WHITE NOISE APPROACH TO STOCHASTIC INTEGRATION

Transcription

1 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON L. ACCARD*-W. AYED**-H. OUERDANE*** Absrac. We exend o whie noise inegrals he scalar ype inegraor ineqaliies inrodced by Accardi, Fagnola and Qaegeber [1] as a generalizaion of he Hdson Parhasarahy basic esimaes on sochasic inegrals. We se hese esimaes as reglariy resls, showing ha some Hida disribions are in fac elemens of he Fock space. We also se hem o prove an analoge reglariy resl for solions of whie noise eqaions wih bonded coefficiens. The whie noise approach o sochasic calcls emerged, beween 1993 and 1995, from he sochasic limi of qanm heory as a new approach [7] o boh classical and qanm sochasic calcls [9], [16]. The main achievemens of he new approach have been: (i) The idenificaion of boh classical and qanm sochasic eqaions wih whie noise Hamilonian eqaions. (ii) The explanaion of he emergence of he niariy condiions of Hdson and Parhsarahy as expression of he symmericiy of he associaed Hamilonian eqaion. (iii) The explici expression of he coefficiens of he sochasic eqaion as (nonlinear) fncions of he coefficiens of he associaed Hamilonian eqaion. (iv) The emergence of a naral nonlinear exension of sochasic calcls. The deep and srprising resls obained in his direcion in he qadraic case sgges ha he compleion of his programme for he higher powers of whie noise is one of he mos challenging and fascinaing problems of conemporary sochasic analysis. None of hese resls cold have been even formlaed in he framework of he sal (classical or qanm) sochasic analysis. However he exciing new developmens emerged from he whie noise approach o sochasic calcls delayed a sysemaic exposiion of is basic analyical ools sch as whie noise inegrals, whie noise eqaions, he heory of disribions on he sandard simplex, he casal normal order,... A large lierare exised in he firs wo of he above menioned direcions, in he framework of Hida s whie noise analysis. However o creae a bridge beween hese resls and hose of classical and qanm sochasic analysis, one needs some reglariy resls, i.e. condiions which assre ha objecs, which a priori are Hida disribions, are in fac vecors in a Hilber space. Sch reglariy condiions will be formlaed here in erms of esimaes on whie noise inegrals. Esimaes of his ype were developed in [3] and hey were srong enogh o prove an exisence heorem for a mlidimensional whie noise inegral eqaion. However, as we shall see in a forhcoming paper [8], hese esimaes were no srong enogh o prove he fndamenal resl of he heory, i.e. he niariy condiion. The main resls of he presen paper are: (i): The exension of he whie noise esimaes of [3] o a larger domain (he maximal algebraic domain inrodced in Secion 1) on he lines of [5]. (ii): The inrodcion of he noion of whie noise adapedness and he proof, nder his assmpion (which reqires a whie noise wih 1 dimensional parameer), of he whie noise analoge of he scalar ype inegraor ineqaliies of Accardi, Fagnola and Qaegeber [1] which, in heir rn, generalize he basic Hdson Parhasarahy esimae on sochasic inegrals (cf. Proposiion 5.6 in [15]). Key words and phrases. Whie noise-maximal algebraic domain, Esimaes on whie noise sochasic inegrals. *:Universià di Roma Tor Vergaa, Cenro Vio Volerra 133 Rome, aly. Mail: accardi volerra.ma.niroma.i. **:Universiy of Tnis El Manar, Deparmen of Mahemaics 16 Tnis, Tnisia. Mail: wided.ayed ipein.rn.n. ***:Universiy of Tnis El Manar, Deparmen of Mahemaics 16 Tnis, Tnisia. Mail: habib.oerdiane fs.rn.n. AMS classificaion: 6H4-6H15. 1

2 L. ACCARD*-W. AYED**-H. OUERDANE*** The presen paper is he firs sep of or plan o complee he programme iniiaed in [3] of giving a sysemaic derivaion of he whie noise niariy condiions. Here we only deal wih he basic esimaes on sochasic inegrals and wih he corresponding exisence heorem for sochasic differenial eqaions. Noaions n his secion, in order o fix or noaions, we review some well known maerial. The erms qanm whie noise and free field are synonyms: he former is more sed in mahemaics, he laer in physics. Since classical whie noise is inclded in qanm whie noise, in he following we will se he erm whie noise o mean he more general (qanm) case. A sandard way o consrc whie noises is hrogh he Fock space. We will consider he scalar Boson Fock space over L (R d ): F(L (R d )) : F : no L sym(r dn ) n F n ; F : C F n : L sym(r dn ), n 1,,... is he Hilber space of sqare inegrable fncions of n-variables in R d, symmeric nder he permaion of heir argmens. f S is a sbspace of L (R d ) we will se he noaion F(S) : n syms no which is clearly a sbspace of F(L (R d )) becase, for each n, n syms F n. The elemens of F n are called n-paricle vecors and he se of n-paricle vecors, for all n N, is also called he se of nmber vecors. For an elemen ψ (n) F n we wrie ψ (n) ψ (n) (s 1,..., s n ), s i R d and, for any permaion π, on {1,..., n} one has: ψ (n) (s 1,..., s n ) ψ (n) (s π(1),..., s π(n) ). So an elemen of he Boson Fock space F is a seqence of fncions ψ {ψ (), ψ (1), ψ (),...} where ψ () C, ψ (n) F n, n 1,,... and ψ ψ (n) L (R dn ) < More explicily ψ n ψ () + n1 R dn ψ (n) (s 1,..., s n ) ds 1...ds n The inner prodc of elemens ψ {ψ (n) } n and φ {φ(n) } n in F is given by (ψ, φ) (ψ (n), φ (n) ) ψ () φ () + ψ (n) (s 1,..., s n )φ (n) (s 1,..., s n )ds 1...ds n R dn n n1 The vecor Φ (1,,,...) is called he vacm vecor. A vecor ψ (ψ (n) ) sch ha here exiss f L (R d ) wih he propery ha, n N, we have (.1) ψ (n) (s 1,, s n ) 1 n! f(s 1 )f(s ) f(s n ) ; a.e for n 1 ψ ( ) 1Φ ψ Φ is called an exponenial vecor wih es fncion f and denoed or ψ(f) (ψ (n) f ).

