ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES

Transcription

1 Communicaion on Sochaic Analyi Vol. 5, No Serial Publicaion ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES TERHI KAARAKKA AND PAAVO SALMINEN Abrac. In hi paper we udy Doob ranform of fracional Brownian moion FBM. I i well known ha Doob ranform of andard Brownian moion i idenical in law wih he Ornein-Uhlenbeck diffuion defined a he aionary oluion of he ochaic Langevin equaion where he driving proce i a Brownian moion. I i alo known ha Doob ranform of FBM and he proce obained from he Langevin equaion wih FBM a he driving proce are differen. However, alo he fir one of hee can be decribed a a oluion of a Langevin equaion bu now wih ome oher driving proce han FBM. We are mainly inereed in he properie of hi new driving proce denoed Y 1. We alo udy he oluion of he Langevin equaion wih Y 1 a he driving proce. Moreover, we how ha he covariance of Y 1 grow linearly; hence, in hi repec Y 1 i more like a andard Brownian moion han a FBM. In fac, i i proved ha a properly caled verion of Y 1 converge weakly o Brownian moion. 1. Inroducion I i well known ha he Ornein-Uhlenbeck diffuion U = {U ; } can be conruced a he unique rong oluion of he Langevin SDE du = αu d + db, 1.1 where α > and B = {B : } i a andard Brownian moion iniiaed from. The oluion of 1.1 can be expreed a U = e x α + e α db, 1.2 where x i he random iniial value of U independen of B. Leing B = {B : } be anoher andard Brownian moion iniiaed from and independen of B. Inroduce for R { B,, B := B,. I i eaily een ha ξ := e α d B Received Mahemaic Subjec Claificaion. 6G15, 6H5, 6G18. Key word and phrae. Fracional Brownian moion, fracional Ornein-Uhlenbeck proce, long range dependence, hor range dependence, covariance kernel, weak convergence. 121

2 122 TERHI KAARAKKA AND PAAVO SALMINEN i a normally diribued random variable wih mean and variance 1/2α. Chooing x = ξ in 1.2 we may wrie he proce wih R a U = e α e α d B and hi decribe a aionary oluion of 1.1. There i alo anoher well known conrucion of he Ornein-Uhlenbeck diffuion. Thi i due o Doob [5] and expree he aionary Ornein-Uhlenbeck diffuion U wih ime axi he whole R a a deerminiic ime change of a andard Brownian moion: U = e α B a, R, 1.3 where α > and a := e 2α /2α. The covariance of U i eaily obained from 1.3 E U U = 1 2α e α,. 1.4 In hi noe we udy fracional Ornein-Uhlenbeck procee. Thee are procee conruced a U above bu now he Brownian moion i replaced wih he fracional Brownian moion FBM. I i known ha he proce obained a he oluion of he Langevin SDE wih FBM a he driving proce doe no coincide wih he proce obained a Doob ranform of FBM. In Cheridio e al. [3] i i proved ha he covariance of he former one behave like he covariance of he incremen proce of FBM. In paricular, if he Hur parameer H i bigger han 1/2 he proce i long range dependen. On he oher hand, he covariance of Doob ranform 1 of FBM decay exponenially and, hence, he proce i hor range dependen for all value of H, 1. Our main conribuion in hi paper i o exrac from Doob ranform he driving proce, o udy i properie and ue i in he Langevin SDE o generae new kind of fracional Ornein-Uhlenbeck procee. In he nex ecion we dicu he baic properie of FBM imporan for our purpoe. To make he paper more readable, we alo recall ome reul from [3]. In he main ecion of he paper he new driving proce i conruced and he oluion of he aociaed Langevin SDE i inroduced. The covariance of he driving proce and alo he covariance of he oluion have kernel repreenaion in cae H > 1/2. I i proved hen ha he driving proce and he oluion are hor range dependen. Moreover, i i een ha i i poible o cale he driving proce o ha i converge weakly o a Brownian moion a he caling parameer end o infiniy. 2. Preliminarie 2.1. Fracional Brownian moion. Le Z = {Z : } be a fracional Brownian moion, FBM, wih elf-imilariy or Hur parameer H, 1, ha i, Z i a cenered Gauian proce wih he covariance funcion EZ Z = 1 2H + 2H 2H In [3] hi ranform i called Lamperi ranform ee Lamperi [9].

