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Home , Malliavin calculus, Stochastic calculus

Stochastic of Variations for Martingales

N. PRIVAULT Equipe d’Analyse et Probabilit´es,Universit´ed’Evry-Val d’Essonne Boulevard des Coquibus, 91025 Evry Cedex, France

The framework of the of variations on the standard Wiener and Poisson space is extended to certain martingales, consistently with other ap- proaches. The method relies on changes of times for the operators. We study the transfer of the structures of stochastic induced by these time- changed operators, in particular the chaotic decompositions.

Key words. Stochastic , Chaotic Calculus, Martingales. Subject Classification. 60G44, 60H05, 60H07.

1 Introduction

The stochastic calculus of variations for the , initiated in Malli- avin 12, aims to obtain conditions for the regularity of the density of Wiener functionals given by the values of diffusion processes. It also developed as an extension to anticipating processes of the Itˆocalculus, by means of the Skoro- hod , cf. Nualart-Pardoux 14, Ust¨unel¨ 25. In the case of point processes we can refer for instance to Bass-Cranston 1, Bichteler et al. 2, Bismut 3, for the absolute continuity of functionals of a Poisson measure, and to Carlen-Pardoux 5, Nualart-Vives 15, Privault 21 for the anticipative stochastic calculus for the standard Poisson process on the positive real line. The chaotic decompositions and the chaotic representation property played a central role in each of these approaches. The aim of this work is to extend those constructions to the case of martingales with a continuous part and a finite number of jump sizes which can be obtained by time changes from independent Wiener and Poisson pro- cesses. This approach is consistent with previous constructions and allows to unify them.

2 Stochastic calculus of variations on the standard Wiener and Poisson spaces

The aim of this section is to review some facts and definitions that will serve as starting points for the next sections. Let (Bt)t∈IR+ and (Nt)t∈IR+ be not

1 necessarily independent Wiener and Poisson processes on the real line, defined B N˜ on a probability space (Ω, F,P ), with associated filtrations Ft and (Ft ), B N˜ B B N˜ N˜ ˜ F = F∞∨F∞, F = F∞ and F = F∞, where Nt = Nt−t is the compensated 2 Poisson process. Let (hk)k∈IN be an orthonormal Hilbert basis of L (IR+), and 2 let (ek)k∈IN be the canonical basis of l (IN). We can define two collections of independent identically distributed Gaussian, resp. exponential random R ∞ variables by ξk = 0 hk(t)dBt and τk = Tk+1 −Tk, k ∈ IN,where Tk = inf{t ≥

0 : Nt = k}, k ≥ 1, is the kth jump time of the Poisson process (Nt)t∈IR+ , and T0 = 0. Let P denote the set of functionals of the form

F = f(ξ0, . . . , ξn, τ0, . . . , τn)

∞ 2n+2 p where f ∈ Cb (IR ), n ∈ IN. We know that P is dense in L (Ω, F,P ), p ≥ 1. Gradient operators

∂B, ∂N˜ : L2(Ω) → L2(Ω) ⊗ l2(IN) are defined as k=n B X ∂ F = ek∂kf(ξ0, . . . , ξn, τ0, . . . , τn), k=0

k=n N˜ X ∂ F = ek∂n+1+kf(ξ0, . . . , ξn, τ0, . . . , τn),F ∈ P. k=0 Such gradient operators can be expressed by an infinitesimal perturbation of B the trajectories of (Bt)t∈IR+ and (Nt)t∈IR+ . Let F ∈ P be F -measurable, i.e. F is a Wiener functional. If we represent by ω = (ξk)k∈IN a trajectory of the Wiener process, then

B d 2 (∂ F, h) 2 = F (ω + εh) | , h ∈ l (IN),F ∈ P. l (IN) dε ε=0

On the other hand, if ω = (τk)k∈IN denotes the sequence of interjump times of the Poisson process and if F ∈ P is F N˜ -measurable, then

N˜ d (∂ F, h) 2 = F (ω + εh) | . l (IN) dε ε=0 Let (k=n ) X U = ukek : u1, . . . , un ∈ P, n ∈ IN , k=0 2 and ( ∞ ) N˜ X U = ukek ∈ U : uk = 0 on {τk = 0}, k ∈ IN . k=0

