# Weak Approximation of SDEs by Discrete-Time Processes

Authors: Henryk Zähle
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008-02-27)

---

## Abstract

We consider the martingale problem related to the solution of an SDE on the line. It is shown that the solution of this martingale problem can be approximated by solutions of the corresponding time-discrete martingale problems under some conditions. This criterion is especially expedient for establishing the convergence of population processes to SDEs. We also show that the criterion yields a weak Euler scheme approximation of SDEs under fairly weak assumptions on the driving force of the approximating processes.


---

## Body

## 1. Introduction

It is well known that a rescaled version of the
classical Galton-Watson process (GWP) with offspring varianceσ
2 weakly
converges to the unique solution of the following one-dimensional stochastic
differential equation (SDE):(1.1)
d
X
t
=
σ
|
X
t
|
d
W
t
,where W is a
one-dimensional Brownian motion (cf. [1]). One might ask whether it is possible to approximate
more general SDEs, driven by a Brownian motion, by generalized GWPs. In
[2] it will be shown
that this is actually possible. In fact, in [2] the solution of the SDE,(1.2)
d
X
t
=
δ
(
t
,
X
t
)
d
t
+
σ
(
t
,
X
t
)
|
X
t
|
d
W
t
,is weakly approximated by two
different types of population-size-dependent GWPs (in the sense of
[3–6]) with immigration, where δ and σ are suitable
nonnegative continuous functions on ℝ
+
×
ℝ. Here the methods of [1] do not apply anymore (cf.
Section 3). In the present article, we establish a general criterion for the
weak approximation of SDEs by discrete-time processes, which is the crux of the
analysis of [2].To be exact, we focus on the following one-dimensional
SDE:(1.3)
d
X
t
=
b
(
t
,
X
t
)
d
t
+
a
(
t
,
X
t
)
d
W
t
,

X
0
=
x
0
,where x
0
∈
ℝ and W is a
one-dimensional Brownian motion. The coefficients a and b are continuous
functions on ℝ
+
×
ℝ satisfying(1.4)
|
a
(
t
,
x
)
|
+
|
b
(
t
,
x
)
|
≤
K
(
1
+
|
x
|
)

∀
t
∈
ℝ
+
,
x
∈
ℝ,for some finite constant (1.4)
K
>
0. We assume that SDE () has a weak solution. It means that there exists a triplet (1.4)
{
X
;
W
;
(
Ω
,
F
,
(
F
t
)
,
ℙ
)
} where (1.4)
(
Ω
,
F
,
(
F
t
)
,
ℙ
) is a filtered
probability space with (1.4)
(
F
t
) satisfying the
usual conditions, (1.4)
W
=
(
W
t
:
t
≥
0
) is an (1.4)
(
F
t
)-Brownian
motion, and (1.4)
X
=
(
X
t
:
t
≥
0
) is a
real-valued continuous (1.4)
(
F
t
)-adapted
process such that (1.4)
ℙ-almost surely,(1.4)X
t
=
x
0
+
∫
0
t
b
(
r
,
X
r
)
d
r
+
∫
0
t
a
(
r
,
X
r
)
d
W
r

∀
t
≥
0.Here the latter is an Ito
^-integral. Moreover, we require the solution to be
weakly unique, which means that any two solutions coincide in law. For
instance, the existence of a unique weak solution is implied by Lipschitz
continuity of b in x (uniformly in t)
and(1.6)|
a
(
t
,
x
)
−
a
(
t
,
x
′
)
|
≤
h
(
|
x
−
x
′
|
)

∀
t
∈
ℝ
+
,
x
,
x
′
∈
ℝ,for some strictly increasing (1.6)
h
:
ℝ
+
→
ℝ
+ with (1.6)
∫
0
0
+
h
−
2
(
u
)
d
u
=
∞. Note that () and Lipschitz continuity of (1.6)
b even imply the
existence of a strongly unique strong solution (Yamada-Watanabe criterion
[]). But the notion
of strong solutions and strong uniqueness is beyond our interest.Our starting point is the fact that any weak solution
of () is a solution of the following martingale problem and vice versa (cf.
[, Section 5.4.B],
or   [, Theorem 1.27]).Definition
A tuple (1.6){
X
;
(
Ω
,
F
,
(
F
t
)
,
ℙ
)
} is said to be a
solution of the ((1.6)
a, (1.6)
b, (1.6)
x
0)-martingale
problem if (1.6)
(
Ω
,
F
,
(
F
t
)
,
ℙ
) is a filtered
probability space with (1.6)
(
F
t
) satisfying the
usual conditions, and (1.6)
X
=
(
X
t
:
t
≥
0
) is a
real-valued continuous (1.6)
(
F
t
)-adapted
process such that(1.6)
M
t
=
X
t
−
x
0
−
∫
0
t
b
(
r
,
X
r
)
d
rprovides a (continuous,
mean-zero) square-integrable (
ℱ
t
)-martingale
with compensator(1.8)
〈
M
〉
t
=
∫
0
t
a
2
(
r
,
X
r
)
d
r
.The solution is said to be
unique if any two solutions coincide in law.In
view of the weak equivalence of the SDE to the martingale problem,
discrete-time processes solving the discrete analogue (Definition2.1) of the (a
,
b
,
x
0)-martingale
problem should approximate weakly the unique solution of SDE (1.3).
Theorem 2.2
below shows that this is true under an additional assumption on the moments of
the increments (condition (2.3)).Note that the characterization of discrete or
continuous population processes as solutions of martingale problems of the form
(1.7)-(1.8), (2.1)-(2.2), respectively, is fairly useful and also common (see, e.g.,
[10–12]). Especially for real-valued discrete-time processes
these characterizations are often easy to see, so that, according to the
criterion, the only thing to check is condition (2.3). Also note that the conditions
of the famous criterion of Stroock and Varadhan for the weak convergence of
Markov chains to SDEs [13, Theorem 11.2.3] are different. In particular, in our
framework we do not insist on the Markov property of the approximating
processes (cf. the discussion at the end of Section 4). Another alternative
approach to the discrete-time approximation of SDEs can be found in the seminal
paper [14], see also
references therein. In [14] general conditions are given, under which the
convergence in distribution (
Y
α
,
Z
α
)
→
(
Y
,
Z
) in the cádlàg
space implies convergence in distribution ∫
Y
α
d
Z
α
→
∫
Y
d
Z of the
corresponding stochastic integrals in the
cádlàg space.In Section3 we will demonstrate that the criterion of
Theorem 2.2   yields an easy proof of the convergence result discussed at the
beginning of the Introduction. Moreover, in Section 4 we will apply our
criterion to obtain a weak Euler scheme approximation of SDEs under fairly weak
assumptions on the driving force of the approximating processes.