3 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 3 Definiion.1. Le S(R dn ) denoe he Schwarz space, i.e. he se of infiniely differeniable complex valed fncions h on R nd sch ha, for any α, β N nd one has: Define he sbspaces of F: h α,β sp x R nd x α D (β) h(x) < F(S(R d )) : D S : {ψ F : n N, ψ (n) S(R dn )} { } N : ψ D S : ψ (n) for almos all n N (1.1.a) (1.1.b) DS : N D S These are he vecors in F for which all componens, wih he excepion of a mos a finie nmber are eqal o zero. They are called finie paricle vecors. We also define: DS 1 : {ψ D S : n ψ (n) < } (1.1.c) D 1 : {ψ F : n1 n1 n ψ (n) < } (1.1.d) Similarly we define D C, D C, D 1 C by replacing in (1.1.a), (1.1.b), (1.1.c) he Schwarz space S(Rnd ) wih he space C(R nd ) L (R nd ) where C(R nd ) denoes he space of coninos fncions on R nd. Remark.1. Each of he 3 spaces D S, D S, D1 S is dense in F. Definiion.. For any s R d, n N and for any ψ D 1 we fix a represenaive ψ (n) (s 1,,, s n ) in he Lebesge class of ψ (n) and we define: a-: he annihilaion densiy a s as he linear operaor: (.) (.3) (.4) (.5) a s : ψ D 1 (a s ψ) (n) (s 1, s,, s n ) n + 1ψ (n+1) (s, s 1, s,, s n ) F (den1) which associaes o ψ D 1 he F valed linear fncional on he sqare inegrable fncions on R d : f n + 1 dsf(s)ψ (n+1) (s, s 1, s,, s n ) : (A f ψ) (n) (s 1,, s n ) A f is called he annihilaion operaor. follows ha A f ψ a s f(s)ψds. R d b-: he creaion densiy a + s as he linear operaor: a + s : ψ F (a + s ψ) (n) (s 1, s,, s n ) 1 n δ(s s i )ψ (n 1) (s 1,, ŝ i,, s n ) n (den) which associaes o ψ D 1 he F valed linear fncional on he sqare inegrable fncions on R d : f 1 n f(s i )ψ (n 1) (s 1,, ŝ i,, s n ) : (A + n f ψ)(n) (s 1,, s n ) A + f is called he creaion operaor and we ge: (A + f ψ)(n) (s 1,, s n ) 1 n δ(s s i )f(s)ψ (n 1) (s 1,, ŝ i,, s n )ds n R d (a + s ψ) (n) f(s)ds. R d

4 4 L. ACCARD*-W. AYED**-H. OUERDANE*** follows ha (.6) A + f ψ R d a + s f(s)ψds. c-: he nmber densiy is defined on D 1 by: n s a + s a s Using (den1) and (den), for any ψ D 1, one has n (n s ψ) (n) (s 1, s,, s n ) δ(s s i )ψ (n) (s, s 1,, ŝ i,, s n ) which allows o inerpre he nmber densiy in he same way as he creaion densiy. Remark.. Taking f χ [,] in.3 and.6, we obain: (.7) and (.8) A A + o o a s ds a + s ds. (den3) (den4) These noaions are hose for whie noise over R, a, a +, since a Brownian moion ( classical or qanm ) is obained by inegraing a whie noise.also, we ge [4] he commaion relaions: [A, A + s ] min(s, ) ; [A, A s ] [A +, A+ s ] (5.5) A Φ (5.6) Proposiion.1. The operaors inrodced in Definiion (.) are well defined and depend only on he seqence (ψ (n) ) (and no on he represenaives of is elemens). Moreover A f : F n F n 1 ; A f F, n 1,, A + f : F n F n+1, n, 1,, The creaion and he annihilaion operaors are adjoin o each oher on D 1, and we have: (.9) A f Φ Proof. For any measrable fncion ψ (n) he righ hand side of (den1) is well defined and he ideniy ϕ (n 1), (A f ψ) (n 1) f(s n ) ψ (n) (s 1,, s n )ϕ (n 1) (s 1,, s n 1 )ds 1 ds n R dn shows ha he Lebesge class of (A f ψ) (n 1) does no depend on he choice of ψ (n) (s 1,, s n ) b only on is Lebesge class ψ (n). The fac ha ψ (ψ (n) ) D 1 and he Schwarz ineqaliy he seqence ((A f ψ) (n) ) defines a coninos linear fncional on F hence a niqe elemen in F. f ψ D 1 and f L (R d ), A + f ψ is in F becase: A + f ψ (A + f ψ)(n) (s 1,..., s n ) ds 1...ds n n1 R dn ( 1 n n1 n1 R dn n f(s i )ψ (n 1) (s 1,, ŝ i,, s n ) R dn n f(s 1 )ψ (n 1) (s,,, s n ) ds 1...ds n ds 1...ds n

5 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 5 f { n ψ (n 1) L (R (n 1)d ) } < n1 The same argmen sed above shows ha he Lebesge class of (a + f )(n) only depends on he classes of f and ψ (n 1). The remaining saemens are proved in a similar way. Definiion.3. We can exend he definiion of nmber operaor; le T ˆ B(L (R d )) be a pre closed linear operaor wih inegral kernel τ, i.e.for all f L (R d ): T f(x) τ(x, y)f(y)dy The nmber operaor N T is defined by: N T τ(x, y)a + x a y dxdy Noice ha, for all f Dom(T ) L (R d ) and for all exponenial vecors, one has: N T dx dyτ(x, y)f(y)a + x dx(t f)(x)a + x A + T f. We allow τ(x, y) o be a disribion: he choice τ(x, y) V (x)δ(x y) allows o inclde all he mliplicaion operaors. Remark.3. The Boson commaion relaions: [a s1, a + s ] δ(s 1 s ).c are inerpreed weakly on D 1 and easily verified on ha domain. 1. The Maximal Algebraic Domain D 1 is no an invarian domain nder he acion of all creaion, annihilaion, nmber and Weyl operaors. There is a nmber of invarian domains which are sefl in differen siaions [9]. n his secion we inrodce he smalles domain conaining he vacm and invarian nder he acion of all hese operaors. We call i maximal algebraic domain ha is he larges domain obainable from he vacm wih prely algebraic operaions on he basic operaors. Definiion 1.1. The maximal algebraic domain denoed by D MAD is by definiion he linear span of he vecors se { } (1.1) DMAD A + f n A + f 1,, /f, f 1,, f n L (R d ), n 1 where A + f f. for f L (R d ) is he creaor operaor, and is he exponenial vecors wih es fncion Since D MAD conains he exponenial vecors i is dense in F. Lemma 1.1. For every f, f 1,, f n L (R d ) one has Dom(A + f n A + f 1 ); n 1. Proof. We will prove by indcion on n 1, ha for any f, f 1,, f n L (R d ), one has A + f n A + f 1 F, so for n 1, since for all f, g L (R d ), we have (see [3]) : [A f, A + g ] f, g

