Malliavin s calculus and applications in stochastic control and finance Warsaw, March, April 2008 Peter Imkeller version of 5.

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1 Malliavin s calculus and applicaions in sochasic conrol and finance Warsaw, March, April 2008 Peer Imkeller version of 5. April 2008 Malliavin s calculus has been developed for he sudy of he smoohness of measures on infinie dimensional spaces. I provides a sochasic access o he analyic problem of smoohness of soluions of parabolic parial differenial equaions. The mahemaical framework for his access is given by measures on spaces of rajecories. In he one-dimensional framework i is clear wha is mean by smoohness of measures. We look for a direc analogy o he smoohness problem in infinie-dimensional spaces. For his purpose we sar inerpreing he Wiener space as a sequence space, o which he heory of differeniaion and inegraion in Euclidean spaces is generalized by exension o infinie families of real numbers insead of finie ones. The calculus possesses applicaions o many areas of sochasics, in paricular finance sochasics, as is underpinned by he recenly published book by Malliavin and Thalmayer. In his course I will repor on recen applicaions o he heory of backward sochasic differenial equaions (BSDE), and heir applicaion o problems of he fine srucure of opion pricing and hedging in incomplee finance or insurance markes. A firs we wan o presen an access o he Wiener space as sequence space. 1 The Wiener space as sequence space Definiion 1.1 A probabiliy space (Ω, F, P ) is called Gaussian if here is a family (X k ) 1 k n or a sequence (X k ) k N of independen Gaussian uni random variables such ha F = σ(x k : 1 k n) resp. σ(x k : k N) (compleed by ses of P -measure 0). 1

2 Example 1: Le Ω = C(R +, R m ), F he Borel ses on Ω generaed by he opology of uniform convergence on compac ses of R +, P he m-dimensional canonical Wiener measure on F. Le furher W = (W 1,, W m ) be he canonical m-dimensional Wiener process defined by he projecions on he coordinaes. Claim: (Ω, F, P ) is Gaussian. Proof: Le (g i ) i N be an orhonormal basis of L 2 (R + ), W j (g i ) = g i (s)dw j s, i N, 1 j m, in he sense of L 2 -limis of Iô inegrals. Then (modulo compleion) we have F = σ(w : 0). Le 0, (a i ) i N a sequence in l 2 such ha Then we have for 1 j m W j 1 [0,] = i N a i g i. n = n lim a i W j (g i ) = a i W j (g i ), i=1 i=1 hence W j is (modulo compleion) measurable wih respec o σ(w j (g i ) : i N). Therefore (modulo compleion) Moreover, due o F = σ(w j (g i ) : i N, 1 j m). E(W j (g i )W k (g l )) = δ jk g i, g l = δ jk δ il, i, l N, 1 j, k m, hence he W j (g i ) are independen Gaussian uni variables. In he following we shall consruc an absrac isomorphism beween he canonical Wiener space and a sequence space. Since we are finally ineresed in infinie dimensional spaces, we assume from now on Assumpion: he Gaussian space considered is generaed by infiniely many independen Gaussian uni variables. 2

3 Le R N = {(x i ) i N : x i R, i N} be he se of all real-valued sequences, and for n N denoe by π n : R N R n, (x i ) i N (x i ) 1 i n, he projecion on he firs n coordinaes. Le B n be he σ-algebra of Borel ses in R n, B N = σ( n N π 1 n [B n ]). Le for n N ν 1 (dx) = 1 2π exp( x2 2 ) dx, ν = P (X n ) n N = i N ν 1, ν n = ν π 1 n. This noaion is consisen for n = 1. We wan o consruc an isomorphism beween he spaces of inegrable funcions on (Ω, F, P ) and (R N, B N, ν). For his purpose, i is necessary o know how funcions on he wo spaces are mapped o each oher. I is clear ha for B N -measurable f on R N we have is F-measurable on Ω. F = f ((X n ) n N ) Lemma 1.1 Le F be F-measurable on Ω. Then here exiss a B N - measurable funcion f on R N such ha F = f ((X n ) n N ). Proof 1. Le F = 1 A wih A = ((X i ) 1 i n ) 1 [B], B B n. Then se f = 1 π 1 n [B]. f is by definiion B N -measurable and we have f((x n ) n N ) = 1 B ((X i ) 1 i n ) = 1 A = F. Hence he assered equaion is verified by indicaors of a generaing se of F which is sable for inersecions. Hence by Dynkin s heorem i is valid for all indicaors of ses in F. 2. By par 1. and by lineariy he claim is hen verified for linear combinaions of indicaor funcions of F-measurable ses. The asserion is sable for monoone limis in he se of funcions for which i is verified. Hence i is valid for all F-measurable funcions by he monoone class heorem. 3

4 Theorem 1.1 Le p 1. Then he mapping L p (R N, B N, ν)) f F = f ((X n ) n N ) L p (Ω, F, P )) defines a linear isomorphism. Proof The mapping is well defined due o F p p = E( f((x n ) n N ) p ) = = f p p, f(x) p ν(dx) (ransformaion heorem) and bijecive due o Lemma 1.1. Lineariy is rivial. Theorem 1.1 allows us o develop a differenial calculus on he sequence space (R N, B N, ν), and hen o ransfer i o he canonical space (Ω, F, P ). For his purpose we are simulaed by he reamen of he one-dimensional siuaion. Quesions of smoohness of probabiliy measures are prevalen. We sar considering hem in he seing of R. 2 Absolue coninuiy of measures on R Our aim is o sudy laws of random variables defined on (Ω, F, P ), i.e. he probabiliy measures P X for random variables X. By means of Theorem 1.1 hese measures correspond o he measures ν f 1 for B N -measurable funcions f on R N. The one-dimensional version of hese measures is given by ν 1 f 1 for B 1 -measurable funcions f defined on R. We firs discuss a simple analyic crierion for absolue coninuiy of measures of his ype. Lemma 2.1 Le µ be a finie measure on B 1. Suppose here exiss c R such ha for all φ C 1 (R) we have φ (x)µ(dx) c φ. Then µ << λ, i.e. µ is absoluely coninuous wih respec o λ, he Lebesgue measure on R. 4

5 Proof Le 0 f be coninuous wih compac suppor, and define Then fdµ = φ(x) = x f(y)dy, φ (x)µ(dx) c φ = c fdλ. By a measure heoreic sandard argumen his inequaliy follows for bounded measurable f. Therefore we conclude for A B 1 which clearly implies µ << λ. µ(a) c λ(a), We aim a applying Lemma 2.1 o he probabiliy measure ν 1 f 1 wih f B 1 -measurable. For his purpose we encouner for he firs ime he cenral echnique of inegraion by pars on Gaussian spaces, which is a he hear of Malliavin s calculus. For reasons of noaional clariy we firs recall he classical echnique of inegraion by pars. Indeed, for g, h C 0 (R) (smooh funcions wih compac suppor) we have ( ) g, h = g (x) h(x) dx = g(x) h (x) dx = g, h. This relaionship can be exended o funcions g, h L 2 (R) which vanish a ± and which possess derivaives in he disribuional sense. Le us for a momen assume his seing and denoe by dg he derivaive in disribuional sense of g, and in he sense of he dualiy (*) wih δh is adjoin operaor. Then for h C 0 (R) we have δh = h, and we can inerpre he dualiy relaionship as dg, h = g, δh. Finally, he operaor δd = d2 dx plays an imporan role in he calculus. 2 Here i is idenical o he negaive of he Laplace operaor. In he jus skeched classical calculus one does no have o disinguish beween d and δ (modulo sign). 5