## 2. Main Result

We will regard discrete-time processes as
continuous-time cádlàg processes. For this reason we denote byD
(
ℝ
) the space
of cádlàg functions from ℝ
+ to ℝ. We equip D
(
ℝ
) with the
topology generated by the Skohorod convergence on compacts and consider it as a
measurable space with respect to its Borel σ-algebra.
Moreover, we set t
n
ϵ
=
n
ϵ
for every n
∈
ℕ
0 and ϵ
>
0.For everyα
∈
ℕ we fix some
ϵ
α
>
0 such that ϵ
α
→
0. For the sake of clarity, we also set t
n
α
=
t
n
ϵ
α
(
=
n
ϵ
α
) for all n
∈
ℕ
0. Now suppose that a
α and b
α are measurable
functions on ℝ
+
×
ℝ such that ∥
a
−
a
α
∥
∞ and ∥
b
−
b
α
∥
∞ converge to 0 as α
→
∞, where ∥
⋅
∥
∞ is the usual
supremum norm. Let (
x
α
)
⊂
ℝ satisfy x
α
→
x
0, and suppose that X
α is a solution
of the following (
ϵ
α
,
a
α
,
b
α
,
x
α
)-martingale
problem for every α
≥
1. Here we write n
α
(
t
) for the largest n
∈
ℕ
0 with t
n
α
≤
t.Definition 2.1.
Suppose thatX
α
=
(
X
t
α
:
t
≥
0
) is a
real-valued process on some probability space (
Ω
,
ℱ
,
ℙ
) whose
trajectories are constant on the intervals [
t
n
α
,
t
n
+
1
α
), n
∈
ℕ
0. Then X
α is called a
solution of the (
ϵ
α
,
a
α
,
b
α
,
x
α
)-martingale
problem if(2.1)
M
t
α
=
X
t
α
−
x
α
−
∑
i
=
0
n
α
(
t
)
−
1
b
α
(
t
i
α
,
X
t
i
α
α
)
ϵ
αprovides a (zero-mean)
square-integrable martingale (with respect to the natural filtration) with
compensator(2.2)
〈
M
α
〉
t
=
∑
i
=
0
n
α
(
t
)
−
1
a
α
2
(
t
i
α
,
X
t
i
α
α
)
ϵ
α
.TheX
α could be
defined on different probability spaces (
Ω
α
,
ℱ
α
,
ℙ
α
). However, we assume without loss of generality that Ω
α
=
D
(
ℝ
), ℱ
α
=
ℬ
(
D
(
ℝ
)
)
, and X
α is the
coordinate process of ℙ
α (each cádlàg
process induces a corresponding law on D
(
ℝ
)). We further
assume that there are some q
>
2 and δ
>
1 such that(2.3)
E
α
[
|
X
t
n
α
α
−
X
t
n
−
1
α
α
|
q
]
≤
C
T
(
1
+
E
α
[
|
X
t
n
−
1
α
α
|
q
]
)
ϵ
α
δfor every α
≥
1 and n
∈
ℕ with t
n
α
≤
T, where C
T
>
0 is some finite
constant that may depend on T. (By an induction on n, (2.3) implies immediately that 𝔼
α
[
|
X
t
n
α
α
|
q
]
<
∞ for all α and n. Lemma 5.1 will provide an even stronger statement.)
The following theorem shows that X
α converges in
distribution to the unique solution of (1.3).Theorem 2.2.
Suppose SDE (1.3) subject to (1.4) has a unique weak
solution, and denote by ℙ the
corresponding law on D
(
ℝ
). Moreover, let ℙ
α be the law (on D
(
ℝ
)) of X
α subject to
(2.1)–(2.3). Then ℙ
α
⇒
ℙ as α
→
∞.Here,⇒ symbolizes weak
convergence. The proof of Theorem 2.2 will be carried out in
Section 5. The
finiteness of the qth moments for
some q
>
2 is not always
necessary, it is true. From time to time the finiteness of the second moments
is sufficient. However, for a general statement involving convenient moment
conditions as (2.3), a weakening of q
>
2 to q
=
2 is hardly
possible. The assumption q
>
2 is common in
the theory of functional, time-discrete approximations of SDEs, SDDEs, and
SPDEs (see, e.g., [12, 15]).


## 3. Example 1: Convergence of Rescaled GWP to (1.1)

As a first application of Theorem2.2, we show that a
rescaled GWP weakly converges to Feller's branching diffusion [16], that is, to the solution
of SDE (1.1). Lindvall [1]
showed this approximation via the convergence of the finite-dimensional
distributions, for which the shape of the Laplace transforms of the transition
probabilities is essential. Here, we will exploit the martingale property of
the Galton-Watson process (with offspring variance σ
2). The latter
is an ℕ
0-valued Markov
process Z
=
(
Z
n
:
n
∈
ℕ
0
) that can be
defined recursively as follows. Choose an initial state Z
0
∈
ℕ and set Z
n
=
∑
i
=
1
Z
n
−
1
N
n
−
1,
i for all n
≥
1, where {
N
n
,
i
:
n
≥
0
,
i
≥
1
} is a family of
i.i.d. ℕ
0-valued random
variables with mean 1 and variance σ
2. In addition, we require that the fourth moment of N
1,1 is finite.
Thereby Z
n has a finite
fourth moment for every n
∈
ℕ
0. Actually, in [1] the finiteness of the fourth moments was not
required. On the other hand, the methods used there break down when considering
a population-size-dependent branching intensity or an additional general
immigration into the system. In contrast, the procedure below still works in
those cases (cf. [2]).SettingZ
t
n
ϵ
ϵ
=
ϵ
Z
n we obtain a
rescaled version, Z
ϵ, of Z. Recall t
n
ϵ
=
n
ϵ, hence Z
ϵ is a process
having ϵ
ℕ
0
=
{
0
,
ϵ
,2
ϵ
,
…
} as both its
index set and its state space. Now pick (
ϵ
α
)
⊂
ℝ
+ such that ϵ
α
→
0, and recall our convention t
n
α
=
t
n
ϵ
α and that ⌊
t
⌋
ϵ denotes the largest element s of ϵ
ℕ
0 with s
≤
t. Regard the process Z
ϵ
α as
continuous-time process, X
α, by setting X
t
α
=
Z
⌊
t
⌋
ϵ
α
ϵ
α, and suppose that X
0
α
=
⌊
x
0
⌋
ϵ
α. The latter requires that Z
0 actually
depends on α. The domain of X
α is denoted by (
Ω
α
,
ℱ
α
,
ℙ
α
). It is easy to see that M
α defined in (2.1)
provides a (zero-mean) square-integrable martingale. Moreover, the compensator
of M
α is given by 〈
M
α
〉
t
=
σ
2
∑
i
=
0
n
α
(
t
)
−
1
X
t
i
α
α
ϵ
α since, in this
case,(3.1)
E
α
[
(
(
M
t
n
α
α
)
2
−
〈
M
α
〉
t
n
α
)
−
(
(
M
t
n
−
1
α
α
)
2
−
〈
M
α
〉
t
n
−
1
α
)
∣
F
t
n
−
1
α
X
α
]
=
0can be checked easily with help
of(3.2)
E
α
[
X
t
n
α
α
∣
X
t
n
−
1
α
α
]
=
X
t
n
−
1
α
α
,