3 ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 123 Noice ha EZ 2 = and EZ 2 1 = 1, and, hence, in paricular Z =. Uing Kolmogorov coninuiy crierion i can be proved ha Z ha a coninuou verion; herefore, we ake Z o be coninuou. In fac, Z i locally Hölder coninuou of exponen α for all α < H. Fracional Brownian moion i H-elf-imilar in he ene {Z α : } d = {α H Z : } for all α >, 2.2 where d = mean ha he righ hand ide and he lef hand ide are idenical in law. Thi follow from 2.1 becaue he covariance funcion deermine a mean zero Gauian diribuion uniquely. Moreover, from 2.1, for 2 > 1 > 2 > 1 E Z 2 Z 1 Z 2 Z 1 = H 1 1 2H 2 2 2H H, and, conequenly, he incremen of Z are poiively correlaed if H > 1/2, negaively correlaed if H < 1/2. Conider now he incremen proce of Z defined a I Z := {Z n+1 Z n : n =, 1, 2,... }. I i eaily een ha I Z i a aionary econd order ochaic proce and, from 2.3, ρ IZ n := E Z 1 Z n+1 Z n = H2H 1n 21 H + On 2H Nex we recall he following definiion ee Beran [1] p. 6 and 42. Definiion 2.1. Le X = {X n : n =, 1, 2,... } be a aionary econd order ochaic proce wih mean zero and e ρ X n := E X i X i+n, where i i an arbirary non-negaive ineger by aionariy, ρ X n doe no depend on i. Then X i called 1 long range dependen if here exi α, 1 and a conan C > uch ha lim n ρ X n/c n α = 1, 2 hor range dependen if lim k k n= ρ Xn exi. From Definiion 2.1 and formula 2.4 i follow ha he incremen proce I Z of he fracional Brownian moion Z i long range dependen if H > 1/2, hor range dependen if H < 1/ Fracional Ornein-Uhlenbeck procee of he fir kind. We replace now he Brownian moion B in 1.1 wih he fracional Brownian moion Z, and conider he SDE du Z,α = αu Z,α d + dz. 2.5

4 124 TERHI KAARAKKA AND PAAVO SALMINEN Analogouly wih 1.2, he oluion can be expreed a U Z,α x = e x α + e α dz 2.6 wih ome random iniial value x. The ochaic inegral exi pahwie a a Riemann-Sielje inegral ee Cheridio e al. [3] and i hold e αu dz u = e α Z αe αu Z u du. 2.7 Furhermore, we inroduce Ẑ, a wo-ided fracional Brownian moion hrough, and conider ξ := e α dẑ. 2.8 To ee ha ξ i well-defined noice fir ha we may exend 2.7 for negaive value on e αu dẑu = e α Ẑ αe αu Ẑ u du. 2.9 To prove ha he limi of he r.h.. of 2.9 a exi we remark ha { } Z o := 2H Ẑ 1/, > i a cenered { Gauian } proce wih he ame covariance kernel } a FBM. Conequenly, Z o, > i idenical in law wih {Ẑ, > and i hold Conequenly lim + Zo = lim Ẑ = + a.. lim Ẑ = lim 2H + 2H Ẑ 1/ = lim + Zo =. 2.1 Hence he limi of he r.h.. of 2.9 exi and ξ i well-defined. The fac lim Ẑ 2H = can alo be een a he rong law of large number of FBM and proved via he Borel-Canelli lemma. For oher proof ha ξ i well-defined, we refer o Garrido- Aienza e al. [6] and Malowki and Schmalfu [1]. Taking in 2.6 x = ξ we wrie he oluion in he form = e α e α dẑ U Z,α Since he incremen of Z are aionary and he ochaic inegral i a Riemann- Sielje inegral i follow ha he proce U Z,α i aionary. The aionary probabiliy diribuion, i.e., he diribuion of ξ, i normal wih mean and variance ee Cheridio e al. [3] Γ2H + 1 inπh π α 2H + x 1 2H 1 + x 2 dx.

