Threshold estimation for jump-diffusions under small noise asymptotics

We consider parameter estimation of stochastic differential equations driven by a Wiener process and a compound Poisson process as small noises. The goal is to give a threshold-type quasi-likelihood estimator and show its consistency and asymptotic normality under new asymptotics. One of the novelties of the paper is that we give a new localization argument, which enables us to avoid truncation in the contrast function that has been used in earlier works and to deal with a wider class of jumps in threshold estimation than ever before.

Suppose that we have discrete data X ε t0 , . . ., X ε tn from (1) for 0 = t 0 < • • • < t n = 1 with t i − t i−1 = 1/n.We consider the problem of estimating the true θ 0 ∈ Θ 0 under n → ∞ and ε → 0. We also define x t as the solution of the corresponding deterministic differential equation with the initial condition x 0 .
There are a lot of estimators for SDEs with small noise except for estimation of jump size density (see, e.g., [8,11,12,22], and references are given in [13]).One may immediately notice that if the background SDEs are driven by Wiener process and a compound Poisson process, then the number of large jumps never go infinity in probability.This means that we would never establish a consistent estimator of jump size density.To avoid this difficulty, we propose to assume that the intensity λ ε of jumps goes to infinity as ε ↓ 0 (λ ε is not necessary to depend on ε as in Remark 2.4), and this type of asymptotics would be the first in many works of literature.Intuitively, this assumption seems natural when we deal with data obtained in the long term by shortening the pitch of observations that is familiar in both cases of ergodic and small noise, so one may agree with our proposal.In practice, the intesity λ ε of jumps should be estimated, and it is possible by Lemma 4.8: Another attempt in this paper is to give a proof with localization argument (as in, e.g., Remark 1 in Sørensen and Uchida [22]) in the entire context.The argument is usually omitted, or instead, Propostion 1 in Gloter and Sørensen [8] is just refered.However, in this paper, we shall give a proof under localization assumptions 2.9 to 2.12, which are more complicated but includes more examples than ever, together with usual localization assumptions 2.6 and 2.7.Indeed, we assume that the sign of c does not change on a compact set, and ψ defined in 2.12 may be not continuous on the whole space R × R × Θ.Thus, we show our main results under the localization argument in the entire of our proof, which is one of the novelties of our paper.
In the ergodic case, threshold estimation for SDEs with Lévy noise is proposed in Shimizu and Yoshida [21], and has been considered so far by various researchers [7,15,20], and other references are given in Amorino and Gloter [1].The examples of jump size density in Shimizu and Yoshida [21] they cannot take gamma distributions as an example of jump size densities, and then Ogihara and Yoshida [15] establish assumptions for examples of jump size density to include gamma distributions.However, the contrast functions in both papers [21,15] are established using truncation and are of complicated form.Our research begins with the creed that we would be able to remove the truncations from our contrast functions when we deal with SDE with small Lévy noise, which is another novelty.Then, we establish our estimator which has the consistency and the asymptotic normality, and includes gamma distributions as an example of jump size densities.As we shall see in Remark 3.1, it is interesting that the range of the rate ρ of our filters D n,ε,ρ k may be disjoint from the range of the rate ρ of the filters {∆X n i > Dh ρ } in Ogihara and Yoshida [15].One can see some numerical examples, e.g., in Shimizu and Yoshida [21], and techniques how to choose the threshold (i.e., v nk /n ρ in this paper) of the filter in our estimators are found in Shimizu [18,19] and Masuda and Uehara [14].
In Section 2, we set up some assumptions and notations.In Section 3, we state our main results, i.e., the consistency and the asymptotic normality of our estimator.In Section 4, we give a proof of our main results.In Section 5, we give some examples of the jump size density for compound Poisson processes in our model.In the Appendix, we state and prove some slightly different versions of well-known results.