V
a
r
α
[
X
t
n
α
α
∣
X
t
n
−
1
α
α
]
=
σ
2
X
t
n
−
1
α
α
ϵ
α
.The formulae in (3.2) are
immediate consequences of the well-known moment formulae for Z (see [17, page 6]) and (
ℱ
t
n
α
X
α
) denotes the
natural filtration induced by X
α. Hence, X
α solves the (
ϵ
α
,
a
,
b
,
x
α
)-martingale
problem of Definition 2.1 with a
(
t
,
x
)
=
|
x
|, b
≡
0 and x
α
=
⌊
x
0
⌋
ϵ
α. It remains to show (2.3). To
this end we state the following lemma.Lemma 3.1.
Assume thatξ
1
,
ξ
2
,
… are independent
random variables on some probability space (
Ω
,
ℱ
,
ℙ
) with 𝔼
[
ξ
i
]
=
0 and sup
i
∈
ℕ
𝔼
[
ξ
i
4
]
<
∞. Let ν be a further
random variable on (
Ω
,
ℱ
,
ℙ
) being
independent of (
ξ
i
), taking values in ℕ and satisfying 𝔼
[
ν
4
]
<
∞. Then there is some finite constant C
>
0, depending only on the second and the fourth moments
of the ξ
i, such that 𝔼
[
(
∑
i
=
1
ν
ξ
i
)
4
]
≤
C
𝔼
[
ν
2
].Proof.
By the
finiteness of the fourth moments the law of total expectation yields(3.3)
E
[
(
∑
i
=
1
ν
ξ
i
)
4
]
=
∑
n
∈
ℕ
∑
i
1
=
1
n
∑
i
2
=
1
n
∑
i
3
=
1
n
∑
i
4
=
1
n
E
[
ξ
i
1
ξ
i
2
ξ
i
3
ξ
i
4
]
ℙ
[
ν
=
n
]
.Since the ξ
i are independent
and centered, the summand on the right-hand side might differ from 0 only if either i
1
=
i
2
=
i
3
=
i
4, or i
1
=
i
2 and i
3
=
i
4
≠
i
1, or i
1
=
i
3 and i
2
=
i
4
≠
i
1, or i
1
=
i
4 and i
2
=
i
3
≠
i
1. Hence,(3.4)
E
[
(
∑
i
=
1
ν
ξ
i
)
4
]
≤
∑
n
∈
ℕ
{
(
n
+
3
n
(
n
−
1
)
)
sup
⁡
i
,
j
∈
ℕ
E
[
ξ
i
2
ξ
j
2
]
}
ℙ
[
ν
=
n
]
≤
4
sup
⁡
i
,
j
∈
ℕ
E
[
ξ
i
2
ξ
j
2
]
E
[
ν
2
]
.This yields the claim of the
lemma with C
=
4
sup
i
,
j
∈
ℕ
𝔼
[
ξ
i
2
ξ
j
2
].With help of Lemma3.1
we obtain(3.5)
E
α
[
|
X
t
n
α
α
−
X
t
n
−
1
α
α
|
4
]
=
E
α
[
|
∑
i
=
1
ϵ
α
−
1
X
t
n
−
1
α
α
(
ϵ
α
N
n
−
1,
i
−
ϵ
α
)
|
4
]
=
E
α
[
|
∑
i
=
1
ϵ
α
−
1
X
t
n
−
1
α
α
(
N
n
−
1,
i
−
1
)
|
4
]
ϵ
α
4
≤
C
E
α
[
(
ϵ
α
−
1
X
t
n
−
1
α
α
)
2
]
ϵ
α
4
≤
C
(
1
+
E
α
[
(
X
t
n
−
1
α
α
)
4
]
)
ϵ
α
2for some suitable constant C
>
0. This shows that (2.3) holds too. Hence, the
assumptions of Theorem 2.2 are fulfilled, and the theorem implies that X
α converges in
distribution to the unique solution of (1.1).


