Stochastic Analysis in Discrete and Continuous Settings

Transcription

1 Nicolas Privaul Sochasic Analysis in Discree and Coninuous Seings Wih Normal Maringales

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3 Sochasic Analysis in Discree and Coninuous Seings Preface This monograph is an inroducion o some aspecs of sochasic analysis in he framework of normal maringales, in boh discree and coninuous ime. The ex is mosly self-conained, excep for Secion 5.7 ha requires some background in geomery, and should be accessible o graduae sudens and researchers having already received a basic raining in probabiliy. Prerequisies are mosly limied o a knowledge of measure heory and probabiliy, namely σ-algebras, expecaions, and condiional expecaions. A shor inroducion o sochasic calculus for coninuous and jump processes is given in Chaper 2 using normal maringales, whose predicable quadraic variaion is he Lebesgue measure. There already exiss several books devoed o sochasic analysis for coninuous diffusion processes on Gaussian and Wiener spaces, cf. e.g. [53, [65, [67, [76, [87, [88, [96, [132, [138, [147, [15, [151. The paricular feaure of his ex is o simulaneously consider coninuous processes and jump processes in he unified framework of normal maringales. These noes have grown from several versions of graduae courses given in he Maser in Imaging and Compuaion a he Universiy of La Rochelle and in he Maser of Mahemaics and Applicaions a he Universiy of Poiiers, as well as from lecures presened a he universiies of Ankara, Greifswald, Marne la Vallée, Tunis, and Wuhan, a he inviaions of G. Walle, M. Arnaudon, H. Körezlioǧlu, U. Franz, A. Sulem, H. Ouerdiane, and L.M. Wu, respecively. The ex has also benefied from consrucive remarks from several colleagues and former sudens, including D. David, A. Joulin, Y.T. Ma, C. Pinoux, and A. Réveillac. I hank in paricular J.C. Breon for numerous suggesions and correcions. Hong Kong, May 29 Nicolas Privaul v February 23, 218 (wih correcions).

4 N. Privaul vi February 23, 218 (wih correcions).

5 Conens Inroducion The Discree Time Case Normal Maringales Sochasic Inegrals Muliple Sochasic Inegrals Srucure Equaions Chaos Represenaion Gradien Operaor Clark Formula and Predicable Represenaion Divergence Operaor Ornsein-Uhlenbeck Semi-Group and Process Covariance Ideniies Deviaion Inequaliies Logarihmic Sobolev Inequaliies Change of Variable Formula Opion Hedging Noes and References Coninuous-Time Normal Maringales Normal Maringales Brownian Moion Compensaed Poisson Maringale Compound Poisson Maringale Sochasic Inegrals Predicable Represenaion Propery Muliple Sochasic Inegrals Chaos Represenaion Propery Quadraic Variaion Srucure Equaions Produc Formula for Sochasic Inegrals vii

6 N. Privaul 2.12 Iô Formula Exponenial Vecors Vecor-Valued Case Noes and References Gradien and Divergence Operaors Definiion and Closabiliy Clark Formula and Predicable Represenaion Divergence and Sochasic Inegrals Covariance Ideniies Logarihmic Sobolev Inequaliies Deviaion Inequaliies Markovian Represenaion Noes and References Annihilaion and creaion operaors Dualiy Relaion Annihilaion Operaor Creaion Operaor Ornsein-Uhlenbeck Semi-Group Deerminisic Srucure Equaions Exponenial Vecors Deviaion Inequaliies Derivaion of Fock Kernels Noes and References Analysis on he Wiener Space Muliple Wiener Inegrals Gradien and Divergence Operaors Ornsein-Uhlenbeck Semi-Group Covariance Ideniies and Inequaliies Momen ideniies for Skorohod inegrals Differenial Calculus on Random Morphisms Riemannian Brownian Moion Time Changes on Brownian Moion Noes and References Analysis on he Poisson space Poisson Random Measures Muliple Poisson Sochasic Inegrals Chaos Represenaion Propery Finie Difference Gradien Divergence Operaor Characerizaion of Poisson Measures Clark Formula and Lévy Processes viii February 23, 218 (wih correcions).

7 Sochasic Analysis in Discree and Coninuous Seings 6.8 Covariance ideniies Deviaion Inequaliies Noes and References Local Gradiens on he Poisson space Inrinsic Gradien on Configuraion Spaces Damped Gradien on he Half Line Damped Gradien on a Compac Inerval Chaos Expansions Covariance Ideniies and Deviaion Inequaliies Some Geomeric Aspecs of Poisson Analysis Chaos Inerpreaion of Time Changes Noes and References Opion Hedging in Coninuous Time Marke Model Hedging by he Clark Formula Black-Scholes PDE Asian Opions and Deerminisic Srucure Noes and References Appendix Measurabiliy Gaussian Random Variables Condiional Expecaion Maringales in Discree Time Maringales in Coninuous Time Markov Processes Tensor Producs of L 2 Spaces Closabiliy of Linear Operaors References Index ix February 23, 218 (wih correcions).

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9 Inroducion Sochasic analysis can be viewed as a branch of infinie-dimensional analysis ha sems from a combined use of analyic and probabilisic ools, and is developed in ineracion wih sochasic processes. In recen decades i has urned ino a powerful approach o he reamen of numerous heoreical and applied problems ranging from exisence and regulariy crieria for probabiliy densiies and soluions of parial differenial equaions by he Malliavin calculus, o funcional and deviaion inequaliies, mahemaical finance, and anicipaive exensions of sochasic calculus. The basic ools of sochasic analysis consis in a gradien and a divergence operaor which are linked by an inegraion by pars formula. Such gradien operaors can be defined by finie differences or by infiniesimal shifs of he pahs of a given sochasic process. Whenever possible, he divergence operaor is conneced o he sochasic inegral wih respec o ha same underlying process. In his way, deep connecions can be esablished beween he algebraic and geomeric aspecs of differeniaion and inegraion by pars on he one hand, and heir probabilisic counerpar on he oher hand. Noe ha he erm sochasic analysis is also used wih somewha differen significaions especially in engineering or applied probabiliy; here we refer o sochasic analysis from a funcional analyic poin of view. Le us urn o he conens of his monograph. Chaper 1 sars wih an elemenary exposiion in a discree seing in which mos of he basic ools of sochasic analysis can be inroduced. The simple seing of he discree case sill capures many imporan properies of he coninuous-ime case and provides a simple model for is undersanding. I also yields non rivial resuls such as concenraion and deviaion inequaliies, and logarihmic Sobolev inequaliies for Bernoulli measures, as well as hedging formulas for coningen claims in discree ime financial models. In addiion, he resuls obained in he discree case are direcly suiable for compuer implemenaion. We sar by inroducing discree ime versions of he gradien and divergence operaors, of chaos expansions, and of he predicable represenaion propery. We wrie he discree ime srucure equaion saisfied by a sequence (X n ) n N of independen Bernoulli random variables defined on he probabiliy space Ω = { 1, 1} N, we consruc he associaed discree 1

