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.

1. Introduction

It is well known that a rescaled version of the
classical Galton-Watson process (GWP) with offspring variance σ2 weakly
converges to the unique solution of the following one-dimensional stochastic
differential equation (SDE):dXt=σ|Xt|dWt,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,dXt=δ(t,Xt)dt+σ(t,Xt)|Xt|dWt,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:dXt=b(t,Xt)dt+a(t,Xt)dWt,X0=x0,where x0∈ℝ and W is a
one-dimensional Brownian motion. The coefficients a and b are continuous
functions on ℝ+×ℝ satisfying|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,(Ft),ℙ)} where (1.4)
(Ω,F,(Ft),ℙ) is a filtered
probability space with (1.4)
(Ft) satisfying the
usual conditions, (1.4)
W=(Wt:t≥0) is an (1.4)
(Ft)-Brownian
motion, and (1.4)
X=(Xt:t≥0) is a
real-valued continuous (1.4)
(Ft)-adapted
process such that (1.4)
ℙ-almost surely,
(1.4)
Xt=x0+∫0tb(r,Xr)dr+∫0ta(r,Xr)dWr∀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)
∫00+h−2(u)du=∞. 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,(Ft),ℙ)} is said to be a
solution of the ((1.6)
a, (1.6)
b, (1.6)
x0)-martingale
problem if (1.6)
(Ω,F,(Ft),ℙ) is a filtered
probability space with (1.6)
(Ft) satisfying the
usual conditions, and (1.6)
X=(Xt:t≥0) is a
real-valued continuous (1.6)
(Ft)-adapted
process such thatMt=Xt−x0−∫0tb(r,Xr)drprovides a (continuous,
mean-zero) square-integrable (ℱt)-martingale
with compensator〈M〉t=∫0ta2(r,Xr)dr.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 (Definition 2.1) of the (a,b,x0)-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 (Yα,Zα)→(Y,Z) in the cádlàg
space implies convergence in distribution ∫YαdZα→∫YdZ of the
corresponding stochastic integrals in the
cádlàg space.

In Section 3 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 by D(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 tnϵ=nϵ
for every n∈ℕ0 and ϵ>0.

For every α∈ℕ we fix some
ϵα>0 such that ϵα→0. For the sake of clarity, we also set tnα=tnϵα(=nϵα) for all n∈ℕ0. Now suppose that aα and bα are measurable
functions on ℝ+×ℝ such that ∥a−aα∥∞ and ∥b−bα∥∞ converge to 0 as α→∞, where ∥⋅∥∞ is the usual
supremum norm. Let (xα)⊂ℝ satisfy xα→x0, and suppose that Xα is a solution
of the following (ϵα,aα,bα,xα)-martingale
problem for every α≥1. Here we write nα(t) for the largest n∈ℕ0 with tnα≤t.

Definition 2.1.

Suppose that Xα=(Xtα:t≥0) is a
real-valued process on some probability space (Ω,ℱ,ℙ) whose
trajectories are constant on the intervals [tnα,tn+1α), n∈ℕ0. Then Xα is called a
solution of the (ϵα,aα,bα,xα)-martingale
problem ifMtα=Xtα−xα−∑i=0nα(t)−1bα(tiα,Xtiαα)ϵαprovides a (zero-mean)
square-integrable martingale (with respect to the natural filtration) with
compensator〈Mα〉t=∑i=0nα(t)−1aα2(tiα,Xtiαα)ϵα.

The Xα could be
defined on different probability spaces (Ωα,ℱα,ℙα). However, we assume without loss of generality that Ωα=D(ℝ), ℱα=ℬ(D(ℝ)), and Xα is the
coordinate process of ℙα (each cádlàg
process induces a corresponding law on D(ℝ)). We further
assume that there are some q>2 and δ>1 such thatEα[|Xtnαα−Xtn−1αα|q]≤CT(1+Eα[|Xtn−1αα|q])ϵαδfor every α≥1 and n∈ℕ with tnα≤T, where CT>0 is some finite
constant that may depend on T. (By an induction on n, (2.3) implies immediately that 𝔼α[|Xtnαα|q]<∞ for all α and n. Lemma 5.1 will provide an even stronger statement.)
The following theorem shows that Xα 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 ℙα be the law (on D(ℝ)) of Xα subject to
(2.1)–(2.3). Then ℙα⇒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 (<xref ref-type="disp-formula" rid="eq1">1.1</xref>)

