THE STOCHASTIC INTEGRAL WITH RESPECT TO THE SUB-FRACTIONAL BROWNIAN MOTION WITH H >

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1 Joral of Maheacal Scece: Advace ad Alcao Vole 6 Nber Page 9-39 E SOCASIC INEGRAL WI RESPEC O E SUB-FRACIONAL BROWNIAN MOION WI > GUANGJUN SEN ad LIAN YAN 3* Deare of Maheac Ea Cha Uvery of Scece ad echology 3 MeLog Rd. Xh Shagha 37 P. R. Cha e-al: gaghe@yahoo.co.c Deare of Maheac Ah Noral Uvery Ea Beg Rd. Wh 4 P. R. Cha 3 Deare of Maheac Dogha Uvery 999 Norh Re Rd. Sogag Shagha 6 P. R. Cha Maheac Sbec Clafcao: G8. Keyword ad hrae: b-fracoal Browa oo Mallav calcl dvergece egral -varao. he Proec-oored by NSFC (874). h wor ored by Key Naral Scece Proec of College ad Uvere of Ah Provce (No. KJA). *Correodg ahor Receved Ocober 7 Scefc Advace Pblher

2 GUANGJUN SEN ad LIAN YAN Abrac We develo a ochac calcl for he b-fracoal Browa oo wh dex > by g he echqe of he Mallav calcl. We eablh eae L axal eqale for he dvergece egral wh reec o b-fracoal Browa oo. We alo dy he varao of he dvergece egral ad geeralze he rel of b-fracoal Browa oo for h dvergece egral.. Irodco Recely he log-rage deedece roery ha becoe a ora aec of ochac odel varo cefc area cldg hydrology elecocao rblece age roceg ad face. he be ow ad o wdely ed roce ha exhb he lograge deedece roery fracoal Browa oo (fb hor). he fb a able geeralzao of he adard Browa oo b exhb log-rage deedece elf-lary ad ha aoary cree. Soe rvey ad colee lerare cold be fod Bag e al. [3] [9] Mhra [] ad Nalar []. O he oher had ay ahor have rooed o e ore geeral elf-lar Gaa rocee ad rado feld a ochac odel. Sch alcao have raed ay ereg heorecal qeo abo elf-lar Gaa rocee ad feld geeral. herefore oe geeralzao of he fb ha bee rodced. owever cora o he exeve de o fb here ha bee lle yeac vegao o oher elf-lar Gaa rocee. he a reao for h he colexy of deedece rcre for elf-lar Gaa rocee whch do o have aoary cree. A a exeo of Browa oo recely Bodec e al. [4] rodced ad ded a raher ecal cla of elf-lar Gaa rocee whch reerve ay roere of he fb. h roce are fro occao e flcao of brachg arcle ye

3 E SOCASIC INEGRAL WI RESPEC wh Poo al codo. h roce called he b-fracoal Browa oo. he o-called b-fracoal Browa oo (b-fb hor) wh dex ( ) a ea zero Gaa roce S { S } wh S ad he covarace C ( ) E[ S S ] [( ) ] (.) for all. For S cocde wh he Browa oo B. S eher a e-argale or a Marov roce le o he Iô aroach o he corco of a ochac egral wh reec o b-fb o vald. A a Gaa roce oble o corc a ochac calcl of varao wh reec o S (ee for exale Aló e al. []). he b-fb ha roere aalogo o hoe of fb (elf-lary log-rage deedece) ad afe he followg eae: [( ) ] ( ) E ( S S ) [ ] [( ) ] ( ). (.) h Kologorov coy crero le ha b-fb ölder coo of order γ for ay γ <. B cree are o aoary. herefore ee ereg o dy b-fb. Fx > ad e : where a ove eger ad. he we have he followg rel for he varao of [4]): S (ee L S S f > (.3) L S S ρ f (.4)