10 N. Privaul muliple sochasic inegrals and prove he chaos represenaion propery for discree ime random walks wih independen incremens. A gradien operaor D acing by finie differences is inroduced in connecion wih he muliple sochasic inegrals, and used o sae a Clark predicable represenaion formula. The divergence operaor δ, defined as he adjoin of D, urns ou o be an exension of he discree-ime sochasic inegral, and is used o express he generaor of he Ornsein-Uhlenbeck process. The properies of he associaed Ornsein-Uhlenbeck process and semi-group are invesigaed, wih applicaions o covariance ideniies and deviaion inequaliies under Bernoulli measures. Covariance ideniies are saed boh from he Clark represenaion formula and using Ornsein-Uhlenbeck semigroups. Logarihmic Sobolev inequaliies are also derived in his framework, wih addiional applicaions o deviaion inequaliies. Finally we prove an Iô ype change of variable formula in discree ime and apply i, along wih he Clark formula, o opion pricing and hedging in he Cox-Ross-Rubinsein discree-ime financial model. In Chaper 2 we urn o he coninuous ime case and presen an elemenary accoun of coninuous ime normal maringales. This includes he consrucion of associaed muliple sochasic inegrals I n (f n ) of symmeric deerminisic funcions f n of n variables wih respec o a normal maringale, and he derivaion of srucure equaions deermined by a predicable process (φ ) R+. In case (φ ) R+ is a deerminisic funcion, his family of maringales includes Brownian moion (when φ vanishes idenically) and he compensaed Poisson process (when φ is a deerminisic consan), which will be considered separaely. A basic consrucion of sochasic inegrals and calculus is presened in he framework of normal maringales, wih a proof of he Iô formula. In his chaper, he consrucion of Brownian moion is done via a series of Gaussian random variables and is pahwise properies will no be paricularly discussed, as our focus is more on connecions wih funcional analysis. Similarly, he noions of local maringales and semimaringales are no wihin he scope of his inroducion. Chaper 3 conains a presenaion of he coninuous ime gradien and divergence in an absrac seing. We idenify some minimal assumpions o be saisfied by hese operaors in order o connec hem laer on o sochasic inegraion wih respec o a given normal maringale. The links beween he Clark formula, he predicable represenaion propery and he relaion beween Skorohod and Iô inegrals, as well as covariance ideniies, are discussed a his level of generaliy. This general seing gives rise o applicaions such as he deerminaion of he predicable represenaion of random variables, and a proof of logarihmic Sobolev inequaliies for normal maringales. Generic examples of operaors saisfying he hypoheses of Chaper 2 can be consruced by addiion of a process wih vanishing adaped projecion o he gradien operaor. Concree examples of such gradien and divergence operaors will be described in he sequel (Chapers 4, 5, 6, and 7), in paricular in he Wiener and Poisson cases. 2 February 23, 218 (wih correcions).

11 Sochasic Analysis in Discree and Coninuous Seings Chaper 4 inroduces a firs example of a pair of gradien and divergence operaors saisfying he hypoheses of Chaper 3, based on he noion of muliple sochasic inegral I n (f n ) of a symmeric funcion f n on R n + wih respec o a normal maringale. Here he gradien operaor D is defined by lowering he degree of muliple sochasic inegrals (i.e. as an annihilaion operaor), while is adjoin δ is defined by raising ha degree (i.e. as a creaion operaor). We give paricular aenion o he class of normal maringales which can be used o expand any square-inegrable random variable ino a series of muliple sochasic inegrals. This propery, called he chaos represenaion propery, is sronger han he predicable represenaion propery and plays a key role in he represenaion of funcionals as sochasic inegrals. Noe ha here he words chaos and chaoic are no aken in he sense of dynamical sysems heory and raher refer o he noion of chaos inroduced by N. Wiener [152. We also presen an applicaion o deviaion and concenraion inequaliies in he case of deerminisic srucure equaions. The family of normal maringales having he chaos represenaion propery, includes Brownian moion and he compensaed Poisson process, which will be deal wih separaely cases in he following secions. The general resuls developed in Chaper 3 are deailed in Chaper 5 in he paricular case of Brownian moion on he Wiener space. Here he gradien operaor has he derivaion propery and he muliple sochasic inegrals can be expressed using Hermie polynomials, cf. Secion 5.1. We sae he expression of he Ornsein-Uhlenbeck semi-group and he associaed covariance ideniies and Gaussian deviaion inequaliies obained. A differenial calculus is presened for ime changes on Brownian moion, and more generally for random ransformaions on he Wiener space, wih applicaion o Brownian moion on Riemannian pah space in Secion 5.7. In Chaper 6 we inroduce he main ools of sochasic analysis under Poisson measures on he space of configuraions of a meric space X. We review he connecion beween Poisson muliple sochasic inegrals and Charlier polynomials, gradien and divergence operaors, and he Ornsein-Uhlenbeck semi-group. In his seing he annihilaion operaor defined on muliple Poisson sochasic inegrals is a difference operaor ha can be used o formulae he Clark predicable represenaion formula. I also urns ou ha he inegraion by pars formula can be used o characerize Poisson measure. We also derive some deviaion and concenraion resuls for random vecors and infiniely divisible random variables. In Chaper 7 we sudy a class of local gradien operaors on he Poisson space ha can also be used o characerize he Poisson measure. Unlike he finie difference gradiens considered in Chaper 6, hese operaors do saisfy he chain rule of derivaion. In he case of he sandard Poisson process on he real line, hey provide anoher insance of an inegraion by pars seing ha fis ino he general framework of Chaper 3. In paricular his operaor can be used in a Clark predicable represenaion formula and i is closely conneced o he sochasic inegral wih respec o he compensaed Poisson process 3 February 23, 218 (wih correcions).

12 N. Privaul via is associaed divergence operaor. The chain rule of derivaion, which is no saisfied by he difference operaors considered in Chaper 6, urns ou o be necessary in a number of applicaion such as deviaion inequaliies, chaos expansions, or sensiiviy analysis. Chaper 8 is devoed o applicaions in mahemaical finance. We use normal maringales o exend he classical Black-Scholes heory and o consruc complee marke models wih jumps. The resuls of previous chapers are applied o he pricing and hedging of coningen claims in complee markes driven by normal maringales. Normal maringales play only a modes role in he modeling of financial markes. Neverheless, in addiion o Brownian and Poisson models, hey provide examples of complee markes wih jumps. To close his inroducion we urn o some informal remarks on he Clark formula and predicable represenaion in connecion wih classical ools of finie dimensional analysis. This simple example shows how analyic argumens and sochasic calculus can be used in sochasic analysis. The classical fundamenal heorem of calculus can be wrien using enire series as f(x) = α n x n n= = α + = f() + and commonly relies on he ideniy x n = n x x nα n n=1 x f (y)dy, yn 1 dy yn 1 dy, x R +. (.1) Replacing he monomial x n wih he Hermie polynomial H n (x, ) wih parameer >, we do obain an analog of (.1) as x H n(x, ) = nh n 1 (x, ), however he argumen conained in (.1) is no longer valid since H 2n (, ), n 1. The quesion of wheher here exiss a simple analog of (.1) for he Hermie polynomials can be posiively answered using sochasic calculus wih respec o Brownian moion (B ) R+ which provides a way o wrie H n (B, ) as a sochasic inegral of nh n 1 (B, ), i.e. H n (B, ) = n H n 1(B s, s)db s. (.2) Consequenly H n (B, ) can be wrien as an n-fold ieraed sochasic inegral wih respec o Brownian moion (B ) R+, which is denoed by I n (1 [, n). 4 February 23, 218 (wih correcions).