6 For he analysis on Gaussian spaces hings are differen. We skech he analogue of a differenial calculus wih respec o dualiy on Gaussian spaces. For g, h L 2 (R, ν 1 ) denoe g h = g(x)h(x) ν 1 (dx). To apply Lemma 1.1 formally o he measure µ = ν 1 f 1 for some B 1 - measurable f, we have o wrie, assuming all operaions are jusified, φ (x)ν 1 f 1 (dx) = φ f(x)ν 1 (dx) = φ f 1 = (φ f) 1 f. Now, as in he classical seing, we wan o ransfer he derivaion o he oher argumen. For his purpose we coninue calculaing for g, h C 0 (R) g h = 1 2π = 1 2π = g (x) h(x) exp( x2 2 ) dx g(x) exp( x2 2 ) d dx = g h + xh. g(x) d [h(x) exp( x2)] dx dx 2 [h(x) exp( x2 2 )] ν 1(dx) So in he seing of Gaussian spaces, if we define as before dg as disribuional derivaive in he generalized sense, is dual operaor on a suiable space of funcions (o be described laer) has o be defined by δh = h + xh. In his sense we have he following dualiy relaionship, compleely analogously o he classical formula dg h = g δh. For he combinaion of he derivaive operaor and is dual we obain his ime he following operaor in he suiable disribuional sense. L = δd = d2 dx 2 + x d dx, 6

7 d will be called Malliavin derivaive, δ Skorokhod inegral, and L Ornsein-Uhlenbeck operaor. The domains of hese operaors will be more precisely defined in he higher dimensional seing. The presen exposiion is given for moivaing he noions o be sudied. Le us reurn o he problem of smoohness of he measure ν 1 f 1. Lemma 2.2 Le g, h L 2 (R, ν 1 ) be such ha dg, δh L 2 (R, ν 1 ). Then we have dg h = g δh. Moreover for f L 2 (R, ν 1 ) such ha δ( 1 df ) L2 (R, ν 1 ) we have ν 1 f 1 << λ. Proof We coninue he above calculaion in he noaion chosen. We have by dualiy and he inequaliy of Cauchy-Schwarz d(φ f) 1 df = φ f δ( 1 df ) = φ f(x) δ( 1 df )(x) ν 1(dx) φ δ( 1 df ) 2. Hence Lemma 1.1 can be applied wih c = δ( 1 df ) 2 which yields he desired absolue coninuiy. Wih his lemma he program for he developmen of Gaussian differenial calculus in finie and infinie dimensional spaces is skeched. We have o develop rigorously in his framework he calculus of he hree operaors. We shall hereby, for breviy, mosly concenrae on he operaors d and δ. One naural orhonormal basis of L 2 (R, ν 1 ) proves o be very useful hereby. 3 Hermie polynomials; orhogonal developmens We coninue denoing d, δ and L he operaors sudied above. They are a leas (and his is he sense in which we use hem) well defined on 7

8 C 0 (R), even, due o he inegrabiliy properies of he Gaussian densiy, on he space of polynomials in one real variable. Definiion 3.1 For n 0 le H n = δ n 1. H n is called Hermie polynomial of degree n. By definiion we have for x R H 0 (x) = 1, H 1 (x) = δ1 = x, H 2 (x) = δx = 1 + x 2, H 3 (x) = δ( 1 + x 2 ) = x 2x + x 3 = x 3 3x. Theorem 3.1 H n is a polynomial of degree n, wih leading coefficien 1. Moreover for n N (i) δh n = H n+1, (ii) dh n = nh n 1, (iii) LH n = n H n. In paricular, H n is an eigenvecor of L wih eigenvalue n. Proof (i): follows by definiion. (ii): We firs briefly invesigae he commuaor of d and δ. In fac, we have for h C 0 (R) (dδ δd) h = d( h +xh) ( h +xh ) = h +xh +h ( h +xh ) = h. This means ha (dδ δd) = id. Wih his in mind we proceed by inducion on he degree n. The claim is clear for n = 1. Assume i holds for n 1. Then dh n = dδh n 1 = δdh n 1 + H n 1 = (n 1)δH n 2 + H n 1 = nh n 1. (iii): LH n = δdh n = nδh n 1 = nh n. 8

9 Corollary 3.1 For g L 2 (R) define he Fourier ransform by ĝ(u) = 1 2π R eiux g(x) dx, u R. Then (H n e x2 2 )(u) = (iu) n e u2 2. Proof Choose u R. Then (H n e x2 2 )(u) = (δ n 1 e x2 2 )(u) = e iu δ n 1 = d n e iu 1 = (iu) n e iu 1 = (iu) n e u2 2. Wih hese preliminaries, we can show ha he Hermie polynomials consiue an orhonormal basis of our Gaussian space in one dimension. Theorem 3.2 ( 1 n! H n ) n 0 is an orhonormal basis of L 2 (R, ν 1 ). Proof 1. Le n, k N, and suppose ha n < k. Then by Theorem 3.1 while H n H k = δ n 1 δ k 1 = d k δ n 1 1 = 0, H n H n = d n δ n 1 1 = n! 1 1 = n!. 2. I remains o show ha (H n ) n 0 is complee in L 2 (R, ν 1 ), i.e. he se of linear combinaions of Hermie polynomials is dense in L 2 (R, ν 1 ). For his purpose, i suffices o show: if φ L 2 (R, ν 1 ) saisfies for all n 0 we have H n φ = 0, hen φ = 0. For z C le F (z) = R φ(v)eivz 1 2 v2 dv. Then we have for k N, R by Cauchy-Schwarz R vk φ(v) e v 1 2 v2 dv [ R φ2 (v)e 1 2 v2 dv 9 R v2k e 2v 1 2 v2 dv] 1 2 <.

10 Hence F may be differeniaed arbirarily ofen under he inegral sign, which implies ha F is an enire funcion. Moreover, we have for k 0 wih x k = k l=0 a l H l (x) F (k) (0) = i k R vk φ(v)e v2 2 dv = i k x k φ = i k k a l H l φ = 0. l=0 This, however, implies ha F = 0, and so by he uniqueness of Fourier ransforms also φ = 0. We now reurn o our arge space, namely R N, he sequence space version of our infinie dimensional Gaussian space. Our ask will be o esablish in his space suiable noions of he operaors d and δ. For his purpose i will be convenien o have again an orhonormal basis of his Gaussian space. We have o define an infinie dimensional exension of Hermie polynomials. Definiion 3.2 For n N le E n = Z n +, le E be he se of sequences in Z + ha vanish excep for finiely many componens. For p = (p 1,, p k, 0, ) E le p = k i=1 p i, = k i=1 p i!. For x R k resp. x R N, and p E k resp. p E le H p (x) = k i=1 H pi (x i ) i N H pi (x i ). resp. H p is called k-dimensional resp. generalized Hermie polynomial. We can exend Theorem 3.2 o he mulidimensional seing. Theorem 3.3 ( 1 H p ) p Ek is an orhonormal basis of L 2 (R k, B k, ν k ), ( 1 H p ) p E is an orhonormal basis of L 2 (R N, B N, ν). Proof 1. For k N and g, h L 2 (R k, ν k ) denoe g h = R k g(x)h(x) ν k(dx). 10

11 Then for p, q E k we have, due o Fubini s heorem H p H q = k H pi H qi. i=1 Hence ( 1 H p ) p Ek is an orhonormal sysem. Moreover, linear combinaions of ensor producs of funcions of one of k variables are dense in L 2 (R k, B k, ν k ). Hence he firs claim follows from Theorem The se n N πn 1 [L 2 (R n, B n, ν n )] is dense in L 2 (R N, B N, ν). Hence, he second asserion follows from he firs. We nex define and sudy Sobolev spaces in finie and infinie dimension, for which we use our knowledge of he orhonormal bases jus acquired. 4 Finie dimensional Gaussian Sobolev spaces Le k N. We consider k-dimensional spaces firs. Before reaing he Gaussian spaces, le us recall he mos imporan facs abou classical Sobolev spaces, i.e. Sobolev spaces wih respec o Lebesgue measure on R k. Definiion 4.1 Le p 1. For f L p (R k ), a R k we say ha f possesses a direcional (generalized) derivaive in direcion a, if here is a funcion d a f L p (R k ) such ha as ε 0. Le 1 ε [f( + εa) f] d af p 0 W p 1 = {f L p (R k ) : f possesses a direcional derivaive in direcion a for any a R k } ( Sobolev space of order (1, p)). By lineariy, i is clear ha if for 1 i k we denoe by e i he ih canonical basis vecor, hen for f W p 1 (R k ) we have d a f = k i=1 a i d ei f, if a = (a 1,, a k ). Le d i = d ei for a canonical basis e 1,, e k of R k. 11

