Statistical inference for discretely sampled stochastic functional differential equations with small noise

Estimating parameters of drift and diffusion coefficients for multidimensional stochastic delay equations with small noise are considered. The delay structure is written as an integral form with respect to a delay measure. Our contrast function is based on a local-Gauss approximation to the transition probability density of the process. We show consistency and asymptotic normality of the minimum-contrast estimator when the dispersion coefficient goes to zero and the sample size goes to infinity, simultaneously.


Introduction
Let (Ω, F , {F t }, P) be a stochastic basis satisfying the usual conditions.We consider a family of d-dimensional stochastic functional differential equations (SFDEs): for ε ∈ (0, 1], where θ 0 = (α 0 , β 0 ) ∈ Θ with Θ = Θ α × Θ β for open bounded convex subsets Θ α and Θ β of R p and R q , respectively; b = (b 1 , . . ., b d ) : R d × R d × Θ → R d and σ = (σ i j ) d×r : R d × R d × Θ β → R d ⊗ R r are known functions; W t = (W 1 t , . . .,W r t ) is an r-dimensional Wiener process.Moreover, φ ε (t) is F 0 measurable R d -valued random variable for each t ∈ [−δ , 0] and ε ∈ [0, 1].Letting C(A; B) be the space of continuous functions from A to B, we also define a functional H : C([0, δ ]; R d ) → R d as follows: for a continuous function where µ is a finite measure on [0, δ ].We call the equation (1.1) stochastic functional delay equations (SFDEs) since the functional H provides a delay structure in the SDE.Especially, when the measure µ is a Dirac measure, the SDE is just called a stochastic delay differential equation (SDDE).
There are many applications of (deterministic) delay differential equations (DDEs) in biology, epidemiology, and physics.For example, Mackey and Glassa [18] consider a homogeneous population of mature circulating cells (white blood cell, red blood cell, or platelet).Hu and Wang [10] take account of the dynamics of controlled mechanical systems, among others.Due to those, their corresponding stochastic versions (SDDEs) of those differential equations also has been well investigated.Volterra [26] considers predator-prey models; Guttorp and Kulperger [7] take the effect of random elements into account, and they change Volterra's model from DDEs to SDDEs; Fu [4] considers a stochastic SIR model with delay for an epidemic model, and also, De la Sen and Ibeas [3] consider SE(Is)(Ih)AR model as a COVID-19 model.From a theoretical point of view, Mohammed [20], and Arriojas [1] have generalized those models to SFDEs.
Turnning our attention to statistical inference for SDDEs and SFDEs, there have been many works so far.Gunshchin and Küchler [6] and Küchler and Kutoyants [14] study the asymptotic behavior of the maximum likelihood type estimators; Küchler and Sørensen [15] consider the pseudo-likelihood estimator for SDDEs; Küchler and Vasil'jev [16] propose a sequential procedure with a given accuracy in the L 2 sense.Moreover, Reiss [22,23] investigate nonparametric inference for affine SDDEs; Ren and Wu [24] consider least squares estimates for path-dependent McKean-Vlasov SDEs from discrete observations.Although all of those are studied in the ergodic context, we are interested in the small noise case; ε → 0, which is useful to justify the validity of the estimators since, in most applications of SFDEs, the ergodicity is often not expected.In this paper, we consider a local-Gauss type contrast function, and show the asymptotic normality of the minimum contrast estimators.
The paper is organized as follows.In Section 2, we make notation and assumptions, and state our main results in Section 3. In Section 4, we provide some numerical studies to support for our results.All the mathematical proofs are put in Section 5.
2 Notation and assumptions 2.1 Notation (N1) X 0 t is the solution of the ordinary differential equations under the true value of the drift parameter for t ∈ [0, 1]: where φ ∈ C([−δ , 0]; R d ) and x 0 is constant.As for the existence and uniqueness of the solution, see the proof of Theorem 3.7 and Remark 3.8 by Smith [19].
(N2) For matrix A, the (i, j)th element is written by A i j , ansd that where A ⊤ is the transpose of A and tr AA ⊤ is the trace of AA ⊤ .
(N3) For multi-index m = (m 1 , . . ., m k ), a derivative operator in z ∈ R k is given by , k and l times continuously differentiable with respect to x, y and θ , respectively.
for universal positive constants C and λ , where for (N6) Denote by G(p) the set of all permutations on {1, . . ., p}.
(N7) For elements b i and [σ σ ⊤ ] i j , we denote by

Assumptions
We make the following assumptions: (A1) There exists a constant K > 0 such that and there exists a constant Moreover, as ε → 0, , for at least one value of t, respectively.
(A6) The matrix is positive definite, where Remark 1.Although the assumption (A4) seems a bit restrictive, it is the same assumption as [A3'] in Gloter and Sørensen [8].

Main theorems
For estimation of θ ∈ Θ in (1.1), we consider the following local-Gauss type contrast function: where and δ n := ⌊nδ ⌋/n.
The consistency of our estimator θ n,ε is given as follows.
Theorem 1. Suppose the assumptions (A1)-(A5).Then we have The next theorem gives the asymptotic normal distribution of θ n,ε .