13 Sochasic Analysis in Discree and Coninuous Seings This allows us o wrie down he following expansion of a funcion f depending on he parameer ino a series of Hermie polynomials, as follows: f(b, ) = β n H n (B, ) n= = β + nβ n n=1 H n 1(B s, s)db s, β n R +, n N. Using he relaion H n(x, ) = nh n 1 (x, ), his series can be wrien as [ f f(b, ) = IE[f(B, ) + IE x (B, ) F s db s, (.3) since, by he maringale propery of (.2), H n 1 (B s, s) coincides wih he condiional expecaion IE[H n 1 (B, ) F s, s <, where (F ) R+ is he filraion generaed by (B ) R+. I urns ou ha he above argumen can be exended o general funcionals of he Brownian pah (B ) R+ o prove ha he square inegrable funcionals of (B ) R+ have he following expansion in series of muliple sochasic inegrals I n (f n ) of symmeric funcions f n L 2 (R n +): F = IE[F + = IE[F + I n (f n ) n=1 n n=1 I n 1 (f n (, )1 { } )db. Using again sochasic calculus in a way similar o he above argumen will show ha his relaion can be wrien under he form F = IE[F + IE[D F F db, (.4) where D is a gradien acing on Brownian funcionals and (F ) R+ is he filraion generaed by (B ) R+. Relaion (.4) is a generalizaion of (.3) o arbirary dimensions which does no require he use of Hermie polynomials, and can be adaped o oher processes such as he compensaed Poisson process, and more generally o he larger class of normal maringales. 5 February 23, 218 (wih correcions).

14 N. Privaul Classical Taylor expansions for funcions of one or several variables can also be inerpreed in a sochasic analysis framework, in relaion o he explici deerminaion of chaos expansions of random funcionals. Consider for insance he classical formula for he coefficiens in he enire series a n = n f x n (x) x= f(x) = n= a n x n n!. In he general seing of normal maringales having he chaos represenaion propery, one can similarly compue he funcion f n in he developmen of as F = n= 1 n! I n(f n ) f n ( 1,..., n ) = IE[D 1 D n F, a.e. 1,..., n R +, (.5) cf. [7, [142. This ideniy holds in paricular for Brownian moion and he compensaed Poisson process. However, he probabilisic inerpreaion of D F can be difficul o find excep in he Wiener and Poisson cases, i.e. in he case of deerminisic srucure equaions. Our aim in he nex chapers will be in paricular o invesigae o which exen hese echniques remain valid in he general framework of normal maringales and oher processes wih jumps. 6 February 23, 218 (wih correcions).

15 Chaper 1 The Discree Time Case In his chaper we inroduce he ools of sochasic analysis in he simple framework of discree ime random walks. Our presenaion relies on he use of finie difference gradien and divergence operaors which are defined along wih single and muliple sochasic inegrals. The main applicaions of sochasic analysis o be considered in he following chapers, including funcional inequaliies and mahemaical finance, are discussed in his elemenary seing. Some echnical difficulies involving measurabiliy and inegrabiliy condiions, ha are ypical of he coninuous-ime case, are absen in he discree ime case. 1.1 Normal Maringales Consider a sequence (Y k ) k N of (no necessarily independen) random variables on a probabiliy space (Ω, F, P), where N = {, 1, 2,...} denoes he se of nonnegaive inegers. Le (F n ) n 1 denoe he filraion generaed by (Y n ) n N, i.e. F 1 = {, Ω}, and F n = σ(y,..., Y n ), n. Recall ha a random variable F is said o be F n -measurable if i can be wrien as a funcion F = f n (Y,..., Y n ) of Y,..., Y n, where f n : R n+1 R is Borel measurable. Assumpion We make he following assumpions on he sequence (Y n ) n N : a) i is condiionally cenered: E [Y n F n 1 =, n, (1.1.1) 7

16 N. Privaul b) is condiional quadraic variaion saisfies: E [ Y 2 n F n 1 = 1, n. Condiion (1.1.1) implies ha he process (Y + + Y n ) n is an F n - maringale, cf. Secion 9.4 in he Appendix. More precisely, he sequence (Y n ) n N and he process (Y + + Y n ) n can be viewed respecively as a (correlaed) noise and as a normal maringale in discree ime. 1.2 Sochasic Inegrals In his secion we consruc he discree sochasic inegral of predicable square-summable processes wih respec o a discree-ime normal maringale. Definiion Le (u k ) k N be a uniformly bounded sequence of random variables wih finie suppor in N, i.e. here exiss N such ha u k = for all k N. The sochasic inegral J(u) of (u n ) n N is defined as J(u) = u k Y k. k= The nex proposiion saes a version of he Iô isomery in discree ime. A sequence (u n ) n N of random variables is said o be F n -predicable if u n is F n 1 -measurable for all n N, in paricular u is consan in his case. Proposiion The sochasic inegral operaor J(u) exends o squareinegrable predicable processes (u n ) n N L 2 (Ω N) via he (condiional) isomery formula E [ J(1 [n, ) u) 2 F n 1 = E [ 1 [n, ) u 2 l 2 (N) F n 1, n N. (1.2.1) Proof. Le (u n ) n N and (v n ) n N be bounded predicable processes wih finie suppor in N. The produc u k Y k v l, k < l, is F l 1 -measurable, and u k Y l v l is F k 1 -measurable, l < k. Hence [ Fn 1 E u k Y k v l Y l = E Fn 1 u k Y k v l Y l k=n = E = k=n k=n l=n u k v k Y 2 k + n k<l k,l=n u k Y k v l Y l + n l<k E [ E [ u k v k Y 2 k F k 1 Fn 1 + n k<l Fn 1 u k Y k v l Y l E [E [u k Y k v l Y l F l 1 F n 1 8 February 23, 218 (wih correcions).

17 = = Sochasic Analysis in Discree and Coninuous Seings + k= n l<k E [E [u k Y k v l Y l F k 1 F n 1 E [ u k v k E [ Y 2 k F k 1 Fn E [u k v k F n 1 k=n [ Fn 1 = E u k v k. k=n n k<l E [u k Y k v l E [Y l F l 1 F n 1 This proves he isomery propery (1.2.1) for J. The exension o L 2 (Ω N) is proved using he following Cauchy sequence argumen. Consider a sequence of bounded predicable processes wih finie suppor converging o u in L 2 (Ω N), for example he sequence (u n ) n N defined as u n = (u n k) k N = ( ) u k 1 { k n} 1 { uk n}, n N. k N Then he sequence (J(u n )) n N is Cauchy and converges in L 2 (Ω), hence we may define J(u) := lim k J(uk ). From he isomery propery (1.2.1) applied wih n =, he limi is clearly independen of he choice of he approximaing sequence (u k ) k N as using Faou s lemma we have [ ( ) 2 [ E lim n J(un ) lim n J(vn ) = E lim n (J(un ) J(v n )) 2 [ lim inf E (J(u n ) J(v n )) 2 n = lim inf [ u E n v n 2 n l 2 (N) =. Noe ha by polarizaion, (1.2.1) can also be wrien as E [ J(1 [n, ) u)j(1 [n, ) v) F n 1 = E [ 1[n, ) u, 1 [n, ) v l 2 (N) F n 1, n N, and ha for n = we ge E [J(u)J(v) = E [ u, v l 2 (N), (1.2.2) and E [ J(u) 2 [ = E u 2 l 2 (N), (1.2.3) 9 February 23, 218 (wih correcions).

