MULTIPLE WIENER-ITÔ INTEGRALS

Transcription

1 IJRRAS Jury 7.pd MULTIPLE WIENER-ITÔ INTEGRALS Hd Ahd Al & Csh Wg Norhws Norl Uvrsy, Lzhou, 737, P R Ch, Uvrsy o Khrou-Sud Norhws Norl Uvrsy, Lzhou, 737, P R Ch Kywords: Sochsc Igrls, row Moo, Mrgls, Wr- Iô Igrls, Hr Polyol, Hoogous Chos, Orhogol ss.. INTRODUCTION I 938, N. wr roducd polyol d hoogous chos hs sudy o sochsc chcs. H dd polyol chos s sus o ly y ulpl grls wh rspc o row oo. Th polyol choss o dr ordr r o orhogol. O h ohr hd, h hoogous choss whch r dd rs o polyol cho o dr ordrs r orhogol. Howvr, wr dd o drcly d hoogous chos s grls. I 95, K.ô roducd w ulpl grls h ur ou o xcly hoogous chos. Th w grls r owdys rrrd o s ulpl Wr- ô grls. Thy r rld o h sochsc grl h K. ô roducd 944. W wll ollow h orgl d o K- ô o d ulpl Wr- ô grls. Th o hs ppr s how w c d ulpl Wr- ô grl or drsc uco. WIENER IDEA Th procss o d h grl cosss o wo sps. Th rs sp s o d h grl or sp ucos. Th scod sp s o pprox L, uco y sp ucos d h l o corrspodg grls. Suppos s sp uco gv y, whr,,, d s s,, s, s, [, [ s, s r pros o [, ]. Th d h doul wr grl o y W,, s s Cosdr spl xpl [,] [,] ovously w hv. w Th doul wr grl hs h spro propry, ly g w g. ITô S IDEA Thr r lso wo sps dg doul wr- ô grl. Th rs sp s o d h grl or odgol sp ucos. Th scod sp s o pprox L, uco y o-dgol sp ucos d h l o h corrspodg grls., pro o To ov h oo o o-dgol sp uco d s cssy l h rvl [, ]. I w h pro o h u squr [,,, [, [, [,.3, Th w g h ollowg R su or h grd 9

2 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls =, Whch s h vlu o h doul wr grl quo.. Hr s h crucl d o K. ô h ppr [8] rov h dgol squrs ro [, quo.3, [, \ [, [, [,.4 Th us h rg squrs h rgh-hd sd o hs quo o or h su ro h crs o h row oo Ovously w hv s s,.5 Hc l s L.6 Th l s dd o h doul wr- ô grl o d.7 Whch s dr ro h vlu quo. L us x or closly h su quo.5 s h doul grl o h ollowg o-dgol sp uco.8 Osrv h s Hc h uco d or ulpl wr ô grl. [, [,, dds s pproxd y squc. DOULE WIENER- ITô INTEGRALS Th oc hs sco s o d h doul wr- ô grl.9,, L, o o-dgol sp ucos h hs s crucl Ths wll do wo sps. Howvr hr s h w crucl oo o o-dgol sp ucos od h d o h prvous sco L D,, ; s sus o, o h or [, [ s, Sp o-dgol sp ucos Do. A o-dgol sp uco o h squr, s dd o uco o h or do h dgol o h squr,. y rcgl hs ppr w wll. [, [, 3

