SDE SOLUTIONS IN THE SPACE OF SMOOTH RANDOM VARIABLES

Pure Mathematics No 11 ISSN 0806–2439 April 2009

SDE SOLUTIONS IN THE SPACE OF SMOOTH RANDOM VARIABLES

YELIZ YOLCU OKUR, FRANK PROSKE, AND HASSILAH BINTI SALLEH

Abstract. In this paper we analyze properties of a dual pair (G,G^∗) of spaces of smooth and generalized random variables on a L´evy white noise space. We show thatG ⊂L²(µ) which shares properties with a Fr´echet algebra contains a larger class of solutions of Itˆo equations driven by pure jump L´evy processes. Further a characterization of (G,G^∗) in terms of theS-transform is given. We propose (G,G^∗) as an attractive alternative to the Meyer-Watanabe test function and distribution space (D∞,D−∞) [W] to study strong solutions of SDE’s.

1. Introduction

Gel’fand triples or dual pairs of spaces of random variables have proved to be very useful in the study of various problems of stochastic analysis. Important applications pertain e.g. to the analysis of the regularity of the solutions of the Zakai equation in non-linear filtering theory, positive distributions in potential theory, the construction of local time of L´evy processes or the Clark-Ocone formula for the hedging of contingent claims in mathematical finance. See e.g. [IkW], [HKPS], [U], [DØP] and the references therein.

The most prominent examples of dual pairs in stochastic and infinite dimensional analysis are ((S),(S)^∗) of Hida and (D^∞,D^−∞) of Meyer and Watanabe. See [HKPS], [W] and [IkW]. The Hida test function and distribution space ((S),(S)^∗) has been e.g. successfully applied to quantum field theory, the theory of stochastic partial differential equations or the construction of Feynman integrals ([HKPS], [HØUZ]). One of the most interesting properties of the distribution space (S)^∗ is that it accommodates the singular white noise which can be viewed as the time-derivative of the Brownian motion. The latter provides a favorable setting for the study of stochastic differential equations (see [Pro]). See also [LP], where the authors derived an explicit representation for strong solutions of Itˆo equations.

From an analytic point of view the pair ((S),(S)^∗) also exhibits the nice feature that it can be characterized by the powerful tool ofS−transform [HKPS]. It is also worth mentioning that test functions in (S) admit continuous versions on the white noise probability space.

However the Brownian motion is not contained in (S) since elements in (S) have chaos expansions with kernels in the Schwartz test function space. Therefore (S) does not seem to be suitable for the study of SDE’s. It turns out that the test function space D^∞is more appropriate for the investigation of solutions of SDE’s than (S), since it carries a larger

Date: April 21, 2009.

Key words and phrases. Strong solutions of jump SDE’s; Malliavin calculus; White noise analysis.

AMS Subject Classification: 60H10, 60H15, 60H40.

1

class of solutions of Itˆo equations. However a severe deficiency of the pair (D^∞,D^−∞) compared to ((S),(S)^∗) is that it lacks the availability of characterization-type theorems.

In this paper we propose a dual pair (G,G^∗) of smooth and generalized random variables on a L´evy white noise space which meets the following two important requirements: A richer class of solutions of (pure jump) L´evy noise driven Itˆo equations belongs to the test function space G. On the other hand (G,G^∗) allows for a characterization-type theorem.

The pair (G,G^∗) has been studied in the Gaussian case by [LiM], [UZ], [PT], [BL]. See also [DØP] and the references therein for the case of L´evy processes. Similarly to the Gaussian case, G is defined by means of exponential weights of the number operator on a L´evy white noise space. The space G comprises the test functions in (S) and is included in the space D^∞,2 ⊃ D^∞. The important question whether G contains a bigger class of Itˆo jump diffusions has not been addressed so far in the literature. We will give an affirmative answer to this problem. Furthermore we will discuss a characterization of (G,G^∗) in terms of the S−transform by using the concept Bargmann-Segal spaces (see [GKS]). We believe that the pair (G,G^∗) could serve as an alternative tool to (D^∞,D^−∞) for the study of L´evy noise functionals. It is conceivable that this pair could be e.g. employed to construct strong solutions of (backward) SDE’s with (functional) discontinuous coefficients. See [Pro] in the Brownian motion case. See also [MP]. The paper is organized as follows: In Section 2 we introduce the framework of our paper, that is we briefly elaborate some basic concepts of a white noise analysis for L´evy processes and give the definitions of the pairs (D^∞,D^−∞), (G,G^∗). In Section 3 we discuss some properties of (G,G^∗) and provide a characterization theorem. In Section 4 we verify that a bigger class of SDE solutions actually lives in G.

2. Framework

In this section, we concisely recall some concepts of white noise analysis of pure jump L´evy processes which was developed in [LP] and [LØP]. This theory presents a framework which is suitable for all pure jump L´evy processes. For general information about white noise theory, see [HKPS], [K], [KU] and [O]. We conclude this section with a discussion of the dual pairs (D^∞,D^−∞), (G,G^∗) and ((S),(S)^∗).

2.1. White noise analysis of L´evy processes. A L´evy process L(t) is defined as a stochastic process on R⁺ which starts in zero and has stationary and independent incre- ments. It is a canonical example of a semimartingale, which is uniquely determined by the characteristic triplet

(2.1) (B_t, C_t,µ) = (aˆ ·t, σ·t, dtν(dz)),

where a, σ are constants andν is theL´evy measure onR0 =R\ {0}. We denote byπ the product measureπ(dt, dz) :=dtν(dz). For more information about such processes, see e.g.

[A], [Be], [Sa], [JS] and [P]. In this paper, we are only dealing with the case of pure jump L´evy processes without drift, i.e. (2.1) with a=σ= 0.