6 6 L. ACCARD*-W. AYED**-H. OUERDANE*** i follows: A + f 1 A + f 1, A + f 1, A f1 A + f 1 f 1 +, A + f 1 A f1 f 1 + A f1, A f1 f 1 + f 1, f < Sppose ha for any f, f 1,, f n L (R d ), we have A + f n A + f 1 L (R d ). To prove ha A + f n+1 A + f n A + f 1 F, we will se he fac: F and le f, f 1,, f n+1 n+1 A fn+1 A + f n+1 A + f n A + f 1 f n+1, f A + f n A + f 1 + f n+1, f j A + f n+1 Aˆ fj A + f 1 F hen: A + f n+1 A + f 1 A + f n A + f 1, A fn+1 A + f n+1 A + f 1 n+1 f n+1, f A + f n A + f 1 + A + f n A + f 1, A + f n+1 Aˆ fj A + f 1 f n+1, f j j1 ( ) n+1 max f j A + j,,n+1 f n A + f 1 A + f n A + f 1 + A + f n+1 Aˆ fj A + f 1 < Now, o prove some propery of invariance of D MAD, we will recall he definiion of he Fock-Weyl represenaion of L (R d ) given in [3], in he following: he Weyl operaor W f is an elemen of he niary grop U(F) wih he srong operaors opology. Moreover W f acs on he exponenial vecor ψ g as: j1 j1 we also have he following properies 1-: -: W f (ψ g ) e 1 f f,g ψ(g + f). W g Φ e 1 g ψ g (W g ) A f W g A f + i f, g 1..a or eqivalenly [A f, W g ] i f, g W g 1..b Lemma 1.. For every g, f 1,, f n L (R d ) and T ˆ B(L (R d )), he linear span of he vecors A + f n A + f 1 ψ g is he smalles vecor sbspace of F conaining he vacm vecor and invarian nder he acion of he operaors (1.) A + f, A f, W h, N T where W h, N T are he Weyl and he nmber operaors. Moreover, on he domain D MAD, one has: (A + f ) A f (N T ) N T

7 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 7 Proof. Le g, f 1,, f n L (R d ), sing (1..a), he prove of he fac ha A f A + f n A + f 1 ψ g D MAD follows from A f A + f n A + f 1 ψ g f, g A + f n A + f 1 ψ g + n f, f j A + f n Aˆ fj A + f 1 ψ g D MAD. To prove ha W f A + f n A + f 1 ψ g D MAD, i is sfficien o prove ha W f A + f n A + f 1 W g Φ D MAD. Hence j1 W f A + f n A + f 1 W g Φ W f A + f n A + f 1 Wf W f W g Φ n [W f A + f i Wf ]W f W g Φ n [A + f i + i f, f i ]e im f,g W (f+g) Φ {j 1,,j α} {1,,n} λ α [A + f j1 A + f jα W (f+g) Φ] D MAD where λ α : ( i ) n α {h 1,...,h n α }{j 1,...,j α} c {1,...,n} f, f i Le f, f 1 (L (R d )), hen sing (1.3) [N T, A + f ] A+ (T f), N T A + T f, we obain: N T A + f 1 A + f 1 N T + A + T f1 A + f 1 A + T f + T f1, f D MAD by indcion, sppose ha for f, g, f 1,, f n+1 belongs o L (R d ) one has N T A + f n A + f 1 ψ g D MAD and sing (1.3), we ge: N T A + f n+1 A + f n, A + f 1 ψ g A + f n+1 N T A + f n A + f 1 ψ g + A + (T f n+1 ) A+ f n A + f 1 ψ g D MAD Corollary 1.1. Denoe by P W he *algebra generaed by he operaors (1.) acing on D MAD which called he polynomial-weyl algebra. We have: D MAD P W Φ Proof. is clear ha D MAD P W Φ The converse inclsion, i.e: P W Φ D MAD follows easily from he relaions (1..a), (1..b) and (.9).

8 8 L. ACCARD*-W. AYED**-H. OUERDANE***. Whie noise sochasic inegrals n his secion we will discss whie noise and sochasic inegrals in R d raher han in R becase exacly he same formlae are valid in he 1-and in he d-dimensional case, b, as we shall see in Secion (3), some esimaes are slighly worse in he non adaped case, compared o he will be called whie noise adaped. We define he operaors: A(F ) < F, A > ds F s a s ; A + (F ) < A +, F > R d ds a + s F s R d where F is a complex valed fncion on R d. The generalizaion of hese inegrals o he case when F in an operaor valed fncion are called righ (resp. lef) sochasic inegrals wih respec o a s (resp. a + s ). Similarly one defines he wo-sided sochasic inegral: R d ds a + s F s a s. is clear ha, he exisence of lef and wo sided sochasic inegral, (also of righ ones if he inegrand process F s is nbonded), reqires some compaibiliy condiions on he domains. Definiion.1. Le L : L(D) be he space of maps F : R d L(D, F) s F s where L(D, F) is he space of linear operaors on he Fock space F densely defined on a domain D, sch ha for any ϕ, ψ D, he maps are locally inegrable. s R d < ψ, F s ϕ >; s F s ψ ϕ, ψ D Elemens of L will be called processes or, if confsion may arise, D-processes. Remark.1. f D D MAD, hen he map s a s is in L, while he map s a + s is no in L.1. Righ annihilaor sochasic inegral. Definiion.. The righ annihilaor sochasic inegral of F L is he operaor: ψ ds F s a s ψ R d (.1.a) where he inegral is mean as a Bochner inegral in he Fock space. is defined for each ψ F sch ha a s ψ is in he domain of F s for each s and he vecor valed fncion s R d F s a s ψ is Bochner inegrable [14]. Lemma.1. The sochasic inegral (.1.a) is defined for all processes F L and for all vecors ψ DC 1 sch ha n dsf s ψ (n) (s,.) < (.1.b) R d n n his case he inegral (.1.a) is eqal o where ψ (n+1) (s,.) is he fncion R d F s a s dsψ n n + 1 R d dsf s ψ (n+1) (s,.) (s 1,..., s n ) R dn ψ (n+1) (s, s 1,..., s n ) (.1.d)

9 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 9 Proof. We sar from he explici form of he righ annihilaor on a vecor ψ D 1 C (.1) (a s ψ) (n) n + 1ψ (n+1) (s,.) Therefore, in he noaion (.1) F s a s ψds ds n + 1Fs ψ (n+1) (s,.) R d R d n (.1.c) Now by assmpion, for each n, he fncion s R d F s ψ (n+1) (s,.) is Bochner inegrable. Moreover, becase of (.1.b), he series on he righ hand side of (.1.c) is absolely convergen. Therefore, one can exchange he series and he inegral. This gives (.1.d). f ψ ; an he exponenial vecors, he explici form of he sochasic inegral (.1.a) is (.) ds F s a s R d ds f(s)f s R d where he righ hand side of (.) is defined on he se of he exponenial vecors wih es fncion in L (R d ) sch ha he vecor valed fncion s f(s)f s is Bochner inegrable... The lef creaor sochasic inegral. Definiion.3. The lef creaor sochasic inegrals of a measrable elemen F L; F : s F s is he operaor: ψ ds a + s F s ψ R d and i is given by he formla for he scalar case: ( ) (n)(s1 (.3) ds a + s F s ψ,..., s n ) 1 n (F si ψ) (n 1) (s 1,..., ŝ i,..., s n ), n 1 R d n where ψ is in he domain of he operaor F s for all s R +. Remark.. This definiion has a meaning for any measrable fncion s F s, he naral domain of he lef creaor sochasic inegral is {ψ : ( ds a + s F s ψ) (n) n1 R d L (R nd ) < } or more explicily, a vecor ψ in D( ds a + s F s ) if and only if ψ D(F s ), s R d R d n (F s1 ψ) (n 1) (s,..., s n ) ds 1...ds n < n1 R dn We wan now o obain esimaes on he norm of ds a + s F s ψ which garanees ha he sochasic R inegral exis. An example of sch esimaes is given by d he following lemma: Lemma.. Le ψ belong o D(F s ) for all s R d. Then one has, for each n N ( ) (n) (.4) ds a + s F s ψ n ds (F s ψ) (n 1) R d R d