B N˜ B N˜ The adjoint operators ∂∗ , ∂∗ of ∂ and ∂ are closable operators

B N˜ 2 2 2 ∂∗ , ∂∗ : L (Ω) ⊗ l (IN) → L (Ω) such that B B E[F ∂∗ (u)] = E[(u, ∂ F )l2(IN)], u ∈ U,F ∈ P,

N˜ N˜ N˜ E[F ∂∗ (u)] = E[(u, ∂ F )l2(IN)], u ∈ U ,F ∈ P. B B At this stage it can be noted that the composition ∂∗ ∂ gives the Ornstein- Uhlenbeck operator on the Wiener space, cf. Watanabe 26, and that the gra- dient ∂B can be used to state a great number of results in that involve the Ornstein-Uhlenbeck operator and the norm of the gradient on the Wiener space. However this construction does not take into account an important property of the Gaussian measure, namely the fact that the adjoint of the gradient on Wiener space can be an extension of the Wiener stochastic 9 B integral, cf. Gaveau-Trauber , which can not be the case for ∂∗ . An analog N˜ property exists on the Poisson space, and can not be verified by ∂∗ as it acts on discrete-time processes. Define two operators

B N˜ 2 2 i , i : l (IN) → L (IR+)

B N˜ by i (ek) = hk and i (ek) = −1]Tk,Tk+1], k ∈ IN,and let

DB = iB ◦ ∂B,DN˜ = iN˜ ◦ ∂N˜ .

Let jB and jN˜ be the adjoint operators of iB and iN˜ , a.s. defined as

B B 2 2 (i (u), v)L (IR+) = (u, j (v))l (IN),

N˜ N˜ 2 2 (i (u), v)L (IR+) = (u, j (v))l (IN),

2 2 B N˜ u ∈ L (IR+), v ∈ l (IN). An explicit description of j and j is given as

Z ∞ Z Tk+1 B N˜ jk (u) = u(t)hk(t)dt, jk (u) = u(t)dt, k ∈ IN. 0 Tk 3 The introduction of iB and jB gives another interpretation of DB as a deriva- 2 tive. For F ∈ P and h ∈ L (IR+), with ω = (ξk)k∈IN,

B 2 (D F, h)L (IR+)

B B d B = (∂ F, j (h)) 2 = F (ω + εj (h)) | l (IN) dε ε=0 d Z ∞   = F (hk(t) + εh(t))dBt |ε=0 dε 0 k∈IN d  Z ·  = F B· + ε hsds |ε=0 . dε 0

Let (k=n ) B X V = i (U) = ukhk : u0, . . . , un ∈ P, n ∈ IN , k=0

B B B N˜ N˜ N˜ N˜ N˜ N˜ and δ = ∂∗ ◦ j , δ = ∂∗ ◦ j . We have j (V) ⊂ U , hence δ is defined on V. Proposition 1 If u ∈ V, then for X = B, N˜,

Z ∞ Z ∞ X X δ (u) = utdXt − Dt utdt. 0 0

Proof. cf. Nualart-Pardoux 14, Privault 21. 2 B B B N˜ N˜ N˜ B N˜ The adjoint operators δ = ∂∗ ◦ j and δ = ∂∗ ◦ j of D and D extend respectively the Wiener and compensated Poisson stochastic on the predictable square-integrable processes: 2 2 X Proposition 2 Let u ∈ L (Ω) ⊗ L (IR+). If u is (Ft )-predictable, then Z ∞ X δ (u) = u(t)dXt, 0 where X = B, N˜. Proof. cf. Carlen-Pardoux 5, Nualart-Pardoux 14. 2 Remark. The operators jB and jN˜ can also be used to give a unified formula- tion of the anticipating Girsanov theorems on the Wiener and Poisson spaces. 2 B Let u :Ω → L (IR+) be F -measurable and satisfy the hypothesis of Th. 6.4. 11 of Kusuoka , and denote again by ω = (ξk)k∈IN a trajectory of the Wiener