We want to work with a white noise measure, which is constructed on the nuclear algebra Se^p(X) as introduced in [LP]. Here X := R×R0. For that purpose, recall that S(R) is the Schwartz space of test functions on R and the space S^p(R) is its dual space, which is

the space of tempered distributions. The spaceSe(X) which is a variation of the Schwartz space on the spaceX is then defined as the quotient algebra

(2.2) S(X) =e S(X)/N_π,

where S(X) is a closed subspace of S(R²), given by

(2.3) S(X) :=

ϕ(t, z)∈ S(R²) :ϕ(t,0) = ( ∂

∂zϕ)(t,0) = 0

and the closed ideal Nπ in S(X) is defined as

(2.4) N_π :={φ ∈ S(X) :kφk_L2(π) = 0}.

The space S(X) is a nuclear algebra with a compatible system of norms given bye

(2.5) kφˆk_p,π:= inf

ψ∈Nπ

kφ+ψ k_p, p≥0,

where k · k_p, p≥0 are the norms of S(R²). Moreover the Cauchy-Bunjakowski inequality holds, that is for all p∈N there exists an M_p such that for all ˆφ,ψˆ∈S(X) we have˜

w wφˆψˆw

wp,π ≤M_pw wφˆw

wp,π

w wψˆw

wp,π. We indicate Se^p(X) as its dual. For further information, see [LP].

Next, we define the (pure jump) L´evy white noise probability measure µ on the Borel sets of Ω =Se^p(X), by means of Bochner-Minlos-Sazonov theorem

(2.6)

Z

Se^p(X)

e^ihω,φidµ(ω) = exp Z

X

(e^iφ−1)dπ

for all φ ∈S(X),e wherehω, φi:=ω(φ) denotes the action ofω ∈Se^p(X) on φ∈S(X).e For ω ∈Se^p(X) and φ∈S(X), define the exponential functionale

˜e(φ, ω) := (e(·, ω)◦l) (φ) = exp

hω, ln(1 +φ)i − Z

X

φ(x)λ⊗ν(dx)

as a function of φ∈ Se_q0(X) for functions φ ∈Se_q0(X) satisfying φ(x)>−1, for all x∈ X.

See [LP].

Denote by S(X)e ^⊗nˆ the n-th completed symmetric tensor product of S(X) with itself.e Since ˜e(φ, ω) is holomorphic in φ around zero for φ(x) > −1, it can be expanded into a power series. Furthermore, there exist generalized Charlier polynomialsC_n(ω)∈(S(X)e ^⊗nˆ )^p such that

(2.7) e(φ, ω) =˜ X

n≥0

1

n!hC_n(ω), φ^⊗ni for φ in a certain neighborhood of zero. One shows that

(2.8) {hC_n(·), φ⁽ⁿ⁾i:φ⁽ⁿ⁾ ∈S(X)e ^⊗nˆ , n∈N0}

is a total set of L²(µ). Further, one observes that for all n, m, φ⁽ⁿ⁾ ∈ S(X)e ^⊗nˆ , ψ^(m) ∈ S(X)e ^⊗mˆ the orthogonality relation

(2.9)

Z

Se^p(X)

hC_n(ω), φ⁽ⁿ⁾ihC_m(ω), ψ^(m)iµ(dω) =δ_n,mn! (φ⁽ⁿ⁾, ψ^(m))_L²_(Xⁿ_,πⁿ₎ holds, where

δn,m =

0, n6=m 1, else

is the Kronecker symbol. Using (2.9) and a density argument we can extend hC_n(ω), φ⁽ⁿ⁾i to act on φ⁽ⁿ⁾ ∈ L²(Xⁿ, πⁿ) for ω a.e. The functionals hC_n(ω), φ⁽ⁿ⁾i can be regarded as an n-fold iterated stochastic integral of functions φ⁽ⁿ⁾ ∈ L²(Xⁿ, πⁿ) with respect to the compensated Poisson random measure

N(dt, dz) =e N(dt, dz)−ν(dz)dt,

whereN(Λ₁,Λ₂) := hω,1_Λ1×Λ₂ifor Λ₁ ∈Rand Λ₂ ∈R such that zero is not in the closure of Λ₂, defined on our white noise probability space

(Ω,F, P) =

Se^p(X),B(Se^p(X)), µ

.

In this setting, a square integrable pure jump L´evy process L(t) can be represented as L(t) =

Z t 0

Z

R0

zNe(dt, dz).

Denote by ˆL²(Xⁿ, πⁿ) the space of square integrable functionsφ⁽ⁿ⁾(t1, z1, . . . , tn, zn) being symmetric in the n-pairs (t₁, z₁), . . . ,(t_n, z_n). Then one infers from (2.7) to (2.9) the L´evy- Itˆo chaos representation property of square L´evy functionals: For all F ∈ L²(µ), there exists a unique sequence of φ⁽ⁿ⁾ ∈L²(Xⁿ, πⁿ) such that

F(ω) =X

n≥0

hCn(ω), φ⁽ⁿ⁾i for ω a.e. Moreover, we have the Itˆo-isometry

(2.10) kF k²_L2(µ)=X

n≥0

n!kφ⁽ⁿ⁾k²_L2(Xⁿ,πⁿ) .

2.2. The spaces (D^∞,D^−∞), (G,G^∗) and ((S),(S)^∗). In our search for appropriate can- didates of subspaces of L²(µ) in which strong solutions of SDE’s live, we shall focus on the Meyer-Watanabe test function and distribution spaces (D^∞,D^−∞) and the dual pair (G,G^∗) of smooth and generalized random variables on the L´evy white noise space.

