Page 150 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.
distribution. However, we can see in the proof that this approximation is also
valid and ̂ still works for the case of non-Gaussian noise.

3. Asymptotic theory for misspecified model of diffusion cesses
Ogihara [3] studied specified model:
Σt, † ≡ (, , ) for some nonrandom ∈ Λ (3.1)
∗
∗

and showed asymptotic mixed normality of ̂ and local asymptotic normality

when ( ) and ( , †) are nonrandom. In machine learning theory including

neural network, we usually consider a model which does not necessarily satisfy
(3.1) (misspecified model).
In the study of a misspecified model, we sometimes face different
asymptotics with a specified model.
For example, Uchida and Yoshida [5] studied ergodic diffusion = ( )
≥0
2
with observations , where ℎ → 0, ℎ → ∞ and ℎ → 0 as → ∞.
ℎ =0
They showed that convergence rate of a maximum-likelihood-type estimator
for parameter in diffusion coefficients is √ℎ , which is different from the rate

√ for the specified model.
In our case, we also face different phenomenon from the specified case.
The maximum-likelihood-type estimator ̂ cannot attain optimal rate of

convergence due to the existence of asymptotic bias. However, we can
construct estimator which attains the optimal rate by modifying the

asymptotic bias.

a. Consistency
In this section, we study results related to consistency of ̂ . Since

convergence of ̂ is not ensured, we characterize convergence by means of a

function (, ′).
Here, we assume some conditions on the latent stochastic process X,Y and
market microstructure noise ∈ , . Let [] = [|{∏ } ] for a random

2
2
variable X and |A| = ∑ || for a matrix A.
,

[A1] 1. There exists a locally bounded function (, ) such that
|(, , ) − (, , )| ≤ (, )(| − | + | − |)
for any , ∈ [0, ], , ∈ and σ ∈ Λ.
2. ∑(, , ) is positive definite for any (, , ) ∈ [0, ] × × ().
3. is locally bounded, that is, there exists an increasing sequence {}

of stopping times such that →∞
= almost surely and
{ ⋀ } 0≤≤ is bounded for each .

4. sup ,, E[(∈ , ) ] < ∞ for any > 0 and

2
sup 0≤<≤ ([| − | ] | − |⁄ ) < ∞.

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