5 ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 125 In cae H = 1/2, he variance equal 1/2α, a i hould. Definiion 2.2. The proce U Z,α given in 2.11 i called he aionary fracional Ornein-Uhlenbeck proce of he fir kind. Nex we recall he aympoic formula for he covariance of U Z,α aken from [3] Theorem 2.3., which i hen applied o derive he range dependence properie of U Z,α. Propoiion 2.3. Le H, , 1] and N = 1, 2,.... Then for fixed R and, EU Z,α U Z,α + = 1 2 N 2n 1 α 2n 2H k 2H 2n + O 2H 2N n=1 k= Propoiion 2.4. The aionary equence {U n Z,α : n = 1, 2,... } and, equivalenly, he proce U Z,α i long range dependen when H > 1/2, and hor range dependen when H < 1/2. Proof. The leading erm of he um in 2.12 i of he order 2H 2. Conequenly, ρ U Z,αn = n 2H 2, n= n= EU Z,α i which, by Definiion 2.1, give he claim. U Z,α i+n 3. Fracional Ornein-Uhlenbeck Procee of he Second Kind 3.1. Definiion and ome baic properie. In hi ecion we derive from Doob ranform of Z a Gauian proce wih aionary incremen. Thi proce i ued a he driving proce in he Langevin SDE. In hi way we conruc a new family of Gauian procee which we call fracional Ornein-Uhlenbeck procee of he econd kind. Thi erminology can be juified by oberving ha in he andard Brownian cae, i.e., H = 1/2, hee procee coincide wih he Ornein-Uhlenbeck diffuion; a alo do he fracional Ornein-Uhlenbeck procee of he fir kind inroduced in Definiion 2.2. Doob ranform of Z i he proce given by n= X D,α := e α Z a, R, 3.1 where α > and a := a, H := H e α/h /α. The covariance of X can be compued from 2.1. Indeed, for > we have EX D,α = 1 2 X D,α 3.2 2H H e α + e α e α 1 e α 2H H. α Since X D,α i a Gauian proce i follow herefrom ha i i aionary. In paricular, uing he elf-imilariy propery of he fracional Brownian moion ee 2.2 i i een ha X D,α i for all normally diribued wih mean and variance H/α 2H.

6 126 TERHI KAARAKKA AND PAAVO SALMINEN Propoiion 3.1. The aionary proce {X D,α : R} i, for all H, 1, hor range dependen. Proof. Formula 3.2 yield for a fixed a and hi implie he reul. EX D,α X D,α = O exp α 1 H/H, 3.3 Conider now he proce Y α defined via Y α := e α dz a, 3.4 where he inegral i a pahwie Riemann-Silje inegral cf. Secion 2.2. I i poible o repreen Y α a Volerra proce w.r. he Brownian moion, by uing e.g. L. Decreuefond and A.S. Üünel [4]. In cae H = 1/2, Y α i, for all α, by Lévy heorem a andard Brownian moion. Uing Y α he proce X D,α can be viewed a he oluion of he equaion dx D,α wih he random iniial value X,α = Z a = Z H α Propoiion 3.2. For all α >, we have = αx D,α d + dy α, 3.5 N, H α 2H. {α H Y α /α : } d = {Y 1 : }. 3.6 Moreover, he proce Y α ha aionary incremen. Proof. Inegraing by par we obain Y α = e α dz a = e α Z a Z a + α e α Z a d 3.7 Uing 2.2 he elf-imilary propery of FBM he claimed ideniy in law 3.6 follow from 3.7. Moreover, he equaliy E Y α 2 Y α 1 Y α 2 Y α 1 = E Y α 2 +h Y α 1 +h Y α 2 +h Y α 1 +h hold for 2 > 1 > 2 > 1 > and h > again by he elf imilariy of FBM and exploiing 3.7. Conequenly, he incremen of Y α are aionary. Inpired by Propoiion 3.2, we conider he Langevin SDE wih Y 1 a he driving proce: du D,γ = γu D,γ d + dy 1, γ >. 3.8 The oluion can be expreed cf a U D,γ = e γ e γ dŷ 1 = e γ e γ 1 dz a, γ >, 3.9 where Ŷ 1 and for he wo ided Y 1 proce and α = 1 in a. To how ha a ochaic inegral erm make ene alo for γ, 1] recall fir ha for all β < H lim Z / β = a.. 3.1