The Meyer-Watanabe test function D^∞ for pure jump L´evy process (see e.g. [Wu1], [Wu2] and [DØP]) is defined as a dense subspace of L²(µ) endowed with the topology

given by the seminorms

(2.11) kF k_k,p= E[|F |^p] +

k

X

j=1

E[kD_·,·^jF k^p_L2(πⁿ)]

!1/p

,

k ∈N,p≥1, with

D_t^j_{1,z1,...,tj,zj}F(ω) :=D_t1,z1D_t2,z2. . . D_tj,zjF(ω)

for F ∈D^∞, where Dt,z stands for the Malliavin derivative in the direction of the (square integrable) pure jump L´evy process L(t), t≥0. D·,· is defined as a mapping

D :D1,2 →L²(µ×π) given by

(2.12) D_t,zF =X

n≥1

n· hC_n−1(·), φ⁽ⁿ⁾(·, t, z)i, if F ∈L²(µ) with chaos expansion

F =X

n≥0

hC_n(·), φ⁽ⁿ⁾i satisfies

(2.13) X

n≥1

n·n!kφ⁽ⁿ⁾k²_L2(πⁿ) <∞.

The domain D^1,2 of D_·,· is the space of all F ∈ L²(µ) such that (2.13) holds. See [Wu1], [Wu2] , [Pri] or [DØP] for further information.

The Meyer-Watanabe distribution spaceD^−∞is defined as the (topological) dual ofD^∞. If one combines the transfer principle from the Wiener space (or Gaussian white noise space) to the Poisson space as devised in [Pri] with the results of [W], one finds that solutions of non-degenerate jump SDE’s exist inD^∞. This is a striking feature which pays off dividends in the analysis of L´evy functionals. However it seems not that easy to set up a characterization-type theorem for (D^∞,D^−∞) in the sense of [PS]. Consequently, other Gel’fand triples have been studied to overcome this deficiency. In [PT] the authors study the pair (G,G^∗) and provide sufficient conditions in terms of theS-transform to characterize (G,G^∗). Using Bargmann-Segal spaces, a complete characterization of this pair (and for a scale of closely related pairs) is obtained by [GKS] in the Gaussian case.

We will show in Section 3 and 4 that (G,G^∗) can be characterized by means of the S-transform on the L´evy noise space and that G contains a richer class of solutions of jump SDE’s. These two properties make (G,G^∗) an interesting alternative to (D^∞,D^−∞) to analyze functionals of L´evy processes.

The test function space G is a subspace of L²(µ) which is constructed by means of exponential weights of the Ornstein-Uhlenbeck or number operator. Denoted by N, this

operator acts on the elements of L²(µ) by multiplying the n-th homogeneous chaos with n∈N0. The space of smooth random variables G is defined as the collection of all

(2.14) f =X

n≥0

hC_n(·), φ⁽ⁿ⁾i ∈L²(µ) such that

kf k²_q:=ke^qNf k²_L2(µ)<∞ for all q ≥0. The latter condition is equivalent to

(2.15) kf k²_q=X

n≥0

n!e^2qnkφ⁽ⁿ⁾ k²_L2(Xⁿ,πⁿ)

for all q ≥ 0. The space G is endowed with the topology given by the family of norms k · k_q, q≥0. Its topological dual is the space of generalized random variables G^∗.

Let us turn our attention to the S-transform which is a fundamental concept of white noise distribution theory and serves as a tool to characterize elements of the Hida test function space (S) and the Hida distribution space (S)^∗. See [HKPS] or [LP] for a precise definition of the pair ((S),(S)^∗). TheS-transform of Φ∈(S)^∗, denoted byS(Φ), is defined as the dual pairing

(2.16) S(Φ)(φ) := hΦ,e(φ,˜ ·)i, φ∈Se_C(X), where k ˜e(φ,·) k²= P∞

n=0 k φ k²ⁿ_p,π and Se_C(X) is the complexification of S(X). Thee S-transform is a monomorphism, that is, if

S(Φ) =S(Ψ) for Φ,Ψ∈(S)^∗ then

Φ = Ψ.

One verifies, e.g. that

(2.17) S(N˙˜(t, z)) =φ(t, z),

N˙˜(t, z) thewhite noise of the compensated Poisson random measure ˜N(dt, dz) in (S)^∗ and φ ∈S˜_C(X). We refer the reader to [HKPS] or [LP] for more information on the Hida test function space (S) and Hida distribution space (S)^∗.

Finally, we give the important definition of theWick orWick-Grassmann product, which can be considered a tensor algebra multiplication on the Fock space. The Wick product of two distributions Φ,Ψ∈(S)^∗, denoted by ΦΨ, is the unique element in (S)^∗ such that

(2.18) S(ΦΨ)(φ) = S(Φ)(φ)S(Ψ)(φ)

for all φ ∈Se_C(X). As an example one finds that

(2.19) hC_n(ω), φ⁽ⁿ⁾i hC_m(ω), ψ^(m)i=hC_n+m(ω), φ⁽ⁿ⁾⊗ψˆ ^(m)i for φ⁽ⁿ⁾ ∈(Se(X))^⊗nˆ and ψ^(m)∈(S(X))e ^⊗mˆ . The latter and (2.7) imply that

(2.20) e(φ, ω) = exp˜ (hω, φi)

for φ∈S(X). The Wick exponential expe (X) of an X ∈(S)^∗ is defined as

(2.21) exp(X) = X

n≥0

1 n!Xⁿ

provided the sum converges in (S)^∗, whereXⁿ =X. . .X.