3 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls No h o-dgol sp uco vshs o h dgol D. Hc h uco, s o o-dgol sp uco. I A [, [ s, s rcgl dso ro h dgol D h A c wr h or o. y g h s,, s, s s h pros o [, ]. Hc A s o-dgol sp uco. Mor grlly suppos A, A, A r rcgls dso ro h dgol D h h uco A s o-dgol sp uco or y,, o R hs c pls h d g r odgol sp ucos, h +g s o-dgol sp uco or y R,. Hc h s o odgol sp uco s vcor spc. For o-dgol sp uco gv y quo. d I. No h h rprso o o-dgol sp uco y quo. s o uqu, u s sly s h I s uquly. Th syrzo ˆ, o uco, s dd y ˆ, =, s, ovously ˆ s syrc uco. I s syrc uco, h ˆ. I grl ˆ h src Iquly c hpp. For xpl, or h uco, o, hv ˆ, d, dds, 3 7 ˆ ˆ, dds 4 Noc h h syrzo opro s lr,. gˆ ˆ gˆ. [, [ s, s, w s o-dgol sp uco, h ˆ s lso o-dgol sp uco. L. l o-dgol sp uco h I ˆ I. Proo Sc I d h syrzo opro r lr sucs o prov h l or h cs Wh, [ s, h syrzo ˆ o s gv y [ s ˆ [, [ s, s [ s, s [, Hc y h do o I quo., s I s I ˆ I I ˆ d h l s provd = Thr or, E I d L.3 I s o-dgol sp uco h E I = ˆ, dds.3 Proo Suppos s rprsd y quo. h I s gv y quo.. Sc h rvls 3

4 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls [, d, r dso wh h xpco o ch r h suo o quo. s [ zro hc E I To prov quo.3, rs ssu h s syrc. I hs cs. h I Thror, w hv L E I 4pqE p q pq or ll. For covc l.4 xd. y osrvg h poso o rvls w c sly s h ollowg plcos: p E q p q p p q q E p q Hc or xd, h suo ovr p q quo.4 rducs o oly o r gv y p d q hror E I E 4 4 = = =, dds. Flly or y o-dgol sp uco w hv I I ˆ y l.. Hc I E I ˆ E ˆ, dds Whch provs quo.3 Sp Approxo y o-dgol sp ucos. Rcll h dos h, dds y l.3 I I o, L -or o uco dd o h squr, E ˆ or y o-dgol sp uco. u ˆ,... Hc E or ll o-dgol sp ucos. Ths quly shows h w c xd h ppg I L provdd h ch uco, ucos. Suppos s uco, D do h s o pos, L. l ro h dgol D. For gv D, dds D C, \ D O h ohr hd, l L c pproxd y squc o o-dgol sp, w c choos.5 sll ough such h d cosdr h rsrco o o C D hvg dsc. A c ro sur hory sys 3

5 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls h hr xss uco o h or A C D W c su up h grls quos.5 d.6 o g,, dds No h h uco vshs o h s wh rcgls A D or ll such h C,, dds.6 D. Hc h uco s o-dgol sp uco L.4 L uco L, h hr xss squc h Do.5 L L, h l I,, dds o o-dgol sp ucos such.7 l I L.8 Is clld h doul wr-ô grl o. W wll lso us Wr-Iô grl I o. I s sly s h Thor.6 l, L, I s wll-dd. h w hv I I ˆ. Hr ˆ s h syrzo o E I ˆ E. Hr s h or o L, 3 I Thor.7 l, L, h ˆ s h syrzo o. Rr.8 Th r grl Hc E x d, o do h doul, ˆ,.9 X, ˆ, ˆ s wr grl. Th E x ds d ˆ Ths show h h sochsc procss X logs o L d, ˆ, d h grl X ds s ô grl. Proo: rs cosdr h cs [, [s, s wh[, [ s,. Th syrzo o s gv y ˆ [, [s, [, [, s s s W y ssu h s s or h ohr cs s s. Jus rchg [ s, wh [,. y h do o I sp O h ohr hd, w hv s, s. 33

