The Existence and Uniqueness of Random Solution to Itô Stochastic Integral Equation

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1 Appled Mhemcs,, 3, 8-84 hp://dx.do.org/.436/m..379 Pulshed Ole July (hp:// The Exsece d Uqueess of Rdom Soluo o Iô Sochsc Iegrl Equo Hmd Ahmed Alff, Csh Wg School of Mhemcs d Iformo Scece, Norhwes Norml Uversy, Lzhou, Ch Eml: hmd@6.com Receved My 8, ; revsed Jue 8, ; cceped Jue 5, ASTRACT The ojecve of hs pper s o emp o pply he heorecl echques of prolsc fucol lyss o swer he queso of exsece d Uqueess of Rdom Soluo o Iô Sochsc Iegrl Equo. Aoher ype of sochsc egrl equo whch hs ee of cosderle mporce o ppled mhemcs d egeers s h volvg he Iô or Iô-oo form of sochsc egrls. Keywords: Iô Iegrl; row Moo; Prolsc Fucol Alyss; ch Spce. Iroduco We shll gve some hsorcl remrks cocerg he developme of hs ype of equo d po ou he essel dfferece ewee hem d oher rdom egrl equos. I 93 N. Weer roduced egrl of he form g d where g deermsc rel-vlued fuco d,, s sclr row moo process. Auhor of [] 944 geerlzed Weer s egrl o clude hose cses where he egrd s rdom. Th s he oed egrl of he form ; d,, g w Whch s referred o s he Iô sochsc egrl or smply he sochsc egrl. Sce h me my scess hve corued o he geerl developme of hs ype of sochsc egrl. For exmple see [- ]. I 946 Auhor of [5] formuled sochsc egrl equo of he form xw ; C f, x; wd g, x; wd where,, (.) ;, s sclr row moo process, d C s cos Resrcos re usully plced o he fucos f d g so h he frs egrl s erpreed s he usul Leesgue egrl of he smple fucos whch c he e reled o he smple egrl of he process, ;,, f x w d he secod egrl s Iô sochsc egrl. The prcpl feure whch dsgushes he ype of equo suded from equo of he Iô ype s he fc h he former cse ech of he egrls volved s erpreed s Leesgue egrl for lmos ll w. Th s, lmos ll smple fucos re Leesgue egrle. Sce he Iô sochsc egrl he lm s ke he me-squre or he proly sese, he heory of such egrls hs ee developed s self-coed d self-cosse. Oe of he m purposes of suseque work coeco wh he Iô sochsc egrl equo hs ee o cosruc Mrkov processes such h her rso proles ssfy gve Kolmogorov equos d o vesge he couy of he processes, mog oher properes of he smple fuco. The mehod of successve pproxmo ws used y Iô d oo o show he exsece d uqueess of rdom soluo o Equo (.).. Prelmres Le ;, e sclr row moo process. I hs seco we shll e cocered wh he egrl ; d w (.) for frly geerl clss of fucos. Ths egrl wll e clled he Iô sochsc egrl s we meoed prevously. As s well kow, lmos ll he smple fucos of he row moo process re of u- Copyrgh ScRes.

2 H. A. ALAFIF, C. S. WAN 8 ouded vro d hece he egrl (.) co e defed s ordry Seljes egrl. Frs we shll defe he egrl (.) for he clss of sep fucos. Th s, fucos of he form w ; w, (.) where, w re mesurle wh respec o he -lger A, d E w f or such f ucos we defe he Iô egrl y w ; d w (.3) 3) E ; w d I vew of Equo (.) s evde h he clss of sep fucos ssfy codos )-3). For he fuco w ; ssfyg codos )-3) we shll defe her orm s follows: ; ; w E w d (.4) For hs cse uhor of [] hs show he followg ) w ; c e pproxmed he me-squre sese w ;. Th s y sequece of sep fucos w ; w ; s ) The sequece of egrls Possesses me-squre lm. Th s here exss w such h (.5) s Now we shll defe he egrl (.) for clss of fu w ; ssfyg codos )-3) y cos w ; d w (.6) As wh he ordry egrls, we shll defe w ; d E w ; w d w ; d lm w ; d, (.7) Now suppose h w ; s y fuco ssfyg he followg codos. ) w ; s produc-mesurle fuco from,, ssumg he usul Leesgue mesure o. ) Fo r ech,,, w ; s mesurle wh respec o -lger A, where A s he smlles -lger o, such h s, s s mesur- le. efo. Le L, where L deoe he collec- of Leesgue mesurle suses of o. efe fuco χ from, y χ ; w f ; w oherwse Lemm. The fuco χ : defed y ;w χ ; w ; wχ where ssfes codos )-3), d χ s s del of he def- fed erler, lso ssfes codos )-3). Proof. The proof s srghforwrd resu o of χ d he fc h ssfes codos )-3). We re ow poso o defe excly wh s me y he expresso ; wd efo. We defe Leesgue-mesurle suse of ; wd y ; wd ; wd for Noe h lemm.4 gurees he expresso o he rgh exss d s well defed efo.3 We shll deoe y C,, L,, P he spce of ll couous fuc- os from, o L,, P orm of C,, L,,P Lemm. Lemm.3 y x w P w sup ; d E E w. We shll defe he ; d E E ; w d Lemm.4 If we defe dsce ewee wo fucos d ech ssfyg codos )-3) y E ; w ; w d d he dsce ewee w ; d y w ; d d Copyrgh ScRes.