18 N. Privaul for all square-inegrable predicable processes u = (u k ) k N and v = (v k ) k N. L 2 (Ω N) be a predicable square- Proposiion Le (u k ) k N inegrable process. We have E [J(u) F k = J(u1 [,k ), k N. Proof. In case (u k ) k N has finie suppor in N i suffices o noe ha [ k Fk E [J(u) F k = E u i Y i + = = = i= k u i Y i + i= k u i Y i + i= k u i Y i i= = J ( u1 [,k ). i=k+1 i=k+1 i=k+1 E [u i Y i F k E [E [u i Y i F i 1 F k E [u i E [Y i F i 1 F k The formula exends o he general case by lineariy and densiy, using he coninuiy of he condiional expecaion on L 2 and he sequence (u n ) n N defined as u n = (u n k ) k N = ( u k 1 { k n}, n N, i.e. )k N [ (J ( ) E u1[,k E [J(u) Fk ) [ 2 (J = lim E ( u n ) 1 [,k E [J(u) Fk ) 2 n [ = lim E (E [J(u n ) J(u) F k ) 2 n [ lim E E [(J(u n ) J(u)) 2 Fk n [ = lim E (J(u n ) J(u)) 2 n =, is a dis- by (1.2.3). Corollary The indefinie sochasic inegral ( J ( u1 [,k ))k N cree ime maringale wih respec o (F n ) n 1. Proof. We have E [ J ( ) [ [ ( ) u1 [,k+1 Fk = E E J u1[,k+1 Fk+1 F k = E [E [J(u) F k+1 F k = E [J(u) F k 1 February 23, 218 (wih correcions).

19 Sochasic Analysis in Discree and Coninuous Seings = J ( u1 [,k ). 1.3 Muliple Sochasic Inegrals The role of muliple sochasic inegrals in he orhogonal expansion of a random variable is similar o ha of polynomials in he series expansion of a funcion of a real variable. In some siuaions, muliple sochasic inegrals can be expressed using polynomials, such as in he symmeric case p n = q n = 1/2, n N, in which he Krawchouk polynomials are used, see Relaion (1.5.2) below. Definiion Le l 2 (N) n denoe he subspace of l 2 (N) n = l 2 (N n ) made of funcions f n ha are symmeric in n variables, i.e. such ha for every permuaion σ of {1,..., n}, f n (k σ(1),..., k σ(n) ) = f n (k 1,..., k n ), k 1,..., k n N. Given f 1 l 2 (N) we le J 1 (f 1 ) = J(f 1 ) = f 1 (k)y k. As a convenion we idenify l 2 (N ) o R and le J (f ) = f, f R. Le k= n = {(k 1,..., k n ) N n : k i k j, 1 i < j n}, n 1. The following proposiion gives he definiion of muliple sochasic inegrals by ieraed sochasic inegraion of predicable processes in he sense of Proposiion Proposiion The muliple sochasic inegral J n (f n ) of f n l 2 (N) n, n 1, is defined as J n (f n ) = f n (i 1,..., i n )Y i1 Y in. (i 1,...,i n ) n I saisfies he recurrence relaion J n (f n ) = n and he isomery formula Y k J n 1 (f n (, k)1 [,k 1 n 1( )) (1.3.1) k=1 11 February 23, 218 (wih correcions).

20 N. Privaul n! 1 n f n, g m l 2 (N) n if n = m, E [J n (f n )J m (g m ) = if n m. Proof. Noe ha we have J n (f n ) = n! f n (i 1,..., i n )Y i1 Y in i 1 < <i n = n! i n = i n 1 <i n (1.3.2) i 1 <i 2 f n (i 1,..., i n )Y i1 Y in. (1.3.3) Noe ha since i 1 < i 2 < < i n and j 1 < j 2 < < j n we have Hence E [Y i1 Y in Y j1 Y jn = 1 {i1 =j 1,...,i n =j n }. E [J n (f n )J n (g n ) = (n!) 2 E f n (i 1,..., i n )Y i1 Y in g n (j 1,..., j n )Y j1 Y jn i 1 < <i n j 1 < <j n = (n!) 2 f n (i 1,..., i n )g n (j 1,..., j n )E [Y i1 Y in Y j1 Y jn i 1 < <i n, j 1 < <j n = (n!) 2 f n (i 1,..., i n )g n (i 1,..., i n ) i 1 < <i n = n! f n (i 1,..., i n )g n (i 1,..., i n ) (i 1,...,i n ) n = n! 1 n f n, g m l 2 (N) n. When n < m and (i 1,..., i n ) n and (j 1,..., j m ) m are wo ses of indices, here necessarily exiss k {1,..., m} such ha j k / {i 1,..., i n }, hence E [Y i1 Y in Y j1 Y jm =, and his implies he orhogonaliy of J n (f n ) and J m (g m ). The recurrence relaion (1.3.1) is a direc consequence of (1.3.3). The isomery propery (1.3.2) of J n also follows by inducion from (1.2.1) and he recurrence relaion. If f n l 2 (N n ) is no symmeric we le J n (f n ) = J n ( f n ), where f n is he symmerizaion of f n, defined as f n (i 1,..., i n ) = 1 f(i σ(1),..., i σn ), i 1,..., i n N n, n! σ Σ n 12 February 23, 218 (wih correcions).

21 Sochasic Analysis in Discree and Coninuous Seings and Σ n is he se of all permuaions of {1,..., n}. In paricular, if (k 1,..., k n ) n, he symmerizaion of 1 {(k1,...,k n )} in n variables is given by and 1 {(k1,...,k n )}(i 1,..., i n ) = 1 n! 1 {{i 1,...,i n }={k 1,...,k n }}, i 1,..., i n N, J n ( 1 {(k1,...,k n )}) = Y k1 Y kn. Lemma For all n 1 we have k N, f n l 2 (N) n. E [J n (f n ) F k = J n (f n 1 [,k n), Proof. This lemma can be proved in wo ways, eiher as a consequence of Proposiion and Proposiion or via he following direc argumen, noing ha for all m =,..., n and g m l 2 (N) m we have: E [ (J n (f n ) J n (f n 1 [,k n))j m (g m 1 [,k m) = 1 {n=m} n! f n (1 1 [,k n), g m 1 [,k m l2 (N n ) =, hence J n (f n 1 [,k n) L 2 (Ω, F k ), and J n (f n ) J n (f n 1 [,k n) is orhogonal o L 2 (Ω, F k ). In oher erms we have E [J n (f n ) =, f n l 2 (N) n, n 1, he process (J n (f n 1 [,k n)) k N is a discree-ime maringale, and J n (f n ) is F k -measurable if and only if f n 1 [,k n = f n, k n. 1.4 Srucure Equaions Assume now ha he sequence (Y n ) n N saisfies he discree srucure equaion: Y 2 n = 1 + ϕ n Y n, n N, (1.4.1) where (ϕ n ) n N is an F n -predicable process. Condiion (1.1.1) implies ha E [ Y 2 n F n 1 = 1, n N, 13 February 23, 218 (wih correcions).