10 1 L. ACCARD*-W. AYED**-H. OUERDANE*** Proof We have ( ) (n) dsa + s F s ψ 1... ds 1...ds n (F si ψ) (n 1) (s 1,..., ŝ i,..., s n ), (F sj ψ) (n 1) (s 1,..., ŝ j,..., s n ) n i,j 1 (Fsi i,j... ds 1...ds n ψ) (n 1) (s 1,..., ŝ i,..., s n ) (F sj ψ) (n 1) (s 1,..., ŝ j,..., s n ) n n ds ds...ds n (F s ψ) (n 1) (s,..., s n ) n ds (F s ψ) (n 1) n R n R d(n 1) R d.3. The normally ordered wo-sided inegral. Definiion.4. n he above noaions, he wo-sided normally ordered inegral of F L: ds a + s F s a s R d is defined,weakly on he maximal algebraic domain: ξ, η D MAD, ξ, ds a + s F s a s η ds a s ξ, F s a s η R d R d n pariclar, on exponenial vecors one has ( ), ds a + s F s a s ψ g ds f(s)g(s), F s ψ g R d R d Lemma.3. For any n N and for any exponenial vecor one has he esimae ( ) (n) (.5) ds a + s F s a s n ds f(s) (F s ) (n 1) R d R d 3. Mlidimensional esimaes on D MAD (C) n his secion, we prove some esimaes on he maximal algebraic domain in he mlidimensional case. Since D MAD is linearly spanned by he vecors of he form A + f n A + f 1, hen i is sefl o have esimaes on he norm of hese vecors. Recall ha F Γ(L (R d )) he symmeric Fock space over L (R d ) and D MAD (C) is he maximal algebraic domain i.e. he linear span of he vecors in he se { } D C A + f n A + f 1,, /f, f 1,, f n C L (R d ) C(R d ), n 1 (4.1.1) Lemma 3.1. We have: a) (3.1) [a s, A + f ] f(s) where f is a es fncion in L (R d ). b) ] (3.) [a s, A + A + fn f1 where f 1,, f n are es fncions in L (R d ). Proof. a) For all ψ D MAD We have: (A + f (ψ))(n) (s 1,, s n ) 1 n n f i (s)a + f n A n f(s i )ψ (n 1) (s 1,, ŝ i,, s n )

11 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 11 Using ha: we obain: (a s ψ) (n) (s 1,, s n ) n + 1ψ (n+1) (s, s 1,, s n ) where s n+1 s. Moreover: hen: (a s A + f (ψ))(n) (s 1,, s n ) n + 1(A + f (ψ))(n+1) (s, s 1,, s n ) n+1 n + 1 f(s i )ψ (n) (s 1,, ŝ i,, s n+1 ) n + 1 (A + f a sψ) (n) (s 1,, s n ) 1 n 1 n n+1 f(s i )ψ (n) (s 1,, ŝ i,, s n+1 ) n f(s i )(a s ψ) (n 1) (s 1,, ŝ i,, s n ) n f(s i ) n(ψ) (n) (s, s 1,, ŝ i,, s n 1 ) ( (a s A + f ψ)(n) (A + f a sψ) (n)) (s 1,, s n ) n+1 f(s i )ψ (n) (s 1,, ŝ i,, s n+1 ) f(s n+1 )ψ (n) (s 1,, s n ) f(s)ψ (n) (s 1,, s n ) ((a s A + f ψ)(n) (A + f a sψ) (n) ) (s 1,, s n ) n f(s i)(ψ) (n) (s, s 1,, ŝ i,, s n 1 ) b) This resl will be proved by indcion on n: le P n be he following propery: ] [a s, A + A + fn f1 n f i (s)a + f n A. Using a) he case n 1 was verified, so we sppose ha P n was verified, hen we will prove P n+1 so: ] [a s, A + A + a fn+1 f1 s A + f n+1 A + f 1 A + f n+1 A + f 1 a s ] [a s, A + A + fn+1 f n A + f 1 + A + f n+1 a s A + f n A + f 1 A + f n+1 A + f 1 a s ] [a s, A ]A + + A + + A [a + fn+1 fn f1 fn+1 s, A + A + fn f1 n f n+1 (s)a + f n A + f 1 + A + f n+1 f i (s)a + f n A n+1 f i (s)a + f n+1 A Proposiion 3.1. (Righ annihilaor esimaes) For each F L and for each bonded sbse R d, i follows :

12 1 L. ACCARD*-W. AYED**-H. OUERDANE*** (3.3) i): for each nmber vecor: ψ A + f n A + f 1 Φ where f 1,, f n are es fncions in L (R d ), we have: ds F s a s ψ c ψ, ϕ J(ψ) ( ) 1 ds F s ϕ where { } J(ψ) : A + f n A Φ, /1 i n D MAD c ψ, : max 1in ( f i, ) (3.1.a) ( f i, : ) 1 f i (s) ds (3.1.b) ii): For each vecor ψ A + f n A + f 1 and for f, f 1,, f n, R d (as in (i) above), (3.3) hold wih J(ψ) {A + f n... Â+ f i, A + f n, 1 i n} (3.1.c) c ψ, max 1in ( f i,, f, ) (3.1.d) so Proof i) Using lemma (3.1) and he fac ha a s Φ, we have for f 1,, f n L (R d ): [ ds F s a s A + f n A + f 1 Φ ds F s a s, A + f n A + f 1 ]Φ ds F s a s A + f n A + f 1 Φ ds F s ds F s n f i (s)a + f n A Φ n f i (s)a + f n A Φ n ds f i (s) F s A + f n A Φ n ( ) 1 ( ) 1 ds f i (s) ds F s A + f n A Φ n ( ) 1 c ψ, ds F s A + f n A Φ and his proves (3.3) where c ψ, is given by (3.1.b). ii) For ψ A + f n A + f 1, where f, f 1,, f n are es fncions and is an exponenial vecor, we have: [ ds F s a s A + f n A + f 1 ds F s a s, A + f n A + f 1 ] + ds F s f(s)a + f n A + f 1 [ n ] ds F s f i (s)a + f n A + f(s)a + f n A + f 1

