HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA

Communicions on Sochsic Anlysis Vol 6, No 4 2012 603-614 Serils Publicions wwwserilspublicionscom THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Absrc We sudy he new sochsic inegrl inroduced by Ayed nd Kuo in [1] Our min resuls re wo Iô formuls h exend he one presened in [1] We generlize he noion of he Iô process ono he clss of insnly independen sochsic processes nd use i in he formulion of he wo Iô formuls we derive 1 Inroducion Le {B : 0}, be Brownin moion nd {F : 0} be filrion such h B is F -mesurble for ech 0 nd B B s is independen of F s for ny 0 s I is well-nown fc h he Iô inegrl is well-defined for {F }-dped sochsic processes {f: b} such h b f 2 d < lmos surely We will denoe he clss of ll such processes by L d Ω,L 2 [,b] The spce L d Ω,L 1 [,b] is defined in similr wy If f L d Ω,L 2 [,b] hs lmos surely coninuous phs, he Iô inegrl is equl o he following limi in probbiliy b fdb lim f i 1 B i, 11 n 0 i1 where n { 0 < 1 < 2 < < n b} is priion of [,b] nd B i B i B i 1 see, for exmple, [5, Theorem 533] One of he crucil heorems in he Iô sochsic clculus is he Iô formul I cn be viewed s sochsic counerpr of he fundmenl heorem of clculus Is mos bsic form is sed below Theorem 11 For funcion f C 2 R, we hve fb b fb b f B db 1 2 b f B d 12 In his pper, we generlize he Iô formul for he new sochsic inegrl inroduced by Ayed nd Kuo [1, 2] We lso inroduce counerpr o he Iô Received 2012-4-19; Communiced by he ediors Aricle is bsed on lecure presened he Inernionl Conference on Sochsic Anlysis nd Applicions, Hmmme, Tunisi, Ocober 10-15, 2011 2000 Mhemics Subjec Clssificion Primry 60H05; Secondry 60H20 Key words nd phrses Brownin moion, Iô inegrl, Iô formul, dped sochsic processes, insnly independen sochsic processes, niciping sochsic processes, sochsic inegrl, niciping inegrl 603

604 HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA processes, nd use i o inroduce mos generl Iô formul for he new inegrl nown o de Thereminderofhispperisorgnizedsfollows InSecions2nd3werecll some bsic fcs bou he new sochsic inegrl nd he Iô formul derived by Ayed nd Kuo [1] in heir originl wor on he new inegrl In Secion 4 we inroduce insnly independen sochsic processes s counerpr of he wellnown Iô processes Secions 5 nd 6 conin our min resuls Theorems 51 nd 61, h is Iô formulsforhe new inegrl We conclude wih someexmples in Secion 7 nd discussion of our resuls in Secion 8 2 The New Sochsic Inegrl Le {B : 0} nd {F : 0} be defined s in Secion 1 We sy h sochsic process {ϕ: 0} is insnly independen wih respec o he filrion {F : 0} if for ech 0, he rndom vrible ϕ nd he σ-field F re independen For exmple ϕb 1 B, [0,1] is insnly independen of F for ny rel mesurble funcion ϕx However, ϕb 1 B is dped for 1 I cn be esily checed h if ϕ is dped nd insnly independen wih respeco{f : 0},henϕisdeerminisic Therefore,hefmilyofinsnly independen sochsic processes cn be regrded s counerpr o he dped processes In [1], Ayed nd Kuo define new sochsic inegrl for dped nd insnly independen processes Suppose h {f: 0} is sochsic process dped o {F : 0} nd {ϕ: 0} is insnly independen wih respec o {F : 0} The new sochsic inegrl of fϕ is defined s b fϕdb lim n 0 i1 f i 1 ϕ i B i, 21 whenever he limi exiss in probbiliy The crucil disincion beween he clssicl Iô definiion nd he one proposed by Ayed nd Kuo is he fc h he evluion poin of he insnly independen process is he righ-endpoin of he sub-inervl, while he evluion poin of he dped process is he lef-endpoin, s in he definiion of he Iô inegrl see Equion 11 This choice of evluion poins ensures h he Iô inegrl is specil cse of he new sochsic inegrl becuse if ϕ 1, hen Equion 21 reduces o Equion 11 For some evluion formuls, he discussion of he ner-mringle propery, nd n isomery formul for he new inegrl, see [6, 7] by Kuo, Se-Tng nd Szozd Applicion of he formuls derived in his pper cn be found in [4] by Khlif e l 3 The Firs Iô Formul for he New Sochsic Inegrl In his secion we recll, very briefly, he firs Iô formul for he new inegrl h ws derived by Ayed nd Kuo in [1]

THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL 605 Theorem 31 Le fx nd ϕx be C 2 -funcions nd θx,y fxϕy x Then he following equliy holds for b, θb,b b θb,b b x B s,b b db s [ 1 2 ] θ 2 x 2B s,b b 2 θ x y B s,b b ds 31 Theorem 31 fcilies compuion of mny inegrls of he new ype, however one of is min drwbcs is he fc h θ cn only be funcion of B nd B b B There re wo exensions h we wish o presen in he forhcoming secions Firs, we will llowforhe evluionprocessesofθ o be n Iô processes nd heir insnly independen counerprs we will lso presen n Iô formul pplicble o inegrls lie 1 0 B 1 db Noe h Theorem 31 cn be pplied o he bove inegrl, only fer decomposing he inegrnd s B 1 B 1 B B 4 Iô Processes nd Their Counerpr An Iô process is sochsic process of he form X X gsdb s γsds, b, 41 where X is n F -mesurble rndom vrible, g L d Ω,L 2 [,b], nd γ L d Ω,L 1 [,b] Observe h if X 0, g 1, nd γ 0, hen X B This gives us n ide on how o find he insnly independen counerpr o he Iô processes Consider Y Y b b hsdb s b χsds, b, 42 where Y b is independen of F b, funcions h L 2 [,b] nd χ L 1 [,b] re deerminisic Noice h if Y b 0, h 1 nd χ 0, hen Y B b B Thus Y is o B b B wh X is o B For convenience, we will ofen use he differenil noion insed of he inegrl one, nd so Equion 41 is equivlen o while Equion 42 is equivlen o dx gdb γsd, dy hdb χd Before we recll he Iô formul for he Iô processes see [5, Theorem 743], we wish o noe h using he differenil noion inroduced bove is very esy when combined wih he fc h db d 0, d 2 0, nd db 2 d For exmple, dx s 2 gs 2 ds nd dy 2 h 2 d

606 HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Finlly, we cn recll he well-nown Iô formul for he Iô processes Theorem 41 Suppose h X is s in Equion 41 nd fx is C 2 -funcion Then for ny b, fx fx f X s dx s 1 2 f X s dx s 2 43 5 The Iô Formul for Insnly Independen Iô Processes As we hve seen, he elemenry Iô formul Equion 12 for he Brownin moion cn be generlized o be pplicble o Iô processes Equion 43 Our gol is o generlize Theorem 31 in similr wy o how Theorem 41 generlizes Theorem 11 in clssicl Iô clculus Th is, we will show h i is possible o chnge fb nd ϕb b B ino fx nd ϕ Y where X is s in Equion 41 nd Y is s in Equion 42 Noe h due o he wy he Iô inegrl cn be compued see Equion 11, i is cler h he process Y is insnly independen of {F : 0} Therefore we cn inegre funcions of he form fx ϕ Y using he new inegrl This observion leds us o he following heorem Theorem 51 Suppose h θx,y fxϕy, where f,ϕ C 2 R Le X be s in Equion 41 nd Y be s in Equion 42 Then for b, θx,y θx,y x X s,y s dx s 1 2 y X s,y s dy s 1 2 x 2X s,y s dx s 2 y 2X s,y s dy s 2 51 Proof Throughou his proof, we will use he sndrd noion inroduced erlier, nmely n { 0 < 1, < n 1 < n } nd X i X i X i 1 To esblish he formul in Equion 51, we begin by wriing he difference θ X,Y θ X,Y in he form of elescoping sum θ X,Y θ X,Y i1 i1 [ θ X i,y i θ X i 1,Y i 1] [ f X i ϕ Y i f X i 1 ϕ Y i 1] 52 Since for > 2, X i Y i o i, we cn use he second order Tylor expnsion of f nd ϕ o obin f X i f X i 1 f X i 1 Xi 1 2 f X i 1 Xi 2, ϕ Y i 1 ϕ Y i ϕ Y i Y i 1 2 ϕ Y i Y i 2 53

THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL 607 Puing Equions 52 nd 53 ogeher, we hve θ X,Y θ X,Y [f X i 1 f X i 1 Xi 12 f X i 1 Xi 2 ϕ Y i i1 f X i 1 ϕ Y i ϕ Y i Y i 1 2 ϕ Y i ] Y i 2 [f X i 1 ϕ Y i X i 12 f X i 1 ϕ Y i X i 2 i1 f X i 1 ϕ Y i Y i 12 f X i 1 ϕ Y i ] Y i 2 [ X i 1,Y i X i 1 x 2 x 2 X i 1,Y i X i 2 X i 1,Y i Y i 1 y 2 y 2 X i 1,Y i ] Y i 2 i1 54 Finlly, s n 0, he expression in Equion 54convergeso he righ-hnd side of Equion 51, hence he heorem holds Argumens similr o he ones in he proof of Theorem 51 cn be used o prove he following corollry I inroduces purely deerminisic pr h depends only on Corollry 52 Suppose h θ,x,y τfxϕy, where f,ϕ C 2 R nd τ C 1 [,b] Le X be s in Equion 41 nd Y be s in Equion 42 Then θ,x,y θ,x,y s s,x s,y s ds x s,x s,y s dx s 1 2 y s,x s,y s dy s 1 2 x 2s,X s,y s dx s 2 y 2s,X s,y s 6 The Iô Formul for More Generl Processes dy s 2 As we hve lredy menioned in Secion 3, upon pproprie decomposiion of he inegrnd, i is possible o use he new definiion of he sochsic inegrl o compue he inegrl of processes h re no insnly independen, for exmple 1 0 B 1dB Our nex gol is o esblish n Iô formul for such processes Noe h using he noion of Equion 42, wih hs 1 nd χs 0, on he inervl [0,1], we hve Y 1 0 1dB B 1 B 0 B 1

608 HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Thus Y o Y is wh B 1 o B 1 B Hence we wish o esblish n Iô formul for θ X,Y, wih X nd Y s defined in Equions 41 nd 42 Keeping in mind h he definiion of he new inegrl does no llow processes h re niciping nd no insnly independen, we hve o impose n ddiionl srucure on funcion θ in order o move freely beween Y nd Y Following he ides of [6, 7], we use funcions whose Mclurin series expnsion hs infinie rdius of convergence Such pproch gives us he ddiionl srucure we need in order o pply he new heory of sochsic inegrion Theorem 61 Suppose h θx,y fxϕy, where f C 2 R, nd ϕ C R hs Mclurin expnsion wih infinie rdius of convergence Le X be s in Equion 41 nd Y be s in Equion 42 Then for b, θx,y θx,y x x y X s,y dx s 1 2 X s,y dx s x 2 X s,y dx s 2 dy s 61 Proof We will derive he formul in Equion 61 symboliclly using he differenil noion inroduced erlier Th is, we need o esblish h dθx,y x X,Y dx 1 2 x 2 X,Y dx 2 X,Y dx dy 2 θ x y To simplify he noion, we will wrie D d θ X,Y Le us consider D d fx ϕ Y ϕ n 0 d fx Y n d fx ϕ n 0 Y Y Y n 62 Applying he binomil heorem nd he fc h Y Y Y llows us o rewrie D s ϕ n 0 n D d fx Y Y n ϕ n 0 n d fx Y n Y 63 Noe h Z fx Y n s produc of Iô processes is n Iô process iself, hence we cn use he Iô formul from Theorem 51 o evlue he differenil

THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL 609 under he sum in Equion 63 We e ηz,y zy o obin η z z,y y,η zz z,y 0,η y z,y zy 1 nd ηz,y yy 1zy 2 which yields d η Z,Y Y dz Z Y 1 dy 1 2 1Z Y 2dY 64 2 Usinghe Iô producruleforiô processes, weesilysee h dz cnbe expressed s dz fx d Y n Y n dfx dfx dy n [ fx n Y n 1 dy 1 ] n 2 n n 1Y dy 2 2 [f X dx 12 ] f X dx 2 Y n n f X Y n 1 dx dy fx n Y n 1 dy 1 2 n n 1fX Y n 2 dy 2 f X Y n dx 65 1 2 f X Y n dx 2 n f X Y n 1 dx dy Puing ogeher Equions 63, 64 nd 65, we see h in order o complee his proof, we hve o evlue D fx 1 2 fx f X ϕ n 0 1 2 f X f X fx fx 1 2 ϕ n 0 ϕ n 0 ϕ n 0 ϕ n 0 ϕ n 0 ϕ n 0 n n Y n 1 n n n n n n Y dy n n 1Y n 2 Y n Y n Y dx n Y n 1 Y n Y dx 2 Y 1 dy 1Y n Y dy 2 Y dx dy Y 2dY 2