4 GUANGJUN SEN ad LIAN YAN P S S f < (.5) N where ρ E( ξ ) ξ ~ ( ). More wor for b-fb ca be fod Bodec e al. [5-7] She ad Ya [] dor [4-7] ad Ya ad She [8] [9]. he a of h aer o develo a ochac calcl wh reec o he b-fb S wh dex > by g he echqe of he Mallav calcl. Ule oe revo wor (ee for ace []) we wll o e he egral rereeao of S a a ochac egral wh reec o a Weer roce. Iead of h we wll rely o he rc Mallav calcl wh reec o S. We eablh eae L axal eqale for he defe egral wh reec o b-fb for > where h defe egral defed a he dvergece egral he fraewor of he Mallav calcl ad dy he varao of he dvergece egral whch geeralze he rel (.4) for h dvergece egral. h oe orgazed a follow. I Seco we revew he bac fac o Mallav calcl ha wll be ed order o defe he ochac egral. I Seco 3 we how he exece of he yerc ochac egral. he yerc egral a he l of yerc Rea a hoe have bee doe Chaer 3 of Nalar [] he ee of he Mallav calcl. We derve axal L eqale for he dvergece egral wh reec o b-fb ad rove he varao of he dvergece egral. δ S o a e erval [ ] eqal o ρ d where ρ a coa deedg o.

5 E SOCASIC INEGRAL WI RESPEC 3. Mallav Calcl for Sb-fB Le ( S [ ]) be a b-fb wh < < defed o he colee robably ace ( Ω F P ). I oble o corc a ochac calcl of varao wh reec o he Gaa roce S whch wll be relaed o he Mallav calcl. Soe rvey ad colee lerare cold be fod Alò e al. [] Nalar []. We recall here he bac defo ad rel of h calcl. he crcal grede he caocal lber ace ( alo ad o be rerodcg erel lber ace) aocaed o he b-fb whch defed a he clore of he lear ace E geeraed by he dcaor fco { [ ] [ ]} wh reec o he calar rodc [ ] [ ] C ( ) [( ) ]. he ag [ ] S ca be exeded o a lear oery bewee ad he Gaa ace aocaed wh S. We wll deoe he oery by ϕ S ( ϕ). For < < we deoe by S he e of ooh fcoal of he for F f ( S ( ϕ ) S ( ϕ )) where f ( R ) ad ϕ. he Mallav dervave of a fcoal C b F a above gve by DF f x ( S ( ϕ ) S ( ϕ )) ϕ ad h oeraor ca be exeded o he clore D ( ) of S wh reec o he or

6 4 GUANGJUN SEN ad LIAN YAN F E F E DF E D F where he -h dervave D defed by erao. he dvergece oeraor δ he ado oeraor of D. Cocreely a rado varable L ( Ω ) belog o he doa of he dvergece oeraor δ ( ybol Do ( δ) ) f E DF c F L ( Ω ) for every F S. I h cae δ ( ) gve by he daly relaoh E ( Fδ ( ) ) E DF for ay F D ad we have he followg egrao by ar: F δ ( ) δ( F) DF (.) for ay Do( δ) F D ch ha F L ( Ω ). I follow ha ) E [ δ( ) ] E E D ( D) where ( D he ado of D he lber ace ad rφ ( r) ddr (.) where φ C ( r) ( r) ( )( r ( r) ) r ad for : [ ] ϕ R we have ϕ 4 [ ] ϕ( ) ϕ( ) φ ( ) φ ( ) ddd d <.

7 E SOCASIC INEGRAL WI RESPEC 5 We deoe by he bace of whch defed a he e of earable fco f o [ ] wh f : f () f ( r) φ ( r) ddr <. (.3) We ca how ha he ace ad a Baach ace for he or E ( δ( ) ) E E D (.4) where ϕ : 4 [ ] ϕ( ) ϕ( ) φ ( ) φ ( ) ddd d <. We alo wll e he oao δ ( ) ds o exre he dvergece egral of a adaed roce. For he dvergece egral we have he followg covergece: If { } a eqece of elee Do ( δ) ch ha L ( Ω; ) ad δ ( ) G L ( Ω) he we have Do( δ) ad δ ( ) G. We alo have he followg eae: Lea.. Le > / ( ). he ad v L [ ] for oe coa C >. v / C v/ L ([ ]) (.5) h eae le he clo ([ ]). L Ideed

8 6 GUANGJUN SEN ad LIAN YAN ϕ α v v / / [ ( ) ]dd α v/ v/ dd α v/ r dr v/ r d dr. he ecod facor he above exreo o a llcave coa eqal o he or of he lef ded fracoal egral I v/. Fally ffce o aly he ardy-llewood eqaly α I f q ( ) L C f L ( ) where < α < < < q < afy α q vale α q ad. wh he arclar A lgh exeo of (.5) le ha ϕ v / C ϕ L ([ ]) v/ L ([ ]) for all ϕ ad v/. ece alyg wce he eqaly yeld he followg rel: If v/ belog o he For ay > v / C v/ L ([ ] ). (.6) we deoe by ( ) D deoe he bace of he Sobolev ace D ( ) whoe elee are ch ha a.. D a.. ad E [ ] E[ D ] <.