12 Definiion 4.2 Le p 1, s N. We define recursively W p s = {f W p 1 : d a f W p s 1 for any a R k } ( Sobolev space of order (s, p)). For f W p s, a 1,, a s R k we define recursively d a1 d a2 d as f. We define he (1, p)-sobolev norm by f 1,p = f p + k d i f p, f W1 p, i=1 and analogous norms for higher derivaive orders. For any p 1, s N we have C 0 (R k ) W p s and for g C 0 (R k ), a = (a 1,, a k ) R k we have d a g = k i=1 a i g x i. Wha is he relaionship of our Sobolev spaces and he weak derivaives or derivaives in disribuional sense encounered above? Definiion 4.3 Le f L 1 loc(r k ), a R k. Then u a L 1 loc(r k ) is called weak derivaive of f in direcion a, if for any φ C 0 (R k ) he equaion is saisfied. f, d a φ = u a, φ Theorem 4.1 Le p 1, f L p (R k. Then he following are equivalen: (i) f W p 1, (ii) for a R k f possesses a weak derivaive u a, and we have u a L p (R k ). In his case, moreover, d a f = u a. 12

13 Proof 1. Le us show ha (i) implies (ii). For his purpose, assume a R k, φ C 0 (R k ). Then for ε > 0 by ranslaional invariance of Lebesgue measure 1 ε [f(x + εa) f(x)]φ(x)dx = 1 f(x)[φ(x εa) φ(x)]dx. ε Now use (i), le ε 0, o idenify he limi as d a f(x)φ(x)dx on he lef hand side, and as f(x)d a φ(x)dx on he righ hand side. Hence for any φ C0 (R k ) d a f, φ = f, d a φ. This means by definiion ha f possesses he weak derivaive d a f which belongs o L p (R k ). 2. Le us now prove ha (ii) implies (i). Fix φ C 0 (R k ), a = (a 1,, a k ) R k. Then by Taylor s formula wih inegral remainder erm and Fubini s heorem we have for any ε > 0 = = 1 R k ε 1 R k = 1 ε = 1 ε = 1 ε [f(x + εa) f(x)]φ(x)dx f(x)[φ(x εa) φ(x)]dx ε R k f(x)[1 ε ε k [ 0 R k i=1 ε [ 0 ε [ 0 ε 0 k i=1 a i φ x i (x ξa)dξ]dx a i φ x i (x ξa)f(x)dx]dξ R k φ(x ξa)u a(x)dx]dξ R k φ(x)u a(x + ξa)dx]dξ ε = R k[1 ε u a(x + ξa)dξ]φ(x)dx. 0 I remains o prove ha 1 ε0 ε u a ( +ξa)dξ converges o u a in L p (R k ). This is cerainly rue provided u a C0 (R k ). Bu for any f, g L p (R k ) we have uniformly in ε > 0 1 ε ε 0 f( + ξa)dξ 1 ε ε 0 g( + ξa)dξ p ε 1 ε f( + ξa) g( + ξa) pdξ 0 = g f p. 13

14 By means of his observaion we can ransfer he desired resul from C 0 (R k ) o L p (R k ), since C 0 (R k ) is dense in L p (R k ). Corollary 4.1 Le e 1,, e k denoe he canonical basis of R k, le (f n ) n N be a sequence in W1 p such ha (i) f n f p 0 as n, (ii) for any 1 i k he sequence (d i f n ) n N converges in L p (R k ). Then f W1 p and f n f 1,p 0 as n. Proof We have o show ha f is weakly differeniable in direcion e i for 1 i k, and d i f = lim n d i f n L p (R k ). For his purpose le u i = lim n d i f n, which exiss due o assumpion (ii). Then by (i) for any φ C0 (R k ), 1 i k f(x)d i φ(x)dx = n lim = lim n f n (x)d i φ(x)dx d i f n (x)φ(x)dx = u i (x)φ(x)dx. This means ha f possesses weak direcional derivaives in direcion e i and d i f = u i L p (R k ). Now Theorem 4.1 is applicable and finishes he proof. Corollary 4.2 Le p 1. Then W1 p is a Banach space wih respec o he norm 1,p, and for any a R k he mapping d a : W1 p L p (R k ) is coninuous. Proof We have o prove ha W p 1 is complee wih respec o 1,p. Le herefore (f n ) n N be a Cauchy sequence in W p 1. Then seing f = lim n f n in L p (R k ), we see ha he hypoheses (i) and (ii) of Corollary 4.1 are saisfied, and i suffices o apply his Corollary. We finally need a local version of Sobolev spaces. Definiion 4.4 For p 1, s N le W p s,loc = {f : f : R k R measurable fφ W p s for φ C 0 (R k ).} ( local Sobolev space of order (s, p)). 14

15 Theorem 4.2 Le p 1, s N. Then f W p s,loc iff for any x 0 R k here exiss an open neighborhood V x0 of x 0 such ha for any φ C 0 (R k ) wih suppor in V x0 we have φf W p s. Proof We only need o prove he only if par of he claim. For any x 0 R k le herefore V x0 be given according o he saemen of he asserion. Then (V x0 ) x0 R is an open covering of k Rk. Then here exiss a locally finie pariion of he uni (φ k ) k N C0 (R k ) which is subordinae o he covering, i.e. such ha (i) 0 φ n 1, for any n N, (ii) for any n N here exiss x 0 (n) such ha supp(φ n ) V x0 (n), (iii) n N φ n = 1, (iv) for any compac se K R k he inersecion of K and supp(φ n ) is non-empy for a mos finiely many n. Now le φ C0 (R k ). Then for any k N (ii) gives supp(φφ k ) V x0 (k) and hus by assumpion φ k φf W p s, k N. Since by (iv) he suppor of φ k φ is non-rivial for a mos finiely many k, (iii) and lineariy yield he desired φf W p s. We now urn o Gaussian Sobolev spaces. Our analysis will again be based on he differenial operaor we know from he above skeched classical calculus. Only he measure wih respec o which we consider dualiy changes from he Lebesgue o he Gaussian measure. Since we hereby pass from an infinie o a finie measure, inegrabiliy properies for funcions and herefore he domains of he dual operaors change. This is why he noion of local Sobolev spaces is imporan. On hese spaces, we can define our operaors locally, wihou reference o inegrabiliy firs. In fac, using Theorem 4.2, and for s N, p 1, 1 j 1,, j s k, f W p s,loc we can define d j1 d j2 d js f 15

16 locally on an open neighborhood V x0 of an arbirary poin x 0 R k by he corresponding generalized derivaive of φf wih φ C 0 (R k ) such ha φ = 1 on an open neighborhood U x0 V x0 of x 0. This gives a globally unique noion, since x 0 is arbirary. Definiion 4.5 Le s N, p 1, 1 j k, f W p s,loc, and denoe by d j he direcional derivaive in direcion of he jh uni vecor in R k according o he preceding remark. Le hen f = (d 1 f,, d k f), δ j f = d j f + x j f, Lf = k δ j d j f = k [ d j d j f + x j d j f]. j=1 j=1 For any 1 r s we define more generally r f = (d j1 d j2 d jr f : 1 j 1, j 2,, j r k). This definiion gives rise o he following noion of Gaussian Sobolev spaces. Definiion 4.6 Le p 1, s N. Then le D p s(r k ) = {f W p s,loc : s r=0 f s,p = s r f p r=0 r f p < }, ( k-dimensional Gaussian Sobolev space of order (s, p)). Remark D p s(r k ) is a Banach space. This is seen by argumens as for he proof of Corollary 4.2. Since our calculus will be based mosly on he Hilber case p = 2, we shall resric our aenion o his case whenever convenien. In his case, our ONB composed of k-dimensional Hermie polynomials as invesigaed in he previous chaper will play a cenral role, and adds srucure o he seing. To ge acquainance wih Gaussian Sobolev spaces, le us 16