As a first application of Theorem 2.2, 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 σ2). The latter
is an ℕ0-valued Markov
process Z=(Zn:n∈ℕ0) that can be
defined recursively as follows. Choose an initial state Z0∈ℕ and set Zn=∑i=1Zn−1Nn−1,i for all n≥1, where {Nn,i:n≥0,i≥1} is a family of
i.i.d. ℕ0-valued random
variables with mean 1 and variance σ2. In addition, we require that the fourth moment of N1,1 is finite.
Thereby Zn has a finite
fourth moment for every n∈ℕ0. 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]).

Setting Ztnϵϵ=ϵZn we obtain a
rescaled version, Zϵ, of Z. Recall tnϵ=nϵ, hence Zϵ is a process
having ϵℕ0={0,ϵ,2ϵ,…} as both its
index set and its state space. Now pick (ϵα)⊂ℝ+ such that ϵα→0, and recall our convention tnα=tnϵα and that ⌊t⌋ϵ denotes the largest element s of ϵℕ0 with s≤t. Regard the process Zϵα as
continuous-time process, Xα, by setting Xtα=Z⌊t⌋ϵαϵα, and suppose that X0α=⌊x0⌋ϵα. The latter requires that Z0 actually
depends on α. The domain of Xα is denoted by (Ωα,ℱα,ℙα). It is easy to see that Mα defined in (2.1)
provides a (zero-mean) square-integrable martingale. Moreover, the compensator
of Mα is given by 〈Mα〉t=σ2∑i=0nα(t)−1Xtiααϵα since, in this
case,Eα[((Mtnαα)2−〈Mα〉tnα)−((Mtn−1αα)2−〈Mα〉tn−1α)∣Ftn−1αXα]=0can be checked easily with help
ofEα[Xtnαα∣Xtn−1αα]=Xtn−1αα,Varα[Xtnαα∣Xtn−1αα]=σ2Xtn−1ααϵα.The formulae in (3.2) are
immediate consequences of the well-known moment formulae for Z (see [17, page 6]) and (ℱtnαXα) denotes the
natural filtration induced by Xα. Hence, Xα solves the (ϵα,a,b,xα)-martingale
problem of Definition 2.1 with a(t,x)=|x|, b≡0 and xα=⌊x0⌋ϵα. It remains to show (2.3). To
this end we state the following lemma.

Lemma 3.1.

Assume that ξ1,ξ2,… are independent
random variables on some probability space (Ω,ℱ,ℙ) with 𝔼[ξi]=0 and supi∈ℕ𝔼[ξi4]<∞. Let ν be a further
random variable on (Ω,ℱ,ℙ) being
independent of (ξi), taking values in ℕ and satisfying 𝔼[ν4]<∞. Then there is some finite constant C>0, depending only on the second and the fourth moments
of the ξi, such that 𝔼[(∑i=1νξi)4]≤C𝔼[ν2].

Proof.

By the
finiteness of the fourth moments the law of total expectation yieldsE[(∑i=1νξi)4]=∑n∈ℕ∑i1=1n∑i2=1n∑i3=1n∑i4=1nE[ξi1ξi2ξi3ξi4]ℙ[ν=n].Since the ξi are independent
and centered, the summand on the right-hand side might differ from 0 only if either i1=i2=i3=i4, or i1=i2 and i3=i4≠i1, or i1=i3 and i2=i4≠i1, or i1=i4 and i2=i3≠i1. Hence,E[(∑i=1νξi)4]≤∑n∈ℕ{(n+3n(n−1))supi,j∈ℕE[ξi2ξj2]}ℙ[ν=n]≤4supi,j∈ℕE[ξi2ξj2]E[ν2].This yields the claim of the
lemma with C=4supi,j∈ℕ𝔼[ξi2ξj2].