## 4. Example 2: Weak Euler Scheme Approximation of (1.3)

As a second application of Theorem2.2, we establish a
weak Euler scheme approximation of SDE (1.3). Our assumptions are partially
weaker than the assumptions of classical results on weak functional Euler
scheme approximations. A standard reference for Euler schemes is the monograph
[18]; see also
references therein. As before we suppose that a and b are continuous
functions on ℝ
+
×
ℝ satisfying
(1.4), and that SDE (1.3) possesses a unique weak
solution. Now let ϵ
>
0, recall the notation introduced in Section 2, and
consider the following stochastic difference equation (weak Euler
scheme):(4.1)
X
t
n
ϵ
ϵ
−
X
t
n
−
1
ϵ
ϵ
=
b
(
t
n
−
1
ϵ
,
X
t
n
−
1
ϵ
ϵ
)
ϵ
+
a
(
t
n
−
1
ϵ
,
X
t
n
−
1
ϵ
ϵ
)
V
t
n
ϵ
ϵ
,

X
t
0
ϵ
ϵ
=
x
ϵ
.Here, (
x
ϵ
) is a sequence
in ℝ satisfying x
ϵ
→
x
0 as ϵ
→
0, and V
ϵ
=
{
V
t
n
ϵ
ϵ
:
n
∈
ℕ
} is a family of
independent centered random variables with variance ϵ and 𝔼
ϵ
[
|
V
t
n
ϵ
ϵ
|
q
]
≤
C
ϵ
q
/
2 for all n
∈
ℕ, ϵ
∈
(
0
,
1
], some q
>
2, and some finite constant C
>
0, where (
Ω
ϵ
,
ℱ
ϵ
,
ℙ
ϵ
) denotes the
domain of V
ϵ. For instance, one may set V
t
n
ϵ
ϵ
=
ϵ
ξ
n where {
ξ
n
:
n
∈
ℕ
} is a family of independent
centered random variables with variance 1 and the qth moment being
bounded uniformly in n. Note that we do not require that the random
variables {
V
t
n
ϵ
ϵ
:
n
∈
ℕ
} are identically
distributed. Below we will see that the independence is necessary neither.By virtue of (1.4), X
t
n
ϵ
ϵ has a finite qth moment if X
t
n
−
1
ϵ
ϵ has. It follows
by induction that the solution X
ϵ
=
(
X
t
n
ϵ
ϵ
:
n
∈
ℕ
0
) of (4.1) is q-integrable,
and hence square integrable. Equation (4.1) is obviously equivalent to the
stochastic sum equation(4.2)
X
t
n
ϵ
ϵ
=
x
ϵ
+
∑
i
=
0
n
−
1
b
(
t
i
ϵ
,
X
t
i
ϵ
ϵ
)
ϵ
+
∑
i
=
0
n
−
1
a
(
t
i
ϵ
,
X
t
i
ϵ
ϵ
)
V
t
i
+
1
ϵ
ϵ
.Suppose that (
ϵ
α
) is an arbitrary
sequence with ϵ
α
∈
(
0
,
1
] and ϵ
α
→
0, set x
α
=
x
ϵ
α and recall our
convention 𝔼
α
=
𝔼
ϵ
α, X
α
≡
X
ϵ
α, t
n
α
=
t
n
ϵ
α. Then it is easy to see that M
α defined in
(2.1) provides a (mean-zero) square-integrable (
ℱ
t
X
α
)-martingale.
Moreover, M
t
n
α
α coincides with
the second sum on the right-hand side of (4.2). Therefore, we also
obtain(4.3)
〈
M
α
〉
t
n
α
=
∑
i
=
1
n
E
α
[
(
a
(
t
i
−
1
α
,
X
t
i
−
1
α
α
)
V
t
i
α
ϵ
α
)
2
∣
F
t
n
−
1
α
X
α
]
=
∑
i
=
0
n
−
1
a
2
(
t
i
α
,
X
t
i
α
α
)
E
α
[
(
V
t
i
+
1
α
ϵ
α
)
2
]
=
∑
i
=
0
n
−
1
a
2
(
t
i
α
,
X
t
i
α
α
)
ϵ
αwhich shows that X
α solves the (
ϵ
α
,
a
,
b
,
x
α
)-martingale
problem of Definition 2.1. For an application of Theorem 2.2 it thus remains to
show (2.3). But (2.3) follows from(4.4)
E
α
[
|
X
t
n
α
α
−
X
t
n
−
1
α
α
|
q
]


≤
2
q
−
1
{
E
α
[
|
b
(
t
n
−
1
α
,
X
t
n
−
1
α
α
)
ϵ
α
|
q
]
+
E
α
[
|
a
(
t
n
−
1
α
,
X
t
n
−
1
α
α
)
|
q
]
E
α
[
|
V
t
n
α
ϵ
α
|
q
]
}


≤
2
q
−
1
{
K
2
q
−
1
(
1
+
E
α
[
|
X
t
n
−
1
α
α
|
q
]
)
ϵ
α
q
+
K
2
q
−
1
(
1
+
E
α
[
|
X
t
n
−
1
α
α
|
q
]
)
C
ϵ
α
q
/
2
}for which we used (4.1), the
independence of X
t
n
−
1
α
α of V
ϵ
α, (1.4), and 𝔼
α
[
|
V
t
n
α
ϵ
α
|
q
]
≤
C
ϵ
α
q
/
2. Hence, Theorem 2.2 ensures that X
α converges in
distribution to the unique solution of SDE (1.3).As mentioned above, the independence of the random variables{
V
t
n
ϵ
ϵ
:
n
∈
ℕ
} is not
necessary. The independence was used for (4.3), (4.4), and the martingale
property of M
α. But these relations may be valid even if the V
t
n
ϵ
ϵ are not
independent. For instance, let {
ξ
n
(
i
)
:
n
,
i
∈
ℕ
} be an array of
independent centered random variables with variance 1 and qth moments
being bounded above by some C
>
0 uniformly in n
,
i, for some q
>
2. Then the martingale property of M
α and the main
statements of (4.3) and (4.4) remain true for V
t
1
ϵ
ϵ
=
ϵ
ξ
1
(
1
) and V
t
n
ϵ
ϵ
=
ϵ
ξ
n
(
f
n
(
V
t
1
ϵ
ϵ
,
…
,
V
t
n
−
1
ϵ
ϵ
)
), n
≥
2, where f
n is any
measurable mapping from ℝ
n
−
1 to ℕ. This follows from the following relations which can
be shown easily with help of the functional representation theorem for
conditional expectations respectively by conditioning(4.5)
E
α
[
V
t
n
α
ϵ
α
∣
F
t
n
−
1
α
X
α
]
=
0
,

E
α
[
(
V
t
i
+
1
α
ϵ
α
)
2
∣
F
t
n
−
1
α
X
α
]
=
ϵ
α
,

1
≤
i
≤
n
−
1
,
E
α
[
|
a
(
t
n
−
1
α
,
X
t
n
−
1
α
α
)
V
t
n
α
ϵ
α
|
q
]
≤
C
ϵ
α
q
/
2
.If the ξ
n
(
i
) are not
identically distributed, then the V
t
n
ϵ
ϵ are typically
not independent. In particular, the approximating process X
ϵ may be non-Markovian.


