Abstract. Let X be a diffusion processes and A be some Borel subset of R. In this
paper, we introduce an estimator for the occupation time Γ(A)t = R0t I{Xs∈A}ds
based on an irregular sample of X and study its asymptotic behavior.
Keywords: Occupation time, diffusion processes, irregular sample.

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JOURNAL OF SCIENCE OF HNUE
Interdisciplinary Science, 2014, Vol. 59, No. 5, pp. 3-16
This paper is available online at
ON DISCRETE APPROXIMATION OF OCCUPATION TIME OF DIFFUSION
PROCESSES WITH IRREGULAR SAMPLING
Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh
Faculty of Mathematics and Informatics, Hanoi National University of Education
Abstract. LetX be a diffusion processes and A be some Borel subset of R. In this
paper, we introduce an estimator for the occupation time Γ(A)t =
∫ t
0 I{Xs∈A}ds
based on an irregular sample of X and study its asymptotic behavior.
Keywords: Occupation time, diffusion processes, irregular sample.
1. Introduction
Let X be a solution to the following stochastic differential equation
dXt = b(Xt)dt+ σ(Xt)dWt, X0 = x0 ∈ R, (1.1)
where b and σ are measurable functions and Wt is a standard Brownian motion defined
on a filtered probability space (Ω,F , (Ft)t>0,P).
For each set A ∈ B(R) the occupation time of X in A is defined by
Γ(A)t =
∫ t
0
I{Xs∈A}ds.
The quantity Γ(A) is the amount of time the diffusionX spends on set A. The problem of
evaluating Γ(A) is very important in many applied domains such as mathematical finance,
queueing theory and biology. For example, in mathematical finance, these quantities are
of great interest for the pricing of many derivatives, such as Parisian, corridor and Eddoko
options (see [1, 2, 9]).
In practice, one cannot observe the whole trajectory of X during a fixed interval.
In other words, we can only collect the values of X at some discrete times, say 0 = t1 <
t2 < . . . Recently, Ngo and Ogawa [10] and Kohatsu-Higa et al. [7] have introduced an
estimate for Γ(A) by using a Riemann sum and they studied the rate of convergence of this
approximation when X is observed at regular points, i.e. {ti = i
n
, i 6 [nt]} for all i > 0
Received December 25, 2013. Accepted June 26, 2014.
Contact Nguyen Thi Lan Huong, e-mail address: nguyenhuong0011@gmail.com
3
Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh
and any n > 0. However, in practice for many reasons we can not observe X at regular
observation points. Thus, in this paper, we will construct an estimation scheme for Γ(A)
based on an irregular sample {Xti , i = 0, 1, . . .} of X and study its asymptotic behavior.
In particular, we first introduce an unbiased estimator for Γ(A) when X is a standard
Brownian motion and provide a functional central limit theorem (Theorem 2.2) for the
error process. It should be noted here that assumption A, which is obviously satisfied
for regular sampling, is the key to construct the limit of the error process for irregular
sampling. We then introduce an estimator for Γ(A) for general diffusion process and show
that its error is of order 3/4.
2. Main results
Throughout out this paper, we suppose that coefficients b and σ satisfy the following
conditions:
(i) σ is continuously differentiable and σ(x) ≥ σ0 > 0 for all x ∈ R,
(ii) |b(x)− b(y)|+ |σ(x)− σ(y)| ≤ C|x− y| for some constantC > 0. (2.1)
The above conditions on b and σ guarantee the continuity of sample path and
marginal distribution ofX (see [11]). We note here that under a more restrictive condition
on the smoothness and boundedness of b, σ and their derivatives, Kohatsu-Higa et al. [7]
have studied the strong rate of approximation of Γ(A) via a Riemann sum as one defined
in [10].
At the nth stage, we suppose thatX is observed at times tni , i = 0, 1, 2, ... satisfying
0 = tn0 < t
n
1 < t
n
2 0 such that
∆n ≤ k0min
i
∆ni , ∀n, (2.2)
where ∆ni = tni − tni−1 and ∆n = max
i
∆ni . We assume moreover that limn→∞∆n = 0.
We denote ηn(s) = tni if tni ≤ s < tni+1.