13 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 13 so: [ ds F s a s A + f n A ˆ + f i A + n ] f 1 ds F s f i (s)a + f n A + f(s)a + f n A + f 1 where c ψ, is as in (3.1.d) [ n ] ds F s f i (s)a + f n A + F s f(s)a + f n A + f 1 [ ds n ] F s f i (s)a + f n A + F s f(s)a + f n A + f1 [ n ] ds f i (s) F s A + f n A + f(s) F s A + f n A + f 1 n ( f i (s) ds ) 1 ( ) 1 ds F s A + f n A ( ) 1 ( ) 1 + f(s) ds ds F s A + f n A + f 1 n ( ) 1 c ψ, ds F s A + f n A ( ) 1 + ds F s A + f n A + f Mlidimensional lef creaor esimae. Lemma 3.. Consider he sochasic inegral A + (F )ψ : ds a + s F s ψ, R d where F L and s, ψ Dom(F s ) sch ha n (F s1 ψ) (n 1) (s,..., s n ) ds 1...ds n <, hen n1 R dn A + (F )ψ where. is he norm defined by: (3.4) ψ : n d F ψ + dsd (F s ψ)(,.) n ψ (n) Proof A + (F )ψ, A+ (F )ψ ds d a + s F s ψ, a + F ψ ds d F s ψ, [a s, a + ]F ψ + d F ψ, F ψ + dsd a F s ψ, a s F ψ. ds d F s ψ, a + a sf ψ

14 14 L. ACCARD*-W. AYED**-H. OUERDANE*** Now consider he inegral: J : ds d a F s ψ, a s F ψ dsd (n + 1) (F s ψ) (n+1) (, σ)(f ψ) (n+1) (s, σ)dσ R nd n Using Cachy Schwars ineqaliy and Lebesge heorem, we ge: J (n + 1) dsd (F s ψ) (n+1) (,.) L (R nd ) n (n + 1) + dsd (F ψ) (n+1) (s,.) L (R nd ) n (n + 1) dsd (F s ψ) (n+1) (,.) L (R nd ) n dsd (F s ψ)(,.) where. is he norm (3.4). Remark 3.1. This esimae in he mlidimensional case is no sefl o prove he convergence of ieraed series, b as i follows, we can find a good esimae in he whie noise adaped and onedimensional case. 4. Whie noise adaped sochasic inegral eqaion n his secion, we will show ha he whie noise adapness condiion gives he opporniy o have more precise resls wih more reglariy. We will generalize some resls proved by Hdson and Parhasarahy in wo direcions: (i) he adapness condiion will be replaced by whie noise adapness [15] (ii) he esimae will be valid no only on he exponenial domain, b on he whole maximal algebraic domain. A firs, we recall ha a sochasic process in F, indexed by R +, is a family (F ) of elemens of L(D, F) saisfying ha for each ψ D, he map F ψ is Borel measrable. Alernaively, a sochasic process indexed by R + can be looked as a map R + F L(D, F) wih he above menioned measrabiliy propery. We remark ha elemens of L can be regarded as sochasic processes. n he nex, we shall only deal wih processes indexed by R +. Definiion 4.1. A process (F ) is said o be whie noise adaped if for any m {a, a +, a a + }, we have: (4.1) ψ D MAD, F s m ψ m F s ψ, s <. Lemma 4.1. f a process (F ) is adaped in he sense of Hdson and Parhasarahy, i is whie noise adaped. Proof. Le (F ) an adaped process in he sense of Hdson and Parhasarahy [15], which means ha for any exponenial vecor ] [, for R + one has: F (F ] ) [

15 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 15 so a s F a s (F ] ) [, s > a s F (F ] ) a s [, s > F a s, s > Then, we conclde ha (F ) is whie noise adaped The Hdson-Parhasarahy esimae. n he following secion we will prove ha he whie noise approach allows o obain a sronger resl. n fac he nex esimae was proved by Hdson and Parhasarahy [15], in he adaped case and here we given a differen proof of he same resl b for general kind of processes, and his sing he whie noise approach. Proposiion 4.1. For all srongly coninos processes (F s ) s on D MAD (C), we have: (4.) ds F s a s e ds F s f(s) so Proof. Le be an exponenial vecor wih es fncion f, hen, we denoe by: () () dsf s a s ds F s a s For fixed, d > and arbirary fncion F (), we se he noaion df () F ( + d) F () (finie difference). n his noaion one has he algebraic ideniy: d (), () d(), () + d(), d() + d(), d() Denoe Re Re +d +d ds F s f(s), () + ds f(s) F s, () + +d +d h( 1, ) : f( 1 )f( ) F 1, F d 1 F 1 f( 1 ), +d d F f( ) +d d 1 d f( 1 )f( ) F 1, F. Since he map s F s is coninos, we dedce ha he map ( 1, ) h( 1, ) is niformly bonded on [, + d] for each R +, in fac: sing his and he ideniy 1 d Re we ge, for d herefore +d h( 1, ) f sp s [,+d] 1 d d (), () F s f(s)ds, () + 1 d F s +d d d (), () Re F f(), () +d d 1 d h( 1, ) d d (), () F f() + ()

16 16 L. ACCARD*-W. AYED**-H. OUERDANE*** Ths, by Gronwall lemma: which is (4.). () e d f() F dimensional, whie noise adaped lef creaor esimae. We need he following preliminary resl, which is re in arbirary dimensions. Lemma 4.. For all ψ in D MAD (C) saisfying (4.3) ψ : n n ψ (n) <. Then for each s 1 R d, he following series: (n + 1) ds ds n+1 ψ (n+1) (s 1, s,, s n+1 ) R dn is finie a.e. n Proof Using he Dini heorem and (4.3), we ge ψ ds 1 (n + 1) ds ds n+1 ψ (n+1) (s 1, s,, s n+1 ) <. R d R dn n Then, we dedce ha for each s 1 in R d we have (n + 1) ds ds n+1 ψ (n+1) (s 1, s,, s n+1 ) < a.e. R dn n. Proposiion 4.. (1-dimensional lef creaor esimaes) For all whie noise adaped processes (F s ) s and all ψ D MAD (C) sch ha he map ( 1, ) F 1 ψ(,.) is coninos for 1, in every inerval of R + for he norm (3.4), we have, for any T < + (4.4) dsa + s F s ψ e (C T,ψ J(ψ) ) F s ψ i ds where J(ψ) is he sbse of F defined by. denoes he cardinaliy, ψ i J(ψ) J(ψ) {ψ i A + f n... Â+ f i, A + f n 1 i n} D MAD, C T,ψ where f,t sp [,T ] f(). Proof. Le ψ A + f n A + f 1, and max c T,ψ i, c T,ψi max ( f j,t, f,t ). ψ i J(ψ) 1jn A (F )ψ ds a + s F s ψ so o esimae A (F )ψ i is sfficien o esimae d A (F )ψ, A (F )ψ for his we obain d A (F )ψ, A (F )ψ da (F )ψ, A (F )ψ + A (F )ψ, da (F )ψ + da (F )ψ, da (F )ψ (4.1.a)