22 N. Privaul hence he hypoheses of he preceding secions are saisfied. Since (1.4.1) is a second order equaion, here exiss an F n -adaped process (X n ) n N of Bernoulli { 1, 1}-valued random variables such ha Y n = ϕ n 2 + X n 1 + Consider he condiional probabiliies ( ϕn ) 2, n N. (1.4.2) 2 p n = P(X n = 1 F n 1 ) and q n = P(X n = 1 F n 1 ), n N. (1.4.3) From he relaion E [Y n F n 1 =, rewrien as ( ϕ n p n we ge p n = 1 2 ( 1 ) ( ( ϕn ) 2 ϕ n + q n ) ( ϕn ) 2 =, n N, 2 ) ( ) ϕ n, q n = 1 ϕ 1 + n, (1.4.4) 4 + ϕ 2 n ϕ 2 n and hence Leing ϕ n = Y n = 1 {Xn =1} qn pn = q n p n, n N, p n q n pn q n qn pn 1 {Xn = 1}, n N. (1.4.5) p n q n Z n = X n + 1 {, 1}, n N, 2 we also have he relaions which yield Y n = q n p n + X n 2 p n q n = Z n p n pn q n, n N, (1.4.6) F n = σ(x,..., X n ) = σ(z,..., Z n ), n N. Remark In paricular, one can ake Ω = { 1, 1} N and consruc he Bernoulli process (X n ) n N as he sequence of canonical projecions on Ω = { 1, 1} N under a counable produc P of Bernoulli measures on { 1, 1}. In his case he sequence (X n ) n N can be viewed as he dyadic expansion of X(ω) [, 1 defined as: 14 February 23, 218 (wih correcions).

23 Sochasic Analysis in Discree and Coninuous Seings X(ω) = n= 1 2 n+1 X n(ω). In he symmeric case p k = q k = 1/2, k N, he image measure of P by he mapping ω X(ω) is he Lebesgue measure on [, 1, see [143 for he non-symmeric case. 1.5 Chaos Represenaion From now on we assume ha he sequence (p k ) k N defined in (1.4.3) is deerminisic, which implies ha he random variables (X n ) n N are independen. Precisely, X n will be consruced as he canonical projecion X n : Ω { 1, 1} on Ω = { 1, 1} N under he measure P given on cylinder ses by P({ɛ,..., ɛ n } { 1, 1} N ) = n k= p (1+ε k)/2 k q (1 ε k)/2 k, {ɛ,..., ɛ n } { 1, 1} n+1. The sequence (Y k ) k N can be consruced as a family of independen random variables given by Y n = ϕ ( n 2 + X ϕn ) 2, n 1 + n N, 2 where he sequence (ϕ n ) n N is deerminisic. In his case, all spaces L r (Ω, F n ), r 1, have finie dimension 2 n+1, wih basis { n { } } qk pk 1 {Y =ɛ,...,y n =ɛ n } : (ɛ,..., ɛ n ), p k q k k= { } n = 1 {X =ɛ,...,x n =ɛ n } : (ɛ,..., ɛ n ) { 1, 1}. An orhogonal basis of L r (Ω, F n ) is given by { Yk1 Y kl = J l ( 1 {(k1,...,k l )}) : k 1 < < k l n, l =,..., n + 1 }. k= Le S n = n k= 1 + X k 2 (1.5.1) 15 February 23, 218 (wih correcions).

24 = N. Privaul n Z k, n N, k= denoe he random walk associaed o (X k ) k N. Remark. In he special case p k = p for all k N, we have J n (1 n [,N ) = K n(s N ; N + 1, p) (1.5.2) coincides wih he Krawchouk polynomial K n ( ; N + 1, p) of order n and parameer (N + 1, p), evaluaed a S N, cf. Proposiion 4 of [119. Le now H = R and le H n denoe he subspace of L 2 (Ω) made of inegrals of order n 1, and called chaos of order n: H n = {J n (f n ) : f n l 2 (N) n }. The space of F n -measurable random variables is denoed by L (Ω, F n ). Lemma For all n N we have L (Ω, F n ) = (H H n+1 ) L (Ω, F n ). (1.5.3) Proof. I suffices o noe ha H l L (Ω, F n ) has dimension ( ) n+1 l, 1 l n + 1. More precisely i is generaed by he orhonormal basis { Yk1 Y kl = J l ( 1 {(k1,...,k l )}) : k 1 < < k l n }, since any elemen F of H l L (Ω, F n ) can be wrien as F = J l (f l 1 [,n l). Hence L (Ω, F n ) and (H H n+1 ) L (Ω, F n ) have same dimension n+1 ( ) n n+1 =, and his implies (1.5.3) since k k= L (Ω, F n ) (H H n+1 ) L (Ω, F n ). As a consequence of Lemma we have L (Ω, F n ) H H n+1. Alernaively, Lemma can be proved by noing ha J n (f n 1 [,N n) =, n > N + 1, f n l 2 (N) n, and as a consequence, any F L (Ω, F N ) can be expressed as N+1 F = E [F + J n (f n 1 [,N n). n=1 16 February 23, 218 (wih correcions).

25 Sochasic Analysis in Discree and Coninuous Seings Definiion Le S denoe he linear space spanned by muliple sochasic inegrals, i.e. { } S = Vec H n (1.5.4) n= { n } = J k (f k ) : f k l 2 (N) k, k =,..., n, n N. k= The compleion of S in L 2 (Ω) is denoed by he direc sum H n. n= The nex resul is he chaos represenaion propery for Bernoulli processes, which is analogous o he Walsh decomposiion, cf. [82. Here his propery is obained under he assumpion ha he sequence (X n ) n N is made of independen random variables since (p k ) k N is deerminisic, which corresponds o he seing of Proposiion 4 in [4. See [4 and Proposiion 5 herein for oher insances of he chaos represenaion propery wihou his independence assumpion. Proposiion We have he ideniy L 2 (Ω) = H n. Proof. I suffices o show ha S is dense in L 2 (Ω). Le F be a bounded random variable. Relaion (1.5.3) of Lemma shows ha E [F F n S. The maringale convergence heorem, cf. e.g. Theorem 27.1 in [71, implies ha (E [F F n ) n N converges o F a.s., hence every bounded F is he L 2 (Ω)- limi of a sequence in S. If F L 2 (Ω) is no bounded, F is he limi in L 2 (Ω) of he sequence (1 { F n} F ) n N of bounded random variables. As a consequence of Proposiion 1.5.3, any F L 2 (Ω, P) has a unique decomposiion F = E [F + n= J n (f n ), f n l 2 (N) n, n N, n=1 as a series of muliple sochasic inegrals. Noe also ha he saemen of Lemma is sufficien for he chaos represenaion propery o hold. 17 February 23, 218 (wih correcions).