2.1. Occupation time of Brownian Motions
We first recall the concept of stable convergence: Let (Xn)n≥0 be a sequence of
random vectors with values in a Polish space (E, E), all defined on the same probability
space (Ω,F , (Ft)t≥0,P) and let G be a sub-σ-algebra of F . We say that Xn converges
G-stably in law to X , denote Xn G−st→ X , if X is an E−valued random vector defined
on an extension (Ω′,F ′,P′) of the original probability space and limn→∞ E(g(Xn)Z) =
E′(g(X)Z), for every bounded continuous functions g : E → R and all bounded
G−measurable random variables Z (see [4, 5, 8]). When G = F we write Xn st→ X
instead of Xn
G−st→ X .
We denote by Lt(a) the local time of a standard Brownian motionB at a, up to and
including t given by
Lt(a) = |Bt − a| − |a| −
∫ t
0
sign(Bs − a)dBs.
4
On discrete approximation of occupation time of diffusion processes with irregular sampling
For each Borel function g defined on R and γ > 0, we set
βγ(g) =
∫
|x|γ|g(x)|dx, λ(g) =
∫
g(x)dx.
In order to study the asymptotic behavior of the estimation error, we need the following
assumption.
Assumption A: There exists a non-decreasing function Ft(x) such that
1
(∆n)3/2
∑
tni 6t
(tni − tni−1)3/2E(Ltni (x)− Ltni−1(x)|Ftni−1)
P−→ Ft(x), ∀ t > 0, x ∈ R. (2.3)
Theorem 2.1 (The approximation of local time). Suppose that g satisfies the following
conditions:
g(x) = o(x) as x→∞, β1(g) <∞, and λ(|g|) <∞. (2.4)
Then for all x ∈ R it holds
∑
tni 6t
√
tni − tni−1g
( Btni−1 − x√
tni − tni−1
)
P−→ λ(g)Lt(x).
Now, we proceed to state the functional central limit theorem for the error process.
First, let us recall the definition of F -progressive conditional martingale (see [5] for more
details). We call extension of B another stochastic basis B˜ = (Ω˜, F˜ , (F˜t), P˜) constructed
as follows: We have an auxiliary filtered space (Ω′ ,F ′, (F ′t)t≥0) such that each σ-field
F ′t− is separable, and a transition probability Qω(dω
′
) from (Ω,F) into (Ω′ ,F ′), and we
set
Ω˜ = Ω× Ω′ , F˜ = F˜ ⊗ F ′, F˜ = ∩s>tFs ⊗F ′s,
P˜(dω, dω
′
) = P(dω)Qω(dω
′
).
A process X on the extension B˜ is called an F -progressive conditional martingale if it is
adapted to F˜ and if for P-almost all ω in the process X(ω, .) is a martingale on the basis
Bω = (Ω′,F ′, (F ′t)t≥0,Qω).
Theorem 2.2. Suppose that B is a standard Brownian motion defined on a filtered space
B = (Ω,F ,Ft, P˜). For each n > 1, t > 0 and K ∈ R we set
Γ˜(K)nt =
∫ t
0
Φ
( Bηn(s) −K√
s− ηn(s)
)
ds,
where Φ is the standard normal distribution function.Then,
(i) Γ˜(K)nt is an unbiased estimator for the occupation time Γ([K,∞))t;
5
Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh
(ii) Γ˜(K)nt
P−→ Γ([K,∞))t;
(iii) Moreover, suppose that A holds then there exists a good extension B˜ of B and a
continuous B-biased F -progressive conditional martingale with independent increment
X ′ on this extension with
〈X ′, X ′〉t = 9
20
√
2pi
Ft(K), 〈X ′, B〉 = 0.
1
(∆n)3/4
(
Γ˜(K)nt − Γ([K,∞))t
) st−→ X ′.
Remark 2.1. AssumptionA make sense. It is is obviously satisfied for regular sampling.
The following condition is sufficient to have (2.3),
lim
n→∞
min
i
∆ni (∆n)
−1 = 1. (2.5)
Moreover, we have Ft(x) = Lt(x).
Indeed, we set γn =
min
i
(∆ni )
max
i
(∆ni )
and
F nt (x) =
1
(∆n)3/2
∑
tni 6t
(tni − tni−1)3/2E(Ltni (x)− Ltni−1(x)|Ftni ).
