Edgeworth expansion for the integrated Levy driven Ornstein-Uhlenbeck process

We verify the Edgeworth expansion of any order for the integrated ergodic Levy driven Ornstein-Uhlenbeck process, applying a Malliavin calculus with truncation over the Wiener-Poisson space. Due to the special structure of the model, the coefficients of the expansion can be given in a closed form.


Introduction
Let (X, Y ) = {(X t , Y t )} t∈R+ be the bivariate model described by where Z = (Z t ) t∈R+ is a non-trivial Lévy process independent of the initial variable X 0 , and the parameter (λ, γ, β, ρ) ∈ (0, ∞) × R × (R\{0}) × R satisfies that β + ρλ = 0. ( The process X is the exponentially ergodic Lévy driven Ornstein-Uhlenbeck (OU) process; we refer to [4] and the references therein for fundamental facts concerning the OU process. The goal of this note is to provide conditions under which the Edgeworth expansion of the expectation E[f (T −1/2 H T )] as T → ∞ is valid, where and f : R → R is a measurable function of at most polynomial growth. The condition (2) will turn to be necessary for the Gaussian limit of L(T −1/2 H T ) to be non-degenerate: as a matter of fact, the necessity of (2) can be seen concisely by the expression so that, if β + ρλ = 0 and (X t − E[X t ]) − (X 0 − E[X 0 ]) = O p (1) as T → ∞, then L(T −1/2 H T ) tends in probability to 0 (See Section 2.2). As is well known, distributional regularity of the underlying model is essential to the validity of the Edgeworth expansion. At first glance, the regularity of the joint distribution L(X, H), which will play an essential role in derivation of the expansion (see Section 3), does not seem enough since we have only one-dimensional random input Z against the two-dimensional objective (X, H). In particular, for pure-jump Z we have to take distributional regularity over the Poisson space into account, rendering the problem mathematically interesting in its own right. In this case, we will execute the Malliavin calculus under truncation, which enables us to successfully pick out a nice event on which the integration by parts formula can apply to ensure distributional regularity; more specifically, our truncation functional will be constructed through two diffusive jumps, so as to make the Malliavin covariance matrix associated with the flow of (X, H) non-degenerate (As will be mentioned in Section 3.4, a single jump is not enough). The Malliavin calculus conveniently enables us to bypass intractable direct estimate of the characteristic function of L(T −1/2 H T ), and results in fairly simple conditions. Our result has the following statistical implication. Suppose that we can directly observe {X t : 0 ≤ t ≤ T }, based on which we want to estimate θ 0 := E[X 0 ] (the mean of the stationary distribution). A natural estimator is then given byθ We easily see that T −1/2 H T = T 1/2 (θ T − θ 0 ) with β = 1 and γ = ρ = 0, hence the consistency, asymptotic normality, and higher order expansion ofθ T are obtained according to our result.
The main result is given in Section 2, followed by the proof in Section 3.

Edgeworth expansion 2.1 Statement of result
We are given a stochastic basis (Ω, F , F = (F t ) t∈R+ , P ), on which our processes are defined.
Assumption 2.1. X is strictly stationary with a stationary distribution F ∈ p>0 L p (P ).
We remark that: under Assumption 2.1 X is exponentially β-mixing and ergodic; Assumption 2.1 is equivalent to Z 1 ∈ p>0 L p (P ). See [4] for details.
Denote by (b, C, Π) the generating triplet of Z in the form where b ∈ R, C ≥ 0, and the Lévy measure Π defined on R is a σ-finite measure satisfying Π({0}) = 0 and 0<|z|≤1 z 2 Π(dz) < ∞. Then the process H of (3) satisfies the k-th cumulant of ξ, with ∂ v denoting the (partial) differentiation with respect to a variable v. Denote by Λ the Poisson random measure associated with jumps of Z. We decompose it as for some Poisson random measures µ ♭ and µ; by the independently scattered property of Λ, such a decomposition is always possible. Correspondingly, we write where ν ♭ and ν stand for the Lévy measures on R + associated with µ ♭ and µ, respectively.
Assumption 2.2. Either one of the following two conditions holds true: (i) C > 0 (no condition is imposed on the jump-part characteristic); (ii) C = 0 and there exists a non-empty open subset of R\{0} on which ν admits a positive C 3density, say g, with respect to the Lebesgue measure.
Let us introduce the notation necessary for the Edgeworth expansion; see [6] for more details. We introduce the r-th cumulant function of T −1/2 H T (r ∈ N, r ≥ 2): Let φ(·; Σ) stand for the one-dimensional centered normal density having variance Σ > 0, then the r-th Hermite polynomial associated with φ(·; Σ) is h r (y; Σ) : The density of Ψ p,T with respect to the Lebesgue measure is given by Now we can state the main result.
Theorem 2.3. Let X, Y, H be given through (1) and (3), and suppose that (2) and Assumptions 2.1 and 2.2 hold true. Fix any positive number Σ 0 such that Then, for any M, K > 0, there exist positive constants M * and δ * such that Most often in practice, the first term in the upper bound in (4) can be quickly vanishing by taking K large; for example, it is the case when f is an indicator function f = 1 A for various A ⊂ R, such as A = (−∞, a], A = [a, b], and so on.
Noticing the explicit solution X t = e −λt X 0 + t 0 e −λ(t−s) dZ s , we can apply the stochastic Fubini theorem to obtain the relation where η(λ, u) = λ −1 (1 − e −λu ); one can consults [2] for a detailed analysis of integrated OU processes, especially in the context of financial econometrics. It follows from (1), (5), and the special relation kλκ [1,4]) that we can express H T as Hence, using the independence between X 0 and Z we obtain By making use of the differential equation with η(λ, 0) = 0 and then integrating the both sides with respect to s over [0, T ], we can proceed as in [5,Section 3] to conclude that where M r,T (j) is given by Thus we can explicitly write down the coefficients of the Edgeworth expansion Ψ p,T up to any order. It is obvious from (6) that χ r,T = O(T −(r−2)/2 ) for r ≥ 2; In particular, F , hence the necessity of the condition (2).