17 compue he operaors defined on he series expansions wih respec o his ONB. For f L 2 (R k, ν k ) we can wrie f = c p (f) p E k wih coefficiens c p (f) R, p E k. Due o orhogonaliy, he Gaussian norm is given by H p f 2 = c p (f) 2 H p E k 2 p H p = c p (f) 2. p E k We also wrie f (c p (f)) o denoe his series expansion. Denoe by P he linear hull of he k-dimensional Hermie polynomials. Plainly, P W p s,loc for any s N, p 1. According o chaper 3, P is dense in L 2 (R k, ν k ). And for funcions in P, he generalized derivaives d j are jus idenical o he usual parial derivaives in direcion j, 1 j k. We firs calculae he operaors on Hermie polynomials. In fac, for p E k, 1 j k we have in he non-rivial cases d j H p = p j H pi H pj 1, δ j H p = H pi H pj +1, LH p = p H p. i j i j Hence for f (c p (f)) P, 1 j k we may wrie d j f = c p (f) p E k δ j f = c p (f) p E k p j i j i j Lf = c p (f) p H p. p E k H pi H pj 1, H pi H pj +1, According o Corollary 4.2 and he calculaions jus skeched, he naural domains of he operaors exending, δ j and L beyond P mus be hose disribuions in R k for which he formulas jus given generae convergen series in he L 2 -norm wih respec o ν k. The mos imporan domain is he one of, he Sobolev space D 2 1(R k ). For f (c p (f)) P 17

18 we have f 2 2 = = k R k f 2 (x)ν k (dx) j=1 = k p 2 j j=1 p E k = k j=1 = R k d jf 2 (x)ν k (dx) c p (f) 2 2 c p (f) 2 p j p E k p c p(f) 2. p E k i j p i!(p j 1)! If in addiion f L 2 (R k, ν k ), we may wrie f (c p (f)) and approximae i by f n = p E k, p n c p(f) P, n N. Hence, according o corollary 4.2, f belongs o D1(R 2 k ) if he following series converges f 2 2 = lim n f n 2 2 = lim n = p E k, p n p c p(f) 2 p c p(f) 2 <. p E k Along hese lines, we now urn o describing Gaussian Sobolev spaces and he domains of our principal operaors for p = 2 by means of Hermie expansions. We sar wih he case k = 1. Theorem 4.3 Le r N, f (c p (f)) L 2 (R, ν 1 ) Wr,loc. 2 Denoe f p = c p(f) H p, p 0. Then he following are equivalen: (i) r f L 2 (R, ν 1 ), (ii) p 0 p r f p 2 2 <, (iii) f Dr(R), 2 (iv) δ r f L 2 (R, ν 1 ). In paricular, Dr(R) 2 is he domain of r, δ r in L 2 (R, ν 1 ). For f, g D1(R) 2 we have f g = f δg. 18

19 Proof 1. We prove equivalence of (i) and (ii). We have f = p 1 and herefore by ieraion Therefore and hence if and only if p c p (f) H p 1 = p 0 r f = p 0 c p+r (f) H p 1. c p+1 (f) H p, r f 2 2 = c p+r (f) 2, f p 2 2 = c p(f) 2, p 0 r f 2 2 = p 0 p 0 and his is he case if and only if (p + r)! f p+r 2 2 < (p + r) r f p+r 2 2 <, p 0 p r f p 2 2 <. 2. We nex prove ha (ii) and (iv) are equivalen. Noe ha δf = p 0 This implies ha c p (f) H p+1, and herefore δ r f = p 0 c p (f) H p+r. δ r f 2 2 = c p (f) 2 (p + r)! = (p + r) (p + 1) c p(f) 2 < p 0 2 p 0 if and only if p 0 p r c p(f) 2 = p 0 p r f p 2 2 <. 3. The equivalence of (i) and (iii) is conained in he definiion. 4. Le f (c p (f)), g (c p (g)) D 2 1(R). Then we have f g = p 0 c p+1 (f) 19 c p (g),

20 whereas This complees he proof. f δg = p 0 c p+1 (f) c p (g). The differenial calculus on Gaussian spaces obeys similar rules as he classical differenial calculus. Theorem 4.4 Le g D 4 1(R), µ = ν 1 g 1. If φ L 4 (R, µ), and φ L 4 (R, µ), we have (φ g) = ( φ) g g. Proof If φ and g belong o C 0 (R), he asserion is clear. To generalize, approximae in P and use Hölder s inequaliy. Theorem 4.5 Le f, g D 4 1(R). Then f g D 2 1(R) and we have (f g) = f g + f g. Proof The asserion is clear for f, g C 0 (R). To generalize, approximae in P and use Hölder s inequaliy. We now urn o arbirary finie dimension k, and inerpre Gaussian Sobolev spaces by convergence properies of Hermie expansions as above. Theorem 4.6 Le f (c p (f)) L 2 (R k, ν k ) W 2 1,loc. Denoe f p = c p (f) H p, p E k. Then he following are equivalen: (i) f = [ k j=1 (d j f) 2 ] 1 2 L 2 (R k, ν k ), (ii) p E k p f p 2 2 <, (iii) f D 2 1(R k ). In paricular, D 2 1(R k ) is he domain of in L 2 (R k, ν k ). Analogous resuls hold for Sobolev spaces of order (r, 2) wih r N. Proof Analogous o he proof of Theorem

21 5 Infinie dimensional Gaussian Sobolev spaces To refer he infinie dimensional seing o he finie dimensional one, we use he following observaion. For n N recall π n : R N R n, (x n ) n N (x k ) 1 k n. Le for n N le C n filraion on (R N, B N ). = σ(π n ) = σ(π 1,, π n ). Then (C n ) n N is a Lemma 5.1 Le p 1, f L p (R N, B N, ν). Then ˆf n = E(f C n ), n N defines a maringale which converges ν-a.s. and in L p o f. Proof This follows from a sandard heorem of discree maringale heory. Le in he following f n : R n R he n-dimensional facorizaion of ˆf n, relaed by f n π n = ˆf n, n N. As a crucial observaion for he definiion of infinie dimensional Sobolev spaces, he maringale propery is essenially no desroyed by he direcional derivaive operaors. Lemma 5.2 Le p > 1, f L p (R N ), (f n ) n N he corresponding sequence according o he above remarks. Suppose ha sup n N f n 1,p <. Then for any j N he sequence (d j f n π n ) n N converges in L p (R N ), o a limi ha we denoe by d j f. Corresponding saemens hold rue for higher order derivaives. Proof Le n N, j N. Then for n j we have E(d j f n+1 π n+1 C n ) = d j f n π n. This means ha (d j f n π n ) n j is a maringale wih respec o (C n ) n j which, due o sup d j f n π n p sup f n 1,p <, n j n N 21

22 is bounded in L p (R N ) and hence converges in L p (R N ), due o p > 1. The preceding Lemmas give rise o he following definiion of Sobolev spaces. Definiion 5.1 Le p 1, s N. Then Ds(R p N ) = {f L p (R N, ν) : f n Ds(R p n ), n N, sup f n s,p < }, n N ( infinie dimensional Sobolev space of order (s, p)), endowed wih he norm f s,p = sup f n s,p, f Ds(R p N ). n N This definiion makes sense, for he following reasons. Theorem 5.1 Le p > 1, s N. Then D p s(r N ) is a Banach space wih he norm s,p. Proof We prove he claim for s = 1. Le (f m ) m N be a Cauchy sequence in D p 1(R N ), and (f m n ) n,m N he corresponding finie dimensional funcions according o he remarks above. Then for m, l N, n N Jensen s inequaliy and he maringale saemen in he preceding proof give he following esimae lim sup fn m fn l 1,p lim f m f l 1,p = 0. m,l m,l D p 1(R n ) being a Banach space for n N, we know ha f n = lim m f m n D p 1(R n ) exiss. Le ˆf n = f n π n. Now le f = lim m f m in L p (R N ). Then by uniform inegrabiliy Moreover E(f C n ) = E( lim m f m C n ) = lim m E(f m C n ) = lim m ˆf m n = ˆf n. sup f n 1,p sup fn m 1,p sup f m 1,p <. n N m,n N m N 22