Hence F nt (x) =
∑
tni 6t
E(Ltni (x)− Ltni−1(x)|Ftni )− Snt (x), where
0 6 Snt (x) =
∑
tni 6t
(
1− (∆
n
i )
3/2
(∆n)3/2
)
E(Ltni (x)− Ltni−1(x)|Ftni )
6 (1− γn)
∑
tni 6t
E(Ltni (x)− Ltni−1(x)|Ftni )
P−→ 0,
since
∑
tni 6t
E(Ltni (x) − Ltni−1(x)|Ftni )
P−→ Lt(x). Thus, Sn(t) P−→ 0, and F nt (x) P−→ Lt(x) as
n→∞.
The condition (2.5) can be localized a little as follows: Suppose that there exists a
sequence of fixed times S1 < S2 < ..., which does not depend on n such that in each
interval (Si, Si+1) the condition (2.5) is satisfied. Then the condition (2.3) also holds.
2.2. Occupation time of general diffusion
In order to study the rate of convergence, we recall the definition of C-tightness.
First, we denote by D(R) the Polish space of all càdlàg function: R+ → R with
Skorokhod topology. A sequence of D(R) -valued random vector (Xn) defined on
(Ω,F , (F)t,P) is tight if inf
K
sup
n
P(Xn /∈ K) = 0, where the infimum is taken over all
6
On discrete approximation of occupation time of diffusion processes with irregular sampling
compact sets K in D(R). The sequence (Xn) of processes is called C-tightness if it is
tight, and if all limit points of the sequence {L(Xn)} are laws of continuous processes
(see [5]).
Denote S(x) =
∫ x
x0
1
σ(u)
du and Yt = S(Xt). For each set A ∈ B(R), A =
m⋃
i=0
[a2i, a2i+1)
where −∞ 6 a0 < a1 < · · · < a2m+1 6 +∞ we introduce the following estimate for
Γ(A)t:
Γ˜(A)nt =
m∑
j=0
∫ t
0
Φ
(S(a2j+1)− S(Xηn(s))√
s− ηn(s)
)
− Φ
(S(a2j)− S(Xηn(s))√
s− ηn(s)
)
ds.
In particular, if A = [K,+∞) then the biased and consistent estimator for the occupation
time
∫ t
0
I{Xs>K}ds is defined by
Γ˜([K,∞))nt =
∫ t
0
Φ
(S(Xηn(s))− S(K)√
s− ηn(s)
)
ds.
Theorem 2.3. For each set A ∈ B(R), A =
n⋃
i=0
[a2i, a2i+1) where −∞ 6 a0 < a1 <
· · · < a2n+1 6 +∞ the sequence of stochastic processes( 1
(∆n)3/4
(
Γ˜(A)nt −
∫ t
0
I{Xs∈A}ds)
)
t≥0
is C-tight.
3. Proofs
We denote (Pt)t>0 a Brownian semigroup given by Ptk(x) =
∫
k(x+ y
√
t)ρ(y)dy,
where ρ(y) = 1√
2pi
e−y
2/2 and k is a Lebesgue integrable function.
3.1. Some preliminary estimates
Throughout this section we denote by K a constant which may change from line
to line. If K depends on an additional parameter γ, we write Kγ . We first recall some
estimates on the semigroup (Pt).
Lemma 3.1 (Jacod [4]). Let k : R→ R be an integrable function. If t > s > 0 and γ > 0
we have:
|Ptk(x)| 6 Kλ(|k|)√
t
, (3.1)
∣∣∣Ptk(x)− λ(k)√
2pit
e−x
2/2t
∣∣∣ 6 Kγ
t
( β1(k)
1 + |x/√t|γ +
β1+γ(k)
1 + |x|γ
)
, (3.2)
∣∣∣Ptk(x)− λ(k)√
2pit
e−x
2/2t
∣∣∣ 6 K
t3/2
(β2(k) + β1(k)|x|). (3.3)
7
Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh
We will need the following estimate.
Lemma 3.2. Let k : R → R be an integrable function. Suppose that the sequence {tni }
satisfies (2.2). Denote
γ1(k, x)
n
t = E
(∑
tni 6t
i>2
(∆ni )
2k(
x+Btni−1√
∆ni
)
)
, γ2(k, x)
n
t = E
(∑
tni 6t
i>2
√
∆ni k(
x+Btni−1√
∆ni
)
)
.