With help of Lemma 3.1
we obtainEα[|Xtnαα−Xtn−1αα|4]=Eα[|∑i=1ϵα−1Xtn−1αα(ϵαNn−1,i−ϵα)|4]=Eα[|∑i=1ϵα−1Xtn−1αα(Nn−1,i−1)|4]ϵα4≤CEα[(ϵα−1Xtn−1αα)2]ϵα4≤C(1+Eα[(Xtn−1αα)4])ϵα2for 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 Xα converges in
distribution to the unique solution of (1.1).

4. Example 2: Weak Euler Scheme Approximation of (<xref ref-type="disp-formula" rid="eq3">1.3</xref>)

As a second application of Theorem 2.2, 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):Xtnϵϵ−Xtn−1ϵϵ=b(tn−1ϵ,Xtn−1ϵϵ)ϵ+a(tn−1ϵ,Xtn−1ϵϵ)Vtnϵϵ,Xt0ϵϵ=xϵ.Here, (xϵ) is a sequence
in ℝ satisfying xϵ→x0 as ϵ→0, and Vϵ={Vtnϵϵ:n∈ℕ} is a family of
independent centered random variables with variance ϵ and 𝔼ϵ[|Vtnϵϵ|q]≤Cϵq/2 for all n∈ℕ, ϵ∈(0,1], some q>2, and some finite constant C>0, where (Ωϵ,ℱϵ,ℙϵ) denotes the
domain of Vϵ. For instance, one may set Vtnϵϵ=ϵξn where {ξn: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 {Vtnϵϵ:n∈ℕ} are identically
distributed. Below we will see that the independence is necessary neither.

By virtue of (1.4), Xtnϵϵ has a finite qth moment if Xtn−1ϵϵ has. It follows
by induction that the solution Xϵ=(Xtnϵϵ:n∈ℕ0) of (4.1) is q-integrable,
and hence square integrable. Equation (4.1) is obviously equivalent to the
stochastic sum equationXtnϵϵ=xϵ+∑i=0n−1b(tiϵ,Xtiϵϵ)ϵ+∑i=0n−1a(tiϵ,Xtiϵϵ)Vti+1ϵϵ.Suppose that (ϵα) is an arbitrary
sequence with ϵα∈(0,1] and ϵα→0, set xα=xϵα and recall our
convention 𝔼α=𝔼ϵα, Xα≡Xϵα, tnα=tnϵα. Then it is easy to see that Mα defined in
(2.1) provides a (mean-zero) square-integrable (ℱtXα)-martingale.
Moreover, Mtnαα coincides with
the second sum on the right-hand side of (4.2). Therefore, we also
obtain〈Mα〉tnα=∑i=1nEα[(a(ti−1α,Xti−1αα)Vtiαϵα)2∣Ftn−1αXα]=∑i=0n−1a2(tiα,Xtiαα)Eα[(Vti+1αϵα)2]=∑i=0n−1a2(tiα,Xtiαα)ϵαwhich shows that Xα solves the (ϵα,a,b,xα)-martingale
problem of Definition 2.1. For an application of Theorem 2.2 it thus remains to
show (2.3). But (2.3) follows fromEα[|Xtnαα−Xtn−1αα|q]≤2q−1{Eα[|b(tn−1α,Xtn−1αα)ϵα|q]+Eα[|a(tn−1α,Xtn−1αα)|q]Eα[|Vtnαϵα|q]}≤2q−1{K2q−1(1+Eα[|Xtn−1αα|q])ϵαq+K2q−1(1+Eα[|Xtn−1αα|q])Cϵαq/2}for which we used (4.1), the
independence of Xtn−1αα of Vϵα, (1.4), and 𝔼α[|Vtnαϵα|q]≤Cϵαq/2. Hence, Theorem 2.2 ensures that Xα converges in
distribution to the unique solution of SDE (1.3).