23 Hence by definiion f D p 1(R N ), and by Faou s lemma f f m 1,p lim inf l f m f l 1,p 0 as m. According o Lemma 5.2, he gradien on he infinie dimensional Gaussian Sobolev spaces is defined as follows. Definiion 5.2 Le p > 1, f D p 1(R N ). Then le f = (d j f) j N ( Malliavin gradien or Malliavin derivaive), where for any j N according o Lemma 5.2 d j f = lim n d j f n π n. Accordingly, for s N we define r f, 1 r s, for f D p s(r N ). Remark The gradien being a coninuous mapping from D1(R p n ) o L p (R n, ν n ) for any finie dimension n, Lemma 5.2 and he definiion of he Malliavin gradien imply, ha is a coninuous mapping from D1(R p N ) o L p (R N, ν). Le us now again resric our aenion o p = 2 and describe Gaussian Sobolev spaces by means of he generalized Hermie polynomials. Firs of all, suppose f = p E c p(f) L 2 (R N, ν). We shall coninue o use he noaion f (c p (f)). Then for n N, we have f n = c (p,0) (f) p E n H p, where we pu (p, 0) = (p 1,, p n, 0, 0, ) for p = (p 1,, p n ) E n. Therefore, we also have ˆf n = c (p,0) (f) p E n H (p,0). Le again P be he linear hull generaed by all generalized Hermie polynomials. As in he preceding chaper, we may calculae he gradien norms for f (c p (f)) D1(R 2 N ). In fac, we have for j N d j f = n lim d j f n π n = n lim = p E c (p,0)(f) p j p E n i j c p (f) p j H pi H pj 1. i j H pi H pj 1 23

24 Furhermore, for f D 2 1(R N ) le us compue he norm of f = [ j N(d j f) 2 ] 1 2 in L 2 (R N, ν). In fac, we have, using he calculaion of gradien norms in he preceding chaper, > sup f n π n 2 2 = f 2 2 n N = sup p c (p,0)(f) 2 n N p E n = p E p c p(f) 2. We herefore obain he following main resul abou he descripion of he infinie dimensional Gaussian Sobolev spaces or order (1, 2). Theorem 5.2 For f L 2 (R N, ν) he following are equivalen: (i) f D1(R 2 N ), (ii) p E p c p(f) 2 <, (iii) f n π n = [ j N(d j f n ) 2 π n ] 1 2 converges in L 2 (R N, ν) o f. Moreover, D1(R 2 N ) is a Hilber space wih respec o he scalar produc (f, g) 1,2 = f g + d j f d j g, f, g D1(R 2 N ). j N For p 2 P is dense in D p 1(R N ). Analogous resuls hold for Sobolev spaces of order (s, 2) wih s N. 6 Absolue coninuiy in infinie dimensional Gaussian space We are now in a posiion o discuss he main resul of Malliavin s calculus in he framework of infinie dimensional Gaussian sequence spaces. The resul is abou he smoohness of laws of random variables defined on he Gaussian space. We sar wih a generalizaion of Lemma 1.1 o finie measures on B d for d N. Lemma 6.1 Le µ B d be a finie measure. Assume here exiss c R such ha for all φ C 1 (R d ) wih bounded parial derivaives, and any 1 j d we have x j φ(x) µ(dx) c φ. Then µ << λ d (d-dimensional Lebesgue measure). 24

25 Proof For simpliciy, we argue for d = 2, and omi he superscrip denoing 2-dimensional Lebesgue measure. 1. Assume ha φ C 1 (R 2 ) possesses compac suppor. We show: [ In fac, we have [ φ 2 dλ] 1 2 [ φ 2 dλ] [ [ 1 2 [ φ 2 dλ + x 1 sup φ(x 1, x 2 ) dx 2 x 1 R φ(x 1, x 2 ) dx 1 dx 2 x 1 x 1 φ 2 dλ + sup x 2 R x 2 φ 2 dλ]. x 2 φ 2 dλ]. φ(x 1, x 2 ) dx 1 ] 1 2 x 2 φ(x 1, x 2 ) dx 2 dx 1 ] Le 0 u be coninuous wih compac suppor, and such ha udλ = 1, define for ε > 0 uε = 1 1 ε u( ε ). Moreover, le 2 ψ ε = u ε ( y)µ(dy) be a smoohed version of µ. Then we obain for h coninuous wih compac suppor, using Fubini s heorem, 3. We show: ψ ε (x)h(x)dx = = = [ [ [ L 2 (R 2 ) g is a coninuous linear funcional. u ε (x y)µ(dy)]h(x)dx u ε (x y)h(x)dx]µ(dy) u(x)h(εx + y)dx]µ(dy) h(y)µ(dy). gdµ R In fac, le φ C 1 (R 2 ) have compac suppor, and le ε > 0. Then by hypohesis and smoohness of ψ ε wih a calculaion as in 2. ψ ε (x)φ(x)dx = x i = [ ψ ε (x) φ(x)dx x i 25 u ε (x y) x i φ(x)dx]µ(dy)

26 = [ = [ c c φ. u ε (x y)φ(x)dx]µ(dy) x i u ε (x y)φ(x)dx]µ(dy) y i u ε (x )φ(x)dx Generalizing his inequaliy o bounded measurable φ, and hen aking φ = sgn(ψ ε ) yields he inequaliy x i ψ ε dλ c for any ε > 0. Now le ε > 0, g L 2 (R 2 ) be given. Then, using 1. and he esimae above ψ ε (x)g(x)dx [ ψ ε (x) 2 dx g(x) 2 dx] [ ψ ε dλ + x 1 c g 2. x 2 ψ ε dλ] 1 2 g 2 Applying his inequaliy in he special case, in which g is coninuous wih compac suppor, and using 2. we ge g(x)µ(dx) c g 2. Finally exend his inequaliy o g L 2 (R 2 ) by approximaing i wih coninuous funcions of compac suppor. This yields he desired coninuiy of he linear funcional. 4. I remains o apply Riesz represenaion heorem o find a square inegrable densiy for µ. We now consider a vecor f = (f 1,, f d ) wih componens in L 2 (R N, B N, ν). Our aim is o sudy he absolue coninuiy wih respec o λ d of he law of f under ν, i.e. of he probabiliy measure ν f 1. For his purpose we plan o apply he crierion of Lemma 6.1. Le φ C 1 (R d ) possess bounded parial derivaives. Then, he inegral ransformaion heorem gives φdν f 1 = x i 26 x i φ fdν.

27 In case d = 1 a his place we use inegraion by pars hidden in he represenaions φ (f) = d(φ f) 1 df. d(φ f) = φ (f)df, Our infinie dimensional analogue of d is he Malliavin gradien. Hence, we need a chain rule for. Theorem 6.1 Le p 2, f D p 1(R N ) d, φ C 1 (R d ) wih bounded parial derivaives. Then φ f D p 1(R N ) and [φ f] = d i=1 x i φ(f) f i. Proof Use Theorem 5.2 o choose a sequence (f n ) n N P d such ha for any 1 i d f i n f i 1,p 0. For each n N we have [φ f n ] = d i=1 x i φ(f n ) f i n. Since is coninuous on D p 1(R N ), and since he parial derivaives of φ are bounded, we furhermore obain ha [φ f] = n lim [φ f n ] = d φ(f) f i i=1 x i in L 2 (R N, ν). This complees he proof. We nex presen a calculaion leading o he verificaion of he absolue coninuiy crierion of Lemma 6.1. We concenrae on he algebraic seps, and remark ha heir analyic background can be easily provided wih he heory of chaper 5. The firs aim of he calculaions mus be o isolae, for a given es funcion φ C 1 (R d ) wih bounded parial derivaives, he expression x i φ(f), 1 i d. Recall he noaion (x, y) = x i y i, x, y l 2. i=1 27