Then
(i) |γ1(k, x)nt | 6 Kλ(|k|)(∆n)3/2
√
t, (3.4)
(ii) |γ2(k, x)nt | 6 Kλ(|k|)
√
t. (3.5)
Moreover, if λ(k) = 0 then
|γ1(k, x)nt | 6 Kβ1(k)(∆n)2k0(1 + log+(
tk0
∆n
)), (3.6)
|γ1(k, x)nt | 6 K(∆n)2(β2(k) + β1(k)|x|), (3.7)
and
|γ2(k, x)nt | 6 Kβ1(k)
√
∆nk0(1 + log+(
tk0
∆n
)), (3.8)
|γ2(k, x)nt | 6 K
√
k0
√
∆n(β2(k) + β1(k)|x|). (3.9)
Proof. From (3.1) and estimates (4.1), (4.2), we obtain (3.4) and (3.6). Furthermore, from
(3.3) in the Lemma 3.1 we get
|γ1(k, x)nt | 6
∑
tn
i
6t, i>2
(∆ni )
2 K
(
tni−1
∆ni
)
3
2
(β2(k) + β1(k)|x|))
6 Kk0(∆n)
5/2(β2(k) + β1(k)|x|))
∫ t
∆n/k0
x−3/2dx
6 K(∆n)
2(β2(k) + β1(k)|x|).
By using analogous arguments as above, we obtain (3.5), (3.8) and (3.9).
Lemma 3.3. Assume that λ(g) = 0 and g satisfies (2.4), then
(i)
1
(∆n)3
E
∑
tni 6t
(tni − tni−1)2g
(
x+Btni−1√
tni − tni−1
)
2
n→∞−−−→ 0.
(ii) E
∑
tni 6t
√
tni − tni−1g
(
x+Btni−1√
tni − tni−1
)
2
n→∞−−−→ 0.
8
On discrete approximation of occupation time of diffusion processes with irregular sampling
Proof. We first note that condition (2.4) implies that λ(g2) <∞. We write
1
(∆n)3
E
∑
tni 6t
(tni − tni−1)2g
(
x+Btni−1√
tni − tni−1
)
2
=
1
(∆n)3
(∑
tni 6t
E((tni − tni−1)4g(
x+Btni−1√
∆ni )
)2)
)
+
+
2
(∆n)3
( ∑
i:tni <t
n
i+16t
E
(
(∆ni )
2g(
x+Btni−1√
∆ni
)(
∑
j:tni <t
n
j 6t
(∆nj )
2g(
x+Btnj−1√
∆nj
))
))
. (3.10)
Using (3.4) and (2.4), the first term of (3.10) is bounded by
∆ng(
x√
∆n1
)2 + (∆n)
1/2Kλ(g2)
√
t→ 0 as n→∞.
Using (3.6), we have
E
( ∑
j:tni <t
n
j 6t
(∆nj )
2g(
x+Btnj−1√
∆nj
)|Ftni−1
)
= E
( ∑
tni <t
n
j 6t
(∆nj )
2g(
y +Btnj−1−tni−1√
∆nj
)
)∣∣∣
y=x+Btn
i−1
≤ Kβ1(k)(∆n)2k0(1 + log+(
(t− ti−1)k0
∆n
)) ≤ Kβ1(k)(∆n)2k0(1 + log+(k0n)).
Thus the second term of (3.10) is bounded by
2
(∆n)3
∑
i:tni 6t
n
i+16t
E
(∆ni )2g(x+Btni−1√
∆ni
)E(
∑
j:tni <t
n
j 6t
(∆nj )
2g(
x+Btnj−1√
∆nj
)|Ftni−1)
≤ Kk0β1(g)(∆n)−1(1 + log+(k0n))
∑
tni <t
n
i+16t
E
(
(∆ni )
2g(
x+Btni−1√
∆ni
)
)
≤ Kk0β1(g)(∆n)−1(1 + log+(k0n))
(
(∆n1 )
2g(
x√
∆n1
) +Kβ1(g)(∆n)
2k0(1 + log+(k0n))
)
,
which tends to 0 as n → ∞ because of condition (2.4). We conclude part (i). In an
analogous manner, applying (3.5), (3.8) and (2.4) we have (ii).