We mention that the following chain of continuous inclusions is valid:

(S),→ G,→L²(µ),→ G^∗ ,→(S)^∗. 3. Properties of the spaces G and G^∗

In the Gaussian case the spaceGhas the nice feature to be stable in the sense of pointwise multiplication of random variables. More precisely, G is a Fr´echet algebra. See [PT] and [LiM]. In the L´evy setting we can show the following:

Theorem 3.1. Suppose that our L´evy measure ν satisfies the moment condition Z

R0

|z |ⁿν(dz)<∞

for all n ∈ N. Let F, G be in G with chaos expansions F = P

n≥0hC_n(·), φ⁽ⁿ⁾i and G = P

n≥0hC_n(·), ϕ⁽ⁿ⁾i. Define K_R={(t, z)∈R×R0 :k(t, z)k< R}, R >0. Assume that

(3.1) sup

n≥0

√

n!kφ⁽ⁿ⁾k_L^∞_(Xⁿ_,πⁿ₎<∞

and

(3.2) sup

n≥0

√

n!kϕ⁽ⁿ⁾k_L^∞_(Xⁿ_,πⁿ₎<∞.

In addition require that there exists aR > 0such that the compact support of φ⁽ⁿ⁾ and ϕ⁽ⁿ⁾ are in (K_R)ⁿ, i.e.,

supp φ⁽ⁿ⁾, supp ϕ⁽ⁿ⁾ ⊆(K_R)ⁿ for all n ≥0. Then

F ·G∈ G.

In particular, let λ₀ = ^ln(π(K₄ ^R)) + ln(4R) + ln(p 2 +√

2) and assume that for λ > 2λ₀, F, G∈ G_λ. Then for all ν > λ₀+ ^λ₂, F ·G∈ Gλ−ν.

Proof. Let F, G ∈ G_λ ⊂ G for some λ ∈ R with F = P

n≥0hC_n(·), φ⁽ⁿ⁾i and G = P

m≥0hC_m(·), ϕ^(m)i. Then,

kF k²_λ=X

n≥0

n!e^2λn kφ⁽ⁿ⁾k²_L2(πⁿ)<∞,

kGk²_λ=X

m≥0

m!e^2λmkϕ^(m)k²_L2(π^m)<∞.

By the product formula in [LS], we get as follows:

hC_n(·), φ⁽ⁿ⁾i · hC_m(·), ϕ^(m)i

=

m∧n

X

k=0

(m∧n)−k

X

r=0

k!r!

m k

n k

m−k r

n−k r

hCm+n−2k−r(·), φ⁽ⁿ⁾⊗ˆ^r_kϕ^(m)i,

where φ⊗ˆ^r_kϕ for φ ∈ Lˆ²(Xⁿ), ϕ ∈ Lˆ²(X^m), 0 ≤ k ≤ m∧n, 0 ≤ r ≤ m∧n −k, is the symmetrization of the functionφ⊗^r_kϕ onX^n−k−r×X^m−k−r×X^r given by

φ⊗^r_kϕ(A, B, Z) :=

Y

z∈Z

p₂(z) Z

X^k

φ(A, Z, Y)ϕ(Y, Z, B)dπ^⊗k(Y)

for (A, B, Z) ∈ X^n−k−r×X^m−k−r×X^r. Here

Q

z∈Zp₂(z)

:= z₁ ·z₂· · ·z_r when Z = ((t₁, z₁),(t₂, z₂),· · · ,(t_r, z_r)). Because of Lemma 3.4 in [LS], we know that

kφ⁽ⁿ⁾⊗ˆ^r_kϕ^(m) k_L²_(R^m+n−2k−r₎

≤R^{r 4} q

(π(K_R))^(m+n−2r)· q

kφ⁽ⁿ⁾ k_L^∞ q

kϕ^(m) k_L^∞ q

kφ⁽ⁿ⁾ k_L²_(Rⁿ₎ q

kϕ^(m) k_L²_(R^m₎. Moreover, using the conditions (3.1) and (3.2), we get as follows:

k hC_n(·), φ⁽ⁿ⁾i · hC_m(·), ϕ^(m)i kλ−ν

≤

m∧n

X

k=0

(m∧n)−k

X

r=0

k!r!

m k

n k

m−k r

n−k r

e(λ−ν)(m+n−2k−r) k hCm+n−2k−r(w), φ⁽ⁿ⁾⊗ˆ^r_kϕ^(m)i k_L²_(µ)

≤

m∧n

X

k=0

(m∧n)−k

X

r=0

k!r!

m k

n k

m−k r

n−k r

e^λ(m+n)e^−ν(m+n)e−(λ−ν)(2k+r)

p(m+n−2k−r)!kφ⁽ⁿ⁾⊗ˆ^r_kϕ^(m)k_L²_(R^m+n−2k−r₎

≤const.k hC_n(·), φ⁽ⁿ⁾i k^1/2_λ k hC_m(·), ϕ^(m)i k^1/2_λ e^−ν(m+n)e^λ2(m+n)2^{m+n m∧n}

X

k=0

k!

m k

n k

p(m+n−2k)!e^{−(λ−ν)2k}

^m∧n X

k=0

(m∧n)−k

X

r=0

R^rp⁴

(π(K_R))^(m+n−2r)r!p

(m+n−2k−r)!

√ n!√

m!p

(m+n−2k)!

,

for ν < λ. From now, without loss of generality we assume thatπ(K_R)> eand R ≥1. It is clear that

r!p

(m+n−2k−r)!

√ n!√

m!p

(m+n−2k)! ≤ r!