17 For he erm da (F )ψ, da (F )ψ, we obain Denoe by: A : WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 17 Re da (F )ψ, A (F )ψ + da (F )ψ, da (F )ψ da (F )ψ, da (F )ψ +d A +d +d +d +d +d + +d +d + d 1 a + 1 F 1 ψ, +d d a + F ψ +d d 1 d a + 1 F 1 ψ, a + F ψ +d d 1 d F 1 ψ, a 1 a + F ψ +d d 1 d F 1 ψ, δ( 1 )F ψ +d d 1 d F 1 ψ, a + a 1 F ψ d 1 F 1 ψ, F 1 ψ +d d 1 d a F 1 ψ, a 1 F ψ hen, we ge: +d +d d 1 d n +d d 1 d a F 1 ψ, a 1 F ψ R nd (a F 1 ψ) (n) (s 1, s,, s n ) (a 1 F ψ) (n) (s 1, s,, s n )ds 1... ds n +d +d d 1 d (n + 1) (F 1 ψ) (n+1) (, s 1, s,, s n ) R nd n where: (F ψ) (n+1) ( 1, s 1, s,, s n )ds 1... ds n +d +d d 1 d g( 1, ) g( 1, ) : (n + 1) (F 1 ψ) (n+1) (, σ)(f ψ) (n+1) ( 1, σ)dσ a.e. R dn n : NF 1 ψ(,.), F ψ( 1,.) The coniniy of he map: ( 1, ) g( 1, )a.e. on [, + d], is a conseqence of he following ineqaliies: ɛ > g( 1 + ɛ, + ɛ) g( 1, ) NF 1 +ɛψ( + ɛ,.), F +ɛψ( 1 + ɛ,.) NF 1 ψ(,.), F ψ( 1,.) N(F 1 +ɛψ( + ɛ,.) F 1 ψ(,.)), F +ɛψ( 1 + ɛ,.) + NF 1 ψ(,.), F +ɛψ( 1 + ɛ,.) F ψ( 1,.) N(F 1 +ɛψ( + ɛ,.) F 1 ψ(,.). F +ɛψ( 1 + ɛ,.) + NF 1 ψ(,.). (F +ɛψ( 1 + ɛ,.) F ψ( 1,.) and of he coniniy of ( 1, ) F 1 ψ(,.) in he norm (3.4). Since g is bonded in [, + d] i follows ha: A β(d), where β >, i.e. A O(d ). n conclsion: 1 d d A (F )ψ, A (F )ψ 1 d Re +d dsa + s F s ψ, A (F )ψ + 1 d +d d 1 F 1 ψ, F 1 ψ + 1 d A

18 18 L. ACCARD*-W. AYED**-H. OUERDANE*** and, if d, we ge d d A (F )ψ, A (F )ψ Re a + F ψ, dsa + s F s ψ + F ψ, F ψ. Le ψ A + f m in D MAD (C) where f, f 1... f m are es fncions in C, and F is a whie noise adaped process, for each, so we obain: since: so: where hen a + F ψ, ds a + s F s a A + f n We denoe by where ds a + s F s ψ ds F ψ, a a + s F s ψ ds F ψ, a + s F s a ψ ds F ψ, a + s F s a ψ ds a + s F s [a, A + f n ] ds + ds a + s F s [ n ds a + s F s dsf()a + f n ] f i ()A + f n... Â+ f i + f()a + f n ds a + s F s a s A + f n... Â+ f i... A + f i [ n ] ds a + s F s f i ()A + f n... Â+ f i + f()a + f n n [ ] ds a + s F s f i ()A + f n... Â+ f i... A + f i + ds a + s F s f()a + f n n ds f i ()a + s F s A + f n... Â+ f i... A + f 1 + ds a + s F s f()a + f n [ n c T,ψ We noe ha ψ i F ψ : ds a + s F s A + f n... Â+ f i + c T,ψ max 1in ( f i,t, f,t ) ds a + s F s A + f n ], d d A (F )ψ, A (F )ψ n F ψ + c T,ψ [ A (F )ψ ] + ds a + s F s A + f n... Â+ f i A (F )ψ F ψ ψ i F ψ A (F )ψ i F ψ {ψ i A + f n... Â+ f i, A + f n, 1 i n} A (F )ψ i A (F )ψ F ψ

19 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 19 and A (F )ψ i ds F s ψ i + c T,ψi j ds A s (F )ψ ij A (F )ψ F ψ where C T,ψ max ψi F c T,ψi. A (F )ψ F ψ ψ i F ψ F s ψ i ds + C T,ψ F ψ ψ i F ψ F s ψ i ds + C T,ψ F ψ by Gronwall lemma, we obain A (F )ψ A (F )ψ F ψ e (C T,ψ F ψ ) ψ i F ψ A s (F )ψ F ψ ds A s (F )ψ F ψ ds F s ψ i ds. So we ge he resl where J(ψ) F ψ. Corollary 4.1. For ψ, we have, sing he fac ha F is whie noise adaped and Gromwall lemma: ds a + s F s e f,t ds [ f + 1] F s. The above esimae is similar o hose proved by Hdson and Parhasaray on he exponenial vecors. Corollary 4.. Le (F ) be a whie noise adaped process sch ha, for each n N and es fncions f, f 1 f n in C, he map ( 1, ) F 1 ψ(,.); ψ A + f n A + f 1 D MAD (C) is coninos nder he norm (3.4). Then he following ineqaliy holds for any T < : (4.5) ds a + s F s a s ψ c T,ψ ds F s ψ i ψ i J(ψ) where J(ψ) D, c T,ψ are defined by { } J(ψ) A + f n A + f 1, A + f n A, : 1 i n c T,ψ n max 1in ( f i,t, f,t )c T,ψ max 1in (e(c T,ψ F )T, e (c T,ψ i F i )T ) where f i,t sp s [,T ] f(s) and c T,ψ, c T,ψi are defined as in (4.1.a)

20 L. ACCARD*-W. AYED**-H. OUERDANE*** Proof. Using lemma (4.1), we ge: ds a + s F s a s ψ [ n ] ds a + s F s F s f i (s)a + f n A + f(s)a + f n A + f 1 + ds a + s F s n F s f i (s)a + f n A ˆ + f i A + f 1 ds a + s F s f(s)a + f n A + f1 [ n n. max ( f i,t ) 1in + f,t ds a + s F s A + f n A + f1 ds a + s F s A + f n A ψ ] max (n f i,t f,t ) max 1in 1in (e(c T,ψ F )T, e (c T,ψ i F i )T ) n [ ds F s A + f n A ] + ds F s A + f n A + f1. where (c T,ψ F ), (c T,ψi F i ) are defined as in (4.1.a). Corollary 4.3. Le (F ) be a whie noise adaped process sch ha, for each n N and es fncions f, f 1 f n in C, he map ( 1, ) F 1 ψ(,.); ψ A + f n A + f 1 D MAD (C) is coninos nder he norm (3.4). Then he following ineqaliies hold for any T < : (4.6) (4.7) a-: b-: where and ds F s a s ψ c ψ, ϕ J(ψ) ( ) 1 ds F s ϕ J(ψ) {A + f n... Â+ f i, A + f n, 1 i n} dsa + s F s ψ c ψ, max 1in ( f i,, f, ) e (C T,ψ J(ψ) ) where J(ψ) is he sbse of F defined by ψ i J(ψ) F s ψ i ds J(ψ) {ψ i A + f n... Â+ f i, A + f n 1 i n} D MAD,. denoes he cardinaliy, C T,ψ where f,t sp [,T ] f(). c-: (4.8) ds a + s F s a s ψ c T,ψ max c T,ψ i, c T,ψi max ( f j,t, f,t ). ψ i J(ψ) 1jn ds ψ i J(ψ) F s ψ i (4.1.a)