For each set A ∈ B(R) where B(R) is Borel σ-algebra on R we denote
Γ(A)nt =
∑
tni 6t
∆ni I{Xtn
i
∈A}.
Lemma 3.4. Suppose that the conditions (2.2) holds and for each set A ∈ B(R) satisfying∫
∂A
dx = 0, then Γ(A)nt
as−→ Γ(A)t.
9
Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh
The proof is similar to one of Proposition 2.1[10] and will be omitted.
Lemma 3.5. Assume that the condition (2.3) holds and the function g satisfies (2.4). Then
for all x ∈ R it holds
1
(∆n)3/2
∑
tni 6t
(
(tni − tni−1)2g(
Btni−1 − x√
tni − tni−1
)
)
P−→ λ(g)Ft(x). (3.11)
Proof. We set gˆ(x) = E(|x+B1| − |x|). Appying the condition (2.3) we write
1
(∆n)3/2
∑
tni 6t
(tni − tni−1)2gˆ(
Btni−1 − x√
tni − tni−1
) =
1
(∆n)3/2
∑
tni 6t
(tni − tni−1)3/2E(Ltni (x)− Ltni−1(x)|Ftni−1)
P−→ Ft(x).
Set g′(x) = g−λ(g)gˆ, then λ(g′) = 0. It follows from Lemma 3.3 and the condition (2.3)
that
1
(∆n)3/2
∑
tni 6t
(tni − tni−1)2gˆ(
Btni−1 − x√
tni − tni−1
)
= λ(g)
1
(∆n)3/2
∑
tni 6t
(tni − tni−1)2gˆ(
Btni−1 − x√
tni − tni−1
) +
1
(∆n)3/2
∑
tni 6t
(tni − tni−1)2g′(
Btni−1 − x√
tni − tni−1
)
P−→ λ(g)Ft(x).
3.2. Proof of Theorem 2.1
We denote gˆ(x) as in Lemma 3.5. From the definition of Lt we have
E(|Btni − x| − |Btni−1 − x||Ftni−1) = E(Ltni (x)− Ltni−1(x)|Ftni−1),
for all x ∈ R. On the other hand, since Btni − Btni−1 is independent of Ftni−1 and it has the
same distribution as
√
∆ni B1, we have
E(|Btni − x| − |Btni−1 − x||Ftni−1) = E(|Btni−1 +Btni − Btni−1 − x| − |Btni−1 − x||Ftni−1)
= E(|y +
√
∆ni B1| − |y|)
∣∣∣
y=Btn
i−1
−x
=
√
∆ni gˆ(
Btni−1 − x√
∆ni
).
Hence, it follows from Lemma 2.14 [3] that
∑
tni 6t
√
tni − tni−1gˆ(
Btni−1 − x√
tni − tni−1
) =
∑
tni 6t
E(Ltni (x)− Ltni−1(x)|Ftni−1)
P−→ Lt(x).
10
On discrete approximation of occupation time of diffusion processes with irregular sampling
Set g′(x) = g − λ(g)gˆ, then λ(g′) = 0. From Lemma 3.3 (ii) one gets
∑
tn
i
6t
√
tni − tni−1g(
Btni−1 − x√
tni − tni−1
)
= λ(g)
∑
tni 6t
√
tni − tni−1gˆ(
Btni−1 − x√
tni − tni−1
) +
∑
tni 6t
√
tni − tni−1g′(
Btni−1 − x√
tni − tni−1
)
P−→ λ(g)Lt(x).
This concludes the proof of Theorem 2.1.
Lemma 3.6. We denote Nnt =
∑
tni 6t
Ni,n andMnt =
∑
tni 6t
Mi,n, where
Ni,n =
1
(∆n)3/4
(
(tni − tni−1)I[K,∞)(Btni−1)−
∫ tni
tni−1
I[K,∞)Bsds
)
,
Mi,n = Ni,n − E(Ni,n|Ftni−1).
Then sequence Mn converges stable to a continuous process defined on an extension of
original probability space. In particular, the sequence (Mn) is C- tigh under probability
measure P.