## 5. Proof of Theorem2.2

Theorem2.2 is an immediate consequence of
Propositions 5.2, 5.5, and the weak equivalence of the
martingale problem to the SDE. For the proofs of the two propositions we note
that there exist K
′
>
0 and α
0
≥
1 such that for
all α
≥
α
0, t
≥
0, and x
∈
ℝ,(5.1)
|
a
α
(
t
,
x
)
|
+
|
b
α
(
t
,
x
)
|
≤
K
′
(
1
+
|
x
|
)
.This is true since we assumed
(1.4) and uniform convergence of a
α and b
α to the
coefficients a and b, respectively. Throughout this section we will
frequently use the well-known inequality |
∑
i
=
1
m
y
i
|
p
≤
m
p
−
1
∑
i
=
1
m
|
y
i
|
p for all m
∈
ℕ,    p
≥
1 and y
1
,
…
,
y
m
∈
ℝ. As a first consequence of (5.1) we obtain
Lemma 5.1.
For every x
∈
ℝ
+ we write ⌊
x
⌋
ϵ for the largest element of ϵ
ℕ
0
=
{
0
,
ϵ
,2
ϵ
,
…
} which is
smaller than or equal to x. Moreover, we assume without loss of generality that ϵ
α
≤
1.Lemma 5.1.
Forq
>
2 and δ
>
1 satisfying (2.3)
and every T
>
0,(5.2)
sup
⁡
α
≥
α
0
E
α
[
sup
⁡
t
≤
T
|
X
t
α
|
q
]
<
∞
.Proof.
First of all, note that for the
proof it actually suffices to requireq
≥
2 and δ
≥
1. Set S
=
sup
α
≥
α
0
|
x
α
|
q and S
t
α
=
𝔼
α
[
max
1
≤
i
≤
n
α
(
t
)
|
M
t
i
α
α
−
M
t
i
−
1
α
α
|
q
]. Using
Proposition A.1 in the appendix and (5.1) we obtain, for all t
>
0 and α
≥
α
0,(5.3)
E
α
[
sup
⁡
i
≤
n
α
(
t
)
|
X
t
i
α
|
q
]


≤
3
q
−
1
{
E
α
[
sup
⁡
i
≤
n
α
(
t
)
|
M
t
i
α
|
q
]
+
S
+
E
α
[
(
∑
i
=
0
n
α
(
t
)
−
1
|
b
α
(
t
i
α
,
X
t
i
α
α
)
|
ϵ
α
)
q
]
}


≤
3
q
−
1
C
q
{
E
α
[
|
∑
i
=
0
n
α
(
t
)
−
1
a
α
2
(
t
i
α
,
X
t
i
α
α
)
ϵ
α
|
q
/
2
]
+
S
t
α
+
S
+
E
α
[
[
∑
i
=
0
n
α
(
t
)
−
1
|
b
α
(
t
i
α
,
X
t
i
α
α
)
|
ϵ
α
]
q
]
}


≤
k
q
{
E
α
[
[
∑
i
=
0
n
α
(
t
)
−
1
(
K
′
(
1
+
|
X
t
i
α
α
|
)
)
2
ϵ
α
]
q
/
2
]
+
S
t
α
+
S
+
E
α
[
[
∑
i
=
0
n
α
(
t
)
−
1
K
′
(
1
+
|
X
t
i
α
α
|
)
ϵ
α
]
q
]
}
,where C
q is independent
of t and α, and k
q
=
3
q
−
1
C
q. By Hölder's inequality we get(5.4)
E
α
[
[
∑
i
=
0
n
α
(
t
)
−
1
(
K
′
(
1
+
|
X
t
i
α
α
|
)
)
2
ϵ
α
]
q
/
2
]


≤
E
α
[
(
∑
i
=
0
n
α
(
t
)
−
1
(
2
K
′
2
(
1
+
|
X
t
i
α
α
|
2
)
)
q
/
2
)
(
∑
i
=
0
n
α
(
t
)
−
1
ϵ
α
(
q
/
2
)
/
(
q
/
2
−
1
)
)
q
/
2
−
1
]


≤
E
α
[
(
∑
i
=
0
n
α
(
t
)
−
1
2
q
/
2
−
1
(
2
K
′
2
)
q
/
2
(
1
+
|
X
t
i
α
α
|
q
)
)
n
α
(
t
)
q
/
2
−
1
ϵ
α
q
/
2
]