21 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 1 where J(ψ) D, c T,ψ are defined by { } J(ψ) A + f n A + f 1, A + f n A, : 1 i n c T,ψ n max 1in ( f i,t, f,t )c T,ψ max 1in (e(c T,ψ F )T, e (c T,ψ i F i )T ) where f i,t sp s [,T ] f(s) and c T,ψ, c T,ψi are defined as in (4.1.a). Proof. a- is obained by aking [, ] in proposiion 3.1. Also he proof of b- and c- are obained by replacing o by in hose of proposiion 4. and corollary 4.1. Corollary 4.4. Le H S a Hilber space called iniial space. The ineqaliies (3.3), (4.4), (4.5) are verified if we ake ψ ψ H S F where is in he iniial space H S and ψ is an elemen of he maximal algebraic domain. Proof. Using he fac ha: F ψ F ψ and he ineqaliies (3.3), (4.4) and (4.5), one has: ( ) 1 F s a s ds ψ c ψ, ds F s ϕ ϕ J(ψ) dsa + s F s ψ e (c T,ψ J(ψ) ) a + s F s a s ds ψ c T,ψ ψ i J(ψ) ds ψ i J(ψ) F s ψ i ds F s ψ i where c ψ,, c T,ψ and c T,ψ are defined in proposiions (3.1), (4.) and Corollary (4.1). 5. Normally ordered whie noise sochasic differenial eqaion 5.1. nrodcion and definiions. The normally ordered whie noise sochasic differenial eqaion: (5.1) U a + E U + F U a + a + G U a + H U U() U where (E ), (F ), (G ) and (H ) be measrable and locally bonded operaors acing on he iniial space H S. The meaning of eqaion (5.1) can be specified in wo differen ways: (i): as an inegral eqaion, i.e. (5.) U U + ds a + s E s U s + ds F s U s a s + ds a + s G s U s a s + ds H s U s where he inegrals on he righ hand side are defined in secion (3.1). (ii): as a weak eqaion in some domain D 1 ( or choice of his domain will be specified laer): (5.3) φ, U ψ a φ, E U ψ + φ, F U a ψ + a φ, E U a ψ + φ, E U ψ where φ and ψ are in D 1. Since also he noion of sochasic inegral reqires he specificaion of some domain, he wo mehods may lead o ineqivalen noions of solion. However he main goal of he heory is o prodce solions which are niary (in pariclar bonded). Wihin his class i can be proved ha, for a large family of coefficien processes (inclding he consan bonded ones, which are he mos sed in applicaions) he wo eqaions lead o he same solion.

22 L. ACCARD*-W. AYED**-H. OUERDANE*** 5.. Whie noise adaped normally ordered whie noise eqaion in R d. Using he above resls, we will prove he exisence and he niqeness of solion of a class of whi noise adaped normally ordered whie noise eqaion by: Definiion 5.1. A whie noise adaped normally ordered whie noise eqaion wih coefficiens is a normally ordered whie noise eqaion as defined above: ] (5.4) U [F 1 U a + a + F U + a + F 3 U a + F 4 U d where (F 1 ), (F ), (F 3 ) and (F 4 ) are, whie noise adaped processes in B(H S ) coninos for he norm operaor opology on B(H S ) Exisence and niqeness of he solion of a WN socahsic eqaion. Theorem 5.1. Consider he normally ordered whie noise eqaion: (5.5) U U + ds F 1 s U s a s + ds a + s F s U s + ds a + s F 3 s U s a s + ds F 4 s U s where (F 1 s ) s, (F s ) s, (F 3 s ) s and (F 4 s ) s are locally bonded coninos processes for he norm operaor opology on B(H S ) sch ha he maps: ( 1, ) F 1 ψ(,.), ( 1, ) F 3 1 ψ(,.) (6.6) are coninos for he norm (3.4) for each ψ D MAD (C) and 1, R +. Under he above condiions, eqaion (5.5) has a niqe whie noise adaped locally bonded coninos solion for all U whie noise adaped srongly coninos process on D MAD (C). Proof. i) Exisence. Define by indcion U U, and U n+1 F 1 s U n s a s ds + For simpliciy, we se he noaion: a + s F s U n s ds + U n+1 F i su n s dm i s a + s F 3 s U n s a s ds + F 4 s U n s ds The seqence (U n ) n N is well defined. n fac U is a srongly coninos process on D MAD (C). Sppose ha k n, U k is a whie noise adaped process on D MAD (C), srongly coninos and sch ha he map : ( 1, ) U k 1 ψ(,.) (1.1) is coninos for he norm (3.4). Using he scalar ype inegraor ineqaliies (3.3), (4.4) and (4.5), we have for any T, and each i {1,, 3, 4}: F 1 s U n s a s dsξ J 1 (ξ) (c T,1,η ) a + s F s U n s ds c T,,η a + s F 3 s U n s a s ds c T,3,η F 4 s U n s ds c T,4,η η J (ξ) η J 1 (ξ) η J 3 (ξ) η J 4 (ξ) ds F 1 s U n s η ds F s U n s η ds F 3 s U n s η ds F 4 s U n s η

23 WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 3 where c T,i,η, J i (ξ), i 1,, 3, 4 are defined as in (3.3), (4.4)and (4.5). Then U n ()ξ dm i 1 F i ( 1 )U n 1 (5.6) 1 ξ (5.7) c T,ξ K T ξ 1 J(ξ) d 1 U n 1 1 ξ 1 where: c T,ξ 4 max c T,i,η ; J(ξ) J i (ξ), K T sp F s. i {1,,3,4},η J(ξ) i s [,T ] Le s now prove ha U n+1 saisfy (1.1). n fac ɛ, ɛ > : U n+1 1 +ɛ ψ( + ɛ,.) U n+1 1 ψ(,.) U n+1 1 +ɛ ψ( + ɛ,.) U n+1 1 ψ( + ɛ,.) sing he ineqaliy 5.6, we ge: {c 1,ψ ψ i 1 +ɛ 1 ds F i su n s ψ i ( + ɛ,.) } 1 + {c 1 1,ψ + U n+1 1 ψ( + ɛ,.) U n+1 1 ψ(,.) + 1 +ɛ ψ j 1 dsf i su n s dm i sψ( + ɛ,.) 1 sing or hypohesis, we have he coniniy of he map 1 s F i su n s ψ i ( + ɛ,.) dsf i su n s dm i sψ( + ɛ,.) ψ(,.) U n+1 1 +ɛ ψ( + ɛ,.) U n+1 1 ψ(,.) ds F i su n s [ψ j ( + ɛ,.) ψ(,.) } 1 so i will be niform bonded on he compac [, + ɛ], hen, we obain ha: 1 +ɛ {c 1,ψ ds U s n ψ i ( + ɛ,.) } 1 < Kɛ, K > 1 ψ i Moreover nder he assmpion, he erms 1 ds U s n [ψ j ( + ɛ,.) ψ(,.) } 1 {c 1 1,ψ ψ j vanish when ɛ,so we ge he resl. Becase of he above hypohesis of he indcion, for all Ms i {a s, a + s, a s a + s } and i {1,, 3, 4}, FsU i s n is a coninos process inegrable wih respec o Ms, i hen we have ha U n+1 is a coninos process on D MAD (C) wih he same propery as U n. follows by indcion ha U n is srongly coninos on D MAD (C) for all n N. We will prove now he basic esimae: (5.8) U n ξ max η J(ξ) η. U. J(ξ) n KT n c n T,ξ (µ ξ(, )) n 1 n! For all n N and all T, where: c T,ξ 4 max c T,i,η, J(ξ) J i (ξ), K T sp F s. i {1,,3,4},η J(ξ) i s [,T ] Using he scalar ype inegraor ineqaliies (3.3), (4.4)and (4.5), we have for each i {1,, 3, 4}: Fs 1 Us n a s dsξ J 1 (ξ) (c T,1,η ) ds Fs 1 Us n η η J 1 (ξ)