(n∧m)!

p(m+n−2k−r)!

p(m+n−2k)! ≤,1

m∧n

X

r=0

R^{r 4} q

(π(K_R))^(m+n−2r) ≤ e^{m+n4 ln(π(KR))}

m∧n

X

r=0

R^r

≤ (m∧n)R^m+ne^{m+n4 ln(π(KR))}<(2R)^m+ne^{m+n4 ln(π(KR))}, and

^m∧n X

k=0

k!

m k

n k

p(m+n−2k)!e^{−(λ−ν)2k}

≤(

√2

√2−1)^1/2( q

2 +√ 2)^m+n. Therefore,

k hC_n(·), φ⁽ⁿ⁾i · hC_m(·), ϕ^(m)i kλ−ν

≤k hC_n(·), φ⁽ⁿ⁾i k^1/2_λ k hC_m(·), ϕ^(m)i k^1/2_λ Hσ_nσ_m, where H is a constant and σ= 4Re−ν+λ/2+ln(π(K_R))/4(p

2 +√

2). Then, kF ·Gkλ−ν

=k

∞

X

m,n=0

hC_n(·), φ⁽ⁿ⁾ihC_m(·), ϕ^(m)i kλ−ν

≤

∞

X

m,n=0

k hC_n(·), φ⁽ⁿ⁾ihC_m(·), ϕ^(m)i kλ−ν

≤H ^∞

X

n=0

σⁿk hC_n(·), φ⁽ⁿ⁾i k^1/2_λ

^∞ X

n=0

σⁿk hC_n(·), ϕ⁽ⁿ⁾i k^1/2_λ

≤H ^∞

X

n=0

σ^4/3n

3/4 ^∞ X

n=0

k hC_n(·), φ⁽ⁿ⁾i k²_λ

1/4 ^∞ X

n=0

σ^4/3n

3/4 ^∞ X

n=0

k hC_n(·), ϕ⁽ⁿ⁾i k²_λ 1/4

≤H( 1

1−σ⁴³)³² kF k^1/2_λ kGk^1/2_λ ,

if σ <1, i.e. ν > ^λ₂ +λ₀, where λ₀ = ^ln(π(K₄ ^R))+ ln(4R) + ln(p 2 +√

2).

Remark 3.2. Note that, in the conditions (3.1) and (3.2), √

n! can be replaced by √³⁸ n!.

Define the spaceD^∞,2 ⊃D^∞ as

(3.3) D^∞,2 = proj lim

k→0 Dk,2

and denote byD^−∞,2 its topological dual. Then it is apparent from the definition ofGthat (3.4) G ⊂D^∞,2 ⊂L²(µ)⊂D^−∞,2 ⊂ G^∗.

IfL(t) is a Poisson process, then a transfer principle to Poisson spaces based on exponential distributions (see [Pri]) gives

G ⊂D^∞ ⊂L²(µ)⊂D^−∞⊂ G^∗.

Finally, we want to discuss the characterization of the spaces G and G^∗ in terms of the S-transform. For this purpose assume a densely defined operator AonL²(X, π) such that

Aξ_j =λ_jξ_j, j ≥1,

where 1< λ₁ ≤λ₂ ≤...and {ξ_j}j≥1 ⊂S(X) is an orthonormal basis ofe L²(X, π). Further we require that there exists aα >0 such thatA^−α/2 is Hilbert-Schmidt. Then let us denote byS the standard countably Hilbert space constructed fromA(see [O]). An application of the Bochner-Minlos theorem leads to a Gaussian measure µ_G onS^p (dual of S) such that

Z

S^p

e^ihω,φiµ_G(dω) = e^{−12kφk2L2(X,π)}

for all ξ ∈ S.It is well-known that each element f inL²(µ_G) has the chaos representation

(3.5) f =X

n≥0

H_n(·), φ⁽ⁿ⁾ ,

for unique φ⁽ⁿ⁾ ∈ L²(Xⁿ, πⁿ), n ≥0, where H_n(ω)∈ (S^⊗nb )^p are generalized Hermite poly- nomials. Comparing (2.14) with (3.5) we observe that the mapping

(3.6) U :L²(µ)−→L²(µ_G)

given by

X

n≥0

C_n(ω), φ⁽ⁿ⁾

7−→X

n≥0

H_n(ω), φ⁽ⁿ⁾

is a unitary isomorphism between the spaces L²(µ) and L²(µ_G). In the following let us denote by S_G the S−transform on the Gaussian Hida distribution space (S)^∗_µ

G which is defined as

(3.7) S_G(φ) =hΦ,ee(φ, ω)i, φ∈(S)^∗_µ

G, where

ee(φ, ω) =ehω,φi−1/2kφk²_L2(X,π)

.

See [HKPS]. Our characterization of (G,G^∗) requires the concept of Bargmann-Segal space (see [Se], [GKS] and the references therein):

Definition 3.3. Letµ_G,¹

2 be the Gaussian measure onS^passociated with the characteristic function C(φ) :=e^−14kφk

2

L2(X,π). Introduce the measure ν on S_C^p given by ν(dz) =µ_G,¹

2(dx)×µ_G,¹

2(dy),

where z =x+iy. Further denote by P the collection of all projections P of the form P z=

m

X

j=1

hz, ξ_jiξ_j, z ∈ S^p

C.

TheBargmann-Segal spaceE²(ν) is the space consisting of all entire functionsf :L²_C(X, µ_G)−→

C such that

sup

P∈P

Z

S^p

C

|f(P z)|ν(dz)<∞.

So we obtain from Theorem 7.1 and 7.3 in [GKS] the following result:

Theorem 3.4. (i) The smooth random variable ϕ belongs to G if and only if S_G(U(ϕ))(λ·)∈E²(ν)

for all λ >0.

