Probability approximation by Clark-Ocone covariance representation ∗

Based on the Stein method and a general integration by parts framework we derive various bounds on the distance between probability measures. We show that this framework can be implemented on the Poisson space by covariance identities obtained from the Clark-Ocone representation formula and derivation operators. Our approach avoids the use of the inverse of the Ornstein Uhlenbeck operator as in the existing literature, and also applies to the Wiener space.


Introduction
The Stein and Chen-Stein methods have been applied to derive bounds on distances between probability laws on the Wiener and Poisson spaces, cf.[6], [7] and [8].The results of these papers rely on covariance representations based on the number (or Ornstein-Uhlenbeck) operator L on multiple Wiener-Poisson stochastic integrals and its inverse L −1 .In particular the bound (1.1) has been derived for centered functionals of a standard real-valued Brownian motion in [6], Theorem 3.1.Here d W is the Wasserstein distance, N is a random variable distributed according to the standard Gaussian law, D is the classical Malliavin gradient and •, • is the usual inner product on L 2 (R + , B(R + ), ), with the Lebesgue measure.
Although the Ornstein-Uhlenbeck operator L has nice contractivity properties as well as an integral representation, it can be difficult to compute in practice as its eigenspaces are made of multiple stochastic integrals.Thus, although the Ornstein-Uhlenbeck operator applies particularly well to functionals based on multiple stochastic integrals, it is of a more delicate use in applications to functionals whose multiple stochastic integral expansion is not explicitly known.This is due to the fact that the operator L is expressed as the composition of a divergence and a gradient operator, on both the Poisson and Wiener spaces.
In this paper we derive bounds on distances between probability laws using covariance representations based on the Clark-Ocone representation formula.In contrast with covariance identities based on the number operator, which relies on the divergence-gradient composition, the Clark-Ocone formula only requires the computation of a gradient and a conditional expectation.In particular, in Corollary 3.4 below we show that (1.1) can be replaced by where F is a functional of a normal martingale such that E[F ] = 0. Here, D denotes a Malliavin type gradient operator having the chain rule of derivation, and {F t } t≥0 is the natural filtration of the normal martingale.
In case D is the classical Malliavin gradient on the Wiener space, the bound (1.2) offers an alternative to (1.1).For example, if F = I n (f n ) is a multiple stochastic integral with respect to the Brownian motion and the symmetric kernels f n satisfy certain integrability conditions the inequality (1.2) gives ) dsdt (1.3) obtained by the multiplication formula for multiple Wiener integrals, where and the symbol • k denotes the canonical symmetrization of the L 2 contraction over k variables, denoted by ⊗ k .On the other hand, by Proposition 3.2 of [6] the inequality (1.1) yields ) .
However, due to its importance, the Wiener case will be the object of a more detailed analysis in a subsequent work.
Here our focus will be on the Poisson space, for which (1.2) provides an alternative to Theorem 3.1 of [7].Several applications are considered in Section 4. This includes functionals of Poisson jump times (T k ) k≥1 of the form f (T k ), for which we obtain the bound

Probability approximation
Poisson multiple stochastic integrals are treated in Proposition 4.3, and comparisons with the results of [7] are discussed.
This paper is organized as follows.In Section 2 we present a general framework for bounds on probability distances based on an abstract integration by parts formula.Next in Section 3 we show that the conditions of this integration by parts setting can be satisfied under the existence of a Clark-Ocone type stochastic representation formula.In Section 4 we apply this general setting to a Clark-Ocone formula stated with a derivation operator on the Poisson space, and consider several examples, including multiple stochastic integrals and other functionals of jump times.In Section 5 we consider the total variation distance between a normalized Poisson compound sum and the standard Gaussian distribution.
We close this section by quoting Stein's lemmas for normal and gamma approximations.The following lemma on normal approximation can be traced back to Stein's contribution [12], see also the recent survey [5], and [6].
x ∈ R, The next lemma on the gamma approximation can be found in e.g.Lemma 1.3-(ii) of [6].In the sequel we denote by Γ(ν/2) a random variable distributed according to the gamma law with parameters (ν/2, 1), ν > 0.
Lemma 1.2.Let h : R → R be a twice differentiable function such that for some c > 0 and a < 1/2.Then, letting Γ ν := 2Γ(ν/2) − ν, the functional equation has a solution f h which is bounded and differentiable on (−ν, ∞), and such that 2 General results

Integration by parts
The main results of this paper will be derived under the abstract integration by parts (IBP) formula (2.1) below.Let T denote a subset of C 1 (R) containing the constant functions.Given F and G two real-valued random variables defined on a probability space (Ω, F, P ) and A ∈ F an event with P (A) > 0, we let denote the covariance of F and G given A. The following general Assumption 2.1 says that the integration by parts formula with weights W 1 and W 2 holds for a random variable F given A on T .
In particular, we note that if the IBP formula (2.1) with weights W 1 and W 2 holds on T for the random variable F given A, then W 1 is centered with respect to P (• | A) if and only if we have φ ∈ T , as follows by taking φ = 1 identically.An implementation of this formula on the Poisson space will be provided in Section 4 via the Clark-Ocone representation formula.