4 process. Then for any F B-measurable and bounded f :Ω → IR,

E[f] = E f ω + jB(u)   B B  B 1 2 ×det2 Il2(IN) + ∂ j (u) exp −δ (u) − | u | 2 , 2 L (IR+) where det2 is the Carleman-Fredholm determinant. If ω = (τk)k∈IN denotes a discrete interjump times trajectory of the Poisson process, then for any F N˜ - measurable and bounded f :Ω → IR,

h  N˜   N˜ N˜   N˜ i E[f] = E f ω + j (u) det2 Il2(IN) + ∂ j (u) exp −δ (u) , provided that jN˜ (u) is F N -measurable and satisfies the hypothesis of Th. 1 in Privault 19. 2 If Y is either a compensated Poisson or Wiener process, define for fp ∈ 2 ◦p L (IR+) symmetric and square-integrable

− − Z ∞ Z tp Z t2 Y Ip (fp) = p! ··· fp(t1, . . . , tp)dY (t1) ··· dY (tp). 0 0 0 If Y = B, such integrals can be expressed with the Hermite polynomials (Hn)n∈IN as p IB(h◦n1 ◦ · · · ◦ h◦nd ) = n ! ··· n !H (ξ ) ··· H (ξ ), n k1 kd 1 d n1 k1 nd kd n1 +···+nd = n, k1 6= · · · 6= kd, where “◦” denotes the symmetric prod- uct. In case Y = N˜, the Charlier polynomials replace the Hermite polynomials, 24 Y 2 2 2 cf. Surgailis . An annihilation operator ∇ : L (Ω) → L (Ω) ⊗ L (IR+) is defined as Y Y Y ∇ In (fn) = nIn−1(fn), n ≥ 1, Y Y 2 2 2 and ∇ c = 0 if c ∈ IR. A creation operator ∇∗ : L (Ω) ⊗ L (IR+) → L (Ω) is defined as

Y Y Y ˆ 2 ◦n 2 ∇∗ In (fn+1) = In+1(fn+1), fn+1 ∈ L (IR+) ⊗ L (IR+), ˆ where fn+1 is the symmetrization of the function fn+1 in its n + 1 variables. The operator LY = δY DY is a number operator, i.e.

Y Y Y 2 ◦n L In (fn) = nIn (fn), n ∈ IN, fn ∈ L (IR+) .

5 Y Proposition 3 The operator ∇∗ coincides with the stochastic integral with respect to Y on the predictable square-integrable processes. Proof. In the Gaussian case, DB = ∇B, and this result is identical to Prop. 2. In the Poisson case, this can be found in Nualart-Vives 15. 2 A consequence of Prop. 2 and 3 is the following formula, cf. Privault 21. For F ∈ Dom(DN˜ ) ∩ Dom(∇N˜ ),

N˜ N˜ N˜ N˜ E[Dt F | Ft− ] = E[∇t F | Ft− ], dt ⊗ dP a.e.

˜ 2 Y For Y = B, N, the space L (Ω, F∞) admits the orthogonal decomposition

2 Y M L (Ω, F∞) = Cn, n≥0

Y 2 ◦n where Cn = {In (fn): fn ∈ L (IR+) } is the nth order chaos, generated Y ˜ by the multiple stochastic integral In . If Y = B, resp. Y = N, we refer to this decomposition as the Wiener-Hermite, resp. Poisson-Charlier chaotic decomposition. We now mention a transfer principle which allows to state on the Poisson space most results of the Malliavin calculus, using the operators N˜ N˜ D and δ . Assume until the end of this section that (Bt)t∈IR+ and (Nt)t∈IR+ are not independent, but linked by the following relation:

ξ2 + ξ2 τ = 2k 2k+1 , k ∈ IN, (1) k 2

which does not change anything to the fact that (Nt)t∈IR+ is a Poisson process, since the half sum of the squares of two Gaussian normal random variables is exponentially distributed. Then the following properties are satisfied, cf. Privault 20. Proposition 4 We have for F ∈ P

B B N˜ N˜ B 2 N˜ 2 δ D F = 2δ D F, and | D F | 2 = 2 | D F | 2 , a.s. L (IR+) L (IR+)

As a consequence, any result that uses the norm of DB and the operator δBDB can be stated on the Poisson space, using the norm of DN˜ and δN˜ DN˜ . Let us now write the Wiener-Hermite chaotic decomposition of F ∈ P, F measurable with respect to F N˜ (i.e. F is a Poisson functional):