As mentioned above, the independence of the random variables {Vtnϵϵ:n∈ℕ} is not
necessary. The independence was used for (4.3), (4.4), and the martingale
property of Mα. But these relations may be valid even if the Vtnϵϵ are not
independent. For instance, let {ξn(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 Mα and the main
statements of (4.3) and (4.4) remain true for Vt1ϵϵ=ϵξ1(1) and Vtnϵϵ=ϵξn(fn(Vt1ϵϵ,…,Vtn−1ϵϵ)), n≥2, where fn is any
measurable mapping from ℝn−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 conditioningEα[Vtnαϵα∣Ftn−1αXα]=0,Eα[(Vti+1αϵα)2∣Ftn−1αXα]=ϵα,1≤i≤n−1,Eα[|a(tn−1α,Xtn−1αα)Vtnαϵα|q]≤Cϵαq/2.If the ξn(i) are not
identically distributed, then the Vtnϵϵ are typically
not independent. In particular, the approximating process Xϵ may be non-Markovian.

5. Proof of Theorem <xref ref-type="statement" rid="thm2.2">2.2</xref>

Theorem 2.2 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 α0≥1 such that for
all α≥α0, t≥0, and x∈ℝ,|aα(t,x)|+|bα(t,x)|≤K′(1+|x|).This is true since we assumed
(1.4) and uniform convergence of aα and bα to the
coefficients a and b, respectively. Throughout this section we will
frequently use the well-known inequality |∑i=1myi|p≤mp−1∑i=1m|yi|p for all m∈ℕ, p≥1 and y1,…,ym∈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 ϵℕ0={0,ϵ,2ϵ,…} which is
smaller than or equal to x. Moreover, we assume without loss of generality that ϵα≤1.

Lemma 5.1.

For q>2 and δ>1 satisfying (2.3)
and every T>0,supα≥α0Eα[supt≤T|Xtα|q]<∞.

Proof.

First of all, note that for the
proof it actually suffices to require q≥2 and δ≥1. Set S=supα≥α0|xα|q and Stα=𝔼α[max1≤i≤nα(t)|Mtiαα−Mti−1αα|q]. Using
Proposition A.1 in the appendix and (5.1) we obtain, for all t>0 and α≥α0,Eα[supi≤nα(t)|Xtiα|q]≤3q−1{Eα[supi≤nα(t)|Mtiα|q]+S+Eα[(∑i=0nα(t)−1|bα(tiα,Xtiαα)|ϵα)q]}≤3q−1Cq{Eα[|∑i=0nα(t)−1aα2(tiα,Xtiαα)ϵα|q/2]+Stα+S+Eα[[∑i=0nα(t)−1|bα(tiα,Xtiαα)|ϵα]q]}≤kq{Eα[[∑i=0nα(t)−1(K′(1+|Xtiαα|))2ϵα]q/2]+Stα+S+Eα[[∑i=0nα(t)−1K′(1+|Xtiαα|)ϵα]q]},where Cq is independent
of t and α, and kq=3q−1Cq. By Hölder's inequality we getEα[[∑i=0nα(t)−1(K′(1+|Xtiαα|))2ϵα]q/2]≤Eα[(∑i=0nα(t)−1(2K′2(1+|Xtiαα|2))q/2)(∑i=0nα(t)−1ϵα(q/2)/(q/2−1))q/2−1]≤Eα[(∑i=0nα(t)−12q/2−1(2K′2)q/2(1+|Xtiαα|q))nα(t)q/2−1ϵαq/2]≤cqtq/2+cqtq/2−1∑i=0nα(t)−1Eα[supj≤i|Xtjαα|q]ϵα,where cq=2q/2−1(2K′2)q/2. Analogously, with c¯q=2q−1K′q,Eα[[∑i=0nα(t)−1K′(1+|Xtiαα|)ϵα]q]≤c¯qtq+cqtq−1∑i=0nα(t)−1Eα[supj≤i|Xtjαα|q]ϵα.Moreover, by (2.3) and (5.1) we obtain, for all t≤T and α≥α0,Stα≤∑i=1nα(t)Eα[|Mtiαα−Mti−1αα|q]≤2q−1∑i=1nα(t)Eα[|Xtiαα−Xti−1αα|q+|bα(ti−1α,Xti−1αα)|qϵαq]≤2q−1∑i=0nα(t)−1{CT(1+Eα[|Xtiαα|q])ϵαδ+Eα[K′(1+|Xtiαα|q)]ϵαq}≤cq,Tt+cq,T∑i=0nα(t)−1Eα[supj≤i|Xtjαα|q]ϵα,where cq,T=2q−1(CT+K′). By all account we have, for all t≤T and α≥α0,Eα[supi≤nα(t)|Xtiα|q]≤kq{S+(cq+c¯q+cq,T)(tq−1∨1)(1+ϵα∑i=0nα(t)−1Eα[supj≤i|Xtjαα|q])}≤(kqS+Cq,T)+Cq,Tϵα∑i=0nα(t)−1Eα[supj≤i|Xtjαα|q],where Cq,T=kq(cq+c¯q+cq,T)(Tq−1∨1). An application of Lemma A.2 yieldsEα[sups≤t|Xsα|q]=Eα[supi≤nα(t)|Xtiα|q]≤(kqS+Cq,T)(1+Cq,Tϵα)nα(t)+(Cq,Tϵα)nα(t)S,where we emphasize that the
constants kq, S, and Cq,T are independent
of t≤T and α≥α0. This proves Lemma 5.1 since limsupα→∞(1+Cq,Tϵα)nα(t) is bounded by exp(tCq,T) (note that nα(t)=⌊t/ϵα⌋1≤t/ϵα).