7 ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 127 becaue Z i Hölder coninuou of order β < H. Nex for T < uing parial inegraion T e γ 1u dz au = e γ 1 Z a e γ 1T Z at γ 1 T e γ 1u Z au du, and by 3.1 he righ hand ide ha a well defined limi a T. Since he incremen proce of Y 1 i aionary i follow ha U D,γ i aionary and, herefore, we have well juified he following Definiion 3.3. The proce U D,γ defined in 3.9 or, equivalenly, via he SDE 3.8 i called he fracional Ornein-Uhlenbeck proce of he econd kind. We conclude hi ecion by characerizing he Hölder coninuiy of Y α and U D,γ. The reul hold for more general ochaic inegral wih repec o Z ee Zähle [12], bu he following imple proof in our pecial cae i perhap worhwhile o preen here. Propoiion 3.4. The ample pah of Y α and U D,γ are locally Hölder coninuou of order β < H. Proof. From 3.5 we have Y α = X D,α X D,α + αx D,α d Conequenly, Y α i coninuou and he Hölder coninuiy properie of Y α and X D,α are he ame. Hence, le T > be given and conider for, < T and β > X D,α X D,α β = e α Z a e α Z a Z a Z a β K T a a β + C T, where K T and C T are random conan which do no depend on and. The claim follow now from he fac ha he pah of FBM are locally Hölder coninuou of order β < H. Similarly, for he proce U D,γ we have, e.g., for > U D,γ e γ U D,γ = Y 1 γe γ e γ Y 1 d, and i follow ha alo U D,γ i Hölder coninuou of order β < H Kernel repreenaion of covariance and hor range dependence. We make now he following aumpion valid hroughou he re of he paper 1/2 < H < 1. In hi cae, a i eaily checked, he covariance of he fracional Brownian moion ha for 2 > 1 and 2 > 1 he kernel repreenaion E Z 2 Z 1 Z 2 Z 1 = H2H 1 u v 2H 2 du dv. In he nex propoiion we derive an analogou repreenaion for he proce Y 1. The reul i formulaed for all value of α >.

8 128 TERHI KAARAKKA AND PAAVO SALMINEN Propoiion 3.5. The covariance of Y α wih 1/2 < H < 1 ha he kernel repreenaion E Y α 2 Y α 1 Y α Y α 1 = Cα, H where 2 > 1, 2 > 1, and α 21 H Cα, H := H2H 1. H The kernel 1 1 e α1 Hu v/h du dv, e αu v/h 21 H e α1 Hu v/h r α,h u, v := Cα, H e αu v/h 21 H i ymmeric, i.e., r α,h u, v = r α,h v, u for all u, v R. Proof. Recall he formula ee Gripenberg and Norro [7] Propoiion 2.2 E fdz gdz R R = H2H 1 fg 2H 2 dd, 3.14 where 1/2 < H < 1 and f, g L 2 R L 1 R. Since H Y α H a := e α dz a = H dz, α a he claim follow by a raighforward applicaion of R Remark 3.6. Noice ha he kernel r α,h i in L 2 [, T ] [, T ] if and only if H > 3/4. Conequenly, for Y 1 we have imilar abolue coninuiy properie a for fracional Brownian moion ee Cheridio [2]. Namely, he meaure induced by he proce {B + Y 1 : }, where Y 1 and he Brownian moion B are aumed o be independen, i aboluely coninuou wih repec o he Wiener meaure. For he nex reul, recall from Propoiion 3.2 ha he incremen of Y α are aionary. Corollary 3.7. The incremen of Y α are poiively correlaed. The incremen proce I Y := {Y α n+1 Y n α ; n =, 1,...} i aionary and hor range dependen. Proof. From 3.12 i follow immediaely ha he incremen are poiively correlaed. Of coure, we may alo deduce from 3.12 he aionariy of he incremen of Y α. To how ha I Y i hor range dependen conider n+1 E Y α 1 Y α n+1 Y n α = du dv r α,h u, v = Cα, He α1 Hn/H du n R dv e α1 Hu v/h 1 e αn/h e αu v/h 2H 1.