≤
c
q
t
q
/
2
+
c
q
t
q
/
2
−
1
∑
i
=
0
n
α
(
t
)
−
1
E
α
[
sup
⁡
j
≤
i
|
X
t
j
α
α
|
q
]
ϵ
α
,where c
q
=
2
q
/
2
−
1
(
2
K
′
2
)
q
/
2. Analogously, with c
¯
q
=
2
q
−
1
K
′
q,(5.5)
E
α
[
[
∑
i
=
0
n
α
(
t
)
−
1
K
′
(
1
+
|
X
t
i
α
α
|
)
ϵ
α
]
q
]
≤
c
¯
q
t
q
+
c
q
t
q
−
1
∑
i
=
0
n
α
(
t
)
−
1
E
α
[
sup
⁡
j
≤
i
|
X
t
j
α
α
|
q
]
ϵ
α
.Moreover, by (2.3) and (5.1) we obtain, for all t
≤
T and α
≥
α
0,(5.6)
S
t
α
≤
∑
i
=
1
n
α
(
t
)
E
α
[
|
M
t
i
α
α
−
M
t
i
−
1
α
α
|
q
]
≤
2
q
−
1
∑
i
=
1
n
α
(
t
)
E
α
[
|
X
t
i
α
α
−
X
t
i
−
1
α
α
|
q
+
|
b
α
(
t
i
−
1
α
,
X
t
i
−
1
α
α
)
|
q
ϵ
α
q
]
≤
2
q
−
1
∑
i
=
0
n
α
(
t
)
−
1
{
C
T
(
1
+
E
α
[
|
X
t
i
α
α
|
q
]
)
ϵ
α
δ
+
E
α
[
K
′
(
1
+
|
X
t
i
α
α
|
q
)
]
ϵ
α
q
}
≤
c
q
,
T
t
+
c
q
,
T
∑
i
=
0
n
α
(
t
)
−
1
E
α
[
sup
⁡
j
≤
i
|
X
t
j
α
α
|
q
]
ϵ
α
,where c
q
,
T
=
2
q
−
1
(
C
T
+
K
′
). By all account we have, for all t
≤
T and α
≥
α
0,(5.7)
E
α
[
sup
⁡
i
≤
n
α
(
t
)
|
X
t
i
α
|
q
]
≤
k
q
{
S
+
(
c
q
+
c
¯
q
+
c
q
,
T
)
(
t
q
−
1
∨
1
)
(
1
+
ϵ
α
∑
i
=
0
n
α
(
t
)
−
1
E
α
[
sup
⁡
j
≤
i
|
X
t
j
α
α
|
q
]
)
}
≤
(
k
q
S
+
C
q
,
T
)
+
C
q
,
T
ϵ
α
∑
i
=
0
n
α
(
t
)
−
1
E
α
[
sup
⁡
j
≤
i
|
X
t
j
α
α
|
q
]
,where C
q
,
T
=
k
q
(
c
q
+
c
¯
q
+
c
q
,
T
)
(
T
q
−
1
∨
1
). An application of Lemma A.2 yields(5.8)
E
α
[
sup
⁡
s
≤
t
|
X
s
α
|
q
]
=
E
α
[
sup
⁡
i
≤
n
α
(
t
)
|
X
t
i
α
|
q
]
≤
(
k
q
S
+
C
q
,
T
)
(
1
+
C
q
,
T
ϵ
α
)
n
α
(
t
)
+
(
C
q
,
T
ϵ
α
)
n
α
(
t
)
S
,where we emphasize that the
constants k
q, S, and C
q
,
T are independent
of t
≤
T and α
≥
α
0. This proves Lemma 5.1 since limsup
α
→
∞
(
1
+
C
q
,
T
ϵ
α
)
n
α
(
t
) is bounded by exp
(
t
C
q
,
T
) (note that n
α
(
t
)
=
⌊
t
/
ϵ
α
⌋
1
≤
t
/
ϵ
α).Proposition 5.2.
If(
ℙ
α
) is tight, then
the coordinate process of any weak limit point, that has no mass outside of C
(
ℝ
), is a solution of the (
a
,
b
,
x
0
)-martingale
problem of Definition 1.1.Proof.
We
consider a weakly convergent subsequence whose limit,ℙ, has no mass outside of C
(
ℝ
). By an abuse of notation, we denote this subsequence
by (
ℙ
α
) either. We
further write X for the
coordinate process of ℙ. Since X is ℙ-almost surely
continuous, we know [19, Theorem 3.7.8] that(5.9)
ℙ
α
∘
π
t
1
,
…
,
t
k
−
1
⟹
ℙ
∘
π
t
1
,
…
,
t
k
−
1for all t
1
,
…
,
t
k
∈
ℝ
+, where π
t
1
,
…
,
t
k
:
D
(
ℝ
)
→
ℝ
k is the usual
coordinate projection. In the remainder of the proof we will show in three
steps that M defined in (1.7)
is square-integrable, provides an (
ℱ
¯
t
X
)-martingale and
has 〈
M
〉 defined in
(1.8) as compensator. Here, (
ℱ
¯
t
X
) denotes the
natural augmentation of the filtration (
ℱ
t
X
) induced by X.
Step 1.
With
help of Fatou's lemma as well as (5.9) and
(5.2) we obtain, for every T
>
0,(5.10)
sup
⁡
t
≤
T
E
[
|
X
t
|
q
]
≤
sup
⁡
t
≤
T
lim
inf
⁡
N
→
∞
lim
⁡
α
→
∞
E
α
[
|
X
t
α
|
q
∧
N
]
≤
sup
⁡
t
≤
T
sup
⁡
α
≥
α
0
E
α
[
|
X
t
α
|
q
]
<
∞
.Taking (1.4) into account we
conclude that M defined in (1.7)
is square-integrable.
Step 2.
We next
show thatM is an (
ℱ
¯
t
X
)-martingale. It
suffices to show that M is an (
ℱ
t
X
)-martingale;
see [20, page 75]. The
latter is true if and only if(5.11)
E
[
(
X
t
+
s
−
X
t
−
∫
t
t
+
s
b
(
r
,
X
r
)
d
r
)
∏
i
=
1
l
h
i
(
X
t
i
)
]
=
0holds for all 0
≤
t
1
<
⋯
≤
t
l
≤
t, s
≥
0, l
≥
1 and bounded h
1
,
…
,
h
l
∈
C
(
ℝ
) (do not confuse t
i and t
i
α). Since X
α solves the (
ϵ
α
,
a
α
,
b
α
,
x
α
)-martingale
problem, we have(5.12)
E
α
[
(
X
t
+
s
α
−
X
t
α
−
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
α
(
t
i
α
,
X
t
i
α
α
)
ϵ
α
)
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
0.We are going to verify (5.11) by
showing that the left-hand side of (5.12) converges to the left-hand side of (5.11)
as α
→
∞. We begin with proving(5.13)
lim
⁡
α
→
∞
E
α
[
X
u
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
E
[
X
u
∏
i
=
1
l
h
i
(
X
t
i
)
]for every u
≥
0, which together with (5.19) below implies the required convergence. To this end we set x
(
N
)
=
(
−
N
∨
x
)
⋀
N for all x
∈
ℝ and N
>
0. The right-hand side of(5.14)
|
E
α
[
X
u
α
,
(
N
)
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
−
E
α
[
X
u
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
|
≤
E
α
[
|
X
u
α
,
(
N
)
−
X
u
α
|
∏
i
=
1
l
∥
h
i
∥
∞
]can be estimated, for every T
≥
u, by(5.15)
sup
⁡
r
≤
T
sup
⁡
α
′
≥
α
0
E
α
′
[
|
X
r
α
′
|
1
|
X
r
α
′
|
>
N
]
∏
i
=
1
l
∥
h
i
∥
∞which tends to 0 as N
→
∞ since {
X
r
α
′
:
r
≤
T
,
α
′
≥
1
} is uniformly
integrable by (5.2). Therefore, we have(5.16)
lim
⁡
N
→
∞
E
α
[
X
u
α
,
(
N
)
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
E
α
[
X
u
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]