26 N. Privaul 1.6 Gradien Operaor We sar by defining he operaor D on he space S of finie sums of muliple sochasic inegrals, which is dense in in L 2 (Ω) by Proposiion Definiion We densely define he linear gradien operaor by k N, f n l 2 (N) n, n N. D : S L 2 (Ω N) D k J n (f n ) = nj n 1 (f n (, k)1 n (, k)), Noe ha for all k 1,..., k n 1, k N, we have 1 n (k 1,..., k n 1, k) = 1 {k / (k1,...,k n 1 )}1 n 1 (k 1,..., k n 1 ), hence we can wrie D k J n (f n ) = nj n 1 (f n (, k)1 {k / } ), k N, where in he above relaion, denoes he firs k 1 variables (k 1,..., k n 1 ) of f n (k 1,..., k n 1, k). We also have D k F = whenever F S is F k 1 - measurable. On he oher hand, D k is a coninuous operaor on he chaos H n since D k J n (f n ) 2 L 2 (Ω) = n2 J n 1 (f n (, k)) 2 L 2 (Ω) (1.6.1) = nn! f n (, k) 2 l 2 (N (n 1) ), f n l 2 (N n ), k N. The following resul gives he probabilisic inerpreaion of D k as a finie difference operaor. Given le and ω = (ω, ω 1,...) { 1, 1} N, ω k + = (ω, ω 1,..., ω k 1, +1, ω k+1,...) ω k = (ω, ω 1,..., ω k 1, 1, ω k+1,...). Proposiion We have for any F S: D k F (ω) = p k q k (F (ω k +) F (ω k )), k N. (1.6.2) Proof. We sar by proving he above saemen for an F n -measurable F S. Since L (Ω, F n ) is finie dimensional i suffices o consider 18 February 23, 218 (wih correcions).

27 Sochasic Analysis in Discree and Coninuous Seings wih from (1.4.6): F = Y k1 Y kl = f(x,..., X kl ), f(x,..., x kl ) = 1 2 l l i=1 q ki p ki + x ki pki q ki. Firs we noe ha from (1.5.3) we have for (k 1,..., k n ) n : D k (Y k1 Y kn ) = D k J n ( 1 {(k1,...,k n )}) = nj n 1 ( 1 {(k1,...,k n )}(, k)) 1 n = 1 {ki }(k) (n 1)! = i=1 n 1 {ki }(k)j n 1 ( 1 {(k1,...,k i 1,k i+1,...,k n )}) i=1 = 1 {k1,...,k n }(k) n i=1 k i k (i 1,...,i n 1 ) n 1 1 {{i1,...,i n 1 }={k 1,...,k i 1,k i+1,...,k n }} Y ki. (1.6.3) If k / {k 1,..., k l } we clearly have F (ω k +) = F (ω k ) = F (ω), hence pk q k (F (ω k +) F (ω k )) = = D k F (ω). On he oher hand if k {k 1,..., k l } we have F (ω k +) = hence from (1.6.3) we ge qk p k F (ω ) k pk = q k l i=1 k i k l i=1 k i k pk q k (F (ω k +) F (ω k )) = 1 2 l 1 = q ki p ki + ω ki 2 p ki q ki, q ki p ki + ω ki 2 p ki q ki, l i=1 k i k l i=1 k i k Y ki (ω) q ki p ki + ω ki pki q ki = D k (Y k1 Y kl ) (ω) 19 February 23, 218 (wih correcions).

28 N. Privaul = D k F (ω). In he general case, J l (f l ) is he L 2 -limi of he sequence E [J l (f l ) F n = J l (f l 1 [,n l) as n goes o infiniy, and since from (1.6.1) he operaor D k is coninuous on all chaoses H n, n 1, we have D k F = lim n D ke [F F n = p k q k lim n (E [F F n (ω k +) E [F F n (ω k )) = p k q k (F (ω k +) F (ω k )), k N. The nex propery follows immediaely from Proposiion Corollary A random variable F : Ω R is F n -measurable if and only if D k F = for all k > n. If F has he form F = f(x,..., X n ), we may also wrie D k F = p k q k (F + k F k ), k N, wih and F + k = f(x,..., X k 1, +1, X k+1,..., X n ), F k = f(x,..., X k 1, 1, X k+1,..., X n ). The gradien D can also be expressed as D k F (S ) = p k q k (F ( S + 1 {Xk = 1}1 {k } ) F ( S 1 {Xk =1}1 {k } ) ), where F (S ) is an informal noaion for he random variable F esimaed on a given pah of (S n ) n N defined in (1.5.1) and S + 1 {Xk = 1}1 {k } denoes he pah of (S n ) n N perurbed by forcing X k o be equal o ±1. We will also use he gradien k defined as k N, wih he relaion k F = X k (f(x,..., X k 1, 1, X k+1,..., X n ) f(x,..., X k 1, 1, X k+1,..., X n )), D k = X k pk q k k, k N, 2 February 23, 218 (wih correcions).

29 Sochasic Analysis in Discree and Coninuous Seings hence k F coincides wih D k F afer squaring and muliplicaion by p k q k. From now on, D k denoes he finie difference operaor which is exended o any F : Ω R using Relaion (1.6.2). The L 2 domain of D, denoed Dom (D), is naurally defined as he space of funcionals F L 2 (Ω) such ha [ E DF 2 l 2 (N) <, or equivalenly by (1.6.1), if F = J n (f n ). n= nn! f n 2 l 2 (N n ) <, n=1 The following is he produc rule for he operaor D. Proposiion Le F, G : Ω R. We have D k (F G) = F D k G + GD k F X k pk q k D k F D k G, k N. Proof. Le F k +(ω) = F (ω k +), F k (ω) = F (ω k ), k. We have D k (F G) = p k q k (F+G k k + F G k k ) ( = 1 {Xk = 1} pk q k F (G k + G) + G(F+ k F ) + (F+ k F )(G k + G) ) ( pk q k F (G G k ) + G(F F ) k (F F )(G k G k ) ) +1 {Xk =1} ( = 1 {Xk = 1} F D k G + GD k F + 1 ) D k F D k G pk q k ( F D k G + GD k F 1 ) D k F D k G. pk q k +1 {Xk =1} 1.7 Clark Formula and Predicable Represenaion In his secion we prove a predicable represenaion formula for he funcionals of (S n ) n defined in (1.5.1). Proposiion For all F S we have 21 February 23, 218 (wih correcions).