Assumptions and notations
Notation.We set the following notations: (N1) Let I x0 be the image of t → x t on [0, 1], and set if c(x, α) = 0 and f α y c(x, α) > 0, 0 otherwise.
(ii.c)There exists δ > 0 such that for any C 1 > 0 and C 2 ≥ 0 the map , and is continous on I δ x0 .Assumption 2.10.There exists δ > 0 such that for (x, y, α) are continuous at every points in I x0 , and there exist δ > 0 and C > 0 such that Assumption 2.12.The functions a, b, c are twice differentiable with respect to µ, σ, α, respectively, on I δ x0 × Θ for some δ, and the families There exists δ > 0 such that for (x, y, α) are continuous at every points x ∈ I x0 , and there exist δ > 0 such that We assume either of the following conditions (i) or (ii): (i) Under Assumption 2.4 (i), there exist constants C > 0, q ≥ 1 and δ > 0 such that sup (ii) Under Assumption 2.4 (ii), we assume the following four conditions: (ii.a)There exists δ > 0 and L > 0 such that if 0 < y 1 ≤ y ≤ y 2 , then as |y| → 0.
Notation.We further introduce the following notations: where n ∈ N, ε > 0.
The goal is to show the asymptotic normality of θn,ε when n → ∞ and ε → 0 at the sametime.In the sequel, we will also assume that λ ε → ∞ as ε ↓ 0 for consistency of θn,ε .Our interest is in a situation where the number of jumps is large and the Lévy noise is small.In practice, λ ε , the intensity of jumps, should be estimated, and it is possible by Lemma 4.8: Theorem 3.1.Under Assumptions 2.1 to 2.10, take ρ as either of the followings: (i) Under Assumption 2.4 (i), take ρ ∈ (0, 1/2).
If θ 0 ∈ Θ and I θ0 is positive definite, then Remark 3.1.If the jump size density {f α } α∈Θ3 is given as the family of probability density functions of normal distribution (see Example 5.1), then the range of ρ in Theorems 3.1 and 3.2 is same as in Shimizu and Yoshida [21] and Ogihara and Yoshida [15].However, if the jump size density {f α } α∈Θ3 is given as the family of probability density functions of gamma distribution (see Example 5.2), then the range of ρ is (0, 1/4) which is different from the range (3/8 + b, 1/2) of ρ in Ogihara and Yoshida [15], where b is the constant defined in the equation (1) in Ogihara and Yoshida [15].Needless to say that this paper deal with a small noise model and they deal with an ergodic model; we note that our estimator is established without truncation, and their estimator is established by using truncation.
, where C depends only on p, a, b, c and f α0 .In particular, when λ ε /n ≤ 1 and λ ε ≥ 1, it holds for p ≥ 2 and k = 1, . . ., n that where C depends only on p, a, b, c and f α0 .
Proof.For any p ≥ 2, we have where C depends only on p.Then, it follows from the Lipschitz continuity of a(•, µ 0 ) that where C depends only on a, and it follows from the Lipschitz continuity of b(•, σ 0 ) and Burkholder's inequality (see, e.g., Theorem 4.4.21 in Applebaum [2]) that where C depends only on p and b, and from the Lipschitz continuity of c(•, α 0 ), it is analogous to the proof of Theorem 4.4.23 in Applebaum [2] that where C depends only on p and c.Here, we have where C depends only on p and f α0 , and where C depends only on p and f α0 .Thus, where C depends only on p, c and f α0 .By using the Burkholder-Davis-Gundy inequality, where C depends only on p and f α0 .From the equations (2) to ( 6), where C depends only on p, a, b, c and f α0 .By Gronwall's inequality, where C depends only on p, a and b.
Proof.Same as the proof of Lemma 4.1, for any p ≥ 2, we obtain where C depends only on p, a, b, c and f α0 .
Proof.Both rates of convergence are obtained immediately from Lemma 4.2.
Lemma 4.4.Under Assumptions 2.2 and 2.3, let a family {g(•, θ)} θ∈Θ of functions from R to R be equicontinuous at every points in Proof.This follows from Lemmas 4.3 and A.2.
Lemma 4.5.Under Assumptions 2.2 and 2.3, let 0 where C depends only on p, a and b, and where C depends only on p, a, b, c and f α0 .
where C depnds only on p, a and b.By using Gronwall's inequality, we obtain where C depnds only on p, a and b.Similarly, where C depnds only on p, a and b.From Lemma 4.1, we have where C depnds only on p, a, b, c and f α0 .We can easily extend this result to the case p ∈ [1, 2) by using Hölder inequality.