(ii) The generalized random variable Φ is an element of G^∗ if and only if there is a λ > 0 such that

S_G(U(ϕ))(λ·)∈E²(ν).

Remark 3.5. The connection between S_G ◦ U and S in (2.16) is given by the following relation: Since

UD

C_n(·), φ^⊗n₁^{b 1}⊗...b ⊗φb ^⊗n_k^{b k}E

= D

H_n(·), φ^⊗n₁^{b 1}⊗...b ⊗φb ^⊗n_k^{b k}E for φ₁, ..., φ_k ∈L²(X, π), n_i ≥1 with n₁+...+n_k=n we find (see (2.19)) that

SD

C_n(·), φ^⊗n₁^{b 1}⊗...bb ⊗φ^⊗n_k^{b k}E

= (S(hC₁(·), φ₁i))ⁿ¹ ·...·(S(hC₁(·), φ_ki))^nk as well as

S_G◦ UD

C_n(·), φ^⊗n₁^{b 1}⊗...b ⊗φb ^⊗n_k^{b k}E

= (S_G◦ U(hC₁(·), φ₁i))ⁿ¹ ·...·(S_G◦ U(hC₁(·), φ_ki))^nk.

We conclude this section with a sufficient condition for a Hida distribution to be an element of G:

Theorem 3.6. Let Q be a positive quadratic form on L²(X, π) with finite trace. Further let Φ be in (S)^∗. Assume that for every >0 there exists a K()>0 such that

|S_G(U(Φ))(zφ)| ≤K()e^|z|2Q(φ,φ)

holds for all φ∈ S and z ∈C for some constant K >0. Then Φ∈ G.

Proof. The proof is a direct consequence from the proof of Theorem 4.1 in [PT].

Example 3.7. Let γ ∈L²(X, π) with γ >−1 and >0. Then Y(t) := exp(

C₁(ω), χ_[0,t]γ ) is the solution of

dY(t) =Y(t⁻) Z t

0

Z

R0

γ(t, u)Ne(dt, du).

So we get

|S_G(U(Y(t)))(zφ)| ≤ exp(

Z

X

χ_[0,t]zφ(x)γ(x)π(dx) )

≤ K() exp(|z|²Q(φ, φ)), where K() =e^1/(4) and

Q(φ, φ) = Z

X

χ_[0,t]γ(x)φ(x)π(dx) 2

.

Thus Y(t)∈ G.

4. Solutions of SDE’s in G

In this section, we deal with strong solutions of pure jump L´evy stochastic differential equations of the type

(4.1) X(t) = x+

Z t 0

Z

R0

γ(s, X(s⁻), z) ˜N(ds, dz)

forX(0) =x∈R, whereγ : [0, T]×R×R0 →Ris bounded and satisfies the linear growth and Lipschitz condition, i.e.,

(4.2) |γ(t, x, z)|< M

(4.3)

Z

R0

|γ(t, x, z)|² ν(dz)≤C(1+|x|²),

(4.4)

Z

R0

|γ(t, x, z)−γ(t, y, z)|² ν(dz)≤K |x−y|²,

for x, y ∈ R, 0 ≤ t ≤ T and some constants C,K and M. Note that since γ satisfies the conditions (4.3) and (4.4), there exists a unique solution X = {X(t), t ∈ [0, T]} with the initial condition X(0) =x. Moreover, the process is adapted and c´adl´ag [A].

If ν(R0) < ∞ (i.e. X(t), t ≥ 0 is compound Poissonian), we will prove that X(t) ∈ G, t≥0. To this end we need some auxiliary results:

Lemma 4.1. Let {Xn}^∞_n=0 be a sequence of random variables converging to X in L²(µ).

Suppose that

sup

n

k |X_n| k_k,2<∞

for some k ≥1. Then X ∈ Dk,2 and D^k_·,·X_n, n ≥0 converges to D^k_·,·X in the sense of the weak topology of L²((λ×ν×µ)^k).

Proof. sup_n k |Xn| kk,2<∞is equivalent to saying that sup

n

k (1 +N)^k2X_n k_k,2<∞.

By weak compactness, there exists a subsequence{X_ni}^∞_i,n=1 such that (1 +N)^k2X_ni) con- verges weakly to some element α ∈ L²(µ×(λ×ν)^k). Then for any Y in the domain of (1 +N)^k2, it follows from the self-adjointness of N that

E

X(1 +N)^k2Y

= lim

n→∞E

X_ni(1 +N)^k2Y

= lim

n→∞E

(1 +N)^k2X_niY

= E

n→∞lim(1 +N)^k2X_niY

= E αY

.

Therefore α = ((1 +N)^k2)^∗X = (1 +N)^k2X. For the proof in Brownian motion case, see

e.g. [N].

For notational convenience, we shall identify from now on Malliavin derivatives of the same order, that is we set D_r,z^NX(t) = D^N_{r1,z1,r2,z2,...,rN,zN}X(t).

Lemma 4.2. Let X(t), 0≤t ≤T be defined as in the equation (4.1). Then X(t)∈D^∞,2, i.e. D^N_·,·X(t) exists for all N ≥1.

We need the following results to prove this lemma:

Proposition 4.3. Let X ∈D1,2 and f be a real continuous function on R. Then f(X) ∈ D1,2 and

(4.5) Dt,zf(X) = f(X+Dt,zX)−f(X).

Proof. See e.g. [DØP].