6 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls s ˆ, s. Hc quo.9 ollows ro quos. d. Nx y h lry o h ppg I d h syrzo opro, quo.9 s lso vld or y odgol sp uco. Flly us h pproxo o xd quo.9 o ucos L, 3. HERMITE POLYNOMIALS Th Hr polyol wll usd o rprs h ulpl wr.l v h Guss sur wh o d vrc.. dv x x dx Do 3. Th Hr plyol x, H o dgr H x H x x H x x,,,,, Ad or grlly ro h rcurrc rlo H x, xh x, ph x, H x, x, N I prculr w hv N d prr ρ> s dd wh, x, x,, x, o ools h rl Hlr spc L v Cosdr h squc. Apply h Gr-schd orhogolzo procdur o hs squc ordr o crsg powr o o orhogol polyols p x, p x,, p x, h Hlr spc L v, whr p x d p x s polyol o dgr wh ldg Coc x T uco, x. Th xpco o,. wh rspc o h sur v s gv y x ρ x Ev,. dx = ρ 3. ρ Th ulplcv rorlzd, x o, x s dd y, x x p, x = Ev,. W c xpd h uco, x s powr srs. Thor 3. L v h Guss sur wh o d vrc ρ. Th h Hr polyols H x,,, r orhogol L v.morovr w hv Proo: For y sr, us quo y quo 3. x x s dv x H x =,! = s H x, dv x!, 3. o g Thror, upo grg oh sds o quo 3.3 w o x s s, s!! = H x, H x,

7 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls s s H x, H x, dv x!!, Sc h l-hd sd s uco o h produc s, h coc o y ly 3.4 s h rgh-hd sd us or H x, H x, dv x 3.5 Hc h Hr polyols r orhogol L v. Ad h quo 3.4 cos H x dv x s!, u w lso hv h powr srs xpso s! s Ovously, w g quo 3. uo coprg h coc o s h ls wo powr srs. Thor 3.3 L v h Guss sur wh d vrc h vry uco v uqu srs xpso h cocs r gv y x H x, dv x,! H x, x! L hs 3.6 Morovr w hv 4. HOMOGENEOUS CHAOS L v h Guss sur o R wh d vrc. y hor 3.3 vry uco h Hlr L hs uqu xpso y h Hr polyols H x,,. [, vshg d l h wr C, s dsol logu o h o-dsol proly spc v L C do h ch spc o rl-vlud couous ucos o ] sur o C. h wr spc spc R,v L 4. l ς,,p proly spc d l X L. Th E Xς covrgs o X Rr 4. Acully s lso ru h oudd rgl h Proo l ς h lro such h ς L s U E Xς covrgs o X los surly. I grl ς. Suppos X s X covrgs los surly o so rdo vrl. U ς ς hr xss S A, R A Uς, such h gv. Sc s h or o L. ς h For splcy, l X EX S L - X 4. 35

8 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls X X X S S E S E S X X S S ES ES X No h ES X ς E S X o g ς ς ς ς 4. ς y h codol Js s quly. Th h xpco ς 4.3 E S X E S X S X A Uς or,,, Morovr, osrv h,,, hs pls h A ς or ll Hc hr xss N such h ς, 4.4 E S S N y pug quos 4., 4.3 d 4.4 o quo 4. w dly s h N. Hc X covrgs o X L s Now w rur o h row oo h wr grl o L, polyol chos o ordr K. o h proly spc,, P. A produc I I X X or ll F. L I I wh,,, L, s clld L R d or, d o h - closur o h lr spc spd y cos uco L d polyol chos o dgr h w hv h clusos Thor 4.3 Th uo L 4.5 L s ds Proo L orhoorl ss or L, d l I, I,, I h ς s lro d w hv o, h or y xd I I I ς h -ld grd y ς F. L L orhogol U E,,, Osrv h I I I s ς -surl. E I I I E I I I E ς Hc Thror, or ll grs,,,, E I I I E ς = No h h rdo vrls, I,, I Morovr For so surl uco I r dpd wh h s sdrd orl dsruo. ς 4.6,,, E I I I o x x x x, x,, x d x R s h sdrd Guss sur o R hus or ll,, H x, H x, H x,,, x x d x R R. Ths quly pls h or y grs,, 36