Proposition 5.2.

If (ℙα) 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,x0)-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 (ℙα) 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] thatℙα∘πt1,…,tk−1⟹ℙ∘πt1,…,tk−1for all t1,…,tk∈ℝ+, where πt1,…,tk:D(ℝ)→ℝk 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¯tX)-martingale and
has 〈M〉 defined in
(1.8) as compensator. Here, (ℱ¯tX) denotes the
natural augmentation of the filtration (ℱtX) 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,supt≤TE[|Xt|q]≤supt≤TliminfN→∞limα→∞Eα[|Xtα|q∧N]≤supt≤Tsupα≥α0Eα[|Xtα|q]<∞.Taking (1.4) into account we
conclude that M defined in (1.7)
is square-integrable.

Step 2.

We next
show that M is an (ℱ¯tX)-martingale. It
suffices to show that M is an (ℱtX)-martingale;
see [20, page 75]. The
latter is true if and only ifE[(Xt+s−Xt−∫tt+sb(r,Xr)dr)∏i=1lhi(Xti)]=0holds for all 0≤t1<⋯≤tl≤t, s≥0, l≥1 and bounded h1,…,hl∈C(ℝ) (do not confuse ti and tiα). Since Xα solves the (ϵα,aα,bα,xα)-martingale
problem, we haveEα[(Xt+sα−Xtα−∑i=nα(t)nα(t+s)−1bα(tiα,Xtiαα)ϵα)∏i=1lhi(Xtiα)]=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 provinglimα→∞Eα[Xuα∏i=1lhi(Xtiα)]=E[Xu∏i=1lhi(Xti)]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|Eα[Xuα,(N)∏i=1lhi(Xtiα)]−Eα[Xuα∏i=1lhi(Xtiα)]|≤Eα[|Xuα,(N)−Xuα|∏i=1l∥hi∥∞]can be estimated, for every T≥u, bysupr≤Tsupα′≥α0Eα′[|Xrα′|1|Xrα′|>N]∏i=1l∥hi∥∞which tends to 0 as N→∞ since {Xrα′:r≤T,α′≥1} is uniformly
integrable by (5.2). Therefore, we havelimN→∞Eα[Xuα,(N)∏i=1lhi(Xtiα)]=Eα[Xuα∏i=1lhi(Xtiα)]uniformlyinα≥α0(and uniformly in u≤T, for every T>0). By (5.9) we further obtain for every N>0,limα→∞Eα[Xuα,(N)∏i=1lhi(Xtiα)]=E[Xu(N)∏i=1lhi(Xti)]since the mapping (x1,…,xl+1)↦xl+1(N)∏i=1lhi(xi) from ℝl+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 giveslimN→∞E[Xu(N)∏i=1lhi(Xti)]=E[Xu∏i=1lhi(Xti)]which along with (5.16) and (5.17)
implies (5.13). It remains to showlimα→∞Eα[∑i=nα(t)nα(t+s)−1bα(tiα,Xtiαα)ϵα∏i=1lhi(Xtiα)]=E[∫tt+sb(r,Xr)dr∏i=1lhi(Xti)].Taking (5.1) and (nα(t+s)−nα(t))ϵα≤s+ϵα into account we