uniformly
in
α
≥
α
0(and uniformly in u
≤
T, for every T
>
0). By (5.9) we further obtain for every N
>
0,(5.17)
lim
⁡
α
→
∞
E
α
[
X
u
α
,
(
N
)
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
E
[
X
u
(
N
)
∏
i
=
1
l
h
i
(
X
t
i
)
]since the mapping (
x
1
,
…
,
x
l
+
1
)
↦
x
l
+
1
(
N
)
∏
i
=
1
l
h
i
(
x
i
) from ℝ
l
+
1 to ℝ is bounded and
continuous. This is the reason why we introduced the truncation x
(
N
). By virtue of (5.10), an application of the dominated
convergence theorem gives(5.18)
lim
⁡
N
→
∞
E
[
X
u
(
N
)
∏
i
=
1
l
h
i
(
X
t
i
)
]
=
E
[
X
u
∏
i
=
1
l
h
i
(
X
t
i
)
]which along with (5.16) and (5.17)
implies (5.13). It remains to show(5.19)
lim
⁡
α
→
∞
E
α
[
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
α
(
t
i
α
,
X
t
i
α
α
)
ϵ
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
E
[
∫
t
t
+
s
b
(
r
,
X
r
)
d
r
∏
i
=
1
l
h
i
(
X
t
i
)
]
.Taking (5.1) and (
n
α
(
t
+
s
)
−
n
α
(
t
)
)
ϵ
α
≤
s
+
ϵ
α into account we
obtain, analogously to (5.16) and
(5.18),(5.20)
lim
⁡
N
→
∞
E
α
[
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
α
(
t
i
α
,
X
t
i
α
α
,
(
N
)
)
ϵ
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]


=
E
α
[
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
α
(
t
i
α
,
X
t
i
α
α
)
ϵ
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]

uniformly
in
α
≥
α
0
,respectively,(5.21)
lim
⁡
N
→
∞
E
[
∫
t
t
+
s
b
(
r
,
X
r
(
N
)
)
d
r
∏
i
=
1
l
h
i
(
X
t
i
)
]
=
E
[
∫
t
t
+
s
b
(
r
,
X
r
)
d
r
∏
i
=
1
l
h
i
(
X
t
i
)
]
.By the uniform convergence of b
α to b and (
n
α
(
t
+
s
)
−
n
α
(
t
)
)
ϵ
α
≤
s
+
ϵ
α, we also have(5.22)
E
α
[
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
α
(
t
i
α
,
X
t
i
α
α
,
(
N
)
)
ϵ
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
E
α
[
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
(
t
i
α
,
X
t
i
α
α
,
(
N
)
)
ϵ
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
+
o
α
(
1
)
.Moreover, we
have(5.23)
E
α
[
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
(
t
i
α
,
X
t
i
α
α
,
(
N
)
)
ϵ
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
E
α
[
∫
t
t
+
s
b
(
r
,
X
r
α
,
(
N
)
)
d
r
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
+
o
α
(
1
)which is a consequence of the
dominated convergence theorem and(5.24)
|
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
(
t
i
α
,
X
t
i
α
α
,
(
N
)
)
ϵ
α
−
∫
t
t
+
s
b
(
r
,
X
r
α
,
(
N
)
)
d
r
|


≤
∫
⌊
t
⌋
ϵ
α
⌊
t
+
s
⌋
ϵ
α
−
ϵ
α
|
b
(
⌊
r
⌋
ϵ
α
,
X
r
α
,
(
N
)
)
−
b
(
r
,
X
r
α
,
(
N
)
)
|
d
r
+
o
α
(
1
)together with the fact that b is bounded and
uniformly continuous on [
0
,
t
+
s
]
×
[
−
N
,
N
]. Finally, we get by (5.9) and the dominated convergence theorem and (5.2)
(5.25)
lim
⁡
α
→
∞
E
α
[
∫
t
t
+
s
b
(
r
,
X
r
α
,
(
N
)
)
d
r
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
∫
t
t
+
s
lim
⁡
α
→
∞
E
α
[
b
(
r
,
X
r
α
,
(
N
)
)
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
d
r
=
∫
t
t
+
s
E
[
b
(
r
,
X
r
(
N
)
)
∏
i
=
1
l
h
i
(
X
t
i
)
]
d
r
=
E
[
∫
t
t
+
s
b
(
r
,
X
r
(
N
)
)
d
r
∏
i
=
1
l
h
i
(
X
t
i
)
]which along with (5.22) and (5.23)
implies(5.26)
lim
⁡
α
→
∞
E
α
[
∑
i
=
n
α
(
t
)
n
α
(
t
+
s
)
−
1
b
α
(
t
i
α
,
X
t
i
α
α
,
(
N
)
)
ϵ
α
∏
i
=
1
l
h
i
(
X
t
i
α
)
]
=
E
[
∫
t
t
+
s
b
(
r
,
X
r
(
N
)
)
d
r
∏
i
=
1
l
h
i
(
X
t
i
)
]
.This, (5.20), and (5.21) ensure
(5.19).
Step 3.
It
remains to show (1.8). By the uniqueness of the Doob-Meyer
decomposition, M has the
required compensator if and only if(5.27)
E
[
(
M
t
+
s
2
−
M
t
2
−
∫
t
t
+
s
a
2
(
r
,
X
r
)
d
r
)
∏
i
=
1
l
h
i
(
X
t
i
)
]
=
0holds for all 0
≤
t
1
<
⋯
≤
t
l
≤
t, s
≥
0, l
≥
1 and bounded h
1
,
…
,
h
l
∈
C
(
ℝ
). Now, the discrete analogue of  (5.27) for 𝔼
α, a
α, and X
α holds.
Proceeding similarly to the proof of (5.11) one can show that the left-hand side
of this equation converges to the left-hand side of (5.27) as α
→
∞. Therefore, we obtain (5.27). For the sake of brevity
we omit the details. It should be mentioned, however, that we now need uniform
integrability of {
(
X
r
α
)
2
:
r
≤
t
+
s
,
α
≥
1
}. This is why we established (5.2) for q being strictly
larger than 2.The assumptions of
Proposition5.2 can be checked with help of the following two
lemmas, where ℚ
α and ℚ refer to any
laws on D
(
ℝ
), and Y
α and Y are the
respective coordinate processes. By an abuse of notation, we denote the
corresponding expectations by ℚ
α and ℚ either. The
first lemma follows from [19, Theorem 3.8.8]
and [19, Theorem 3.8.6(b) ⇒ (a)] along with
Prohorov's theorem. Lemma 5.4 is more or less standard and can be proved with
help of the continuity criterion 3.10.3 in
[19]; we omit the details.Lemma 5.3.
Assume
that(
Y
t
α
) is tight in ℝ for every
rational t
≥
0. Let m
>
0, γ
>
1, and assume for every T
>
0 that there is
some finite constant C
T
>
0 such that for
all α
≥
1 and t
,
h
≥
0 with 0
≤
t
−
h and t
+
h
≤
T,(5.28)
ℚ
α
[
|
Y
t
−
h
α
−
Y
t
α
|
m
/
2
|
Y
t
α
−
Y
t
+
h
α
|
m
/
2
]
≤
C
T
h
γ
.Then (
ℚ
α
) is tight.Lemma 5.4.
Letm
>
0, γ
>
1, and assume for every T
>
0 that there is
some finite constant C
T
>
0 such that for
all α
≥
1 and 0
≤
t
′
≤
t
′
′
≤
T,(5.29)
lim sup
⁡
α
→
∞
ℚ
α
[
|
Y
t
′
α
−
Y
t
′
′
α
|
m
]
≤
C
T
(
t
′
′
−
t
′
)
γ
.Then if ℚ
α
⇒
ℚ, the limit ℚ has no mass
outside of C
(
ℝ
).Proposition 5.5.
(
ℙ
α
) is tight and
each limit point has no mass outside of C
(
ℝ
).Proof.
Letq
>
2 and δ
>
1 satisfy (2.3).
Using techniques as in the proof of Lemma 5.1 we can find a finite constant C
q
>
0 such that for
every 0
≤
t
′
≤
t
′
′ and α
≥
α
0,(5.30)
E
α
[
|
X
t
′
α
−
X
t
′
′
α
|
q
]
≤
C
q
{
ϵ
α
q
/
2
E
α
[
(
∑
i
=
n
α
(
t
′
)
n
α
(
t
′
′
)
−
1
(
1
+
|
X
t
i
α
α
|
)
2
)
q
/
2
]
+
ϵ
α
q
E
α
[
(
∑
i
=
n
α
(
t
′
)
n
α
(
t
′
′
)
−
1
(
1
+
|
X
t
i
α
α
|
)
)
q
]