30 F = E [F + = E [F + N. Privaul E [D k F F k 1 Y k (1.7.1) k= Y k D k E [F F k. k= Proof. The formula is obviously rue for F = J (f ). Given n 1, as a consequence of Proposiion above and Lemma we have: J n (f n ) = n = n = n = J n 1 (f n (, k)1 [,k 1 n 1( ))Y k k= J n 1 (f n (, k)1 n (, k)1 [,k 1 n 1( ))Y k k= E [J n 1 (f n (, k)1 n (, k)) F k 1 Y k k= E [D k J n (f n ) F k 1 Y k, k= which yields (1.7.1) for F = J n (f n ), since E [J n (f n ) =. By lineariy he formula is esablished for F S. For he second ideniy we use he relaion E [D k F F k 1 = D k E [F F k which clearly holds since D k F is independen of X k, k N. Alhough he operaor D is unbounded we have he following resul, which saes he boundedness of he operaor ha maps a random variable o he unique process involved in is predicable represenaion. Lemma The operaor L 2 (Ω) L 2 (Ω N) is bounded wih norm equal o one. F (E [D k F F k 1 ) k N Proof. Le F S. From Relaion (1.7.1) and he isomery formula (1.2.2) for he sochasic inegral operaor J we ge E [D F F 1 2 L 2 (Ω N) = F E [F 2 L 2 (Ω) (1.7.2) F E [F 2 L 2 (Ω) + (E [F )2 = F 2 L 2 (Ω), 22 February 23, 218 (wih correcions).

31 Sochasic Analysis in Discree and Coninuous Seings wih equaliy in case F = J 1 (f 1 ). As a consequence of Lemma we have he following corollary. Corollary The Clark formula of Proposiion exends o any F L 2 (Ω). Proof. Since F E [D F F 1 is bounded from Lemma 1.7.2, he Clark formula exends o F L 2 (Ω) by a sandard Cauchy sequence argumen. Le us give a firs elemenary applicaion of he above consrucion o he proof of a Poincaré inequaliy on Bernoulli space. Using (1.2.3) we have hence Var (F ) = E [ F E [F 2 ( ) 2 = E E [D k F F k 1 Y k k= [ = E (E [D k F F k 1 ) 2 k= [ E E [ D k F 2 F k 1 k= [ = E D k F 2, k= Var (F ) DF 2 L 2 (Ω N). More generally he Clark formula implies he following. Corollary Le a N and F L 2 (Ω). We have and F = E [F F a + E [ F 2 = E [ (E [F F a ) 2 + E k=a+1 [ E [D k F F k 1 Y k, (1.7.3) k=a+1 (E [D k F F k 1 ) 2. (1.7.4) Proof. From Proposiion and he Clark formula (1.7.1) of Proposiion we have E [F F a = E [F + a E [D k F F k 1 Y k, k= 23 February 23, 218 (wih correcions).

32 N. Privaul which implies (1.7.3). Relaion (1.7.4) is an immediae consequence of (1.7.3) and he isomery propery of J. As an applicaion of he Clark formula of Corollary we obain he following predicable represenaion propery for discree-ime maringales. Proposiion Le (M n ) n N be a maringale in L 2 (Ω) wih respec o (F n ) n N. There exiss a predicable process (u k ) k N locally in L 2 (Ω N), (i.e. u( )1 [,N ( ) L 2 (Ω N) for all N > ) such ha M n = M 1 + n u k Y k, n N. (1.7.5) k= Proof. Le k 1. From Corollaries and we have: hence i suffices o le M k = E [M k F k 1 + E [D k M k F k 1 Y k = M k 1 + E [D k M k F k 1 Y k, u k = E [D k M k F k 1, k, o obain n n M n = M 1 + M k M k 1 = M 1 + u k Y k. k= k= 1.8 Divergence Operaor The divergence operaor δ is inroduced as he adjoin of D. Le U L 2 (Ω N) be he space of processes defined as { n } U = J k (f k+1 (, )), f k+1 l 2 (N) k l 2 (N), k =,..., n, n N. k= We refer o Secion 9.7 in he appendix for he definiion of he ensor produc l 2 (N) k l 2 (N), k. Definiion Le δ : U L 2 (Ω) be he linear mapping defined on U as δ(u) = δ(j n (f n+1 (, ))) = J n+1 ( f n+1 ), f n+1 l 2 (N) n l 2 (N), for (u k ) k N of he form 24 February 23, 218 (wih correcions).

33 Sochasic Analysis in Discree and Coninuous Seings u k = J n (f n+1 (, k)), k N, where f n+1 denoes he symmerizaion of f n+1 in n + 1 variables, i.e. f n+1 (k 1,..., k n+1 ) = 1 n+1 f n+1 (k 1,..., k k 1, k k+1,..., k n+1, k i ). n + 1 i=1 From Proposiion 1.5.3, S is dense in L 2 (Ω), hence U is dense in L 2 (Ω N). Proposiion The operaor δ is adjoin o D: E [ DF, u l2 (N) = E [F δ(u), F S, u U. Proof. We consider F = J n (f n ) and u k = J m (g m+1 (, k)), k N, where f n l 2 (N) n and g m+1 l 2 (N) m l 2 (N). We have E [ D J n (f n ), J m (g m+1 (, )) l2 (N) = ne [ J n 1 (f n (, )), J m (g m (, )) l 2 (N) = ne [ J n 1 (f n (, )1 n (, )), J m (g m (, )) l2 (N) = n1 {n 1=m} E [J n 1 (f n (, k)1 n (, k))j m (g m+1 (, k)) k= = n!1 {n 1=m} 1 n (, k)f n (, k), g m+1 (, k) l 2 (N n 1 ) k= = n!1 {n=m+1} 1 n f n, g m+1 l 2 (N n ) = n!1 {n=m+1} 1 n f n, g m+1 l 2 (N n ) = E [J n (f n )J m ( g m+1 ) = E [F δ(u). The nex proposiion shows ha δ coincides wih he sochasic inegral operaor J on he square-summable predicable processes. Proposiion The operaor δ can be exended o u L 2 (Ω N) wih δ(u) = u k Y k D k u k δ(ϕdu), (1.8.1) k= k= provided ha all series converges in L 2 (Ω), where (ϕ k ) k N appears in he srucure equaion (1.4.1). We also have for all u U: E[ δ(u) 2 = E[ u 2 l 2 (N) + E D k u l D l u k (D k u k ) 2. (1.8.2) k,l= k l k= 25 February 23, 218 (wih correcions).

34 N. Privaul Proof. Using he expression (1.3.3) of u k = J n (f n+1 (, k)) we have δ(u) = J n+1 ( f n+1 ) = fn+1 (i 1,..., i n+1 )Y i1 Y in+1 (i 1,...,i n+1 ) n+1 = fn+1 (i 1,..., i n, k)y i1 Y in Y k k= (i 1,...,i n ) n n fn+1 (i 1,..., i n 1, k, k)y i1 Y in 1 Y k 2 k= (i 1,...,i n 1 ) n 1 = u k Y k D k u k Y k 2 k= k= k= = u k Y k D k u k ϕ k D k u k Y k. k= k= Nex, we noe he commuaion relaion 1 ( ) D k δ(u) = D k u l Y l Y l 2 D l u l = l= l= l= ( Y l D k u l + u l D k Y l X ) k D k u l D k Y l pk q k l= ( Y l 2 D k D l u l + D l u l D k Y l 2 X k pk q k D k Y l 2 D k D l u l ) = δ(d k u) + u k D k Y k X k D k u k D k Y k D k u k D k Y k 2 pk q k ( Xk = δ(d k u) + u k + 2Y k D k Y k X ) k D k Y k D k Y k D k u k pk q k pk q k = δ(d k u) + u k 2Y k D k u k. On he oher hand, we have δ(1 {k} D k u k ) = Y l 1 {k} (l)d k u k Y l 2 D l (1 {k} (l)d k u k ) l= l= = Y k D k u k Y k 2 D k D k u k = Y k D k u k, 1 See A. Manei, Maserarbei Sochasisches Kalkül in diskreer Zei, Saz 6.7, February 23, 218 (wih correcions).