3 8 H. A. ALAFIF, C. S. WAN E The. For he proof of he Lemms see []. Lemm.5 Le xw ; ; wd,, The xw ; C,, L,, P For he proof see [4]. 3. O Iô Sochsc Iegrl Equo I hs seco we shll vesge sochsc egrl equo of he ype where xw ; xw ; k, ; w f, x; w d ; w d (.) s he ukow rdom process defed for d w. We shll plce he followg resrcos o he rdom fucos whch cosue he sochsc egrl Equo (.). ') k, ; w s eleme of L,, P d k, ; w: L,,P s couous where, :. ') x w ; f xw, ; s operor o he se S wh vlues he ch spce ssfyg, ; f, y; w f x w ; yw ; xw for x w ;, yw ; S. 3') Codos )-3) of seco hold. Thus wh he gve ssumpos he frs egrl of (.) c e erpreed s Leesgue egrl d he secod s Iô sochsc egrl. We shll ow proceed o se d prove heorem cocerg he ehvor of he Iô egrl. More precsely, f we show h he Iô egrl s eleme of he spce CC, L,, P, we c pply he heory of dmssly o Equo (.) o show he exsece of rdom soluo. y rdom soluo o Equo (.) we me rdom fuco xw ; from o L,, P such h for ech, xw ; ssfes he egrl equo P-.e. sh owg h h e Iô egrl s eleme of CC, L,,P wll mke fesle he ssumpo h we wsh o mke h he egrl s eleme of, ch spce coed he opologcl spce meoed For covee we shll deoe he Iô egrl y Theorem. For hw ; ; wd, C, h ; w C, L,, P Proof Fx The ; ; d χ, ; d hw w w Thus ; ;, d E x, ; w d Eh w E x w y lemm.3. Hece Ehw ;. Therefore for fxed, hw ; L,, P. Now le. To sh ow h h ; w h; w L,, P, s suffce o show h χ, χ, c e mde rrrly smll. Th s, we mus show h E χ ;,, ; d w χ w C e mde rrrly smll. Choose. Cosder he oegve fuco q; w E ; w. y codo 3) q ; w s egrle over. Hece here exss such h for every se of Leesgue mes- q ; w d. Thus ure less h, E χ w, w χ, E χ, ; w χ w, E ; w d q; wd ; ; d ; d Sce for N d d sce he Leesgue mesure of he ervl, s s legh, we coclude h he Leesgue mesure of, s less h. Hece q ; wd Implyg h h; w o L,, P d he I hs seco we shll sudy he exsece d uque- s couous from proof s complee. Sce we hve show h hw ; CC, L,, P, we c coclude h he sochsc egrl Equo (.) possesses u que rdom soluo 4. O Iô-oo-Type Sochsc Iegrl Equos Copyrgh ScRes.