obtain, analogously to (5.16) and
(5.18),limN→∞Eα[∑i=nα(t)nα(t+s)−1bα(tiα,Xtiαα,(N))ϵα∏i=1lhi(Xtiα)]=Eα[∑i=nα(t)nα(t+s)−1bα(tiα,Xtiαα)ϵα∏i=1lhi(Xtiα)]uniformlyinα≥α0,respectively,limN→∞E[∫tt+sb(r,Xr(N))dr∏i=1lhi(Xti)]=E[∫tt+sb(r,Xr)dr∏i=1lhi(Xti)].By the uniform convergence of bα to b and (nα(t+s)−nα(t))ϵα≤s+ϵα, we also haveEα[∑i=nα(t)nα(t+s)−1bα(tiα,Xtiαα,(N))ϵα∏i=1lhi(Xtiα)]=Eα[∑i=nα(t)nα(t+s)−1b(tiα,Xtiαα,(N))ϵα∏i=1lhi(Xtiα)]+oα(1).Moreover, we
haveEα[∑i=nα(t)nα(t+s)−1b(tiα,Xtiαα,(N))ϵα∏i=1lhi(Xtiα)]=Eα[∫tt+sb(r,Xrα,(N))dr∏i=1lhi(Xtiα)]+oα(1)which is a consequence of the
dominated convergence theorem and|∑i=nα(t)nα(t+s)−1b(tiα,Xtiαα,(N))ϵα−∫tt+sb(r,Xrα,(N))dr|≤∫⌊t⌋ϵα⌊t+s⌋ϵα−ϵα|b(⌊r⌋ϵα,Xrα,(N))−b(r,Xrα,(N))|dr+oα(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)
limα→∞Eα[∫tt+sb(r,Xrα,(N))dr∏i=1lhi(Xtiα)]=∫tt+slimα→∞Eα[b(r,Xrα,(N))∏i=1lhi(Xtiα)]dr=∫tt+sE[b(r,Xr(N))∏i=1lhi(Xti)]dr=E[∫tt+sb(r,Xr(N))dr∏i=1lhi(Xti)]which along with (5.22) and (5.23)
implieslimα→∞Eα[∑i=nα(t)nα(t+s)−1bα(tiα,Xtiαα,(N))ϵα∏i=1lhi(Xtiα)]=E[∫tt+sb(r,Xr(N))dr∏i=1lhi(Xti)].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 ifE[(Mt+s2−Mt2−∫tt+sa2(r,Xr)dr)∏i=1lhi(Xti)]=0holds for all 0≤t1<⋯≤tl≤t, s≥0, l≥1 and bounded h1,…,hl∈C(ℝ). Now, the discrete analogue of (5.27) for 𝔼α, aα, and Xα 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 {(Xrα)2:r≤t+s,α≥1}. This is why we established (5.2) for q being strictly
larger than 2.

The assumptions of
Proposition 5.2 can be checked with help of the following two
lemmas, where ℚα and ℚ refer to any
laws on D(ℝ), and Yα and Y are the
respective coordinate processes. By an abuse of notation, we denote the
corresponding expectations by ℚα 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 (Ytα) 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 CT>0 such that for
all α≥1 and t,h≥0 with 0≤t−h and t+h≤T,ℚα[|Yt−hα−Ytα|m/2|Ytα−Yt+hα|m/2]≤CThγ.Then (ℚα) is tight.

Lemma 5.4.