Limit theorems
We make a version of Lemma 2.2 in Shimizu and Yoshida [21] in the sequel, and remark that the estimations for C n,ε,ρ k,1 and D n,ε,ρ k,1 , the notations of which are denoted in (N8), are essentially important (the estimation for C n,ε,ρ k,0 may be useless).
where κ is given in (N9), and C depends only on p, a, b, c, f α0 and v 1 .
Proof.We only give a proof under Assumption 2.4 (i), because the same argument still works under Assumption 2.4 (ii).Same as in the proof of Lemma 2.2 in Shimizu and Yoshida [21, Section 4.2], it follows that Also, it follows from and Lemmas 4.2, 4.5 and A.2 that where C depends only on p, a, b, c, f α0 and v 1 .Here, note that κ := 4v 2 /c 1 ≥ 4v nk /c 1 .The other inequalities follow from Lemma 4.5.
In the proof of Proposition 3.3 (ii) in Shimizu and Yoshida [21], the intensity of the Poisson process driving on the background is constant, although we assume the intensity λ ε goes to infinity.So, we prepare the following lemma.
as n → ∞ and ε → 0. This condition seems to be natural when we consider the asymptotic normality for our estimator (see, e.g., the condition (B2) in Sørensen and Uchida [22]).
Proof of Lemma 4.12.Let δ > 0 be a sufficiently small number satisfying the conditions of the statement and where c 1 and c 2 are the constants from Assumption 2.6.In this proof, we may simply write the maps (y, θ) → g(X ε t k−1 , y, θ) =: g k (y, θ) and (y, θ) → ∂g ∂y (x, y, θ) and we denote the following event by Dn,ε,ρ under either of the conditions (i.c) or (ii.e), we obtain from Lemma 4.10 that for any non-random r n,ε > 0 as n → ∞, ε → 0 λ ε → ∞ and ελ ε → 0, uniformly in θ ∈ Θ.Thus, it is sufficient to show that .
Lemma 4.13.Let ρ ∈ (0, 1/2).Under Assumptions 2.2 to 2.6, suppose that for θ ∈ Θ are continuous at every points in I x0 , and that there exist δ > 0, C > 0 and q ≥ 0 such that Then, Proof.It follows from Lemma 4.4 and the continuity of (11) that for each θ ∈ Θ as n → ∞, ε → 0, λ ε → ∞ and ελ ε → 0. Thus, Lemma 9 in Genon-Catalot and Jocod [6] shows us that for each Then, by the same argument in the proof of Lemma 4.10, it follows from Lemma 4.3 that To say the uniformity of this convergence in θ ∈ Θ, put and we shall use Theorem 5.1 in Billingsley [3] with the state space C(Θ), same as in the proofs of Propositions 3.3 and 3.6 in Shimizu and Yoshida [21]. 1 From ( 12), we obtain for j = 1, . . ., d.The above equalities hold from the fact that V N λε τ k and 1 J n,ε k,1 are independent.Hence, for any closed ball B r of radius r > 0 centered at zero in W 1,∞ (Θ), we obtain from Markov's inequality that sup n,ε where C is defined as (12).From Rellich-Kondrachov's theorem (see, e.g., Theorem 9.16 in Brezis [4]), it follows that the balls B r , r > 0 are compact in C(Θ), and so from Theorem 5.1 in Billingsley [3] that {χ n,ε } is relatively compact in distribution sense as in the Billingsley's book.Since for each θ ∈ Θ {χ n,ε (θ)} converges to zero in probability, all convergent subsequences of {χ n,ε } converges to zero in probability.Analogously, all subnet of {χ n,ε } has a subsequence convergent in probability to zero, and so {χ n,ε } converges to zero in probability as n → ∞, ε → 0, λ ε → ∞ and ελ ε → 0.
From Lemma 4.11, we remain to prove that as n → ∞, ε → 0, λ ε → ∞ and ελ ε → 0, uniformly in θ ∈ Θ.At first, by using Lemma 4.4, we have as n → ∞, ε → 0, λ ε → ∞ and ελ ε → 0, and as n → ∞, ε → 0, λ ε → ∞ and ελ ε → 0. Thus, by Lemma 9 in Genon-Catalot and Jocod [6], we obtain From the equidifferentiablities of g on I δ x0 for some δ > 0, the uniform tightness is shown by the same argument in the proof of Lemma 4.13.At second, we shall see This convergence is obtained from Lemma A.3 and the following estimate: is bounded in probability, it follows from Lemmas 4.1, 4.8 and 4.9 and the linarity of b that