9 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls u ollows ro hor 3.3 h h collco H x, H x, H x,,,,, Is orhogol ss or L R, los vrywhr wh rspc o. Th y quo 4.6 w hv E ς, los surly or y. O ohr hd y l 4. provs h ssro h h uo E ς covrgs o L s. Hc s ds L, Hc los surly. Ths Do 4.4 l ogv gr. Th ls o h Hlr spc r clld hoogous choss o ordr. Th spc,, r ll -dsol. Th hoogous choss o ordr r Guss rdo vrls. Thor 4.5 h spc L s h orhogol drc su o h spcs o hoogous choss o ordr, ly L = Ech uco Morovr. s h or o L Thor 4.6 L ucos L, H L hs uqu hoogous chos xpso p do h orhogol proco o d L oo,,, r ozro orhogol, r ogv grs, h p I I, I ; H I ;. I prculr w hv For y ozro uco L, I H I ; = p 4.9 Rr 4.7 I ollows ro h hor h I ; H I ; H I ; wh 5. ORTHONOMAL ASIS FOR HOMOGENEOUS CHAOS L L, H d or grlly r ll hoogous choss o ordr.. For covc, w wll lso us h oo ~ o do h wr grl o, ly ~ I. I hs sco w x orhoorl ss or h spc L, or squc o ogv grs wh su, d H,, H ; H ; H ; 5.!!! 37

10 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls No h x; H d hr r oly ly y o zro ' s. Hc h produc hs quo s produc o oly ly y cors. L 5. For y xd gr, h collco o ucos H,, ; Is sus o. Morovr, h lr spc spd y hs collco o ucos s ds. Proo h rs ssro ollows ro hor 4.6. To prov h scod ssro, oc h h s rgus s hos h proo o hor 4.3 show h ollowg plco E H,,, 5. Now suppos s orhogol o E H or ll such ' s. Hr, H,, or ll,, wh h p d,, p s h orhogol proco o p,, L oo. Morovr, osrv h or y H ;,, 38 ohrws,, H 5.3 Th or y,,, w c us h ssupo o d quo 5.3 o show h = E p H E H, H p, H, p H,,,,,,,,,, s h r produc o h Hlr spc L. Thror, y h plco quo 5., w coclud h. Ths shs h proo or h l Thor 5. For y xd gr, h collco o ucos H,, ; 5.4 Is orhoorl ss or h spc o hoogous chos o ordr. Thor 5.3 h collco o ucos H,, ;,,,, 5.5 Is orhoorl ss or h Hlr spc L. Evry,,,,,,,,,, E H H dp L hs uqu srs xpso H MULTIPLE WIENER- ITô ITOINTEGRALS T, h rs hs sco s o d h ulpl wr- ô grl L,,, For T L T. D,,, T ; such h. Th ssl d s lrdy gv h cs or h doul wr- ô grl. W wll sply ody h rgus d oo sco o h cs 3 L h dgol s ot. A sus o T o h or [, [, [, s clld rcgl. Sp O-dgol sp ucos A sp uco o T s uco o h or

11 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls,,,, [ [, [,,,, A o-dgol sp uco s sp uco wh coc ssyg h codo,, p q or so p q 6. Th collco o o-dgol sp ucos s vcor spc. For o-dgol sp uco gv y quo 6. d I,,, 6.3, p p p p,,, Th I s wll-dd,.. dos o dpd o how s rprsd y quo 6. Morovr, h ppg I s lr o h vcor spc o o-dgol sp ucos. Th syrzo,, o uco,, s dd y ˆ,,,,,,! h suo s ovr ll pruos o h s,,, L 6. s o-dgol sp uco, h I I ˆ [, [, [, Proo: No h I d h syrzo opror r lr. Hc sucs o prov h l or h cs h rvls[,,, r dso. Th w hv O h ohr hd, h syrzo ˆ o s gv y ˆ! ˆ 6. I 6.4 [, [, [, Thror I ˆ Osrv h! = or y pruo orovr, hr r! pruos o h s,,, I ollows h I ˆ = 6.5! Equos 6.4 d 6.5 provs h ssro o h l L 6. I s o-dgol sp uco, h EI d E I! T,, dd ˆ, d 6.6 Proo: L o-dgol sp uco gv y quo 6. h I s gv y quo 6.3. Sc h uco sss h codo quo 6. h coc [,,, 3,, us whvr h rvls,,[,,,[ r o dso. O h ohr hd wh hs rvl r dso, h 39