Let m>0, γ>1, and assume for every T>0 that there is
some finite constant CT>0 such that for
all α≥1 and 0≤t′≤t′′≤T,lim supα→∞ℚα[|Yt′α−Yt′′α|m]≤CT(t′′−t′)γ.Then if ℚα⇒ℚ, the limit ℚ has no mass
outside of C(ℝ).

Proposition 5.5.

(ℙα) is tight and
each limit point has no mass outside of C(ℝ).

Proof.

Let q>2 and δ>1 satisfy (2.3).
Using techniques as in the proof of Lemma 5.1 we can find a finite constant Cq>0 such that for
every 0≤t′≤t′′ and α≥α0,Eα[|Xt′α−Xt′′α|q]≤Cq{ϵαq/2Eα[(∑i=nα(t′)nα(t′′)−1(1+|Xtiαα|)2)q/2]+ϵαqEα[(∑i=nα(t′)nα(t′′)−1(1+|Xtiαα|))q]+∑i=nα(t′)nα(t′′)−1(1+Eα[|Xtiαα|q])ϵαδ∧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≤Cq{ϵαq/2(∑i=nα(t′)nα(t′′)−1Eα[(1+|Xtiαα|)2q/2])(∑i=nα(t′)nα(t′′)−11q/(q−2))q/2−1+ϵαq(∑i=nα(t′)nα(t′′)−1Eα[(1+|Xtiαα|)q])(∑i=nα(t′)nα(t′′)−11q/(q−1))q−1+∑i=nα(t′)nα(t′′)−1(1+Eα[|Xtiαα|q])ϵαδ∧q}≤Cq,T{ϵαq/2(nα(t′′)−nα(t′))q/2+ϵαq(nα(t′′)−nα(t′))q+ϵαδ∧q(nα(t′′)−nα(t′))}≤C¯q,T(ϵα(nα(t′′)−nα(t′)))γ≤C¯q,T((t′′−t′)+ϵα)γ,where
Cq,T,C¯q,T>0 are some finite
constants being independent of t′,t′′≤T, and α≥α0. Then Lemma 5.4 ensures that any weak limit point of (ℙα) 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 (ℙα) we use Hölder's inequality to getEα[|Xt−hα−Xtα|q/2|Xtα−Xt+hα|q/2]≤Eα[|Xt−hα−Xtα|q]1/2Eα[|Xtα−Xt+hα|q]1/2.If h≥ϵα/2, then (5.31) implies that both
factors on the right-hand side of (5.32) are bounded by C¯q,T(3h)γ/2. If h<ϵα/2, then at least one of these factors vanishes since Xα is constant on
intervals of length ϵα. Hence,Eα[|Xt−hα−Xtα|q/2|Xtα−Xt+hα|q/2]≤(C¯q,T23γ)hγfor all α≥α0 and t,h≥0 with t+h≤T. That is, (5.28) holds with m=q. Therefore, Lemma 5.3 ensures tightness of (ℙα).

AppendixAuxiliary Results

Here we give two auxiliary results. We first recall a
square function inequality for martingales. Let M=(Mn:n∈ℕ0) be an (ℱn)-martingale on
some probability space (Ω,ℱ,ℙ). The corresponding compensator is given by 〈M〉n=∑i=1n𝔼[(Mi−Mi−1)2∣ℱi−1].

Proposition A.1 (see [<xref ref-type="bibr" rid="B6">21</xref>, Theorem 2.11]).

For every q>0 there is some
finite constant Cq>0 depending only
on q such thatE[max1≤i≤n|Mi|q]≤Cq(E[〈M〉nq/2]+E[max1≤i≤n|Mi−Mi−1|q]).

The second result is a Gronwall lemma for functions with discrete domain. It
can be proven by means of iterating (A.2) n-times. We omit
the proof since it is more or less well known.

Lemma A.2.

Suppose g is a mapping
from ℕ0 to ℝ¯+=[0,∞] with g(0)<∞. If there are finite constants c0,c1≥0 such thatg(n)≤c0+c1∑i=0n−1g(i)∀n≤N,theng(n)≤c0(1+c1)n+c1ng(0)<∞∀n≤N.

Acknowledgment

The author thanks a referee for revealing a flaw of
the original manuscript.

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