35 Sochasic Analysis in Discree and Coninuous Seings hence δ(u) 2 L 2 (Ω) = E[ u, Dδ(u) l 2 (N) [ = E u k (u k + δ(d k u) 2Y k D k u k ) k= where we used he equaliy [ = E[ u 2 l 2 (N) + E D k u l D l u k 2E u k Y k D k u k k,l= k,l= k= = E[ u 2 l 2 (N) + E D k u l D l u k 2 (D k u k ) 2, E [u k Y k D k u k = E [ p k 1 {Xk =1}u k (ω k +)Y k (ω k +)D k u k + q k 1 {Xk = 1}u k (ω k )Y k (ω k )D k u k k= = p k q k E [ (1 {Xk =1}u k (ω k +) 1 {Xk = 1}u k (ω k ))D k u k = E [ (D k u k ) 2, k N. In he symmeric case p k = q k = 1/2 we have ϕ k =, k N, and δ(u) = u k Y k D k u k. k= The las wo erms in he righ hand side of (1.8.1) vanish when (u k ) k N is predicable, and in his case he Skorohod isomery (1.8.2) becomes he Iô isomery as in he nex proposiion. Corollary If (u k ) k N saisfies D k u k =, i.e. u k does no depend on X k, k N, hen δ(u) coincides wih he (discree ime) sochasic inegral δ(u) = k= Y k u k, (1.8.3) k= provided he series converges in L 2 (Ω). If moreover (u k ) k N is predicable and square-summable we have he isomery E [ δ(u) 2 [ = E u 2 l 2 (N), (1.8.4) and δ(u) coincides wih J(u) on he space of predicable square-summable processes. 27 February 23, 218 (wih correcions).

36 N. Privaul 1.9 Ornsein-Uhlenbeck Semi-Group and Process The Ornsein-Uhlenbeck operaor L is defined as L = δd, i.e. L saisfies LJ n (f n ) = nj n (f n ), f n l 2 (N) n. Proposiion For any F S we have LF = δdf = Y k (D k F ) = pk q k Y k (F + k F k ), k= k= Proof. Noe ha D k D k F =, k N, and use Relaion (1.8.1) of Proposiion Noe ha L can be expressed in oher forms, for example LF = k F, k= where k F = (1 {Xk =1}q k (F (ω) F (ω )) k 1 {Xk = 1}p k (F (ω+) k F (ω))) = F (1 {Xk =1}q k F (ω ) k + 1 {Xk = 1}p k F (ω+)) k = F E [F Fk c, k N, and F c k is he σ-algebra generaed by {X l : l k, l N}. Le now (P ) R+ = (e L ) R+ denoe he semi-group associaed o L and defined as P F = e n J n (f n ), R +, n= on F = J n (f n ) L 2 (Ω). The nex resul shows ha (P ) R+ admis an n= inegral represenaion by a probabiliy kernel. Le q N defined by : Ω Ω R + be N q N ( ω, ω) = (1 + e Y i (ω)y i ( ω)), ω, ω Ω, R +. i= Lemma Le he probabiliy kernel Q ( ω, dω) be defined by [ dq ( ω, ) E F N (ω) = q N ( ω, ω), N 1, R +. dp 28 February 23, 218 (wih correcions).

37 Sochasic Analysis in Discree and Coninuous Seings For F L 2 (Ω, F N ) we have P F ( ω) = F (ω)q ( ω, dω), ω Ω, n N. (1.9.1) Ω Proof. Since L 2 (Ω, F N ) has finie dimension 2 N+1, i suffices o consider funcionals of he form F = Y k1 Y kn wih k 1 < < k n N. By Relaion (1.4.5) we have for ω Ω, k N: E [ Y k ( )(1 + e Y k ( )Y k (ω)) ( ) ( ) qk = p k 1 + e qk pk Y k (ω) q k 1 e pk Y k (ω) p k p k q k q k = e Y k (ω), which implies, by independence of he sequence (X k ) k N, E [ Y k1 Y kn q N (ω, ) = E = [ Y k1 Y kn N i=1 (1 + e Y ki (ω)y ki ( )) N E [ Y ki ( )(1 + e Y ki (ω)y ki ( )) i=1 = e n Y k1 (ω) Y kn (ω) = e n J n ( 1 {(k1,...,k n )})(ω) = P J n ( 1 {(k1,...,k n )})(ω) = P (Y k1 Y kn )(ω). Consider he Ω-valued saionary process (X()) R+ = ((X k ()) k N ) R+ wih independen componens and disribuion given by P(X k () = 1 X k () = 1) = p k + e q k, (1.9.2) P(X k () = 1 X k () = 1) = q k e q k, (1.9.3) P(X k () = 1 X k () = 1) = p k e p k, (1.9.4) k N, R +. P(X k () = 1 X k () = 1) = q k + e p k, (1.9.5) 29 February 23, 218 (wih correcions).

38 N. Privaul Proposiion The process (X()) R+ = ((X k ()) k N ) R+ is he Ornsein-Uhlenbeck process associaed o (P ) R+, i.e. we have P F = E [F (X()) X(), R +, (1.9.6) for F bounded and F n -measurable on Ω, n N. Proof. By consrucion of (X()) R+ in Relaions (1.9.2)-(1.9.5) we have ) P(X k () = 1 X k ()) = p k (1 + e qk Y k (), P(X k () = 1 X k ()) = q k (1 e Y k () where Y k () is defined by (1.4.6), i.e. hus Y k () = q k p k + X k () 2 p k q k, k N, p k pk dp(x k ()( ω) = ɛ X())(ω) = ( 1 + e Y k (ω)y k ( ω) ) dp(x k ( ω) = ɛ), ε = ±1. Since he componens of (X k ()) k N are independen, his shows ha he law of (X (),..., X n ()) condiionally o X() has he densiy q n ( ω, ) wih respec o P: dp(x ()( ω) = ɛ,..., X n ()( ω) = ɛ n X())( ω) = q n ( ω, ω)dp(x ( ω) = ɛ,..., X n ( ω) = ɛ n ). Consequenly we have E [F (X()) X() = ω = Ω (ω)qn ( ω, ω)p(dω), (1.9.7) hence from (1.9.1), Relaion (1.9.6) holds for F L 2 (Ω, F N ), N. The independen componens X k (), k N, can be consruced from he daa of X k () = ɛ and an independen exponenial random variable τ k via he following procedure. If τ k >, le X k () = X k () = ɛ, oherwise if τ k <, ake X k () o be an independen copy of X k (). This procedure is illusraed in he following equaliies: P(X k () = 1 X k () = 1) = E [ [ 1 {τk >} + E 1{τk <}1 {Xk =1} q k ), = e + p k (1 e ), (1.9.8) 3 February 23, 218 (wih correcions).