28 For 1 i, k d le Then we have for 1 k d ( (φ f), f k ) = σ ik = ( f i, f k ). j=1 = d j (φ f)d j f k j=1 1 i d = 1 i d x i φ(f)d j f i d j f k x i φ(f)σ ik. We now assume ha he marix σ is (almos everywhere) inverible. Then, denoing is inverse by σ 1 we may wrie x i φ(f) = 1 k d = 1 k d j=1 ( (φ f), f k σ 1 ki ) d j (φ f)σ 1 ki d j f k. We nex assume, ha he dual operaor δ j of d j, which is defined in he usual way on P, is well defined and he series appearing is summable. Then we have x i φ(f)dν = = 1 k d j=1 φ f[ d j (φ f)σ 1 ki d j f k dν 1 k d j=1 δ j (d j f k σ 1 ki )]dν. The righ hand side can be esimaed by c φ wih c = 1 k d j=1 δ j (d j f k σ 1 ki ) 2 in L 2 (R N, ν). I can be seen (in analogy o Theorem 4.3) ha his series makes sense under he hypoheses of he following main heorem. Theorem 6.2 Suppose ha f = (f 1,, f d ) L 2 (R N, ν) saisfies (i) f i D 4 2(R N ) for 1 i d, (ii) σ ik = ( f i, f k ), 1 i, k d, is ν-a.s. inverible and σ 1 ki D 4 1(R N ) for 1 i, k d. Then we have ν f 1 << λ d. 28

29 Proof Approximae f by polynomials, use coninuiy properies of he operaors. 7 The canonical Wiener space: muliple inegrals We now reurn o he canonical Wiener space. The ransfer beween sequence and canonical space is provided by he isomorphism of chaper 1. We briefly recall i. Le (g i ) i N be an orhonormal sequence in L 2 (R + ). Then i is given by T : L p (R N, B N, ν) L p (Ω, F, P ), f f ((W (g i ) i N ), where W (g i ) is he Gaussian sochasic inegral of g i for i N. I will be consruced in he following chaper. For simpliciy we confine our aenion o he canonical Wiener space in one dimension, i.e. Ω = C(R +, R), F he (compleed) Borel σ-algebra on Ω generaed by he opology of uniform convergence on compac ses in R +, P Wiener measure on F. In he approach of differenial calculus on Gaussian sequence spaces, in he Hilber space seing he mos imporan ool proved o be he Hermie expansions of funcions in L 2 (R N, ν). In he seing on he canonical space, hey can be given a differen inerpreaion which we shall now develop. According o our isomorphism, he objecs corresponding o generalized Hermie polynomials on he canonical space are given by i=1 H pi (W (g i )), p E. We shall inerpre hese objecs as ieraed Iô inegrals. To do his, we use he abbreviaion B 1 + for he Borel ses of R +. Definiion 7.1 For m N le E m = {f f : R m + R, f = n i 1,,i m =1 a i1 i m 1 Ai1 A im, (A i ) 1 i n B 1 + p.d., a i1 i m = 0 in case i k = i l for some k l}. Remark For f L 2 (R + ) of he form f = n a i 1 Ji, (J i ) 1 i n p.d. inervals in R + i=1 29

30 le W (f) = n a i W (J i ) = n a i (W i W si ), i=1 if J i =]s i, i ], 1 i n. Then by Iô s isomery we have i=1 W (f) 2 2 = f 2 2. Since E 1 is dense in L 2 (R + ), we can exend he linear mapping f W (f) o L 2 (R + ). Therefore, in paricular for A B 1 + wih finie Lebesgue measure, W (A) = W (1 A ) is defined. I will be used for he definiion of he following muliple sochasic inegrals. In his chaper, he scalar produc, will be wih respec o Lebesgue measure on R m + wih unspecified ineger m. Definiion 7.2 For f = n i1,,i m =1 a i1 i m 1 Ai1 A i m E m le I m (f) = n i 1,,i m =1 a i1 i m W (A i1 ) W (A im ). The addiiviy of B 1 + A W (A) R implies ha I m is well defined. We sae some elemenary properies of I m. Denoe by S m he se of all permuaions of he numbers 1,, m. Lemma 7.1 Le m, q N, f E m, g E q. (i) I m E m is linear, (ii) if f(1,, m ) = 1 m! σ S m f( σ(1),, σ(m) ) ( symmerizaion of f), hen I m (f) = I m ( f), (iii) E(I m (f)i q (g)) = m! f, g, if m = q, and 0 oherwise. Proof 1. (i) follows from he addiiviy of he map A W (A). 2. (ii) is a direc consequence of he fac ha in he definiion of I m he produc W (A i1 ) W (A im ) is invarian under permuaions of he facors. 30

31 3. By (ii), we may assume ha f, g are symmeric. By choosing common subdivisions, we may furher assume ha f = g = n i 1,,i m =1 n i 1,,i q =1 a i1 i m 1 Ai1 A i m, b i1 i q 1 Ai1 A i q. Now if m q, by he assumpions ha (A i ) 1 i n consiss of p.d. Borel ses, and ha coefficiens vanish if wo of he indices coincide, E(I m (f)i q (g)) = 0 is eviden. Assume m = q. Then, again by hese wo assumpions and symmery E(I m (f)i m (g)) = n n i 1,,i m =1 j 1,,j m =1 = m! 2 i 1 < <i m a i1 i m b j1 j m E( m p=1 a i1 i m b j1 j m E( m j 1 <,j m = m! 2 a i1 i m b i1 i m i 1 < <i m = m! f, g. m p=1 λ(a ip ) W (A ip )W (A jp )) p=1 W (A ip )W (A jp )) To exend I m beyond he space E m of elemenary funcions, we proceed as for m = 1. Lemma 7.2 E m is dense in L 2 (R m +) for any m N. Proof We may assume m 2, he asserion being known for m = 1. Due o sandard resuls of measure heory, i is enough o show: for A 1,, A m B 1 + wih finie Lebesgue measure, and ε > 0, here exiss f E m such ha 1 A1 A m f 2 < ε. Le δ > 0 o be deermined laer. Choose B 1,, B n B 1 + wih finie Lebesgue measure, pairwise disjoin, and such ha for any 1 j n we have λ(b j ) < δ, and such ha any A i can be represened by a finie union of some of he B j. Then we have 1 A1 A m = n i 1,,i m =1 31 b i1 i m 1 Bi1 B im,

32 where b i1 i m = 1 if B i1 B im A 1 A m, and 0, if no. Le I = {(i 1,, i m ) : i k i l for k l}, and J = {1,, n} m \ I. Then by definiion f = b i1 i m 1 Bi1 B E i m m and we have (i 1,,i m ) I 1 A1 A m f 2 2 = b 2 m i 1 i m (i 1,,i m ) J p=1 n λ(b ip ) m(m 1) 2 i=1 i=1 m(m 1) δ( n λ(b i )) m 1 2 i=1 Finally, we have o choose δ small enough. λ(b i ) 2 ( n λ(b i )) m 2 Using Lemma 7.2, we may now exend I m o L 2 (R m +). Definiion 7.3 The linear and coninuous exension of I m E m o L 2 (R m +) which exiss according o Lemma 7.2 is called muliple Wiener-Iô inegral of degree m and also denoed by I m. Properies of he elemenary inegral are ransferred in a sraighforward way. Theorem 7.1 Le m, q N, f L 2 (R m +), g L 2 (R q +). Then (i) I m L 2 (R m +) is linear, (ii) we have I m (f) = I m ( f), (iii) E(I m (f)i q (g)) = m! f, g, if m = q, and 0 oherwise, (iv) I 1 (f) = W (f), f L 2 (R + ). Noaion We wrie I m (f) = = R m + R m + f( 1,, m )dw 1 dw m f( 1,, m )W (d 1 ) W (d m ). 32