Lemma 4.4. Let X(t), 0 ≤ t ≤ T be defined as in the equation (4.1). Then, the N-th Malliavin derivative of X(t) can be written as

D_r,z^NX(t) = Z t

r

Z

R0

N

X

k=0

N k

(−1)^kγ(s,

N−k

X

i=0

N −k i

Dⁱ_r,zX(s⁻), ξ) ˜N(ds, dξ)

+N

N−1

X

k=0

N −1 k

(−1)^kγ(r,

N−k−1

X

i=0

N −k−1 i

Dⁱ_r,zX(r⁻), z) (4.6)

for N ≥1 and D⁰_r,zX(t) :=X(t).

Proof. We will prove the equality (4.6) by using induction. ForN = 1, Dr,zX(t) =

Z t r

Z

R0

1

X

k=0

1 k

(−1)^kγ(s,

1−k

X

i=0

1−k i

D_r,zⁱ X(s⁻), ξ) ˜N(ds, dξ) +γ(r, X(r⁻), z)

= Z t

r

Z

R0

γ(s, X(s⁻) +D_r,zⁱ X(s⁻), ξ)−γ(s, X(s⁻), ξ) N˜(ds, dξ) +γ(r, X(r⁻), z).

Let us assume that it holds for N ≥1. Then,

D^N+1_r,z X(t) =D_r,z(D_r,z^NX(t))

=Dr,z

"

Z t r

Z

R0

N

X

k=0

N k

(−1)^kγ(s,

N−k

X

i=0

N −k i

Dⁱ_r,zX(s⁻), ξ) ˜N(ds, dξ)

+N

N−1

X

k=0

N−1 k

(−1)^kγ(r,

N−k−1

X

i=0

N −k−1 i

D_r,zⁱ X(r⁻), z)

#

= Z t

r

Z

R0

N

X

k=0

N k

(−1)^kγ(s,

N−k

X

i=0

N−k i

Dⁱ_r,zX(s⁻) +

N−k

X

i=0

N −k i

Dⁱ⁺¹_r,z X(s⁻), ξ) ˜N(ds, dξ)− Z t

r

Z

R0

N

X

k=0

N k

(−1)^kγ(s,

N−k

X

i=0

N −k i

Dⁱ_r,zX(s⁻), ξ) ˜N(ds, dξ)

+

N

X

k=0

N k

(−1)^kγ(r,

N−k

X

i=0

N −k i

D_r,zⁱ X(r⁻), z)

+N

"_N−1 X

k=0

N −1 k

(−1)^kγ(r,

N−k−1

X

i=0

N −k−1 i

Dⁱ_r,zX(r⁻) +

N−k−1

X

i=0

N −k−1 i

Dⁱ⁺¹_r,z X(r⁻), z)−

N−1

X

k=0

N−1 k

(−1)^kγ(r,

N−k−1

X

i=0

N −k−1 i

D_r,zⁱ X(r⁻), z)

# .

Note that,

N−k

X

i=0

N −k i

Dⁱ_r,zX(s⁻) +

N−k

X

i=0

N −k i

Dⁱ⁺¹_r,z X(s⁻) =

N−k+1

X

i=0

N −k+ 1 i

Dⁱ_r,zX(s⁻)

and,

N−k−1

X

i=0

N −k−1 i

Dⁱ_r,zX(r⁻)+

N−k−1

X

i=0

N −k−1 i

Dⁱ⁺¹_r,z X(r⁻) =

N−k

X

i=0

N −1 i

Dⁱ_r,zX(r⁻).

Hence,

D_r,z^N+1X(t) = Z t

r

Z

R0

N

X

k=0

N k

(−1)^kγ(s,

N−k+1

X

i=0

N −k+ 1 i

D_r,zⁱ X(s⁻), ξ) ˜N(ds, dξ)

+ Z t

r

Z

R0

N

X

k=0

N k

(−1)^k+1γ(s,

N−k

X

i=0

N −k i

Dⁱ_r,zX(s⁻), ξ) ˜N(ds, dξ)

+

N

X

k=0

N k

(−1)^kγ(r,

N−k

X

i=0

N −k i

Dⁱ_r,zX(r⁻), z)

+N

"_N−1 X

k=0

N −1 k

(−1)^kγ(r,

N−k

X

i=0

N −k i

Dⁱ_r,zX(r⁻), z)

+

N−1

X

k=0

N −1 k

(−1)^kγ(r,

N−k−1

X

i=0

N −k−1 i

D_r,zⁱ X(r⁻), z)

#

= Z t

r

Z

R0

N

X

k=0

N k

(−1)^kγ(s,

N−k+1

X

i=0

N −k+ 1 i

D_r,zⁱ X(s⁻), ξ) ˜N(ds, dξ)

+ Z t

r

Z

R0

N+1

X

k=1

N k−1

(−1)^kγ(s,

N−k+1

X

i=0

N −k+ 1 i

D_r,zⁱ X(s⁻), ξ) ˜N(ds, dξ)

+

N

X

k=0

N k

(−1)^kγ(r,

N−k

X

i=0

N −k i

Dⁱ_r,zX(r⁻), z)

+N

"_N−1 X

k=0

N −1 k

(−1)^kγ(r,

N−k

X

i=0

N −k i

Dⁱ_r,zX(r⁻), z)

+

N

X

k=1

N −1 k−1

(−1)^kγ(r,

N−k

X

i=0

N −k i

D_r,zⁱ X(r⁻), z)

#

= Z t

r

Z

R0

γ(s,

N+1

X

i=0

N + 1 i

D_r,zⁱ X(s⁻), ξ) ˜N(ds, dξ)

+ Z t

r

Z

R0

N

X

k=1

N k

+

N k−1

(−1)^kγ(s,

N−k+1

X

i=0

N −k+ 1 i

D_r,zⁱ X(s⁻), ξ) ˜N(ds, dξ)