24 4 L. ACCARD*-W. AYED**-H. OUERDANE*** a + s F s U n s ds c T,,η a + s F 3 s U n s ab s ds c T,3,η F 4 s U n s ds c T,4,η η J (ξ) η J 3 (ξ) η J 4 (ξ) ds F s U n s η ds F 3 s U n s η ds F 4 s U n s η where c T,i,η, J i (ξ), i 1,, 3, 4 are defined as in (3.3), (4.4)and (4.5). Then U n ()ξ dm i 1 F i ( 1 )U n 1 1 ξ c T,ξ K T d 1 U n 1 1 ξ 1 where: c T,ξ K T ξ 1 J(ξ) d 1 ( 1 ξ 1 J(ξ) ) dm i F i U n ξ 1 c T,ξ 4 max c T,i,η ; J(ξ) J i (ξ), K T sp F s i {1,,3,4},η J(ξ) i s [,T ] becase of he condiions on F i, hen: U n ()ξ c T,ξ K T Using he same ineqaliy again c T,ξ K T J(ξ) ξ 1 J(ξ) ξ J(ξ) d 1 1 d 1 1 An n-fold ieraion of he same argmens gives s he esimae: U n ()ξ c n T,ξ Kn T J(ξ) n 1 ξ n J(ξ) ) dm i (F i U n ξ 1 d U n 1 ξ 1 d 1 d max η J(ξ) η. U c n T,ξ Kn T J(ξ) n n 1 n! n 1 d n U ξ n Therefore he series n U n converges in he srong opology on D MAD (C) niformly on bonded inervals of R +. This implies ha i defines a process U srongly coninos on D MAD (C). We show ha U is a solion of (5.5). By he inegraor of scalar ype esimae we have, for all n N, n FsU i s n dmsξ i dmsf i s i Us k ξ c T,ξ K T Us k η ds k hen, we obain for all s [, ], k N, η J(ξ) η J(ξ) U k s η max η J(ξ) η U J(ξ) k K k+1 T c k T,ξ k 1 k! kn+1 hen he series k U k (s)η converges in he srong opology on he Fock space niformly on bonded inervals of R +. follow from Lebesge heorem ha lim n dm i s ( kn Fs i k U k s ) dm i sf i su s

25 This ogeher wih he ideniy WHTE NOSE APPROACH TO STOCHASTC NTEGRATON 5 kn+1 k U k ξ U ξ + dms i ( kn Fs i implies ha U verifies he sochasic differenial eqaion (5.5). ii)niqeness will be sfficien o prove ha all bonded coninos process Z, R + saisfying he following whie noise sochasic differenial eqaion: Z F i sz s dm i s ms be zero. n fac, for all ψ D MAD (C), applying or esimaes, we have: Z ψ c T,ψ K T Z s η ds η J(ψ) applying again he esimaes o he inegral in he righ-hand side (n - 1) imes and comping he ieraed inegral as we did before, we obain: k U k s ) ξ Z ψ sp st Z s max η J(ψ) η J(ψ) n KT n c n 1 T,ψ n n! Since his is re for all n N, i follows ha Z ψ for all [, T ]. References [1] L.Accardi, F.Fagnola and J.Qaegeber: A Represenaion Free Qanm Sochasic Calcls, A represenaion free Qanm Sochasic Calcls, Jorn. Fnc. Anal. 14 (1) (199) Volerra preprin N. 18 (199). [] L. Accardi, F. Fagnola: Sochasic negraion, in Qanm Probabiliy and Aplicaion, Lecre Noes in Mah, Vol. 133, pp, 6-19, Springer-Verlag, New York, (1988). [3] L.Accardi,.V.Vlovich and Y.G.L: A Whie Noise Approach o Classical and Qanm Sochasic Calcls, Volerra Preprin 375, Rome, Jly [4] L.Accardi: Qanm Probabiliy: An nrodcion o Some Basic deas and Trends, Modelos Esocasicos 16, Sociedad Mahemaica Mexicana, 1. [5] Accardi L., Qaegeber J.: o algebras of Gassian qanm fields, Jorn. Fnc. Anal. 85 (1988) [6] L.Accardi, Y.G.L.,.V.Volovich: Qanm heory and is sochasic limi. Springer Verlag () [7] Accardi L., L Y.G., Volovich.: Nonlinear exensions of classical and qanm sochasic calcls and essenially infinie dimensional analysis, in: Probabiliy Towards ; L. Accardi, Chris Heyde (eds.) Springer LN in Saisics 18 (1998) 1 33 Proceedings of he Symposim: Probabiliy owards wo hosand, Colmbia Universiy, New York, 6 Ocober (1995) [8] L. Accardi, W. Ayed and H. Oerdiane: Whie Noise Flows, in preparaion. [9] S. Aal and J. M. Lindsay: Qanm Sochasic Calcls wih maximal operaor domains.the annales of probabiliy, (4), Vol 3. [1] Ph. Biane: Calcl Sochasiqe non-commaif, Séminaire de Probabiliés XXX. Lecre Noe in Mah(168), Springer, Berlin. [11]. Gelfand and N. Vilenkin: Generelized fncions vol.1. Academic Press. [1] T. Hida: Brownian Moion. Springer, 199. [13] Hdson, R. L. and Parhasarahy, K. R.: Qanm ô s formla and sochasic evolion, Comm. Mah. Phys 93(1984), [14] H. Ko: Whie Noise Disribion Theory. C.R.C Press, Boca Raon, New York, London, Tokyo, (1996). [15] K. R. Parhasarahy: An nrodcion o Qanm Sochasic Calcls.(199)Birkhäser Verlag. Basel.Boson. Berlin. [16] P. A. Meyer: Qanm Probabiliy for Probabiliss, (1993), Springer, Berlin. [17] L. Schwarz: Theorie des disribions. Hermann, Paris. [18] K. Yosida: Fncional Analysis. Springer, (1978).