33 We nex aim a explaining he relaionship beween generalized Hermie polynomials and muliple Wiener-Iô inegrals. For his purpose we will need a recursive relaionship beween Hermie polynomials of differen degrees. Remark Recall he definiion of Hermie polynomials in one variable, given by H n = δ n 1. Moreover, we may compue for n N xh n = (d + δ)h n = nh n 1 + H n+1, or H n+1 = xh n nh n 1. For echnical reasons, we need he following operaion of conracion. Definiion 7.4 Le m N. Suppose f L 2 (R m +), g L 2 (R + ). Then for 1,, m, R + f g( 1,, m, ) = f( 1,, m ) g() (ensor produc), f 1 g( 1,, m 1 ) = R + f(1,, m )g( m ) d m (conracion). The recursion relaion for Hermie polynomials will emerge from he recursion relaion beween Wiener-Iô inegrals saed in he following Lemma. Lemma 7.3 Le m N, f L 2 (R m +), g L 2 (R + ). Then we have I m (f)i 1 (g) = I m+1 (f g) + mi m 1 (f 1 g). Proof 1. By lineariy and densiy of E m in L 2 (R m +) we may assume ha f = 1 A1 A m, g = 1 A0 or g = 1 A1, where (A i ) 0 i m B 1 + is a collecion of p.d. Borel ses wih finie Lebesgue measure. 2. The case g = 1 A0 is rivial. Then he second erm on he righ hand side of he claimed formula vanishes, and he oher wo erms are obviously idenical by definiion of he elemenary inegral. 33

34 3. Le now g = 1 A1. For ε > 0 choose a collecion of p.d. ses B 1,, B n B 1 + such ha A 1 = n i=1b i, and for any 1 i n we have λ(b i ) < ε. Then I m (f)i 1 (g) = W (A 1 ) 2 W (A 2 ) W (A m ) = i j W (B i )W (B j )W (A 2 ) W (A m ) + 1 i n [W (B i ) 2 λ(b i )]W (A 2 ) W (A m ) + λ(a 1 )W (A 2 ) W (A m ). a) We now prove ha he firs erm on he righ hand side of our formula is close o I m+1 (f g). In fac, le h ε = i j 1 Bi B j A 2 A m E m+1. Then h ε f g 2 2 n λ(b i ) 2 λ(a 2 ) λ(a m ) i=1 ελ(a 1 ) λ(a m ). b) Le us nex prove ha he second erm on he righ hand side is negligeable in he limi ε 0. In fac, denoe i by R ε. Then, since for 1 i n he variance of W 2 (B i ) λ(b i ) is given by cλ(b i ) 2 wih some consan c, we obain E(Rε) 2 c n λ(b i ) 2 λ(a 2 ) λ(a m ) ɛ cλ(a 1 )λ(a 2 ) λ(a m ). i=1 c) To evaluae he las erm, noe ha Therefore 1 A1 A m 1 1 A1 = 1 1 A2 A m m λ(a 1 ). λ(a 1 )W (A 2 ) W (A m ) = mi m 1 ( 1 A1 A m 1 1 A1 ), and we obain he desired recursion formula. This finally pus us in a posiion o derive he relaionship beween Hermie polynomials and ieraed sochasic inegrals. 34

35 Theorem 7.2 Le m N, h L 2 (R + ) be such ha h 2 = 1. Denoe by h m he m-fold ensor produc of h wih iself. Then we have H m (W (h)) = I m (h m ). Le H 0 = R, H m = I m (L 2 (R m +)), m N. Then: (H m ) m N is a sequence of pairwise orhogonal closed linear subspaces of L 2 (Ω, F, P ) and we have L 2 (Ω, F, P ) = m=0h m. In paricular, for any F L 2 (Ω, F, P ) here exiss a sequence (f m ) m 0 of funcions f m L 2 (R m +) such ha F = m=0 I m (f m ). The represenaion wih symmeric f m is λ m -a.e. unique, m N. Proof 1. We firs have o prove: H m (W (h)) = I m (h m ). This is done by inducion on m. For m = 1, he formula is clear from H 1 = x, I 1 (h) = W (h). Now assume i is known for m. Then Lemma 7.3 and he recursion formula for Hermie polynomials given above combine o yield, remembering ha h 2 = 1, I m+1 (h m+1 ) = I m (h m )I 1 (h) mi m 1 (h m 1 h) = I m (h m )I 1 (h) mi m 1 (h m 1 ) = H m (W (h))h 1 (W (h)) mh m 1 (W (h)) = H m+1 (W (h)). 2. Le L 2 s(r m +) be he linear space of symmeric funcions in L 2 (R m +). Then by Theorem 7.1 I m ( f) 2 2 = m! f 2 2, hence H m = I m (L 2 s(r m +)) is closed. Orhogonaliy is also a consequence of Theorem

36 3. Le (g i ) i N be an orhonormal basis of L 2 (R + ), F L 2 (Ω, F, P ). Le f L 2 (R N, B N, ν) be such ha T (f) = F. Assume f (c p (f)), according o he noaion of chaper 3. For m 0, define f m = p E, p =m c p (f) i N g p i i, considered as a funcion of m variables. Then f m L 2 (R m +) and wih he help of Lemma 7.3 we see I m (f m ) = = = p E, p =m p E, p =m p E, p =m c p (f) I m ( g p i i ) i N c p (f) c p (f) i N i N I pi (g p i i ) H pi (W (g i )). Summing his expression over m yields he desired F = m=0 The remaining claims are obvious. I m (f m ). 8 The canonical Wiener space: Malliavin s derivaive In his chaper we shall invesigae he analogue of he gradien we encounered in he differenial calculus on he sequence space. Fix again an orhonormal basis (g i ) i N of L 2 (R + ), and recall he isomorphism T : L 2 (R N, B N, ν) L 2 (Ω, F, P ), f f((w (g i ) i N ). Of course, every permuaion of he orhonormal basis funcions gives anoher orhonormal basis. So here we encouner a problem of coordinae dependence of our objecs of sudy. How can we define Malliavin s derivaive on he canonical space in a boh consisen and basis independen way? According o Theorem 5.2 f = (d j f) j N 36

37 akes values in l 2. The corresponding objec on he side of he canonical space is L 2 (R + ). I is herefore plausible if we se Definiion 8.1 For n N le C p (R n ) denoe he se of smooh funcions he parial derivaives of which possess polynomial growh. Le S = {F F = f(w (h 1 ),, W (h n )), h 1,, h n L 2 (R + ), f C p (R n ), n N}. For F = f(w (h 1 ),, W (h n )) S, 0 le D F = n i=1 x i f(w (h 1 ),, W (h n )) h i (). To see if his is a good candidae for he definiion of he Malliavin gradien in he seing of he canonical space, le us verify in deail he independence on he specific represenaion of funcionals. Le h 1,, h n L 2 (R + ), and g 1,, g m L 2 (R + ) orhonormal, such ha he linear hulls of he wo sysems are idenical, and such ha wih f C p (R n ), g C p (R m ) we have For 1 i n wrie Then, denoing f(w (h 1 ),, W (h n )) = g(w (g 1 ),, W (g m )). h i = m h i, g j g j. j=1 Γ = ( h i, g j ) 1 i n,1 j m, we obviously have f Γ = g. Therefore m j=1 x j (f Γ)(W (g 1 ),, W (g m ))g j = m = n n j=1 i=1 h i, g j g j i=1 x i f(w (h 1 ),, W (h n )) x i f(w (h 1 ),, W (h n ))h i. This proves ha he definiion of D is independen of he represenaion of funcionals in S. 37