+ Z t

r

Z

R0

(−1)^N+1γ(s, X(s⁻), ξ) ˜N(ds, dξ) +

N

X

k=0

N k

(−1)^kγ(r,

N−k

X

i=0

N −k i

D_r,zⁱ X(r⁻), z)

+N

"

γ(r,

N

X

i=0

N i

D_r,zⁱ X(r⁻), z) + (−1)^Nγ(r, X(r⁻), z) +

N−1

X

k=1

N −1 k−1

+

N−1 k

(−1)^kγ(r,

N−k

X

i=0

N −k i

Dⁱ_r,zX(r⁻), z)

#

= Z t

r

Z

R0

γ(s,

N+1

X

i=0

N + 1 i

D_r,zⁱ X(s⁻), ξ) ˜N(ds, dξ) + Z t

r

Z

R0

(−1)^kγ(s, X(s⁻), ξ) ˜N(ds, dξ)

+ Z t

r

Z

R0

N

X

k=1

N+ 1 k

(−1)^kγ(s,

N−k+1

X

i=0

N −k+ 1 i

Dⁱ_r,zX(s⁻), ξ) ˜N(ds, dξ)

+

N

X

k=0

N k

(−1)^kγ(r,

N−k

X

i=0

N −k i

Dⁱ_r,zX(r⁻), z) +N

N

X

k=0

N k

(−1)^kγ(r,

N−k

X

i=0

N −k i

Dⁱ_r,zX(r⁻), z)

= Z t

r

Z

R0

N+1

X

k=0

N + 1 k

(−1)^kγ(s,

N−k+1

X

i=0

N −k+ 1 i

D_r,zⁱ X(s⁻), ξ) ˜N(ds, dξ)

+ (N + 1)

N

X

k=0

N k

(−1)^kγ(r,

N−k

X

i=0

N −k i

Dⁱ_r,zX(r⁻), z).

Now, we are ready to prove Lemma 4.2.

Proof. Let us consider the Picard approximationsX_n(t) to X(t) given by

(4.7) X_n+1(t) =x+

Z t 0

Z

R0

γ(s, X_n(s⁻), z) ˜N(ds, dz),

for n ≥0 and X₀(t) =x. We want to show by induction on n that X_n(t) belongs toDN,2

and

ϕ_n+1,N(t)≤k₁+k₂

N

X

j=1

Z t 0

ϕ_n,j(u)du,

for all n ≥0,N ≥1 and t∈[0, T] where ϕ_n+1,N(t) := sup

0≤r≤tE

"

Z

R^N0

sup

r≤s≤t

|D_r,z^N X_n+1(s)|² ν(dz). . . ν(dz)

#

<∞.

Note that

D^N_r,zX_n+1(t) = Z t

r

Z

R0

N

X

k=0

N k

(−1)^kγ(s,

N−k

X

i=0

N −k i

D_r,zⁱ X_n(s⁻), ξ) ˜N(ds, dξ)

+N

N−1

X

k=0

N −1 k

(−1)^kγ(r,

N−k−1

X

i=0

N −k−1 i

D_r,zⁱ X_n(r⁻), z), (4.8)

with D_r,z⁰ X_n(s⁻) := X_n(s⁻). See (4.4) for a proof. Then, by Doob’s maximal inequality, Fubini’s theorem, Itˆo isometry, (4.3) and (4.4), we get

(4.9)

N

X

j=1

E

"

Z

R^j₀

sup

r≤s≤t

D^j_r,zX_n+1(s)2

(ν(dz))^j

#

=

N

X

j=1

E

"

Z

R^j₀

sup

r≤s≤t

Z s r

Z

R0

j

X

k=0

j k

(−1)^kγ(u,

j−k

X

i=0

j−k i

D_r,zⁱ Xn(u⁻), ξ) ˜N(du, dξ)

+j

j−1

X

k=0

j −1 k

(−1)^kγ(r,

j−k−1

X

i=0

j−k−1 i

D_r,zⁱ X_n(r⁻), z)

!²

(ν(dz))^j





≤2

N

X

j=1

Z

R^j₀

E

sup

r≤s≤t

Z s r

Z

R0

j

X

k=0

j k

(−1)^kγ(u,

j−k

X

i=0

j−k i

D_r,zⁱ Xn(u⁻), ξ) ˜N(du, dξ) 2

(ν(dz))^j

+2

N

X

j=1

E



 Z

R^j0

sup

r≤s≤t

j

j−1

X

k=0

j−1 k

(−1)^kγ(r,

j−k−1

X

i=0

j−k−1 i

D_r,zⁱ X_n(r⁻), z)

!²

(ν(dz))^j





≤8

N

X

j=1

Z

R^j₀

E Z t

r

Z

R0

j

X

k=0

j k

(−1)^kγ(u,

j−k

X

i=0

j−k i

Dⁱ_r,zX_n(u⁻), ξ) ˜N(du, dξ)2

(ν(dz))^j

+ 2

N

X

j=1

E Z

R^j₀

j

j−1

X

k=0

j−1 k

(−1)^kγ(r,

j−k−1

X

i=0

j−k−1 i

D_r,zⁱ Xn(r⁻), z) 2

(ν(dz))^j

= 8

N

X

j=1

Z

R^j0

E



 Z t

r

Z

R0

j

X

k=0

j k

(−1)^kγ(u,

j−k

X

i=0

j−k i

D_r,zⁱ X_n(u⁻), ξ)

!²

ν(dξ)du



(ν(dz))^j

+ 2

N

X

j=1

j²E



 Z

R^j0

j−1

X

k=0

j−1 k

(−1)^kγ(r,

j−k−1

X

i=0

j −k−1 i

Dⁱ_r,zX_n(r⁻), z)

!²

(ν(dz))^j