Total variation distance
The total variation distance between two real-valued random variables Z 1 and Z 2 with laws P Z1 and P Z2 is defined by where B(R) and B b (R) stand for the families of Borel and bounded Borel subsets of R, respectively.The following bounds on the total variation distance d T V (F | A , N ) between the law of F given A and the law of N hold under Assumption 2.1.
Theorem 2.1.Let A ∈ F be such that P (A) > 0 and assume that the IBP formula (2.1) (2.2) (2.3) where µ(dx) = (dx of the sum of the Lebesgue measure and the law of F given A), cf.[11] or Corollary 1.10 of [2].Lemma 1.1 and the integration by parts formula (2.1) show that for any n ≥ 1 we have Dividing first this inequality by P (A) > 0 and then taking the limit as n goes to infinity, the Dominated Convergence Theorem shows that 2) By (2.4) and the integration by parts formula, for any n ≥ 1, we have The claim follows arguing exactly as in case (1) above.

Wasserstein distance
The Wasserstein distance between the laws of Z 1 and Z 2 is defined by where Lip(1) denotes the class of real-valued Lipschitz functions with Lipschitz constant less than or equal to 1.We have the following upper bound for the Wasserstein distance between a centered random variable F and N .
Theorem 2.2.Assume that the IBP formula (2.1) holds for F given A with W 1 = F , on the space T of twice differentiable functions whose first derivative is bounded by 1 and whose second derivative is bounded by 2. Then we have provided Proof.Using the bound (2.33) in [7] and the IBP formula (2.1), we have

Gamma approximation
Here we use the distance where where the random variable Γ ν is defined in Lemma 1.2.
Proof.Let h ∈ H be arbitrarily fixed.Since h is bounded above by 1, there exist c > 0 and a < 1/2 such that |h(x)| ≤ ce ax , ∀ x > −ν (take c > 1 and 0 < a < 1/2 so small that 1 < ce −aν ).Let f h be solution of (1.4) (its existence is guaranteed by Lemma 1.2).By the IBP formula (2.1) on C 1 b (R) for the centered random variable F given A with W 1 = F , we have The claim follows by dividing the above inequality by P (A) > 0 and then taking the supremum over all functions h ∈ H.

Integration by parts via the Clark-Ocone formula
In this section we consider an implementation of the the IBP formula (2.1) of Assumption 2.1, based on the Clark-Ocone formula for a real-valued normal martingale (M t ) t≥0 defined on a probability space (Ω, F, P ), generating a right-continuous filtration (F t ) t≥0 .In other words, (M t ) t≥0 is a square integrable martingale with respect to the natural filtration and the filtration is right-continuous.Let be the Lebesgue measure on R + .
In this section we assume the existence of a gradient operator with domain Dom(D), defined by DF = (D t F ) t≥0 and satisfying the following properties: (i) D satisfies the Clark-Ocone representation formula (ii) D satisfies the chain rule of derivation for all φ ∈ T ⊆ C 1 (R), cf.e.g.§ 3.6 of [10] (here T contains the constant functions).This condition will be satisfied in both the Wiener and Poisson settings of Section 4. In addition we will assume that for any F ∈ Dom(D) and φ ∈ T we have φ(F ) ∈ Dom(D).

Integration by parts
We now implement the IBP formula (2.1) for functionals in the domain of D, based on the Clark-Ocone representation formula (3.1).Note that IBP formulas of the form (2.1) can also be obtained by the Ornstein-Uhlenbeck semigroup, cf.e.g.Proposition 2.1 of [3].
where ϕ F,G is the function Proof.By Lemma 3.1 and the properties of the gradient operator, for any φ ∈ T and F, G ∈ Dom(D), we have (3.5)

Normal and gamma approximation
We now apply Theorems 2.1 and 2.2 using the Clark-Ocone formula (3.3).For any and note that by Jensen's inequality where ϕ F is defined in Letting •, • denote the usual inner product on L 2 (R + ), from Proposition 3.3 we also have the following corollary: EJP 18 (2013), paper 91. and Proof.The first inequality follows by Proposition 3.3 and the Cauchy-Schwarz inequality.The second inequality follows by the triangle inequality noticing that by the Itô isometry and the Clark-Ocone formula we have The counterpart of this statement for the Wasserstein and d H distances is proved similarly.
In this work our main focus will be on the Poisson space, and in Section 4, we shall compare the upper bound on the Wasserstein distance with the bound obtained in [7] on the Poisson space.