X B F = E[F ] + In (fn). n≥1

6 The Laguerre polynomials are defined as

k=n X (−x)k L (x) = Ck , x ∈ IR , n n k! + k=0 they are orthogonal for the exponential density. Proposition 5 The space of Poisson functionals in the 2nth Wiener chaos C2n is the completion of the vector space generated by

(i=d k=n ! Y Xi  n  IB i h◦2k ◦ h◦2ni−2k 2ni k 2ki 2ki+1 i=1 k=0 n o : n + ··· + n = , k 6= · · · 6= k , d ∈ IN , 1 d 2 1 d and \ 2  N˜  C2n+1 L Ω, F = {0}, n ∈ IN.

B For even n, if In (fn) is a Poisson functional it can be represented in terms of multidimensional Laguerre polynomials as the linear combination

B X In (fn) = g(k1, . . . , kp)Ln1 (τk1 ) ··· Lnp (τkp ).

n1 + ··· + np = n k1 6= · · · 6= kp

Proof. The proof relies on the following relation between the Laguerre and Hermite polynomials:

k=n x2 + y2  (−1)n X n! n!L = p(2k)!(2n − 2k)!H (x)H (y), n 2 2n k!(n − k)! 2k 2n−2k k=0

8 2 cf. Erd´elyi , and the fact that the set {Ln(τk): k, n ∈ IN} is total in L (Ω, F).

2 A consequence of this proposition is that the projection on the 2nth order Wiener chaos of a Poisson functional can be represented as a discrete stochastic integral, defined below. 2 ◦n 2  N˜  Definition 1 We define Jn : l (IN) → L Ω, F as a linear functional with ◦n J (e◦n1 ◦ · · · ◦ e p ) = n ! ··· n !L (τ ) ··· L (τ ), (2) n k1 kp 1 p n1 k1 np kp n1 + ··· + np = n, k1 6= · · · 6= kp.

7 2 ◦n 2 N˜ The maps Jn : l (IN) → L (Ω, F ,P ) are linear, bounded, and Jn(gn) is orthogonal to Jm(gm) for n 6= m. Moreover, each square-integrable Poisson functional has the decomposition

X 2 ◦k F = Jn(gn), gk ∈ l (IN) , k ∈ IN, (3) n∈IN referred to as the Poisson-Laguerre chaotic decomposition. On the Wiener space, it is known that the notions of derivation defined by infinitesimal per- B B B B turbations and by annihilation coincide, i.e. D = ∇ and δ = ∇∗ . The sit- uation is different on the Poisson space, and more generally for point processes. The operators DN˜ and δN˜ are related to the Wiener-Hermite decomposition by the isometry property Prop. 4, and to the Poisson-Laguerre decomposition by the relations, cf. Privault 21:

k=n−1 N˜ X n! 2 ◦n (∂ J (g ), e ) 2 = J (g (∗, i . . . , i)), g ∈ l (IN) , n n i l (IN) k! k n n k=0

N˜ 1 2 ◦n 2 ∂∗ Jn(gn+1) = Jn+1(ˆgn+1) − nJn(ˆgn+1), gn+1 ∈ l (IN) ⊗ l (IN) 1 whereg ˆ denotes the symmetrization of g and gn+1 is the contraction defined as 1 gn+1(k1, . . . , kn) =g ˆn+1(k1, . . . , kn, kn), k1, . . . , kn.

3 A generalization to certain martingales by change of time

Consider a real square-integrable martingale on a filtered probability space (Ω, (Ft), F∞,P ) with a continuous part X0 and independent compensated sums X1,...,Xd of jumps of sizes z1, . . . , zd ∈ IRrespectively:

k=d X M = X0 + Xk − νk. k=1

Let ν0 =< X0 > denote the of X0. Assume also that (Ft) is generated by M, that the processes X0,X1,...,Xd are independent, and that 10 limt→∞ νk(t) = ∞, k = 0, . . . , d, P -a.s. We know, cf. Ikeda-Watanabe , that there is a B and N 1,...,N d independent standard Poisson processes such that X = B , Xk = z N k , k = 1, . . . , d. Let us call (F 0), 0 ν0 k νk t 1 d 1 d (Ft ),..., (Ft ) the filtrations generated by (Bt)t∈IR+ ,(Nt )t∈IR+ ,..., (Nt )t∈IR+ . −1 −1 Denote by ν0 , ··· , νd the right-continuous inverses of ν0, . . . , νd: −1 νk (t) = inf{s ≥ 0 : νk(s) > t}, k = 0, . . . , d. 8 We make the following hypothesis: −1 −1 0 d (H1) The processes ν0 , . . . , νd are respectively (Ft ),..., (Ft )-adapted. −1 −1 (H2) The trajectories of ν0 , . . . , νd are continuous and strictly increasing. Two consequences of those hypothesis are stated in the lemma below. 0 d Lemma 1 We have F∞ = F∞ ∨ · · · ∨ F∞, and

0 k−1 k k+1 d F −1 ⊂ F∞ ∨ · · · ∨ F∞ ∨ Ft ∨ F∞ ∨ · · · ∨ F∞, k = 0, . . . , d. νk (t)

Proof. It is sufficient to prove this result for d = 0. In this case, M = Bν0 implies F ⊂ F 0 since (F ) is generated by M, and F = F 0 . We also t ν0(t) t ∞ ∞ 0 0 S −1 have Fν−1(t) ⊂ F −1 . If A ∈ F −1 , then A = s∈Q A ∩ {ν0 (t) ≤ 0 ν0(ν0 (t)) ν0(ν0 (t)) −1 0 0 s ≤ ν0 (u)}, u > t. Hence A ∈ Ft by right-continuity of the filtration (Ft ). 2 We define an extension of the notion of Cameron-Martin space as

( k=d Z ∞ ) d+1 2 X 2 H = u : IR+ → IR measurable : | u |H = uk(t)dνk(t) < ∞ , k=0 0 and denote by H its equivalence classes for | · |H . Similarly, let

2 n d+1 2 o L (M) = u :Ω × IR+ → IR measurable : E[| u |H ] < ∞ , and denote by L2(M) the equivalence classes of L2(M). No adaptedness re- quirement is made on the elements of L2(M). We now turn to the definition of the gradient by change of time. If X is a with jumps of size X−ν 2 2 z ∈ IRand compensator ν, written as X = Nν , define i : l (IN) → L (IR+) as X−ν N˜ it (u) = ziν(t)(u) t ∈ IR+.

If X is a continuous martingale with quadratic variation ν, written as X = Bν , let X B it (u) = iν(t)(u), t ∈ IR+.

If M = M0 + ··· + Md is a sum of d + 1 independent martingales of the above M 2 d+1 2 d+1 type, define i : l (IN, IR ) → L (IR+, IR ) as   M M1 Md it (u) = it (u0), . . . , it (ud) ,

M u = (u0, . . . , ud), t ∈ IR+. The operator i is easily extended to stochastic processes. Definition 2 Define an operator DM : P → L2(M) by DM = iM ◦ D.

9 We notice that

k=d M 2 B 2 X 2 N k 2 | D F | =| D F | 2 + z | D F | 2 ,F ∈ P, (4) H L (IR+) k L (IR+) k=1

M 2 2 M hence D : L (Ω) → L (M) is closable. We call ID2,1 its domain.

Remark. This definition is consistent with that of Bismut 3, Decreusefond 6. For instance, if M = X1 − ν1 and F ∈ P with F = f(T1,...,Tn),

k=n −1 Z ν1 (Tk) M X 2 (D F, h)L (IR+) = − ∂kf(T1,...,Tn) h(t)dν1(t), k=1 0

M 2 which means that (D F, h)L (IR+) is defined by perturbation of the kth jump −1 time ν1 (Tk) of X1 into

−1 ! Z ν1 (Tk) −1 ν1 (1 + εh(t))dν1(t) . 0

2 M 2 d+1 2 d+1 M Let j : L (IR+, IR ) → l (IN, IR ) be the random dual operator of i with respect to (·, ·)H , satisfying

M (j (u), v)l2(IN,IRd) = (i(u), v)H . We have   M B −1 N˜1 −1 N˜d −1 j (u) = j (u0 ◦ ν0 ), j (u1 ◦ ν1 ), . . . , j (ud ◦ νd ) .