Proof. We will prove the lemma in the following steps:
Step 1. A simple calculation using properties of Brownian motion yiels
E(Ni,n|Ftni−1) =
1
(∆n)3/4
(
∆ni I{Btn
i−1
>K} −
∫ ∆ni
0
Φ(
Btni−1 −K√
u
)du
)
=
( ∆ni
(∆n)3/4
1√
2pi
∫ +∞
Btn
i−1
−K√
∆n
i
(1− (Bt
n
i−1
−K)2
∆ni t
2
)e−t
2/2dt
)
I{Btn
i−1
>K}
−
( ∆ni
(∆n)3/4
1√
2pi
∫ Btni−1−K√
∆n
i
−∞
(1− (Bt
n
i−1
−K)2
∆ni t
2
)e−t
2/2dt
)
I{Btn
i−1
<K}.
We set g1(x) =
( ∫∞
x
(1 − x
2
t2
)e−t
2/2dtI{x>0} −
∫ x
−∞(1 −
x2
t2
)e−t
2/2dtI{x<0}
)2
. We have
∫
R
g1(x)dx =
7
√
2pi
20
, and g1(x) 6 min{pi2 , x−2e−x
2} for any x ∈ R. Hence it follows
from Lemma 3.5 that
∑
tni 6t
(
E(Ni,n|Ftni−1)
)2
=
∑
tni 6t
1
2pi
1
(∆n)3/2
(∆ni )
2g1(
Btni−1 −K√
∆ni
)
P−→ 7
√
2pi
20
Ft(K). (3.12)
11
Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh
Step 2. Next, by Markov property and Fubini theorem, we have
E
[( ∫ tni
tni−1
I{Bs>K}ds
)2
|Ftni−1
]
=
∫ ∆ni
0
∫ ∆ni
0
E
(
I{Bs>−r}I{Bu>−r}
)
duds|r=Btn
i−1
−K .
A direct calculation of the expectation E
(
I{Bs>−r}I{Bu>−r}
)
yields, that if r 6 0
E(
∫ ∆ni
0
I{Bs>−r}ds)
2 =
∆ni
pi
∫ 1
0
z3/2√
1− z exp (−
r2
2∆ni (1− z)
)dz,
and if r > 0 then
E(
∫ ∆ni
0
I{Bs>−r}ds)
2
= (∆ni )
2
(
1−
∫ 1
0
1
pi
√
z(1 − z) exp (−
r2
2z∆ni
)dv
)
+
(∆ni )
2
pi
∫ 1
0
z3/2√
1− z exp (−
r2
2∆ni (1− z)
)dz.
We have
E(N2i,n|Ftni−1) =
(∆ni )
2
(∆n)3/2
I{Btn
i−1
>K} − 2∆
n
i
(∆n)3/2
I{Btn
i−1
>K}E(
∫ tni
tni−1
I{Bs>K}ds|Ftni−1)
+
1
(∆n)3/2
E(
∫ tni
tni−1
I{Bs>K}ds)
2|Ftni−1).
Hence
E(N2i,n|Ftni−1) =
(∆ni )
2
(∆n)3/2
{ 1
pi
∫ 1
0
z3/2√
1− z exp (−
(Btni−1 −K)2
2∆ni (1− z)
)dzI{Btn
i−1
<K}
+ 2
∫ 1
0
(1− Φ( 1√
∆ni
√
u
(Btni−1 −K)))duI{Btni−1>K}
+
1
pi
∫ 1
0
z3/2√
1− z exp (−
1
2∆ni z
(Btni−1 −K)2)dzI{Btni−1>K}
− 1
pi
∫ 1
0
1√
(1− z)z exp (−
1
2∆ni z
(Btni−1 −K)2)dzI{Btni−1>K}}.
Set
g2(x) =
1
pi
∫ 1
0
z3/2√
1− z exp (−
x2
2(1− z))dzI{x<0}
+ {2
∫ 1
0
(1− Φ( x√
u
))du+
1
pi
∫ 1
0
z3/2√
1− z exp (−
x2
2z
)dz
− 1
pi
∫ 1
0
1√
z(1− z) exp (−
x2
2z
)dz}I{x>0}.