Analysis on the Poisson space
In this section we apply the results of Section 3 to functionals of a standard Poisson process (N t ) t≥0 with jump times (T k ) k≥1 defined on an underlying probability space (Ω, F, P ).We let denotes the space of continuously differentiable functions such that f and its partial derivatives have polynomial growth, i.e. for any i ∈ {0, 1, . . ., d} there exist α (see e.g.Definition 7.2.1 in [10] p. 256).We recall that the gradient operator is closable (see [10] p. 259).We shall continue to denote by D its minimal closed extension, whose domain Dom(D) coincides with the completion of S with respect to the norm EJP 18 (2013), paper 91.By Proposition 7.2.8 in [10] p. 262 the operator D satisfies the Clark-Ocone representation formula, i.e. for any F ∈ Dom(D) we have where (F t ) t≥0 is the filtration generated by (N t ) t≥0 .We note that the gradient D satisfies the chain rule on the set T of real-valued functions which have polynomial growth and are continuously differentiable with bounded derivative, i.e. for any g ∈ T and F ∈ Dom(D) we have g(F ) ∈ Dom(D) and Dg(F ) = g (F )DF , cf.Lemma 6.1 in the Appendix.
Before turning to some concrete examples of Poisson functionals we note that identi- 2) amounts to finding the predictable representation of the random variable F .For example if F = X T is the terminal value of the solution (X t ) t∈[0,T ] to the stochastic differential equation Corollary 3.4 shows that e.g.
, provided the terminal value X T belongs to Dom(D) and E[X 0 ] = 0.In particular, the domain condition can be achieved under a usual Lipschitz condition on σ(•, x), x ∈ R, and a usual sub-linear growth condition on σ(t, •), t ∈ [0, T ].We refer the reader to e.g.Proposition 3.2 of [4] for an explicit solution of (4.3) which is suitable for Ddifferentiation when σ(t, x) vanishes at t = T , for any x ∈ R.

Approximation of Poisson jump times functionals
. (4.5) By the formula in [10] p. 261 we have Finally, by Proposition 3.3 we deduce .
The inequalities concerning d W and d H can be proved similarly.

Example -Linear Poisson jump times functionals
where the latter inequality follows by the Cauchy-Schwarz inequality.So (4.4) and (4.5) recovers the classical Berry-Esséen bound.For the gamma approximation we take e.g.

Example -Quadratic Poisson jump times functionals
Proposition 4.1 can also be applied to quadratic functionals of Poisson jump times.Consider first the normal approximation, take e.g.
Recall that if X is gamma distributed with parameters a and b, then cf. the Appendix.Note that this upper bound is asymptotically equivalent to (3+ 1 √ 3 )/ √ k as k → ∞, and so we recover the Berry-Esséen bound.

Approximation of multiple Poisson stochastic integrals
We present some applications of Corollary 3.4 to Poisson functionals.For n ≥ 1, we denote by ) when f n is not symmetric, where fn denotes the symmetrization of f n in n variables (see e.g.Section 6.2 in [10]).As a convention we identify L 2 (R 0 + ) with R, and let Moreover, we shall adopt the usual convention n of weakly differentiable functions be defined as the completion of the symmetric functions where and We recall the multiplication formula for multiple Poisson stochastic integrals, cf.e.g.Proposition 4.5.6 of [10].For symmetric functions of n + m − k − l variables.We denote by In particular, letting 1 1 {(t1,...,tn)<t} denote the function 1 1 [0,t] n (t 1 , . . ., t n ), for any symmetric function f n ∈ S 1,2 n , we have we have EJP 18 (2013), paper 91. and 2) Proof.By Lemma 4.2 we have I n (f n ) ∈ Dom(D) and So by Lemma 2.7.2 p. 88 of [10] and the definition of ∂ 1 f n[t , we have Using the first equality above and the isometry formula for multiple Poisson stochastic integrals (see Proposition 2.7.1 p. 87 in [10]) we have We conclude by Corollary 3.4, noticing that I n (f n ) is a centered random variable (see [10] pp.87-88), ) dsdt.