12 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls corrspodg produc hs xpco. Hc w hv I ˆ E. No h I I y l 6.. Hc w y ssu h s syrc provg quo 6.6 h cs,,...,,,, For y pruo. Thus I. I quo 6.3 c rwr s I!,,, hror,,...,,, E I!, E, or splcy o oo, w hv od h dcs s, c. osrv h or xd s o dcs w hv E I ollows h p p = p,,,, ;, ohrws! E I!,,,!,, p p,, p = d d,, T p p p Whch provs quo 6.6 sc s ssud o syrc Sp : Approxo y o-dgol sp ucos. Rcll h s D dd rlr hs sco. Th s D c rwr s D D, whch s h D s uo o h rscos o -dsol hyprpls wh D. Hc h lsgu sur o D s zro. L 6.3 L uco T o o-dgol sp ucos such h Now suppos L T L. Th hr xss squc l,,,,,, ddd 6.7 T. Choos squc L 6.3 such squc xss. Th y h lry o I d l 6. E I! ˆ ˆ I L L Hc h squc Th vlu 6.8. o o-dgol sp ucos covrgg o L T! s, L. I s Cuchy L D I I l L. y, L 6.8 I s wll-dd, ly, dos o dpd o h chos o h squc usd quo 4

13 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls Do 6.4 L L T d s dod y T. Th l I quo 6.8 s clld h ulpl Wr- ô grl o,,, No h dd sco. I s sply h Wr grl I o d Ovously, ls 6. d 6. c xdd o ucos L T I s h doul Wr- ô grl o Usg h pproxo l 6.3 d h do o h ulpl Wr- ô grl. Thor 6.5 l L T, ˆ h w hv I I. Hr ˆ s h syrzo o I E 3 I E! ˆ. Hr s h or o L T Th x hor gvs quly o wr ulpl Wr- Iô grl s rd Iô grl. I s usul or copuo. Thor 6.6 l L T, T h,,, =!,,, ˆ ˆ s h syrzo o Proo I succs o prov h hor or h cs h s h chrcrsc uco o rcgl h s dso ro h s D. y l 6. w y ssu h s o h or Th h ulpl Wr- ô grl o s gv y O h ohr hd, o h Hc w g [, [,,,, = T [, ˆ o h rgo!. ˆ,, =! [, [, [, F 6.9 whch s -surl d c rgrdd s cos sochsc procss or gro o h rvl [, wh rspc o d. Hc w c rp h ov rgus o g ˆ,, =! 6. Th hor ollows ro quos 6.9 d 6. Do 6.7 l g,, g L, h sor produc g g s dd o h uco g g, g g 4

14 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls Th sor produc Thor 6.8 l Th s h s rpd, s,,, ozro orhogol ucos L, I d l H I ;, posv grs. 6.. I prculr, or y ozro L,, I HI ; Proo. W rs prov quo 6.. Th cs = s ovous. Th cs = s sy o show h I For grl gr, w us hcl duco. Suppos quo 6. s vld or. h y hor 6.6 d d T X! X = X s gv y y hor 6.6 d h duco o X! [, ] Thror, w hv h quly T! H ; ds 6. = H ; ds 6.3 O h ohr hd, w c pply ô s orul o H x, o g dh, ds = H H d H d x x u w hv h ollowg ds H x; H x; x H Thus w o, x; p H x; x dh ; ds = H ; ds Whch, upo gro ovr,, gvs h quly H I ; = H ; ds 6.4 Equos 6.3 d 6.4 shows h quo 6. s vld or. Hc y duco, quo 6. holds or y posv gr. 4

15 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls Now w prov quo 6.. ~ L I. For y rl urs r r,,, r pply h quo To g I xp = HI ;! r r r r O ohr hd, y h quo I xp ~. r = H ~ ;! ~ H I ; Wh d r! r r Hr ;! Th w pply wh w hv lrdy provd quo 6. o xp ~ r H h rgh hd sd o g 6.5 ~ r r r 6.6! T Equo 6. ollow y coprg h coc o 6.6 Thor 6.9 L h EI I g or y L T Proo I sucs o prov h hor or d g o h ollowg or: [ g [ s,, s [ [ s,, s [, [ s, s h rvls ssy h codo, s s s s s s s s Th I I g s gv y r r r h rgh hd sds o quos 6.5 d d g L T. I I g = s s 6.7 Now, pu hs pos d s oghr o or crsg s o pos r wh r. Th ch cor h rs produc o quo 6.7 c rwr s su o crs o o h - rvls. Hc, upo ulplyg ou h cors, ch r h rs produc. Slrly, ch r h scod produc. us o h or 6.8 quo 6.7 us o h or 6.9 I s sy o s h h produc o quos 6.8 d 6.9 hs xpco o cus. Thus w c 43