38 If h L 2 (R + ) is anoher funcion, we have by definiion D F, h = m in paricular for i N D F, g i = i=1 x i f(w (g 1 ),, W (g m )) g i, h, x i f(w (g 1 ),, W (g m )) = d i f(w (g 1 ),, W (g m )). We herefore may inerpre D F, g i as direcional derivaive in direcion of g i, and we have by Parseval s ideniy DF, DF = m DF, g i 2 = m (d i f) 2 (W (g 1 ),, W (g m )) i=1 i=1 = f 2 (W (g 1 ),, W (g m )). Analogously, higher derivaives are relaed o each oher. So we see ha he isomorphism T also maps DF, DF o f 2. Consequenly, we can jus ransfer he definiions of Gaussian Sobolev spaces o he seing of he canonical Wiener space. Definiion 8.2 Le p 2, s N. For F L 2 (Ω, F, P ) denoe by f L 2 (R N, B N, ν) he funcion for which we have F = f((w (g i ) i N ). Then le D p s = {F f D p s(r N )} ( canonical Gaussian Sobolev space of order (s, p)), wih he norm For F = T (f) D p s, 1 r s, le D r F = j 1,,j r =1 F s,p = f s,p. d j1 d jr f(w (g i ) i N )g j1 g jr ( canonical Malliavin derivaive of order r). Remark From our knowledge of sequence spaces we can easily derive ha D p s is a Banach space wih respec o he norm s,p for p 2, s N, and ha for F D p s we have F s,p = s D r F, D r F p. r=0 38

39 We know ha D p s is he closure of S wih respec o he norm s,p. Turning o p = 2, we know ha D 2 1 is a Hilber space wih respec o he scalar produc (F, G) 1,2 = E(F G) + E( DF, DG ), F, G D 2 1. Moreover, we know ha D is a closed operaor, defined on D 2 1, which is coninuous as a mapping from D 2 1 o L 2 (Ω, F, P ). Le us now invesigae how D operaes on he decomposiion ino Wiener-Iô inegrals. Theorem 8.1 Le F = m=0 I m (f m ) L 2 (Ω, F, P ) be given, f m symmeric for any m 0. Then we have In his case we have F D 2 1 if and only if D F = m=1 (for P λ-a.e. (ω, ) Ω R + ). m=1 m m! f m 2 2 <. m I m 1 (f m (, )) Proof 1. Suppose ha wih respec o an orhonormal basis (g i ) i N of L 2 (R + ) we have F = T (f) wih f (c p (f)). As before, for m 0 le We inerpre i N g p i i f m = p E, p =m c p (f) i N g p i i. as k j=1 g p ij i j, if i 1,, i k are precisely hose indices for which p i1,, p ik > Le now p E such ha p = m, and le 0. Then D I m ( i N g p i i ) = D I pi (g p i i ) i N = D H p ((W (g i ) i N ) = p i i N j i = I m 1 ( H pj (W (g j ))H pi 1(W (g i ))g i () p i i N j i 39 g p j j g p i 1 i g i ()).

40 Hence by closedness of D, symmery of f m and p = m, he desired formula D I m (f m ) = mi m 1 (f m (, )) follows. 3. For n N le now F n = n m=0 I m (f m ). By he closedness of he operaor D and he remarks above, we know ha F D 2 1 if and only if (F n ) n N is Cauchy in D 2 1. Now we know from he firs par of he proof ha DF n = n m=1 Le n, m N, n m be given. Then E( D(F n F m ), D(F n F m ) ) = m I m 1 (f m (, )). n k=m+1 = n k=m+1 k 2 R + (k 1)! f k (, ), f k (, ) d k 2 (k 1)! f k 2 2. Hence (DF n ) n N is a Cauchy sequence in D 2 1 if and only if k=0 kk! f k 2 2 <. In his case, he series wih he desired represenaion converges. We need some rules o be able o calculae wih he Malliavin gradien D. Theorem 8.2 Le p 2, d N, φ C 1 (R d ) wih bounded parial derivaives, le F = (F 1,, F d ) (D1) p d. Then φ F D1 p and we have Dφ F = d φ(f )DF i. x i i=1 Proof The proof of Theorem 6.1 ranslaes. Wih he following properies we prepare a sudy of he dual operaor of D. 40

41 Theorem 8.3 Le F S, h L 2 (R + ). Then we have E( DF, h ) = E(F W (h)). Proof We may assume ha F = f(w (g 1 ),, W (g n )), h = g 1 wih respec o an orhonormal sysem g 1,, g n L 2 (R + ). In his case we have by dualiy of d 1 and δ 1 E( DF, h ) = E( x 1 f((w (g 1 ),, W (g n ))) = f (1, 0,, 0) This complees he proof. = f δ 1 1 = f H (1,0, ) = E(f(W (g 1 ),, W (g n ))W (g 1 )) = E(F W (h)). Theorem 8.4 Le F, G S, h L 2 (R + ). Then we have E(G DF, h ) = E(F GW (h) F DG, h ). Proof Apply Theorem 8.3 o he funcion F G. 9 The canonical Wiener space: Skorokhod s inegral In his chaper we dedicae a more careful sudy o he dual operaor (in he sense of Hilber space heory) of he Malliavin gradien han in he Gaussian sequence spaces. In he seing of he canonical Wiener space, his operaor urns ou o be a sochasic inegral. So far we know ha is densely defined and linear. D : D 2 1 L 2 (Ω R + ) 41

42 Definiion 9.1 Le dom(δ) = {u L 2 (Ω R + ) : here is c R such ha for any F D 2 1 we have E( DF, u ) c F 2 }. For u dom(δ) he mapping F E( DF, u ) can be exended o a coninuous linear funcional. Hence by Riesz represenaion we may find δ(u) L 2 (Ω) such ha E( DF, u ) = E(F δ(u)), F D 2 1. Since D is densely defined, δ(u) is unique for any u dom(δ). Definiion 9.2 For u dom(δ) he uniquely deermined random variable δ(u) L 2 (Ω) is called Skorokhod inegral of u. Noaion We wrie δ(u) = R + u δw. Why is his operaor called inegral? To answer his quesion, we firs ask how elemenary processes are inegraed. Definiion 9.3 Le S L2 (R + ) = {u u = n F i h i, F i S, h i L 2 (R + ), n N}. i=1 Lemma 9.1 Le u = n i=1 F i h i S L2 (R + ). Then we have δ(u) = n [F i W (h i ) DF i, h i ]. i=1 Proof By lineariy of δ we may assume ha u = F h wih F S and h L 2 (R + ). Then for G S by means of Theorem 8.4 E( u, DG ) = E(F h, DG ) = E(F GW (h) G h, DF ) = E(G[F W (h) DF, h ]). Hence we have δ(u) = F W (h) h, DF. 42

43 This complees he proof. Recall now he sandard Wiener filraion (F ) 0, which for 0 is given by he P -compleion F of σ(w s : s ). Lemma 9.1 yields he elemenary Iô inegral, if F i is F i -measurable, h i = 1 ]i, i+1 ], where 0 = 0 < 1 < < n, if DF i, h i = 0, 1 i n 1. This is indeed he case, as we will show now. Lemma 9.2 Le F D 2 1, A B 1 +, F A = σ(w (1 B ) : B A, λ(b) < ). Then we have E(F F A ) D 2 1 and (in L 2 (Ω R + )). D E(F F A ) = E(D F F A )1 A () Proof 1. We firs consider F = f(w (h 1 ),, W (h n )) S. By seing g(x 1,, x n, y 1,, y n ) = f(x 1 + y 1,, x n + y n ), x 1,, y n R, we can wrie Le F = g(w (h 1 1 A ),, W (h n 1 A ), W (h 1 1 A c),, W (h n 1 A c)). Q = P (W (h 1 1 A c),, W (h n 1 A c)) 1. Then by independence of F A and he vecor (W (h 1 1 A c),, W (h n 1 A c)) we have E(F F A ) = Hence E(F F A ) S and D (E(F F A ) = g(w (h 1 1 A ),, W (h n 1 A ), y 1,, y n )dq(y 1,, y n ). n i=1 x i g(w (h 1 1 A ),, W (h n 1 A ), y 1,, y n ) dq(y 1,, y n )h i ()1 A () = E(D F F A )1 A (). 2. I remains o approximae F D 2 1 by sandard argumens. Theorem 9.1 Le u L 2 (Ω R + ) be (F )-adaped. Then u dom(δ) and δ(u) = 43 R + u dw (Iô inegral).