Proof of Theorem 2.3
We will apply [6,Theorem 1]. In order to ensure distributional regularity necessary for the Edgeworth expansion, we will make use of a Malliavin calculus with an effective truncation functional. The main idea of the proof is in principle similar to that of [5,Section 4] treating the stochastic volatility model, where X expresses the latent positive volatility process. However, the OU process X in the present model can take negative values too, and, as such, the way of constructing a truncation functional is essentially different from that of [5]. To save space, we will sometimes omit the technical details, referring to the pertinent parts of [3,5].
Let us briefly overview the fundamental device. By means of [6, Theorem 1], it suffices to verify the following conditions: [A1] X is strongly mixing with exponential rate; [A2] sup t∈[0,T ] H t L p+1 (P ) < ∞ for each T ∈ R + ; [A3] there exist positive constants t 0 , a, a ′ and B, and a truncation functional ψ : (Ω, F ) → ([0, 1], B([0, 1])) such that 0 < a, a ′ < 1, 4a ′ < (a − 1) 2 and that As was mentioned in Section 2, Assumption 2.1 ensures [A1] and [A2] (see (6)), so that it remains to verify [A3], which is a version of conditional Cramér conditions. Although it may be difficult in general to verify [A3], we will be able to construct a specific truncation ψ which significantly simplify the task. We also note that the condition (Ã ′ − 4) of [3, p. 60 and p.130] (smoothness of the coefficients, and integrability under cut-off through an auxiliary function) is indispensable. We will mention this point in Section 3.2

Transformation of the Poisson random measure
In order to execute a Malliavin calculus of [3], we introduce a transformation of the absolutely continuous part of the Poisson random measure.

Assumption 2.2 assures the existence of a bounded domain
for which the Lévy density g of ν satisfies that inf z∈E0 g(z) > 0.
Without loss of generality, we may and do suppose that c 1 , c 2 > 0: if ν(R + ) ≡ 0, then take −Z as Z anew. We introduce the change of variables z * = z * (z) = g + (z) through z * = z * (z) = c2 z g(v)dv for z ∈ E 0 ; obviously, g + is strictly decreasing on E 0 . Let g − denote the strictly decreasing inverse function of g + defined on E = (g + (c 2 ), g + (c 1 )).
Let µ * denote the integer-valued random measure defined by for each t ∈ R + , a 1 , a 2 ∈ R such that a 1 < a 2 , and for any measurable function h on R + × R + ; in particular, The bivariate process (X, H) satisfies the stochastic differential equation where J(z * ) :

Completion of the proof under Assumption 2.2 (i)
Suppose that C > 0. It follows from (8) that, in the matrix sense, where H k := t 0 0 e kλs ds and χ := ρ + β/λ. The determinant of the rightmost side is which is positive as soon as t 0 λ = 0 and β + ρλ = 0. Thus S(t 0 ,v) is bounded from below by a positive-definite matrix, hence the non-degeneracy of U (t 0 ,v) follows from (9) without any non-trivial truncation functional; simply let ψ ≡ 1 in [A3]. Thus we have obtained the non-degeneracy of the Malliavin covariance matrix (i.e. enough integrability of {detU (t 0 ,v)} −1 ), which corresponds to [5,Lemma 6]. We further notice the following.
• Following the same argument as in [5, pp.1184-1185], we see that there exists a random variable for every B > 0.
After all, we have deduced the analogous assertions to [5,Lemmas 6,7 and 8], completing the proof of Theorem 2.3 under Assumption 2.1 and Assumption 2.2 (i).

Construction of a truncation functional
It remains to prove Theorem 2.3 under Assumptions 2.1 and 2.2 (ii); then, in order to verify distributional regularity we have to make an effective use of jumps. We will construct the truncation functional ψ in an explicit way through two diffusive jumps.
The proof of Theorem 2.3 is thus complete.