ s ds = θ ( K(, s) ds σ = 1 σ ( = θ σ C ( H) 1 H B σ 1 σ K(, s)(θ + σ 1 ψ s ) ds ( K(, s)ψ s ds ( 3 H, 3 ) H σ 1 σ ( K(, s)ψ s ds. Se B 1 = B ( 3 H, 3 H). Using equaliies (6) and (7) for he likelihood raio (5), aking he logarihms, differeniaing wih respec o θ, and equaing he derivaive o License or copyrigh resricions may apply o redisribuion; see hp://

4 14 NADIYA RUDOMINO-DUSYAS KA zero, we obain ha a he poin = C B 1 ( H) s 1/ H d W s σ ( ) + s [θ H 1 C B 1 ( H) s 4H σ + C B 1 ( H) s 1 H σ ( s K(,u)ψ u du s ] ds =, or s 1/ H d W s + θ C B 1 H + σ 1 K(,s)ψ s ds =. σ σ his implies anoher esimae for he parameer θ: (14) θ = σ s1/ H d W s σ 1 K(,s)ψ s ds C B 1 H. Now we subsiue expression (6) for he funcion ψ ino relaion (14) and obain θ σ = C B 1 H s 1/ H 1 d W s + C B 1 H K(,s)(θ + σ ϕ s ) ds [ σ ] = θ + C B 1 H K(,s)ϕ s ds s 1/ H d W s. Recall ha s1/ H d W s = K(,s) dbh s.hus K(,s)ϕ s ds K(,s) dbs H ( = K(,s)ϕ s ds K(,s) d he second esimae of he parameer θ is given by or (15) θ = θ θ = θ σ C B 1 H B H s σ C B 1 ( H) ) + ϕ s ds K(,s) d B H s, s1/ H d W s H H = K(,s) d B s H. he srong consisency of he esimae θ is also clear. Now we compare he esimaes θ 1 and θ. Firs we compue he variances of he remainder erms in formulas (11) and (15) and compare σ 1 σ 1 and C B 1 ( H) H. Since H ( 1 ;1), i is obvious ha here exiss a number N such ha σ 1 < σ 1 C B 1 ( H) H. License or copyrigh resricions may apply o redisribuion; see hp://

5 PROPERIES OF MAXIMUM LIKELIHOOD ESIMAES 143 for all >N. his means ha he variance of he deviaion of he esimae θ 1 from he rue value is smaller han ha of he corresponding deviaion of he esimae θ from he rue value. I his sense, he esimae θ 1 is beer han θ.. Local asympoic normaliy and asympoic efficiency of he esimae of he shif parameer in a linear Brownian diffusion model Consider a pure linear Brownian model dx = 1 ( ] 1 α θx d + cx dw, X = = X, θ R, [,], α, 1. Pu Θ = (, ), θ Θ. According o Definiion.1 in [5], a family of measures P θ () is locally asympoically normal (LAN) a he poin θ Θas if (16) Z,θ (u) = dp { θ+a(,θ)u() uξ,θ 1 dp θ () u + ζ (u, θ) for some funcion A(, θ) and any number u R, whereξ,θ N(, 1) as wih respec o he measure P θ (), and ζ (u, θ) P,, for all numbers u R. We say in his case ha he LAN propery holds for he family of measures P θ () as a he poin θ. heorem 1. he LAN propery holds for he family of measures P θ () as a any poin θ Θ. Proof. We change he probabiliy measure P θ () forhemeasurep (). hen he shif θx d disappears and we obain X = X + θ ( α X s ds + c X s dw s = X + c X s d W s + θ ) c α s. his change of measure ransforms he Wiener process W +θ/(c α ) ino he new process Ŵ,and ϕ s ds = θ/(c α ), ha is, ϕ = θ/(c α ). Consider he likelihood raio corresponding o his change of measure { dp θ () dp () ϕ s dŵs 1 { ϕ θ s ds c α dŵs 1 θ (c α ) ds { θ c α Ŵ 1 θ (c α ). Now we consider he linear model wih parameer θ shifed by A()u. he likelihood raio for such a change of measure is of he form { P θ+a()u () 1 dp () c α (θ + 1 A()u)Ŵ (c α ) (θ + A()u) and dp θ+a(,θ)u () = dp ( ) 1 θ+a(,θ)u() dpθ () dp θ () dp () dp () { 1 c α (θ + 1 A()u)Ŵ (c α ) (θ + A()u) θ c α Ŵ 1 θ (c α ) { ua() c α Ŵ 1 u A () (c α ) A()uθ (c α ). License or copyrigh resricions may apply o redisribuion; see hp://