9 ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 129 The inegral erm ha a poiive finie limi a n. Indeed, Lebegue dominaed convergence heorem yield lim n Conequenly, and, hence, du ρ Y αn := E compleing he proof. lim E N dv e α1 Hu v/h 1 e αn/h e αu v/h 2H 1 = du dv e α1 Hu v/h. Y α 1 Y α n+1 Y n α = O e α1 Hn/H 3.15 Y α N Y α 1 = ρ Y αn < n= Nex we udy he aympoic behaviour of he variance and covariance of Y α. For hi, i i pracical o rewrie he ymmeric kernel r α,h in 3.13 a wih r α,h, = k α,h k α,h x := Cα, H e α1 Hx/H 1 e αx/h 2H Propoiion 3.8. The following formula hold: E Y α Y α 2 = 2 x k α,h x dx, 3.18 E Moreover, and Y α Y α = + α lim EY Y α x k α,h x dx 3.19 x k α,h x dx xk α,h x dx. E Y α 2 = O a, 3.2 = k α,h xdx + Proof. We apply 3.12 o obain 3.18: E Y α Y α 2 = du = 2 = 2 du u x k α,h xdx dv r α,h u, v dv r α,h u, v x k α,h x dx.

10 13 TERHI KAARAKKA AND PAAVO SALMINEN Puing here = and uing k α,h x dx < and x k α,h x dx < yield 3.2. Furhermore, raighforward compuaion produce formula 3.19 from Likewie formula 3.21 i obain fairly eaily. We omi he deail. Remark 3.9. The hor range dependence propery of Y α alo follow from 3.21 ince recall ha Y α = ρ Y αn = lim E Y α N Y α 1 < +. N n= Propoiion 3.1. The covariance of U D,γ ha he kernel repreenaion E U D,γ U D,γ = H2H 1H 2H 2 e γ+ e γ 1+ 1 H u+v du dv. e u/h e v/h 21 H Proof. A in he proof of Propoiion 3.5, we ue alo here formula However, now we need an exended verion due o Pipira and Taqqu [11] aing ha 3.14 hold rue for funcion f and g aifying f g 2H 2 dd < Conider R R U D,γ = e γ e γ 1 dz a 3.23 a = H γ 1H e γ γ 1H dz, where we have made he change of variable = He /H. To check ha condiion 3.22 i valid for f = g = γ 1H 1,a i i enough o conider a a = pr γ 1H p r 2H 2 dpdr = 2a 2γH = 2a 2γH = a2γh γh a ua v γ 1H a u a v 2H 2 dudv u du u γ 1H dv v γ 1H u v 2H 2 du u γ 1H u γ+1h 1 Bea1 + γ 1H, 2H 1 Bea1 + γ 1H, 2H 1 <,

11 ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 131 where Bea and for he Bea funcion, i.e. Beaa, b = x a 1 1 x b 1 dx, a >, b >. To verify he claimed kernel repreenaion i now a raighforward compuaion uing formula Recall from Corollary 3.7 ha he incremen proce of Y 1 i hor range dependen, and ha if Y 1 i ued a he driving proce in he Langevin equaion he oluion i he proce U D,γ. In he nex propoiion we how ha alo U D,γ i hor range dependen. Formula 3.24 can be compared wih he correponding formula 3.3 for X D,α. In fac, 3.3 wih α = 1 i 3.24 wih γ = 1, a i hould. Propoiion The rae of decay of he covariance of U D,γ i exponenial. More preciely, E U D,γ = O exp min{γ, 1 H/H}, a U D,γ In paricular, he aionary proce U D,γ i hor range dependen. Proof. Wihou lo of generaliy, we may ake = and, hence, conider E U D,γ U D,γ = H2H 1H 2H 2 e γ = 1 + 2, where, for ome fixed T >, and Clearly, 1 := H2H 1H 2H 2 e γ T 2 := H2H 1H 2H 2 e γ T 1 = O exp γ For he inegral erm in 2 we have T du dv e γ 1+ 1 H u+v e u/h e v/h 21 H = T e γ 1+ 1 H u+v du dv e u/h e v/h 21 H e γ 1+ 1 H u+v du dv e u/h e v/h 21 H e γ 1+ 1 H u+v du dv e u/h e v/h. 21 H a +. du dv eγ+1 1 H u e γ 1+ 1 H v. 1 e v u/h 21 H For u, v T,, 1 1 e v u/h 21 H 1 e T/H 21 H, and, conequenly, formula 3.24 hold.