9 E SOCASIC INEGRAL WI RESPEC 7 We have he clo We deoe by D ( ) D ( ) Do( δ). L he e of rocee ( ) D ch ha L : E L E D ([ ]) L ([ ] ) <. Fro (.5) ad (.6) we oba D ( ) C L ad a a coeqece he ace L Do( δ). By Meyer eqale (ee Nalar [] for exale) f > a roce D ( ) o he doa of he dvergece L ( Ω) ad we have ece f L we have E δ ( ) C ( E E D ). belog E δ ( ) C ( E E D ) C. L ([ ]) L ([ ] ) L (.7) 3. he Sochac Iegral wh Reec o he Sb-fB wh > I he cae of a ordary Browa oo he adaed rocee ([ ] Ω) L belog o he doa of he dvergece oeraor ad o h e he dvergece oeraor cocde wh he Iô ochac egral. Acally he dvergece oeraor cocde wh a exeo of he Iô ochac egral rodced by Sorohod [3]. We ca a whch ee he dvergece oeraor wh reec o a b-fb S ca be erreed a a ochac egral. Noe ha he dvergece oeraor rovde a oery bewee he lber ace aocaed wh he

10 8 GUANGJUN SEN ad LIAN YAN b-fb S ad he Gaa ace ( S ) ad gve re o a oo of ochac egral he ace of deerc fco clded. Le rodce he yerc egral a he l of yerc Rea a hoe have bee doe Chaer 3 of Nalar [] he fraewor of he Mallav calcl. Coder a earable roce { [ ]} ch ha d < aroxag eqece of rocee a... Le defe he ( ) ( d) ( ] () (3.) where ad. Le S ( d)( S S ). (3.) Defo 3.. he yerc egral of a roce wh egrable ah wh reec o he b-fb defed a he l robably of he eqece S a f ex. We deoe h l by ds. he followg heore gve ffce codo for he exece of he yerc egral ad rovde a rereeao of he dvergece oeraor a a ochac egral. heore 3.. Le { [ ]} be a ochac roce he ace D ( ). Soe alo ha a.. D ( ( ) ) dd <. (3.3)

11 E SOCASIC INEGRAL WI RESPEC 9 he he yerc egral ex ad we have ( ) ( ( ) ) dd D ds δ (3.4) where ( ). Proof. he roof of he heore wll be decooed o wo e: Se. We cla ha e (3.5) for oe ove coa. e I fac we have ha ( ) ( ) ( ( ) )dd ( ) dd f where ( ) ( ]( ) ( ] () ( ( ) ). θ σ σ θ σ θ d d f herefore order o how (3.5) ffce o chec ha ( ) ( ( ) ). e f Noce ha ( ( ) ) [( )( )]. S S S S E d d θ σ θ σ θ σ h for ( ] ( ) ( ( ) ) ( ) f

12 GUANGJUN SEN ad LIAN YAN 3 where we e he eae (.) ad for ( ] ( ] wh < we ge ( ) ( ( ) ) f ( ( ) ) [( ) ( ) ( ) ] [ ( ) ( ) ] ( ) ( ). ax ece (3.5) hold wh ( ax e [ ( ) ( ) ] ( ) ( ) ). We ca fd a eqece of e rocee ch ha. he ( ) e ad leg fr ad he we have l a.. ad by doaed covergece heore we ge [ ]. l E

13 E SOCASIC INEGRAL WI RESPEC 3 I a lar way we ca how ha D d D for oe coa > d ad a a coeqece [ ]. l D D E ece he or of he ace ( ). D Se. By (.) we have ( )( ) ( ) ( ) A S S d δ (3.6) where [ ]. d D A By Se ( ) δ coverge o ( ) δ ( ) Ω L a ed o fy. I order o ed he heore ffce o rove ha ( ) A coverge alo rely o ( ( ) ). dd D I fac we ca rewre ( ) ( ) dd g D A where ( ) [ ]() ( ( ) ). σ σ σ d g By doaed covergece heore ffce o rove ha ( ) ( ( ) ) f g (3.7)