6 144 NADIYA RUDOMINO-DUSYAS KA Se A() =c α /.hen dp θ+a(,θ)u () dp θ () { u Ŵ 1 u uθ c α. Since Ŵ/ N(, 1) and uθ /(c α ) as for and α> 1,he above definiion implies he LAN propery for he family P θ () as and a any poin θ Θ. Now we are able o prove he asympoic efficiency of he esimae {θ, >. According o he definiion inroduced in he monograph [5], an esimae {θ, > of a parameer θ is asympoically efficien under he LAN propery for he cos funcion ω(a 1 (, θ)x) ahepoinθ if lim lim δ sup θ θ <δ E Pθ () ω ( A 1 (, θ)(θ θ ) ) = E ω(n(, 1)). Le ω W,whereW is he class of funcions defined on Θ and saisfying he condiions: 1) w(u), w() =, w is a Borel funcion coninuous a zero and no idenically zero; ) w(u) =w( u); 3) he se {u: w(u) <c is convex for any c>. he esimae {θ,> is asympoically efficien for he cos funcion w ( A 1 (, θ)x ) W p, where W p W is he class of funcions of W ha have a polynomial dominan. Consider he maximum likelihood esimae of he parameer θ in a linear Brownian model θ = c α Ŵ = c α ( W + 1 ) c α θ = θ + c α W. o prove he asympoic efficiency of he esimae θ we use heorem 1.3 in [5]. According o his heorem, he esimae θ is asympoically efficien in he sense menioned above if he following condiions hold: (B1) he limi lim A 1 (, θ )A(, θ 1 )=B(θ 1,θ ) exiss, and he convergence is uniform in θ i Θ; (B) ζ (θ) :=A 1 (, θ)( θ θ) N(, 1) uniformly in θ i Θas wih respec o he measure P θ (); (B3) random variables A 1 (, θ)( θ θ) N, N>, are P θ ()-inegrable for any θ Θ uniformly in > (N). Condiions (B1) and (B3) hold in he case under consideraion, since A() = c α does no depend on θ. Now we check condiion (B): ζ (θ) =A 1 (, θ) ( θ θ ) = c α 1 c α W = W N(, 1). hus he esimae θ is asympoically efficien as. License or copyrigh resricions may apply o redisribuion; see hp://

7 PROPERIES OF MAXIMUM LIKELIHOOD ESIMAES Local asympoic normaliy and asympoic efficiency of he esimae of he shif parameer in a linear fracional-brownian diffusion model Consider a pure linear fracional-brownian model dx = 1 α θx d + X db H, X = = X, θ R, [,], α (1 H, 1]. I will be clear from he furher argumen ha i is sufficien o consider he case of ( α 1 H, 1 ) in his model. In his case ϕ = θ/ α.hen δ s ds = K(, s) θ α ds = θ α C δ =(θ/ α )C B 1 1 H ( H). herefore s 1/ H ( s) 1/ H ds = θ = α s1/ H dŵs C B 1 H, θ α C B 1 H, where In oher words, Pu Θ = (, ), θ Θ. Ŵ s = W s θ α C H B 1 3 H s3/ H. θ = θ α s1/ H dw s C B 1 H. heorem. he LAN propery holds for he family P θ () as a any poin θ Θ. Proof. We replace he probabiliy measure P θ () wih he measure P (). As a resul, he shif θx d disappears. he corresponding likelihood raio is given by { dp θ () dp () s H 1/ δ s dŵs 1 s H 1 δs ds { θc B 1 ( H) α s 1/ H dŵ 1 α (θc B 1 ( H)) H H Now we consider he linear model wih parameer θ shifed by A()u. Pu K := C B 1 ( H);. hen P θ+a()u () dp () { (θ + A()u)K α s 1/ H dŵ 1 H ((θ + A()u)K). α H License or copyrigh resricions may apply o redisribuion; see hp://

8 146 NADIYA RUDOMINO-DUSYAS KA he likelihood raio for his model is of he form dp θ+a(,θ)u () = dp ( ) 1 θ+a(,θ)u() dpθ () dp θ () dp () dp () { K (θ + A()u) s 1/ H α dŵ 1 H ((θ + A()u)K) α H { K α A()u θ K α s 1/ H dŵ 1 u A () K α s 1/ H dŵ 1 H (θk) H H K A()uθ H α Se A() := α H/(K 1 H ). hen he likelihood raio akes he form { dp θ+a(,θ)u () u s1/ H dŵs 1 dp θ () 1 H u uθk1 H H α. H Since s1/ H dŵs N(, 1) 1 H H and uθk 1 H α H as, he LAN propery holds for he family P θ () as a any poin θ Θ. H H Now we check he asympoic efficiency of he esimae θ. Consider condiions (B1) (B3). wo of hem, (B1) and (B3), are eviden. o check (B) we use he following relaions: ζ (θ) =A 1 (, θ)( θ θ) = K1 H α α s1/ H dw s H C B 1 H = s1/ H dw s N(, 1). 1 H H herefore, he esimae θ of he parameer θ is asympoically efficien as. Bibliography 1. M. Zähle, Inegraion wih respec o fracional funcions and sochasic calculus. I, Probab. heory Relaed Fields 111 (1998), MR 99j:673. Yu. Mishura and N. Rudomino-Dusyaska, Consisency of drif parameer esimaes in fracional Brownian diffusion models, heory Soch. Processes 7(3) (1), no. 3 4, A. Kukush, Yu. Mishura, and E. Valkeila, Saisical Inference wih Fracional Brownian Moion, Preprin, urku Universiy, L. Decreusefond and A. S. Üsünel, Sochasic analysis on fracional Brownian moion, Poenial Analysis 1 (1998), MR b: I. A. Ibragimov and R. Z. Khas minskiĭ, Saisical Esimaion: Asympoic heory, Nauka, Moscow, 1979; English ransl., Springer-Verlag, Berlin Heidelberg New York, MR 81h:64 Deparmen of Probabiliy heory and Mahemaical Saisics, Faculy of Mechanics and Mahemaics, Kyiv Naional aras Shevchenko Universiy, Academician Glushkov Avenue 6, Kyiv , Ukraine address: nadiya rudomino@homail.com. Received /JUN/ ranslaed by YU. MISHURA License or copyrigh resricions may apply o redisribuion; see hp://