4 H. A. ALAFIF, C. S. WAN 83 ess of rdom soluo o sochsc egrl equ- o of he form x; w f, x; wd, x; wd (3.) where,. As efore, he frs egrl s Leesgue egrl, whle he secod s Iô-ype sochsc egrl defed wh respec o sclr row moo process,,. Recll h C,, L,, PCC, L,, P, We shll defe he operors W d W from C,, L,, P C,, L,, P y d o wx; w x ; wd ; ; d (3.) wx w x w (3.3) Noe h vew of lemm.5 xw ; C,, L,, P. Is cler h W d W re ler operors. Theorem 3. The operors W d W defed y (3.) d ( 3.3) respecvely, re couous operors from C,, L,, P o C,, L,, P. Lemm 3. Le T e couous operor from CC, L,, P o self. If d re ch spces sroger h C C d he pr (, ) s dmss- le wh respec o T. The T s couous operor from o. Proof of heorem 3. The fc h W s couous operor from C,, L,, P o C,, L,, P follows from lemm 3.. From (3.3) we hve wx w ; d Pw x ; wd L,, P Furhermore d ; d x w P w wx w ; d sup x; w L,, P L,, P Therefore w L,, P sup x ; d xw (; ) d x w ; wxw xw ; ; Thus W d W re couous operors from,,,, P o C,, L,,. C L P A Exsece Theorem We shll ssume h le mm 3. holds wh respec o he operors W d W. Therefore here exs posve coss K d K less h oe such h wx w ; k x; w d wx w ; k xw ; The followg heorem gves suffce codos for he exsece of uque rdom soluo, secod order sochsc process, o he Iô-oo sochsc egrl Equo (3.). Theorem 3. Cosder he sochsc egrl equo (3.) uder he followg codo: ) d re ch spces C,, L,, P whch re sroger h C,, L,, P such h, s dmssle wh respec o he operors W d W ) ) x w ; f, xw ; s operor o ; : ; d ; w S xw xw x Wh vlues ssfyg fxw, ; fyw, ; xw ; yw ; ) xw ; xw, ; ssfyg s operor o S o, xw ; yw, ; xw ; yw ; where d re coss. The here exss uque rdom soluo o Equo (3.) provded h k k. A d,, f k k Proof. efe operor U from he se S o s follows Ux; w f, x; wd, x; wd We eed o show h U s corco operor o S d h US S. Le xw ;, yw ; S. The Ux; w Uy; w ecuse s ch spce. Furher, we hve Ux; w Uy; w f, x ; w f, y ; w d, ;, ; d x w y w k f, x; w f, y; w k, x; w, y; w k k xw ; yw ; xw ; yw ; Copyrgh ScRes.

5 84 H. A. ALAFIF, C. S. WAN Thus U s corco operor. For y eleme S we hve Ux; w, ; d, ; d, ;, ; f x w x w k f x w k x w ; ; k xw k xw k f, k, xw S follows h Ux; w k k f,, Sce ; from he ssumpos he heorem. Thus he exsece d uqueess of rdom soluo o Equo (3.) follow from he ch fxed-poheorem. Theorem 3.4 (S. ch s fxed-po prcple) ([]). If T s corco operor o complee merc spce H. he here exss uque po x H for whch T x x. 5. Cocluso We vesged he exsece d uqueess of Iô sochsc egrl equo y pplyg he heorecl echques of prolsc fucol lyss. I fc uhor of [] refers o prolsc fucol lyss s eg cocered wh he pplcos d exesos of he mehods of fucol lyss o he sudy of he vrous coceps, processes, d srucures whch rse he heory of proly d s pplcos. Flly o develop d ufy he he ory of sochsc or rdom equos see [3-5]. REFERENCES [] K. Io, Sochsc Iegrl, Proceedgs of he Imperl Acdemy, Vol., No. 8, 944, pp do:.379/p/ [] J. L. oo, Sochsc Processes, Wley, New York, 953, pp [3] Y. yk, Mrkov Processes, Acdemc Press, New York, 964, pp [4] A. Jzwsk, Sochsc Processes d Flerg Theory. Mhemcs Scece d Egeerg, Vol. 64, Acpp demc Press, New York, 97, [5] K. Io, O Sochsc Iegrl Equo, Proceedgs of he Jp Acdemy, Vol., No., 946, pp do:.379/pj/ [6] H. P. Mcke, Sochsc Iegrls, Acdemc Press, New York, 969, pp. -5. [7] T. L. Sy, Moder Noler Equos, Mcrow- Hll, New York, 967, pp [8] L. khm d A. V. Skorokhod, Iroduco o he Theory of Rdom Process-Suders, Phldehph, Pe- sylv, 969, pp [9] R. L. Sroovch, A New Represeo for Sochsc Iegrls d Equos, Jourl of SL Corol, Vol. 4, 966, pp [] E. Wog d M. Zk, O he Relo ewee Ordry d Sochsc fferel Equos, Ierol Jourl of Egeerg Scece, Vol. 3, No., 965, pp do:.6/-75 (65)945-5 [] I. P. Nso, Theory of Fucos of Rel Vrle, Vol. II, Ugr, New York,. [] A. T. hruch-red, O he Theory of Rdom Equ- Vol. os, Proceedgs of Sympos Appled Mhemcs, Vol. 6, 964, pp [3]. Adom, Rdom Operor Equos Mhemcl Physcs, Jourl of Mhemcl Physcs,, No. 3, 97, pp do:.63/ [4]. Adom, Ler Rdom Operor Equos Mhemcl Physcs III, Jourl of Mhemcl Physcs, Vol., No. 9, 97, pp do:.63/ [5]. Adom, Theory of Rdom Sysems, Trscos of he fourh Prgue Coferece o Iformo Theory, Sscl ecso Fucos, Rdom Processes, Prgue, 3 Augus- Sepemer 965, pp. 5-. Copyrgh ScRes.