Remark 3.4. The point of the definition of C γ, δ is that it is written independently of ε. As a
result, if θ ∈ C γ, δ, then π
η
, s
η
, ε
η
associated with the drift coefficient b
η
= b
θ
η
2
and the diffusion coefficient c
η
= c
θ
η is also an element of C γ, δ for any η 0. In fact π
η
= π
θ
and s
η
= s
θ
. On the other hand, ε
η
= ηε
θ
, so that Theorem 3.3 implies, with a slight abuse of notation, E[ f Z
η T
] = E[1 + p
η
N f Z − logD − Σ
η
2 + p
Σ
η
N ] + O η
2
14 as
η → 0.
Remark 3.5. Given
θ ∈ C , Condition 3.2 does not hold for any δ 0 only when considering vicious examples such as the case
p π
θ
s
′ θ
ϕρ
′
is not continuous at any point of U; a sufficient condition for Condition 3.2 to hold with some
δ 0 is that p
π
θ
s
′ θ
ϕρ
′
is continuous at some point of U. If Condition 3.2 holds with some
δ 0, then it holds with any ˆ δ ∈ 0, δ] as well.
3.2 Examples
Lemma 3.6. Let θ ∈ C . If there exist γ
+
, γ
−
∈ [0, ∞
2
such that κ
±
2γ
±
, lim sup
v →±∞
1 + ϕv
2
e
γ
±
|v|
c
θ
v
2
∞ 15
with κ
+
= − lim sup
v →∞
b
θ
v c
θ
v
2
, κ
−
= lim inf
v →−∞
b
θ
v c
θ
v
2
, then there exists
δ 0 such that for any δ ∈ 0, δ
∧ 1, Condition 3.1 holds for θ = π, s, ε with γ = γ
+
, γ
−
and δ. Proof: This is shown in a straightforward manner by 3.
Example 3.7. Consider
dZ
t
= r
t
− 1
2 V
t
dt + p
V
t
ρdW
1 t
+ p
1 − ρ
2
dW
2 t
dV
t
= ξη
−2
µ − V
t
dt + η
−1
|V
t
|
ν
dW
1 t
for positive constants ξ, µ, η 0, ρ ∈ −1, 1 and ν ∈ [12, ∞. We assume ξµ 12 if ν = 12.
Then, the scale function s
V
of V satisfies s
V
0, ∞ = R, so that we can apply Itˆo’s formula to X = logV to have
dX
t
= η
−2
ξµe
−X
t
− ξ − e
−21−νX
t
2dt + η
−1
e
−1−νX
t
dW
1 t
. In this scale,
ϕx = expx2, so that we can take any open set as U ⊂ R. We fix ξ, µ, ν, ρ arbitrarily. In the light of Remark 3.4, it suffices to verify Conditions 3.1 and 3.2 only when
η = 1. It is trivial that Condition 3.2 holds with a sufficiently small
δ 0. If ν = 12, then 15 also holds with
κ
+
= ∞, κ
−
= ξµ − 1
2 ,
γ
+
= 2, γ
−
= 0.
772
If ν ∈ 12, 1, then it holds with
κ
±
= ∞, γ
+
= 3 − 2ν, γ
−
= 0. If
ν = 1, then it holds with κ
+
= ξ + 1
2 ,
κ
−
= ∞, γ
+
= 1, γ
−
= 0 provided that
ξ 32. Unfortunately, 15 does not hold if ν ∈ 1, 118]. If ν 118, it then holds with
κ
+
= 1
2 ,
κ
−
= ∞ γ
+
= 3 − 2ν
+
, γ
−
= 2ν − 2. Note that the case
ν = 12 corresponds to the Heston model. In this case, we have a more explicit expression of the asymptotic expansion formula; we have 14 with
p
η
z = ηρ
2 ξ
1 − z
2
+ 1
Σ
1 2
η
z
3
− 3z , Σ
η
= µT. This is due to the fact that the ergodic distribution of the CIR process is a gamma distribution.
Example 3.8. Here we treat 4. In order to prove the validity of the singular expansion in the form 14 for Z
η T
= logS
η T
, it suffices to show that there exist γ, δ and η 0 such that Conditions 3.1
and 3.2 hold for θ = π, s, ε ∈ C associated with
b
θ
x = m − x − ην p
2Λx, c
θ
x = ν p
2 for any
η ∈ 0, η ], in the light of Remark 3.4. Here we fix m ∈ R and ν ∈ 0, ∞. Suppose that
there exists γ
+
, γ
−
∈ [0, ∞
2
such that lim sup
x →±∞
e
−γ
±
|x|
ϕ
2
x ∞ and that Λ is locally bounded on R with
λ
∞
:= lim inf
|x|→∞
Λx x
−∞. Then we have
−sgnv b
θ
v c
θ
v
2
→ ∞, as
|v| → ∞ uniformly in η ∈ 0, η ] with, say, η
= 1 ∧ |12νλ
∞
∧ 0|. Hence, by Lemma 3.6, there exists
δ ∈ 0, 1 such that Condition 3.1 holds for any η ∈ 0, η ] with γ = γ
+
, γ
−
and δ. By, if necessary, replacing
δ, η with a smaller one, Condition 3.2 also is verified for any η ∈ 0, η
] provided that there exists a non-empty open set U such that
ϕ is continuously differentiable on U. Consequently, by Theorem 3.3, we have 14 for 4 if
|ρ| 1 and |ϕ
′
| 0 on U in addition. The obtained estimate of error O
η
2
is a stronger result than one obtained by Fouque et al. [9][10] and Alòs [1].
773
Example 3.9. Here we treat a diffusion which is not geometrically mixing. Consider the stochastic differential equation
dX
t
= − 1
η
2
1 2
+ ξ tanhY
t
coshY
t 2
dt + 1
η 1
coshX
t
dW
t
with ξ 12 and η 0. Putting Y
t
= sinhX
t
, we have dY
t
= − 1
η
2
ξY
t
1 + Y
2 t
dt + 1
η dW
t
This stochastic differential equation has a unique weak solution which is ergodic. A polynomial lower bound for the
α mixing coefficient is given in Veretennikov [19] which implies in particular that X = sinh
−1
Y is not geometrically mixing for any ξ. Now, let us verify Conditions 3.1 and 3.2 for 1 with
bx = −
1 η
2
1 2
+ ξ tanhx
coshx
2
, cx = 1
η 1
coshx for any
η 0. In the light of Remark 3.4, it suffices to deal with the case η = 1. Since − lim
|x|→∞
sgnx bx
cx
2
= 1
2 + ξ,
we have 15 if there exists µ ≥ 0 such that
sup
|x|→∞
e
−µ|x|
ϕx
2
∞, 1
2 + ξ 4 + 2µ.
Condition 3.2 also is satisfied with a sufficiently small δ 0 under the condition on ϕ and ρ stated
in the beginning of this section.
4 Edgeworth expansion
In this section, we present basic results of the Edgeworth expansion which play an essential role in the proof of Theorem 3.3 given in the next section. In Section 4.1, we give a validity theorem for
the classical iid case with a brief introduction to the Edgeworth expansion theory. The theorem is applied to a non-iid case by the regenerative approach in Section 4.2 to establish a general validity
theorem for regenerative functionals including additive functionals of ergodic diffusions.
4.1 The Edgeworth and Gram-Charlier expansions