The adjoint δM of DM is defined below. Let

VM = iM (U) (k=n ) X 0 d i = (ukhk ◦ ν0, . . . , ukhk ◦ νd): uk ∈ P, i = 0, . . . , d, n ∈ IN . k=0

Definition 3 We define δM : VM → L2(Ω) by

k=d M B −1 X N˜ k −1 δ (u) = δ u0 ◦ ν0 + zkδ uk ◦ νk . k=1

10 It is clear that

 M   M  M E F δ (u) = E (D F, u)H ,F ∈ P, u ∈ V , hence δM is closable and adjoint of DM . Moreover, the operator δM coincides with the stochastic integral with respect to M on square-integrable predictable processes: 2 k Proposition 6 If u = (u0, . . . , ud) ∈ L (M) with uk (Ft )-predictable, k = 0, . . . , d, then

Z ∞ k=d Z ∞ M 0 X δ (u) = u (t)dX0(t) + uk(t)d(Xk(t) − νk(t)). 0 k=1 0

0 B Proof. In the case d = 0. If u0 is (Ft )-predictable, then u0 ◦ ν0 is (Ft )- predictable and Z ∞ Z ∞ M B δ (u) = δ (u0 ◦ ν0) = u0(ν0(t))dBt = u0(t)dMt. 0 0

2 2 In particular, if u ∈ L (M) is (Ft)-predictable with

"k=d Z ∞ # X 2 E u(t) dνk(t)) < ∞, k=0 0 then Z ∞ M δ (u) = u(t)dMt. 0 Proposition 7 In the anticipative case, we have for u ∈ VM :

δM (u) = k=d Z ∞ X Z ∞ u0(t)dX0(t) + uk(t)d(Xk(t) − dνk(t)) 0 k=1 0 Z ∞ k=d Z ∞ X0 −1 X Xk−νk −1 − Dt (u0(ν0 (t)))dν0(t) − Dt (uk(νk (t)))dνk(t). 0 k=1 0

Proof. We apply Prop. 1. 2

11 2 ◦n0 2 ◦n1 2 ◦nd For fn0,...,nd ∈ L (IR+) ◦l (IN) ◦· · ·◦l (IN) , define the Wiener-Hermite-

Laguerre integral Jn(fn0,...,nd ) as

J (f ◦ f ◦ · · · ◦ f ) = IB (f )J 1 (f ) ··· J d (f ). n n0 n1 nd n0 n0 n1 n1 nd nd

Where J k (f ) is defined as in Def. 1, for the Poisson process N˜ . Then any nk nk k square-integrable functional on Ω has the decomposition X F = Jn(fn0,...,nd ),

n0,...,nd≥0

2 ◦n0 2 ◦n1 2 ◦nd fn0,...,nd ∈ L (IR+) ◦ l (IN) ◦ · · · ◦ l (IN) . Next, we notice that this decomposition is preserved by the operator δM DM : M M B B Pk=d 2 N k N k Proposition 8 We have δ D = δ D + k=1 zkδ D . Consequently,

k=d ! M M X 2 δ D Jn(fn0,...,nd ) = nkzk Jn(fn0,...,nd ), (5) k=0

2 ◦n0 2 ◦n1+···+nd where n = n0 + ··· + nd, and fn0,...,nd ∈ L (IR+) ◦ l (IN) . (We let z0 = 1).

4 Absolute continuity results

The aim of this section is to show that from Prop. 4, absolute continuity results can be obtained with DM and δM as consequences of their Wiener space counterparts. Sobolev spaces are defined as follows. Let k · kp,k be M k/2 M the norm defined as k F kp,k=k (I + L ) F kp, and denote by IDp,k the completion of P with respect to k · kp,k, and

M \ M ID∞ = IDp,k. p>1,k∈ZZ

A number of results in stochastic analysis can be expressed with the operators δM and DM . For the sake of simplicity, those results are stated in the case of IR-valued functionals, but can also be obtained in the finite dimensional vector-valued case. Theorem 1 (Meyer 13) There exists two constants A, B > 0 such that for any F ∈ P, M 2 M A k| D F |H kp≤k F kp,k≤ B k| D F |H kp .