Single Poisson stochastic integrals
In the particular case n = 1, the space S 1,2 EJP 18 (2013), paper 91. and The following result is a simple consequence of Proposition 4.3 for n = 1.
Corollary 4.4.For any f ∈ S 1,2 1 , we have Note that Corollary 3.4 of [7] states that for any f ∈ L 2 (R + ), which shows that
2. Take Note that g k is continuous and piecewise differentiable (with a piecewise continuous derivative) and so g k is weakly differentiable.We shall show later on that g k ∈ S 1,2 1 .We have and hence by Corollary 4.4 we get whereas by Corollary 3.4 of [7] we have Note that 16/(3π) < √ 3.

Normal approximation of the compound Poisson distribution
In this section we present an application of formula (2.2) to the compound Poisson distribution.Let (Z k ) k≥1 be a sequence of real-valued i.i.d.random variables independent of a Poisson distributed random variable N n with parameter n ≥ 1.We assume that EJP 18 (2013), paper 91.
Z 1 has moments of any order and that its distribution has a continuously differentiable density p Z1 (z) with respect to the Lebesgue measure, such that lim z→±∞ |z| p p Z1 (z) = 0 for all p ≥ 1.We also assume that d dz log p Z1 (z) = p Z1 (z)/p Z1 (z) has at most polynomial growth.Consider the sequence , n ≥ 1.
It is well-known that F n −→ N , in law, as n −→ ∞.In the following we are going to upper bound the total variation distance between F n and N .The following lemma applies the IBP formula of [1] to each F n | N n = m, m, n ≥ 1.
Lemma 5.1.Let m, n ≥ 1 be fixed integers.Under the foregoing assumptions on the law of the jump amplitude Z 1 , we have that the IBP formula (2.1) and W 2 = 1.
We have the following bound for the total variation distance.
Proposition 5.2.Under the foregoing assumptions on the law of the jump amplitude Z 1 , we have .
follows if we check that Dg(F (n ) ) −→ g (F )DF in L 2 (P ⊗ ).By the boundedness of g , for a positive constant C > 0 we have . This latter quantity tend to zero as n −→ ∞.
m ≥ 1.We have (F (m) ) m≥1 ⊂ S and F (m) converges to F = I n (f n ) in L 2 (P ).Indeed, by the isometry formula for multiple Poisson stochastic integrals (see e.g.Proposition 6.2.4 in [10]), we have and this latter term tends to 0 as m goes to infinity by the Dominated Convergence Theorem.Moreover, each of the remaining terms EJP 18 (2013), paper 91.
as m goes to infinity by the isometry formula for multiple Poisson stochastic integrals, and the two terms in (6.3) and (6.4) converge to 0 since f n ∈ C 1 c ([0, ∞) n ).
In order to complete the proof of the first part of the lemma by closability, given f n ∈ S 1,2 n we choose a sequence (f ) converging to f n for the norm (4.9) and we define the sequence of functionals (F (m) ) m≥1 in Dom(D) by Then we note that by the isometry formula for multiple Poisson stochastic integrals and the convergence of f to f n with respect to the norm • 1,2 .Finally, using again the isometry formula for multiple Poisson stochastic integrals, we have EJP 18 (2013), paper 91.

Probability approximation
Proof of (4.8).We have We shall provide an upper bound for both these addends.We have Now, consider the other term.We have Collecting all these inequalities leads to (4.8).

From ( 3 . 1 )Lemma 3 . 1 .
the gradient operator D satisfies the following covariance identity, cf.e.g.Proposition 3.4.1 in [10], p. 121.For any F, G ∈ Dom(D) we have we consider the gradient on the Poisson space defined as

Proposition 4 .
1 can be applied to linear functionals of Poisson jump times.Consider first the normal approximation.Take e.g.f (x) = (x − k)/ √ k, i.e. f (T k ) = (T k − k)/ √ k, k ≥ 1, and note that T k /k is gamma distributed with mean 1 and variance 1/k.All hypotheses of Proposition 4.1 are satisfied and we have

Proposition 4 . 3 . 1 )
Part (2) of the next proposition proposes an alternative to the Gamma bound of Theorem 2.6 of[8].For any symmetric function f n ∈ S1,2 0 as k goes to infinity.Next we consider a couple of examples for comparison with Corollary 4.4.

Theorem 2.3. Let
The following upper bound for the d H -distance between the centered random variable F given A and a centered gamma random variable holds under the IBP formula (2.1) of F be a P (• | A)-centered, a.s.(−ν, ∞)-valued random variable.Given A ∈ F such that P (A) > 0, assume that the IBP formula (2.1) holds for F given A on One easily sees that all the assumptions of Proposition EJP 18 (2013), paper 91.