Page 147 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.
stocks, and so on) make it difficult to capture the stock microstructure. On the
other hand, high-frequency data contains huge information and therefore
machine learning methods such as neural network or support vector machine
is useful to identify the structure of the diffusion coefficient. In this approach,
we need to consider a theory of misspecified model.
We will study asymptotic theory of a maximum-likelihood-type estimator
for misspecified model. In this setting, the original maximum-likelihood-type
−1
estimator cannot attain the optimal convergence rate ⁄ 4 due to the
asymptotic bias. We construct a new estimator which attains the optimal rate
by using a bias correction and show the asymptotic mixed normality.

2. Parametric estimation under misspecified settings
Let (Ω, , ) be a probability space with a filtration = {}0 ≤ ≤ for

some > 0. We consider a −dimensional −adapted process = {}0 ≤
≤ satisfying an integral equation:

= + ∫ ∫ ,† , ∈ [0, ] (2.1)

0

0 0

Where { } a − dimensional standard F-Wiener process,
0≤ ≤

{ } and = { } 0≤ ≤ are ℝ and ℝ ⨂ ℝ -valued F-progressively
−
0≤ ≤
†
,†
measurable processes, respectively.
We assume that the observations of processes occur in a nonsynchronous
manner and are contaminated by market microstructure noise, that is, we

̃
observe the vectors { }0 ≤ ≤ , = 1,2 , where {J , }1 ≤ ≤ , ∈ are

,
positive integer-valued random variables, { , } J , are random times,
=0
{∈ , } ∈ + ,1≤≤2 is an independent identical distributed random sequence, and

̃

= , +∈ , . (2.2)

Let denote the transpose operator for matrices (and vectors). We
consider estimation of covariance matrix ∑ ,† = of the diffusion
,† ,†
coefficient by using functional Σ(t,Xt,σ) with a − dimensional càdlàg

̃
stochastic process Xt and a parameter σ. We observe possibly noisy data: =

,

, + for 0 ≤ j ≤ and 1 ≤ ≤ , where { }1 ≤ ≤ , ∈ are positive
, ,

integer-valued random variables, { , } , are random times, and
=0
,
{ } , are random variables possibly equal to zero.
=0
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