12 26 M Theorem 2 (Watanabe ) Let F ∈ ID∞ such that

M −1 p | D F |H ∈ ∩p>1L (Ω, F). Then F has a C∞ density with respect to the Lebesgue measure. 4 M M Theorem 3 (Bouleau-Hirsch ) Let F ∈ ID2,1 such that | D F |H > 0 a.s. Then F has a density with respect to the Lebesgue measure. Those results are directly obtained from their Wiener space counterparts, using the isometry property, cf. Prop. 4, and relations (4), (5) .

5 Clark formula and chaotic calculus

Functionals in L2(Ω) have a Wiener-Hermite-Charlier chaotic decomposition. If Y1,...,Yp are compensated Poisson or Wiener processes, with Yi independent 2 ◦p of Yj if Yi 6= Yj, i, j ∈ {1, . . . , p}, define for fp ∈ L (IR+) symmetric and square-integrable

− − Z ∞ Z tp Z t2 Y1,...,Yp Ip (fp) = p! ··· fp(t1, . . . , tp)dY1(t1) ··· dYp(tp). 0 0 0

2 With the notation Y0 = B, Yk(t) = Nk(t) − t, k = 1, . . . , d, any F ∈ L (Ω) can be written as X X Yθ(1),...,Yθ(n) F = E[F ] + In (fθ),

n≥1 θ∈Θn where Θn is the set of all applications from {1, . . . , n} into {0, . . . , d}, cf. Del- lacherie et al. 7. We can define an operator

M 2 2 2 d+1 ∇ : L (Ω) → L (Ω) ⊗ L (IR+, IR ) by ∇M F = iM ◦ ∇F . The adjoint of this operator is given by

i=d M B −1 X N˜ i −1 ∇∗ (u) = ∇∗ (u0 ◦ ν0 ) + zi∇∗ (ui ◦ νi ). i=1 This operator extends the stochastic integral with respect to M on the pre- 2 M M dictable processes in L (M) as in Prop. 6, and the composition ∇∗ ∇ satisfies

k=n ! M M Yθ(1),...,Yθ(n) X 2 Yθ(1),...,Yθ(n) ∇∗ ∇ In (fn) = zθ(k) In (fn). k=0 The Clark formula can also be written with the operator DX or ∇X .

13 M B Proposition 9 Let F ∈ Dom(D ). We have if F ∈ F∞: Z ∞ X0 F = E[F ] + E[Dt F | Ft]dX0(t). 0 k If F is F∞ measurable, ∞ Z k X −νk F = E[F ] + E[Dt F | Ft− ]d(Xk(t) − dνk(t)). 0 M k If F ∈ Dom(∇ ) and is F∞-measurable, Z ∞ Xk F = E[F ] + E[∇t F | Ft− ]d(Xk(t) − dνk(t)). 0 Proof. We do the proof for d = 0. Let us write the Clark formula on Wiener space, cf. Ocone 16: Z ∞ B 0 F = E[F ] + E[Dt F | Ft ]dBt, 0 and notice that by definition of the adapted projection, cf. Revuz-Yor 23, E[DB F | F 0 ] = E[DBF | F 0](ν (t)). ν0(t) ν0(t) · · 0 The same argument holds in case d > 0 for the predictable projection. 2

⊗∞ Remark. Consider the space Ω = ([−1, 1], dx/2) . Denote by θk the kth canonical projection, and by Y the process X Yt = 1[2k+1+θk,∞[, t ∈ IR+, k≥0 with compensator X 1 dν(t) = 1 (t)dt. 2k + 2 − t [2k,2k+1+θk[ k≥0

The hypothesis H2 is not fulfilled. If, instead of dνt we use dt as a compensator, it can be shown, cf. Privault 18 that the definition of the gradient should be ˜ X DF = −((1 − θk)1]2k,2k+1+θk] − (1 + θk)1]2k+1+θk,2k+2])∂kf(θ0, . . . , θn), k≥0 ˜ 2 2 which is a variation of Def. 2. Let δ denote the adjoint of D˜ in L (B)⊗L (IR+). The spectral decomposition of δ˜D˜ is then given by the Legendre polynomials evaluated in the θk’s, instead of the Laguerre or Hermite polynomials as in Prop. 8.