14 3 GUANGJUN SEN ad LIAN YAN for oe coa f >. If ( ] he g ( ) ( ( ) ) ( ) [( ) ( ) ] ( )( ). ( ) O he oher had for ( ] ( ] wh < we ge g ( ) ( ( ) ) K λ [ ] [( λ) ( λ) ]. ( ) ( ) ece (3.7) hold wh f ax( λ λ K λ [ ] [( ) ( ) ] ( ) ( ) ). h colee he roof of he heore. Rear 3.. Soe ha { [ ]} be a ochac roce he ace ( ) D ch ha codo (3.3) hold. he for ay [ ] he roce [ ] alo belog o ( ) D ad afe (3.3). herefore by heore 3. we ca defe he defe egral ds [ ]() ds ad he followg eqaly hold:

15 E SOCASIC INEGRAL WI RESPEC 33 ds δ( [ ]) Dr ( r ( r) ) drd. he ecod ad h exreo a roce wh abolely coo ah ha ca be ded by ea of al ehod. herefore order o dedce L eae of ds we ca redce or aaly o he roce δ ( [ ]). I he followg we wll eablh L axal eae for h dvergece roce. We wll ae e of he oao δs δ( [ ]). : heore 3.. Le > { [ ]} be a ochac roce L. he we have E( δs C E d E [ ] ) [ ( D d) dr] where he coa C > deed o. Proof. Le < γ < g he eqaly cγ ( θ) γ r γ θ ( θ r ) d we have r δs cγ ( ( r) ( r ) dr ) δs γ γ c γ γ r ( r) ( ( r ) δs ) dr γ r γ c ( r ) S γq γ δ dr ( ( r) dr )q γ c ( r ) δs dr γ r where we e he ölder eqaly ad he fac γ <.

16 34 GUANGJUN SEN ad LIAN YAN ece by (.7) ad ölder eqaly ag o acco ha < γ we have E [ ] δs cγ E r γ ( r ) δs dr c γ r γ ( ( r ) E d) dr r γ cγ E ( ( r ) Dθ dθd) dr C[ E d E ( Dr d) dr ] where he coa C > deed o. I he la we coder ow he oo of -varao for a coo roce X { X < }. Fx > ad we le where a ove eger ad. Defe V ( X; ) : X X where >. he roce X of -varao o [ ] f V ( X; ) coverge L ( Ω) a ed o fy. We deoe by V ( X; ) he -varao of X o he erval [ ]. I o dffcl o rove he followg rel (ee for exale Gerra ad Nalar [8]): Le X { X } [ ] ad Y { Y } [ ] be ochac rocee defed o ( Ω F P ) ch ha EV ( X; ) ad EV ( Y ; ) are fe. he

17 E SOCASIC INEGRAL WI RESPEC 35 E( V ( X ) V ( Y ; ) ) ; ( E[ V ( X Y ; )]) [( EV ( X; )) ( EV ( Y ; )) ]. (3.8) By alyg (.7) ad (3.8) oe ca oba he followg lea: Lea 3.. Le v L X δs Y v δs for ay [ ]. We have where L E( V ( X; ) V ( Y ; ) ) C v ( v ) C a coa deedg o. L L By Lea 3. eay o ee ha f X δs L for ay [ ] he E( V ( X; ) ) C. (3.9) L Coder he followg ace of rado varable: D b : { F D : F boded ad E( D F d) < }. Le ϕ be he e of boded e rocee F [ ) b where N D for ad < < < F. I obvo ha ϕ dee L. he followg heore dy he varao for he defe egral δs ad geeralze rel (.4) for he dvergece egral.