12
On discrete approximation of occupation time of diffusion processes with irregular sampling
We have |g2(x)| 6 Kmin{1, |x|−1e−x2/2+x−2} for all x ∈ R, and
∫ +∞
−∞ g2(x)dx =
2
√
2
5
√
pi
.
Therefore, applying Lemma 3.5 we get
∑
tni 6t
E(N2i,n|Ftni−1) =
1
(∆n)3/2
∑
tni 6t
(∆ni )
2g2(
Btni−1 −K√
∆ni
)
P−→ 2
√
2
5
√
pi
Ft(K). (3.13)
Step 3. It follows from step 1 and step 2 that
∑
tn
i
6t
E(M2i,n|Ftni−1) =
∑
tn
i
6t
(
E(N2i,n|Ftni−1)− (E(Ni,n|Ftni−1))2
)
P−→ 9
20
√
2pi
Ft(K).
(3.14)
Step 4. Since
E(Mi,n(Btni −Btni−1)|Ftni−1) = −
1
(∆n)3/4
∫ tni
tni−1
E((Bs −Btni−1)I{Bs>K}|Ftni−1)ds.
From Markov’s property, we get
E
(∑
tni 6t
∣∣E(Mi,n(Btni −Btni−1)|Ftni−1)∣∣) 6 1(∆n)3/4
∑
tni 6t
∫ ∆ni
0
√
z√
2pi
E exp (−Bt
n
i−1
−K)2
2z
)dz
6
1
(∆n)3/4
∫ ∆n1
0
√
z√
2pi
dz +
1
(∆n)3/4
∑
tni 6t, i>2
∫ ∆ni
0
dz
∫ +∞
−∞
√
z
2pi
√
tni−1
exp (−(x−K)
2
2z
)dx
6 (∆n)
3/4 +
1
(∆n)3/4
∆ni
∑
tni 6t, i>2
∆ni√
tni−1
6 (∆n)
3/4 + (∆n)
1/4
√
t.
Therefore, ∑
tni 6t
E(Mi,n(Btni −Btni−1)|Ftni−1)
P−→ 0. (3.15)
Step 5. We have
∑
tni 6t
E(M4i,n|Ftni−1) 6 16
∑
tni 6t
E(N4i,n|Ftni−1). Moreover, Markov
property yields
∑
tni 6t
E(M4i,n|Ftni−1) 6
16
(∆n)3
∑
tni 6t
E
(
∆ni I{r>0} −
∫ ∆ni
0
I{Bs>−r}ds
)4
|r=Btn
i−1
−K
6
16
(∆n)3
∑
tni 6t
E
(∫ ∆ni
0
I{Bs>r}ds
)4
|r=|Btn
i−1
−K| 6 16
∑
tni 6t
∫ ∆ni
0
P(Bs > r)ds|r=|Btn
i−1
−K|.
13
Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh
Hence
E|
∑
tn
i
6t
E(M4i,n|Ftni−1)| 6 16
∑
tn
i
6t
∫ ∆ni
0
E
(
1− Φ( |Bt
n
i−1
−K|√
s
)
)
ds ≤ 16∆n + 128
3
√
∆n
√
t.
Therefore, ∑
tni 6t
E(M4i,n|Ftni−1)
P−→ 0 khi n→∞. (3.16)
Step 6. We have that Mn = (Mi,n,Fni ) is a martingale and under probability measure
P, any martingale with respect to Ft orthogonal to B is constant. Hence, from (3.14),
(3.15), (3.16) and applying Theorem IX.7.28 [5] we obtain thatMn converges stably to a
continuous function defined on an extension of original probability space. In particular,
(Mn) is C-tight under probability measure P.
3.3. Proof of Theorem 2.2
(i) Since Γ˜(K)nt =
∫ t
0
Φ
( Bηn(s) −K√
s− ηn(s)
)
ds =
∑
i>1
E
( ∫ tni ∧t
tni−1∧t
I{Bs>K}ds|Btni−1
)
, we
have E(Γ˜(K)nt ) =
∫ t
0
I{Bs>K}ds = Γ([K,+∞))t. Thus Γ˜(K)nt is an unbiased estimator
of Γ([K,+∞))t.
(ii) Moreover, we have
Γ([K,+∞))t − Γ˜(K)nt =
∑
tni 6t
(
∆ni I{Btn
i−1
>K} − E(
∫ tni