12 132 TERHI KAARAKKA AND PAAVO SALMINEN 3.3. Weak convergence of Y 1 o Brownian moion. In Propoiion 3.8 i i proved ha he growh of he variance of Y 1 i aympoically linear a + ee 3.2. Thi ugge ha Y 1, when properly caled, behave aympoically like a andard Brownian moion. We give he precie aemen in he nex propoiion formulaed for arbirary α >. Propoiion For a > define Z a,α := 1 a Y α a,, and le B = {B : } denoe a andard Brownian moion ared from. Then a a + where weakly {Z a,α : } weakly {σb : }, and for weak convergence in he pace of coninuou funcion and σ = σα, H i a non-random quaniy depending only on α and H ee Proof. We how fir ha he finie dimenional diribuion of Z a,α converge o he finie dimenional diribuion of σb. Since Z a,α i a Gauian proce wih mean zero i i enough o verify he convergence of he covariance funcion. From 3.19 in Propoiion 3.8 we have for > E Z a,α Z a,α = 1 a E Y α a Y a α a = 1 a a x k α,h x dx + a x k α,h x dx a a a a xk α,h x dx wih k α,h defined in Leing here a + yield, afer ome imple compuaion, lim E Z a,α Z a,α = 2 k α,h x dx = κα, H, a where κα, H := 2 Cα, H H Bea1 H, 2H 1. α Since EB B = for > we have proved he convergence of finie dimenional diribuion of Z a,α of he finie dimenional diribuion of σb wih σ = σα, H = κα, H To prove ighne, i i enough o verify ee, e.g., Lamperi [8] ha here exi a conan C which migh depend on α and H uch ha for all a > and > 2 := E Z a,α Z a,α C.

13 ON FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 133 We have by formula 3.18 in Propoiion 3.8 = 1 2 a E Y α a Y a α = 2 1 a a a x k α,h x dx C wih, e.g., C = 2 k α,h x dx. Thi complee he proof. Acknowledgemen. We hank Zhan Shi, Eko Valkeila and Marc Yor for dicuion and commen on an early verion of hi paper. Thank alo o he anonymou referee for commen which improved he paper. Reference 1. Beran, J.: Saiic for Long Memory Procee, Chapman & Hall, New York, Cheridio, P.: Repreenaion of Gauian meaure ha are equivalen o Wiener meaure. In Azéma J., Émery M., Ledoux M. and Yor M., edior Séminaire de Probabilié XXXVII number 1832 in Springer Lecure Noe in Mahemaic, 81 89, Berlin, Heidelberg, New York, Cheridio, P., Kawaguchi, H., and Maejima, M.: Fracional Ornein-Uhlenbeck procee, Elecr. J. Prob Decreuefond, L. and Üünel, A.S.: Sochaic analyi of he fracional Brownian moion. Poenial Analyi Doob, J. L.: The Brownian movemen and ochaic equaion, Ann. Mah Garrido-Aienza, M. J., Kloeden, P., and Neuenkirch, A.: Dicreizaion of aionary oluion of ochaic yem driven by fracional Brownian moion, Appl. Mah. Opim Gripenberg, G. and Norro, I.: On he predicion of fracional Brownian moion, J. Appl. Probab Lamperi, J.: On convergence of ochaic procee, Tranacion, Amer. Mah. Soc Lamperi, J.: Semi-able Markov procee, Z. Wahrcheinlichkeiheorie verw. Gebiee , Univeriy of California Pre, Berkeley. 1. Malowki, B. and Schmalfu, B.: Random dynamical yem and aionary oluion of differenial equaion driven by he fracional Brownian moion, Proc. Soc. Anal. and Appl Pipira, V. and Taqqu, M.: Inegraion queion relaed o fracional Brownian moion, Probab. Theory Rela. Field Zähle, M.: Inegraion wih repec o fracal funcion and ochaic calculu, I., Probab. Theory Relaed Field Terhi Kaarakka: Deparmen of Mahemaic, Tampere Univeriy of Technology, FIN-3311 Tampere, Finland addre: erhi.kaarakka@u.fi Paavo Salminen: Mahemaical Deparmen, Åbo Akademi Univeriy, FIN-25 Åbo, Finland addre: phalmin@abo.fi