16 IJRRAS Jury Al & Wg Mulpl Wr-ô Iogrls coclud ro quo 6.7 h EI I g 7. WIENER- ITô THEOREM Thor 7. Wr- ô hor Th spc L c dcoposd o h orhogol drc su L = rprsd y cosss o ulpl Wr- ô grls o ordr. Ech uco Ad h ollowg quly holds L c uquly I, Lsy T 7. Lsy T! Do 7. L I,.Th vrol drvv o s dd o I, 7. h rgh-hd sd s udrsood o wh. I prculr I Thor 7.3 L L. Assu h ll vrol drvvs xs d hv xpco. Pu,,, E h h Wr- ô xpso o s gv y I! 7.3. REFERENCES []. row, R. : A r ccou o croscopcl osrvos d h ohs o Ju, July d Augus, 87, o h prcls cod h poll o pls, d o h grl xsc o cv olculs orgc d orgc ods; phl. Mg []. Chug, K. L.: A cours proly hory. Scod do, A cdc prss, 974. [3]. Chug, K. L. d Wlls, R. J.: Iroduco o sochsc gro.scod do,rhäusr, 99. [4]. Frd, A.: Sochsc Drl Equos d Applcos. Volu II, A cdc prss, 976. [5]. Iô, K.: Sochsc grl, Proc. Ip. Acd, Toyo [6]. Iô, K.: O sochsc grl quo; Proc. Ip. Acd. Toyo [7]. Iô, K.: O Sochsc Drl Equos. Mor, Ar. Mh. Soc., vol. 4, 95. [8]. Iô, K.: Mulpl Wr grl; J. Mh. Soc. Jp [9]. Iô, K.: Exso o sochsc grls; Proc. Irol Syp. Sochsc Drl Equo, K. Iôd , ouy. []. Iô, K.: Iroduco o proly Thory. Crdg Uvrsy Prss, 978. []. Iô, K.: Slcd Pprs. D. W. Srooc d S. R. S. Vrdh ds., sprgr-vrlg. []. Arold, L.: Sochsc Drl Equos: Thory d Applcos. Joh Wly & Sos, 974. [3]. lc, F. d Schols, M.: Th prcg o opos d corpor lls; J. polcl Ecooy [4]. Cro. R. H. d Mr, W. T.: Trsoro o Wr grls udr rslos; Als o Mhcs [5]. Iyg, S. d Kwd, Y.dor: Ecyclopdc Dcory o Mhcl socy o Jp. Eglsh rslo rvwd y K. O. My, MIT Prss, 977. [6]. Jcod, J.: Clcul Sochsqu Prolès d Mrgls. Lcur Nos Mh., Sprgr-vrlg. [7]. Kc, M.: O dsruos o cr Wr ucols; Trs. Ar. Mh. Soc [8]. Kllpur, G.: Sochsc Flrg Thory. Sprgr-vrlg, 98. [9]. Kgh, F.. : Essls o row Moo. Ar. Mh. Soc., 98. []. Kopp, P.: Mrgls d sochsc grls. Crdg Uvrsy Prss, 984. []. Kuo, H. H.: Guss Msurs ch spc. Lcur Nos Mh. 463, sprgr-vrlg, 975. []. Mc, H. P.: Sochsc Igrls, A cdc Prss, 969. [3]. Hu-Hsug Kuo.: Iroduco o sochsc gro. Acd-r ppr, 5. [4]. Mévr, M. Ad Pllul, J.: Sochsc Igro. Acdc Prss, 98. [5]. Novov, A. A. : O dy or sochsc, Igrls; Thory o proly d Ipplcos