18 36 GUANGJUN SEN ad LIAN YAN heore 3.3. Le < < ad L X δs for ay [ ]. We have a ed o fy V Ω ( X; ) ρ d L ( ) (3.) N where ρ E( ξ ) ξ ~ ( ). Proof. Sce eqece ϕ dee L for ay L we ay fd a F ϕ [ ) ch ha L a. Se X δs he we have E V ( X; ) ρ d E V ( X; ) V ( X ; ) E V ρ E By Lea 3. we oba ha a ed o fe ( X ; ) ρ d ( ) d. L E V ( X ) V ( X ) C ; ; L ( ). L Alyg he ea vale heore o f ( x) x wh < < ad ölder eqaly we have ha a ed o fe

19 E SOCASIC INEGRAL WI RESPEC 37 ( )d E ρ ( )d E ρ ( ). ρ L L L I order o fh he roof eogh o how ha for ay fxed ( ). ; l ρ d X V E Who lo of geeraly we coder ha for large eogh ( ) ( ). Le. F D b By he eqaly ( ) ( ) F F δ δ DF we have ha ( ) [ ( ) ]ddr r r D S S F X r. : Z Y By he raglar eqaly we have ha ( ) d X V E ; ρ ( ) ( ) ( ) d Y E V Y V X V E ; ; ; ρ. : b a Alyg Lea 3. o a we have ha a ed o fe ( [ ( )]) [( ( )) ( ( )) ]. ; ; ; Y EV X EV Z E V a

20 38 GUANGJUN SEN ad LIAN YAN Sce { Z } [ ] ha boded varao ad hece ha zero varao ad by (3.9) EV ( X ; ) ad EV ( Y ; ) are boded a ed o fe. Ug (.4) we have ha a ed o fe b E V ( Y ; ) ρ d ω Ω I ( F ( ω) ) E[ ( S S ) ρ ] : I where { : [ )}. h colee he roof. Referece [] E. Alò O. Maze ad D. Nalar Sochac calcl wh reec o Gaa rocee A. Probab. 9 () [] E. Alò ad D. Nalar Sochac egrao wh reec o he fracoal Browa oo Sochac ad Sochac Reor 75 (3) 9-5. [3] F. Bag Y. B. Øedal ad. Zhag Sochac Calcl for Fracoal Browa Moo ad Alcao Srger-Verlag 8. [4]. Bodec L. G. Goroza ad A. alarczy Sb-fracoal Browa oo ad relao o occao e Sa. Probab. Le. 69 (4) [5]. Bodec L. G. Goroza ad A. alarczy Fracoal Browa dey roce ad elf-ereco local e of order J. heore. Probab. 69 (4) [6]. Bodec L. G. Goroza ad A. alarczy L heore for occao e flcao of brachg ye : Log-rage deedece Sochac Proce. Al. 6 (6) -8. [7]. Bodec L. G. Goroza ad A. alarczy Soe exeo of fracoal Browa oo ad b-fracoal Browa oo relaed o arcle ye Elec. Co. Probab. (7) 6-7. [8] J. M. E. Gerra ad D. Nalar he varao of he dvergece egral wh reec o he fracoal Browa oo for > ad fracoal Beel rocee Sochac Proce. Al. 5 (5) 9-5. [9] Y. Iegral raforao ad acave calcl for fracoal Browa oo Meor Aer. Mah. Soc. 75(85) (5).

21 E SOCASIC INEGRAL WI RESPEC 39 [] Y. Mhra Sochac calcl for fracoal Browa oo ad relaed rocee Lec. Noe Mah. 99 (8). [] D. Nalar Mallav Calcl ad Relaed oc d edo Srger New Yor 6. [] G. She ad L. Ya Rear o b-fracoal Beel rocee o aear Aca Maheaca Scea (). [3] A. V. Sorohod O a geeralzao of a ochac egral heory Probab. Al. (975) [4] C. dor Soe roere of he b-fracoal Browa oo Sochac 79 (7) [5] C. dor Soe aec of ochac calcl for he b-fracoal Browa oo A. Uv. Bcre Maheaca (8) [6] C. dor Ier rodc ace of egrad aocaed o b-fracoal Browa oo Sa. Probab. Le. 78 (8) -9. [7] C. dor O he Weer egral wh reec o a b-fracoal Browa oo o a erval J. Mah. Aal. Al. 35 (9) [8] L. Ya ad G. She O he collo local e of b-fracoal Browa oo Sa. Probab. Le. 8 () [9] L. Ya ad G. She Iô forla for he b-fracoal Browa oo o aear Co. Soch. Aal. (). g