610 HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA fx Σ 1 dy 1 2 fx Σ 2 dy 2 f X Σ 3 dx 1 2 f X Σ 3 dx 2 f X Σ 1 dx dy fx Σ 4 dy fx 1 2 Σ 5 dy 2 66 In order o simplify D in Equion 66, we need o evlue he 5 sums denoed bove by Σ i, wih i {1,2,,5} Σ 1 The firs sum is given by Σ 1 ϕ n 0 n n Y n 1 Y 67 Noe h for n he expression under he sum is equl o zero, so we hve ϕ n n 1 0 n Σ 1 n Y n 1 Y n1 Now, since 1 n n 1 n 1!, we ge Σ 1 n1 ϕ n 0 n 1! n 1 n 1 n 1 Y n 1 nd pplicion of he binomil heorem yields ϕ n 0 Σ 1 Y Y n 1 n 1! n1 Y, Since, by definiion, Y Y Y nd ϕ n 0 n1 n 1! xn 1 ϕ x, we obin Σ 2 The second sum we hve o evlue is ϕ n 0 n Σ 2 Σ 1 ϕ Y 68 n n 1Y n 2 Y 69 Due o he n nd n 1 fcors, he erms wih n nd n 1 do no conribue o he sum, hence ϕ n n 2 0 n Σ 2 n n 1Y n 2 Y n2 Since 1 n n n 1 1 n 2 n 2!, we hve Σ 2 n2 ϕ n 0 n 2! n2 n 2 n 2 Y n 2 Using he binomil heorem, we obin ϕ n 0 Σ 2 Y Y n 2 n 2! Y

THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL 611 Using he fcs h ϕ x ϕ n 0 n2 n 2 xn 2 nd Y Y Y we ge Σ 2 ϕ Y 610 Σ 3 Using he sme resoning s previously, we cn wrie he nex sum ppering in Equion 66 s ϕ n 0 n Σ 3 Y Y n ϕ n 0 Y Y n ϕ Y 611 Σ 4 Now, we evlue he following sum Σ 4 ϕ n 0 j0 n Y n Y 1 Noice h subsiuion j n ogeher wih he fc h n n j n j yields ϕ n 0 n Σ 4 n jy j Y n j 1 n j j0 ϕ n 0 n n jy j Y n j 1, j nd his is he sme sum we hve evlued in Equion 67, hence Σ 4 ϕ Y 612 Σ 5 The ls sum needed is ϕ n 0 Σ 5 n 1Y n Y 2 Using he subsiuion j n nd he fc h n n j n j gin, we obin ϕ n 0 n Σ 5 n jn j 1Y j Y n j 2 n j ϕ n 0 n n jn j 1Y j Y n j 2 j And his sum ppers in Equion 69, hus Σ 5 ϕ Y 613

612 HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Now, puing ogeher Equions 66, 68, 610, 611, 612 nd 613 we obin D fx Σ 1 dy 1 2 fx Σ 2 dy 2 f X Σ 3 dx 1 2 f X Σ 3 dx 2 f X Σ 1 dx dy fx Σ 4 dy 1 2 fx Σ 5 dy 2 fx ϕ Y dy 1 2 fx ϕ Y dy 2 f X ϕ Y dx 1 2 f X ϕ Y dx 2 f X ϕ Y dx dy fx ϕ Y dy 1 2 fx ϕ Y dy 2 Since dy dy, we hve D fx ϕ Y dy 1 2 fx ϕ Y dy 2 f X ϕ Y dx 1 2 f X ϕ Y dx 2 f X ϕ Y dx dy fx ϕ Y dy 1 2 fx ϕ Y dy 2 x X,Y dx 1 2 x 2 X,Y dx 2 X,Y dx dy, 2 θ x y which complees he proof As wih Corollry 52, we cn esily deduce, h if we hd componen of θ h is deerminisic nd depends only on, he following Corollry will hold Corollry 62 Suppose h θ,x,y τfxϕy, where τ C 1 R, f C 2 R, nd ϕ C R hs Mclurin expnsion wih infinie rdius of convergence Le X be s in Equion 41 nd Y be s in Equion 42 Then θ,x,y θ,x,y s,x s,y ds s,x s,y dx s 1 x 2 x y s,x s,y dx s 7 Exmples x 2 s,x s,y dx s 2 dy s 614 To illusre he usge of he new Iô formuls inroduced in previous secions, we will esblish simpler versions of Theorems 51 nd 61 for funcions of Brownin moion Le X B, g 1, nd γ 0, so h he process X becomes B

THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL 613 Moreover, le Y b 0, h 1, nd χ 0, so h he process Y becomes B b B Using he bove in Theorem 51 gives us he following corollry Corollry 71 Suppose h θx,y fxϕy wih f,ϕ C 2 R Then θb,b b B θb,b b B Similrly, Theorem 61 becomes x B s,b b B s db s 1 2 y B s,b b B s db s 1 2 x 2B s,b b B s ds y 2B s,b b B s ds Corollry 72 Suppose h θx,y fxϕy, where f C 2 R, nd ϕ C R hs Mclurin expnsion wih infinie rdius of convergence Then θb,b b B θb,b b B 1 2 x 2B,B b B ds x B,B b B db s x y B,B b B ds Exmple 73 Applying Corollry72 on he inervl [0,1] o funcion θx,y x n1 n1 ym, wih m,n N, we obin 1 0 B n Bm 1 db Bmn1 1 n1 B m 1 1 1 0 n B n 1 2 B 1 mb d This shows how we cn express he inegrl of n niciping process in erms of rndom vrible nd Riemnn inegrl of sochsic process 8 Conclusions We hve derived wo Iô formuls for Iô processes nd heir insnly independen counerprs Our resuls re pplicble in vriey of siuions nd exend he resul of Ayed nd Kuo [1] Below, we compre he wo formuls derived in his pper s Theorems 51 nd 61 Noice h here re 4 min componens o he Iô formul in he seing of he new sochsic inegrl, nmely, x,y,fx nd ϕy In boh formuls derived bove, x is n Iô process nd f C 2 R This ssumpions re nurl becuse he new sochsic inegrl is n exension of he sochsic inegrl of Iô, nd puing ϕy 1 shows h our formuls re in fc exensions of he clssicl resul cied erlier s Theorems 11 nd 31 The min differences beween Theorem 51 nd Theorem 61 re he properies of y nd ϕy In Theorem 51, he process subsiued for y is n insnly independen process h rises in similr wy s he Iô processes do in he heory of dped processes This llows us o wor wih funcions ϕ C 2 R In Theorem 61, we hve derived formul h llows us o wor wih rndom vribles h rise from he insnly independen Iô processes in he plce of y However, his exension comes price of n ddiionl smoohness condiions on he funcion ϕ Th is ϕ hs o hve infinie rdius of convergence of is

614 HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Mclurin series expnsion In mny pplicions, such requiremen should no be oo resricive Oneofhepplicionsofheformulsesblishedinhispperissoluionof clss of liner sochsic differenil equions wih niciping iniil condiions This will be presened in forhcoming pper by Khlif e l [4] References 1 Ayed, W nd Kuo, H-H: An exension of he Iô inegrl, Communicions on Sochsic Anlysis 2 2008, no 3, 323 333 2 Ayed, W nd Kuo, H-H: An exension of he Iô inegrl: owrd generl heory of sochsic inegrion, Theory of Sochsic Processes 1632 2010, no 1, 1 11 3 Iô, I: Sochsic inegrl, Proceedings of he Imperil Acdemy 20 1944, no 8, 519 524 4 Khlif, N, Kuo, H-H, Ouerdine, H nd Szozd, B: Liner sochsic differenil equions wih niciping iniil condiions, pre-prin, 2012 5 Kuo, H-H: Inroducion o sochsic inegrion, Universiex, Springer, 2006 6 Kuo, H-H, Se-Tng, A nd Szozd, B: A sochsic inegrl for dped nd insnly independen sochsic processes, in Advnces in Sisics, Probbiliy nd Acuril Science Vol I, Sochsic Processes, Finnce nd Conrol: A Fesschrif in Honour of Rober J Ellio eds: Cohen, S, Mdn, D, Siu, T nd Yng, H, 53 71, World Scienific, 2012 7 Kuo, H-H, Se-Tng, A nd Szozd, B: An isomery formul for new sochsic inegrl, Acceped for he Proceedings of Inernionl Conference on Qunum Probbiliy nd Reled Topics, My 29 June 4, 2011, Levico, Ily, o pper in QP PQ, Vol 32 Hui-Hsiung Kuo: Deprmen of Mhemics, Louisin Se Universiy, Bon Rouge, LA 70803, USA E-mil ddress: uo@mhlsuedu URL: hp://wwwmhlsuedu/ uo Anuw Se-Tng: Deprmen of Mhemics, Fculy of Science, King Mongu s Universiy of Technology Thonburi, Bngo, Thilnd E-mil ddress: nuwse@much Benedy Szozd: The TN Thiele Cenre for Mhemics in Nurl Science, Deprmen of Mhemicl Sciences, & CREATES, School of Economics nd Mngemen, Arhus Universiy, DK-8000 Arhus C, Denmr E-mil ddress: szozd@imfud URL: hp://homeimfud/szozd