⁢
+
∑
i
=
n
α
(
t
′
)
n
α
(
t
′
′
)
−
1
(
1
+
E
α
[
|
X
t
i
α
α
|
q
]
)
ϵ
α
δ
∧
q
}
.Applying Hölder's inequality to
each of the first two summands on the right-hand side, using (5.2) and setting γ
=
(
q
/
2
)
⋀
δ, we may continue with(5.31)
≤
C
q
{
ϵ
α
q
/
2
(
∑
i
=
n
α
(
t
′
)
n
α
(
t
′
′
)
−
1
E
α
[
(
1
+
|
X
t
i
α
α
|
)
2
q
/
2
]
)
(
∑
i
=
n
α
(
t
′
)
n
α
(
t
′
′
)
−
1
1
q
/
(
q
−
2
)
)
q
/
2
−
1



+
ϵ
α
q
(
∑
i
=
n
α
(
t
′
)
n
α
(
t
′
′
)
−
1
E
α
[
(
1
+
|
X
t
i
α
α
|
)
q
]
)
(
∑
i
=
n
α
(
t
′
)
n
α
(
t
′
′
)
−
1
1
q
/
(
q
−
1
)
)
q
−
1
+
∑
i
=
n
α
(
t
′
)
n
α
(
t
′
′
)
−
1
(
1
+
E
α
[
|
X
t
i
α
α
|
q
]
)
ϵ
α
δ
∧
q
}
≤
C
q
,
T
{
ϵ
α
q
/
2
(
n
α
(
t
′
′
)
−
n
α
(
t
′
)
)
q
/
2
+
ϵ
α
q
(
n
α
(
t
′
′
)
−
n
α
(
t
′
)
)
q
+
ϵ
α
δ
∧
q
(
n
α
(
t
′
′
)
−
n
α
(
t
′
)
)
}
≤
C
¯
q
,
T
(
ϵ
α
(
n
α
(
t
′
′
)
−
n
α
(
t
′
)
)
)
γ
≤
C
¯
q
,
T
(
(
t
′
′
−
t
′
)
+
ϵ
α
)
γ
,where
C
q
,
T
,
C
¯
q
,
T
>
0 are some finite
constants being independent of t
′
,
t
′
′
≤
T, and α
≥
α
0. Then Lemma 5.4 ensures that any weak limit point of (
ℙ
α
) has no mass
outside of C
(
ℝ
). At this point, it is essential that we required q and δ to be strictly
larger than 2, 1, respectively.
Toward the verification of tightness of(
ℙ
α
) we use Hölder's inequality to get(5.32)
E
α
[
|
X
t
−
h
α
−
X
t
α
|
q
/
2
|
X
t
α
−
X
t
+
h
α
|
q
/
2
]
≤
E
α
[
|
X
t
−
h
α
−
X
t
α
|
q
]
1
/
2
E
α
[
|
X
t
α
−
X
t
+
h
α
|
q
]
1
/
2
.If h
≥
ϵ
α
/
2, then (5.31) implies that both
factors on the right-hand side of (5.32) are bounded by C
¯
q
,
T
(
3
h
)
γ
/
2. If h
<
ϵ
α
/
2, then at least one of these factors vanishes since X
α is constant on
intervals of length ϵ
α. Hence,(5.33)
E
α
[
|
X
t
−
h
α
−
X
t
α
|
q
/
2
|
X
t
α
−
X
t
+
h
α
|
q
/
2
]
≤
(
C
¯
q
,
T
2
3
γ
)
h
γfor all α
≥
α
0 and t
,
h
≥
0 with t
+
h
≤
T. That is, (5.28) holds with m
=
q. Therefore, Lemma 5.3 ensures tightness of (
ℙ
α
).



---

*Source: 275747-2008-03-23.xml*