14 References

1. R.J. Bass and M. Cranston. Malliavin calculus for jump processes and applications to . Annals of Probability 17, 507 (1984). 2. K. Bichteler, J.B. Gravereaux, and J. Jacod. Malliavin Calculus for Processes with Jumps. Stochastics Monographs. Gordon and Breach, 1987. 3. J.M. Bismut. Calcul des variations stochastique et processus de sauts. Z. Warsch. Verw. Gebiete 63, 147 (1983). 4. N. Bouleau and F. Hirsch. Dirichlet Forms and Analysis on Wiener Space. de Gruyter, Berlin/New York, 1991. 5. E. Carlen and E. Pardoux. Differential calculus and on Poisson space. In Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, pages 63–73. Kluwer, 1990. 6. L. Decreusefond. Une autre formule d’int´egration par parties. In P. Florchinger, editor, Proceedings of a Second Workshop on Stochas- tic Analysis. Birk¨auser,1994. 7. C. Dellacherie, B. Maisonneuve, and P.A. Meyer. Probabilit´eset Poten- tiel, volume 4. Hermann, 1992. 8. A. Erd´elyi. Higher Transcendental Functions, volume 2. McGraw Hill, New York, 1953. 9. B. Gaveau and P. Trauber. L’int´egralestochastique comme op´erateurde dans l’espace fonctionnel. Journal of 46, 230 (1982). 10. N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffu- sion Processes. North-Holland, 1989. 11. S. Kusuoka. The nonlinear transformation of Wiener measure on Banach space and its absolute continuity. Journal of the Faculty of Science of Tokyo University, Section IA, Mathematics 29, 567 (1982). 12. P. Malliavin. Stochastic calculus of variations and hypoelliptic oper- ators. In Intern. Symp. SDE. Kyoto, pages 195–253, Tokyo, 1976. Kinokumiya. 13. P.A. Meyer. Transformations de Riesz pour les lois gaussiennes. In S´eminaire de Probabilit´eXVIII, Lecture Notes in Mathematics, pages 179–193. Springer Verlag, 1984. 14. D. Nualart and E. Pardoux. Stochastic calculus with anticipative inte- grands. and Related Fields 78, 535 (1988). 15. D. Nualart and J. Vives. Anticipative calculus for the Poisson process based on the Fock space. In S´eminaire de Probabilit´ede l’Universit´e de Strasbourg XXIV, volume 1426 of Lecture Notes in Mathematics.

15 Springer Verlag, 1990. 16. D. Ocone. A guide to the stochastic calculus of variations. In H. Korezli- oglu and A.S. Ust¨unel,editors,¨ Stochastic Analysis and Related Topics, volume 1316 of Lecture Notes in Mathematics. Springer-Verlag, 1988. 17. N. Privault. Calcul des variations stochastique pour les martingales. To appear in the Comptes Rendus de l’Acad´emiedes Sciences de Paris, S´erie I. 18. N. Privault. Calcul des variations stochastiques pour la mesure de densit´e uniforme. To appear in Potential Analysis. 19. N. Privault. Girsanov theorem for anticipative shifts on Poisson space. To appear in Probability Theory and Related Fields. 20. N. Privault. A transfer principle from Wiener to Poisson space and applications. To appear in the Journal of Functional Analysis. 21. N. Privault. Chaotic and variational calculus in discrete and continuous time for the Poisson process. Stochastics and Stochastics Reports 51, 83 (1994). 22. N. Privault. In´egalit´esde Meyer sur l’espace de Poisson. Comptes Ren- dus de l’Acad´emiedes Sciences de Paris, S´erieI 318, 559 (1994). 23. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer-Verlag, 1994. 24. D. Surgailis. On multiple Poisson stochastic integrals and associated Markov semi-groups. Probability and Mathematical 3, 217 (1984). 25. A.S. Ust¨unel.¨ The Itˆoformula for anticipative processes with non- monotonous time via the Malliavin calculus. Probability Theory and Related Fields 79, 249 (1988). 26. S. Watanabe. Lectures on Stochastic Differential Equations and Malli- avin Calculus. Tata Institute of Fundamental Research, 1984.

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