Page 148 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.
We can arbitrarily set the explaining variable Xt and the function (, , ).
For example, we set other stock price processes, a price process of stock index,
accumulated volume of stock trade, Yt itself or some combination of those.
Let be a càdlàg stochastic process and Σ(t,x,σ): [0,T]×O×clos(Λ) → R ⊗
γ

γ
R be some known continuous function, where O ⊂ R X be an open set and the
γ
parameter space ⊂ ℝ be a bounded open set with ∈ ℕ.

Let Πn = ({ , } { , } ), then we assume that ℱ , (Πn) ∈ℕ and

,
,
{∈ , } ,, are mutually independent. Let

= ℱ ⋁ ({Πn} ) ⋁ ( ∩ { , ≤ }; ∈ {∈ , }, ∈ {1,2}, ∈ ℤ , ∈ ℕ),

+

where ℋ ⋁ ℋ denotes the minimal σ-field which contains σ-fields ℋ and
1
1
2
,
ℋ . Moreover, we assume that 1 , is -measurable, E[∈ , ] = 0 and
2
0

{
≤}
) ] = , where 1A is the indicator function for a set A and is positive
E[(∈ , 2 ,∗ ,∗
0
constant for 1 ≤ ≤ .
We consider a maximum-likelihood-type estimator of σ based on a quasi-
likelihood function. Construction is based on that of Ogihara [3]. Let
{ } and { } be sequences of positive numbers satisfying ≥ 1, ℓ ∈
∈ℕ

∈ℕ
⁄
⁄
ℕ, → ∞, ℓ ⁄ −1 3− → ∞ and 1 2− ⁄ ℓ → ∞ as → ∞ for some > 0.

By technical issues, we construct a quasi-log-likelihood function by
dividing whole observation interval [0, ] into disjoint local intervals

{ −1 , )} ℓ . Here the sequence { } ∈ is the order of sampling

=1
frequency, that is,

0 < - lim ( −1 , ) < ∞

→∞

Almost surely for 1 ≤ ≤ .

We prepare several notations: = ℓ −1 , = −1, = #{ ∈ ℕ; , <

0

,

}, = − −1 − 1, , = [ , + −1 , +1+ ) and () = {2 1 , 2 −

−1
1 {| 1 , 2 |=1} } , where is Kronecker’s delta. For an interval = [, ), we

1 , 2 =1
denote || = − . denotes a unit matrix of size .

̂
Let us consider an observable approximation of s −1 defined by

̃

̂
= (#{; , [ −1 , )} −1 ∑ ) .

,
j; ϵ[s m-1 ,s m )
1≤≤

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