Simulations
We consider the following 2 -dimensional SFDE: (2) = 6X (1) −δ , X In this example, the estimator is given explicitly as follows: In the experiments, we generate discrete samples {X t k } n k=1 and {X −i/n } nδ n i=1 by the Euler-Maruyama method (see Buckwar [2]).We show means and standard deviations of estimators through 1000 times replications according to several values of (n, ε) in Tables 1-3, which illustrate the consistency of our estimator.
We also show the results of normal Q-Q plot in the ideal case where (n, ε) = (10000, 0.01) in Figures 1-4, which illustrate the asymptotic normality of each marginal of θ n,ε .Moreover, Figure 5 shows that the distribution of the bilinear form of the estimator follows the χ 2 (4)-distribution, which illustrates the (joint) asymptotic normality of θ n,ε .

Proofs
We first establish some preliminary lemmas.The idea of the proof of Lemma 1 is due to that of Lemma 2.2.1 by Nualart [21].
Lemma 1. Suppose that (A1) and (A2) hold true.Then there exists a strong solution Proof.Let and for n ≥ 0, First, we show that there exists a strong solution {X ε t }.From Lemma 2.2.1 of Nualart [21], it suffices to show that for any p ≥ 2. By a recursive argument, we can show that the inequality (5.1) holds.By using the Burkholder-Davis-Gundy inequality and (A1), where C p and C ′ p are constants depending only on p.For (5.2), by applying the Burkholder-Davis-Gundy inequality and (A1) again, we have Consequently, we have the inequality (5.2) by (5.1).Finally, we shall prove that the solution of (1.1) is unique.We assume that X ε t is the solution of (1.1).Then, it follows by (A1) that Hence, it follows from Gronwall's inequality that The proof is completed.
For Lemma 2, we shall use the notations: In Lemma 2, the proof ideas follow Long et al. [17] Lemma 2. Suppose that (A1) and (A2) hold true.Then, it follows for p ≥ 1, Proof.Since it is easy to see that as ε → 0 and n → ∞.We shall prove only the case where δ ≥ 1 because the proof for δ ≤ 1 is almost the same.It follows from (A1) and the Burkholder-Davis-Gundy inequality that where C p is a constant depending only on p.It holds from Gronwall's inequality and (A2) that as ε → 0. From the continuity of X 0 t , the proof is completed.
(ii) From Lemma 2 and the continuity of X 0 t , we have as ε → 0 and n → ∞.
Lemma 4. Suppose the conditions (A1) and (A2).Then it holds that Proof.From (A1) and (A2), we find that For the following Lemmas, we shall use the notations: for all a > 0.
Lemma 5. Suppose the conditions (A1) and (A2).For p ≥ 1 and where C p is constant depending only on p, and Φ ε p (•) is a function: Proof.In the same way as Lemma 6 in Kessler [13], we have where C p is constant depending only on p.We find that Next, we consider three cases: (b1) In the same way as (5.3), we find that (5.4) (b2) In the same way as (5.3), where Cp is constant depending only on p.
(ii) We find that It follows from the Lipschitz condition on b in (A1) that From Lemma 5, we have (5.7) From the same argument of (5.7), it holds that It is satisfied from the proof of (i) that Therefore, (5.9) It follows from (5.7)-(5.9)that Therefore, (iii) From the same argument as the proof of (ii), it hold that and (iv) It follows from the same argument as the proof of (ii) that Proof.(i) From lemma 2, Lemma 3 and Taylor's formula, we find that sup From the Lipschitz condition on b in (A1) it holds that sup which converges to zero as ε → 0 and n → ∞ by Lemma 2 and Lemma 3.
Next, we have Let We want to prove that u i n,ε (θ ) → 0 in P as ε → 0 and n → ∞, uniformly in θ ∈ Θ.Therefore, it is sufficient to check the pointwise convergence and the tightness of the sequence {u i n,ε (•)}.For the pointwise convergence, by the Chebyshev's inequality, the linear growth condition on σ in (A1) and itô's isometry, which converges to zero as ε → 0 and n → ∞ with fixed m.For the tightness, by using Theorem 20 in Appendix I of Ibragimov and Has'minskii [11], it is adequate to prove the following two inequalities: for θ , θ 1 , θ 2 ∈ Θ, where 2l > p + q.The proof of (5.12) is analogous to moment estimates in (5.11) by replacing itô's isometry with the Burkholder-Davis-Gundy inequality, so we omit the detail here.For (5.13), by using Taylor's formula and the Burkholder-Davis-Gundy inequality, we get Combining (5.10) and arguments above, we have that converges to zero in probability as ε → 0 and n → ∞.Therefore, the proof is complete.
Proof of Theorem 1.Following the proof of Theorem 1 in Sørensen and Uchida [25], the consistency follows from the two properties: where as ε → 0 and n → ∞, uniformly in θ ∈ Θ.About (5.15), from Lemma 7(i) and Lemma 8(ii), as ε → 0 and n → ∞, it holds that Finally, we prove the asymptotic normality of θ n,ε .For the proof, we shall use the notations: In Theorem 2, the proof ideas mainly follow Sørensen and Uchida [25].
• Funding: The study was funded by JSPS KAKENHI Grant Number 21K03358.
• Conflict of Interest: Author Hiroki Nemoto declares that he has no conflict of interest.
Author Yasutaka Shimizu has received research grants from JSPS.
• Ethical approval: This article does not contain any studies with human participants performed by any of the authors.