Proof of Theorem 3.1
Proof of Theorem 3.1.It follows from that , and from Lemmas 4.11, 4.14 and 4.15 that , Also, it follows from Lemmas 4.12 and 4.13 that . Thus, by using usual argument (see, e.g., the proof of Theorem 1 in Sørensen and Uchida [22]), the consistency of θn,ε holds under Assumption 2.1.

Examples
This section is devoted to give some examples of densities which satisfies Assumptions 2.9 to 2.12.For simplicity, suppose that c(x, α) is an enough smooth postive function on I δ x0 × Θ 3 , and derivatives of c are uniformly continuous.Let D + is the interior of the common support of {f α } α∈Θ3 , i.e., The values of these functions may be undefined if (x, y, α) ∈ I δ x0 × ∂D + × Θ 3 .Otherwise their values are equal to zero.

Examples under Assumption 2.4 (i)
Example 5.1 (Normal distribution).Let Θ 3 be a smooth open convex set which is compactly contained in R × R + × R d3−2 , and let f α be of the form Then, for (x, y, α) ∈ I δ x0 × R + × Θ 3 and j = 3, . . ., d 3 .Furthermore, the derivatives of c and log c with respect to α are bounded on I δ x0 × Θ 3 , and so for (x, y, α) where C is a constant not depending on (x, y, α).Thus, Assumptions 2.9 to 2.12 are satisfied.

A Appendix
In this section, we state and prove some slightly different versions of well-known results.More precisely, we prepare Lemma A.2 as localization of the continuous mapping theorem.Lemma A.3 is a slightly different version of Lemma 9 in Genon-Catalot and Jacod [6].Lemma A.1.Let X be a Banach space, and let {g θ } θ∈Θ be a family of functions from X to R, and let T g θ be the composition operator on L ∞ ([0, 1]; X ) generated by g θ , i.e., Suppose that y • is a version of a function of C([0, 1]; X ) in L ∞ ([0, 1]; X ), and that {g θ } θ∈Θ is equicontinuous at every points in Image(y • ) := {y t | t ∈ [0, 1]}.Then, there is a neighborhood N y• of y • in L ∞ ([0, 1]; X ) such that {T g θ } θ∈Θ is a family of operators from N y• to L ∞ ([0, 1]), and is equicontinuous at y • .
Since Image(y This implies the conclusion.
We prepare the following lemma as localization of the continuous mapping theorem.This implies the conclusion.
Remark A.1.By the proof of Lemma A.2, it also follows that for any C 1 > 0, where C 2 depends only on C 1 , g and Image(y • ).Lemma A.3.Suppose that (X , • ) is a Banach space, {(n, ε)} is a directed set and {G n,ε i } i is a filtration for each n, ε.Let χ n,ε i , U be X -valued G n,ε i -measurable random variables.Proof.Since for any η, η > 0 we obtain Thus, the assertions (i) and (ii) follows.
Remark A.2.When X = R, this lemma can be shown by the same argument in the proof of Lemma 9 in Genon-Catalot and Jacod [6].However, the argument does not work in general, since we may not have Lenglart's inequality (e.g., Lemma 3.30 in Jacod and Shiryaev [10]) when X is a Banach space.Remark A.3.We have an immediate consequence from this lemma that