Weak approximation of SDEs by discrete-time processes

Weak approximation of SDEs by discrete-time processes

Henryk Z¨ ahle

Fakult¨at f¨ur Mathematik Preprint Nr. 2008-01

Technische Universit¨at Dortmund April 2008 Vogelpothsweg 87

Weak approximation of SDEs by discrete-time processes

Henryk Z¨ahle^∗

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.

Key words and phrases. Stochastic differential equation, martingale problem, Doob-Meyer de- composition, discrete-time process, weak convergence, Galton-Watson process, Euler scheme MSC 2000 subject classification. 60H10, 60J80, 60F17

∗Postal address: Dortmund University of Technology, Faculty of Mathematics, Vogelpothsweg 87, D-44227 Dortmund, Germany; Email: henryk.zaehle@math.uni-dortmund.de

1 Introduction

It is well-known that a re-scaled version of the classical Galton-Watson process (GWP) with offspring varianceσ²weakly converges to the unique solution of the following one-dimensional stochastic differential equation (SDE)

dXt=σp

|X_t|dWt (1)

whereW is a one-dimensional Brownian motion (cf. [14]). One might ask whether it is possible to approximate more general SDEs, driven by a Brownian motion, by generalized GWPs. In [21] it will be shown that this is actually possible. In fact, in [21] the solution of the SDE

dX_t=δ(t, X_t)dt+σ(t, X_t)p

|X_t|dW_t (2)

is weakly approximated by two different types of population-size-dependent GWPs (in the sense of [5], [8], [10], [11]) with immigration, whereδ andσ are suitable nonnegative continu- ous functions onR+×R. Here the methods of [14] 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 [21].

To be exact, we focus on the following one-dimensional SDE

dX_t=b(t, X_t)dt+a(t, X_t)dW_t, X₀ =x₀ (3) where x₀ ∈ R and W is a one-dimensional Brownian motion. The coefficients a and b are continuous functions onR+×Rsatisfying

|a(t, x)|+|b(t, x)| ≤ K(1 +|x|) ∀t∈R+, x∈R (4) for some finite constant K > 0. We assume that SDE (3) has a weak solution. It means that there exists a triplet{X;W; (Ω,F,(F_t),P)}where (Ω,F,(F_t),P) is a filtered probability space with (F_t) satisfying the usual conditions,W = (W_t:t≥0) is an (F_t)-Brownian motion and X = (Xt : t ≥ 0) is a real-valued continuous (F_t)-adapted process such that P-almost surely,

Xt=x0+ Z t

0

b(r, Xr)dr+ Z t

0

a(r, Xr)dWr ∀t≥0.

Here the latter is an Itˆo-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 ofbinx (uniformly int) and

|a(t, x)−a(t, x⁰)| ≤h(|x−x⁰|) ∀ t∈R+, x, x⁰ ∈R (5) for some strictly increasingh :R+ → R+ with R0+

0 h⁻²(u)du =∞. Note that (5) and Lips- chitz continuity ofbeven imply the existence of a strongly unique strong solution (Yamada- Watanabe criterion [20]). But the notion of strong solutions and strong uniqueness is beyond our interest.

Our starting point is the fact that any weak solution of (3) is a solution of the following martingale problem and vice versa (cf. Section 5.4.B of [9], or Theorem 1.27 of [1]).

Definition 1.1 A tuple{X; (Ω,F,(F_t),P)}is said to be a solution of the (a,b,x₀)-martingale problem if(Ω,F,(F_t),P)is a filtered probability space with(F_t) satisfying the usual conditions andX = (Xt:t≥0)is a real-valued continuous (F_t)-adapted process such that

Mt=Xt−x0− Z t

0

b(r, Xr)dr (6)

provides a (continuous, mean-zero) square-integrable(F_t)-martingale with compensator hMi_t=

Z t 0

a²(r, X_r)dr. (7)

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 pro- cesses solving the discrete analogue (Definition 2.1) of the (a,b,x0)-martingale problem should approximate weakly the unique solution of SDE (3). Theorem 2.2 below shows that this is true under an additional assumption on the moments of the increments (condition (10)).

Note that the characterization of discrete or continuous population processes as solutions of martingale problems of the form (6)-(7) respectively (8)-(9) is fairly useful and also common (see e.g. [16], [17], [19]). Especially for real-valued discrete-time processes these characteri- zations are often easy to see, so that, according to the criterion, the only thing to check is condition (10). Also note that the conditions of the famous criterion of Stroock and Varadhan for the weak convergence of Markov chains to SDEs ([18] 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 [13], see also refer- ences therein. In [13] general conditions are given under which the convergence in distribution (Yα, Zα)→(Y, Z) in the c´adl`ag space implies convergence in distribution R

YαdZα →R Y dZ of the corresponding stochastic integrals in the c´adl`ag 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 c´adl`ag processes. For this reason we denote by D(R) the space of c´adl`ag 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 Borelσ-algebra. Moreover we sett_n=nfor everyn∈N0 and >0.

For every α ∈ N we fix some _α > 0 such that _α → 0. For the sake of clarity we also set t^α_n = t_n^α (= n_α) for all n ∈ N0. Now suppose a_α and b_α are measurable functions on

R+ ×R such that ka−a_αk_∞ and kb−b_αk_∞ converge to 0 as α → ∞, where k.k_∞ is the usual supremum norm. Let (x_α) ⊂R satisfyx_α →x₀, and suppose that X^α is a solution of the following (α, aα, bα, xα)-martingale problem for everyα ≥1. Here we writenα(t) for the largestn∈N0 witht^α_n≤t.

Definition 2.1 SupposeX^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^α_n, t^α_n+1),n∈N0. ThenX^α is called solution of the (α, aα, bα, xα)-martingale problem if

M_t^α=X_t^α−xα−

nα(t)−1

X

i=0

bα(t^α_i, X_t^αα

i) α (8)

provides a (zero-mean) square-integrable martingale (w.r.t. the natural filtration) with com- pensator

hM^αi_t=

nα(t)−1

X

i=0

a²_α(t^α_i, X_t^αα

i) α. (9)

The X^α could be defined on different probability spaces (Ωα,F_α,Pα). However we assume without loss of generality Ωα =D(R), F_α=B(D(R)) and thatX^α is the coordinate process of Pα (each c´adl`ag process induces a corresponding law on D(R)). We further assume that there are someq >2 and δ >1 such that

Eα

h

|X_t^αα n−X_t^αα

n−1|^qi

≤ CT

1 +Eα

h

|X_t^αα n−1|^qi

^δ_α (10)

for every α ≥ 1 and n ∈ N with t^α_n ≤ T, where C_T > 0 is some finite constant that may depend onT. (By an induction on n, (10) implies immediately that Eα[|X_t^αα

n|^q]<∞ for all αand n. Lemma 5.1 will provide an even stronger statement.) The following theorem shows thatX^α converges in distribution to the unique solution of (3).

Theorem 2.2 Suppose SDE (3) subject to (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 X^α subject to (8)-(10). ThenPα ⇒P as α→ ∞.

Here⇒symbolizes weak convergence. The proof of Theorem 2.2 will be carried out in Section 5. The finiteness of the q-th 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 (10), a weakening of q >2 toq = 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. [15], [19]).

3 Example 1: Convergence of re-scaled GWP to (1)

As a first application of Theorem 2.2, we show that a re-scaled GWP weakly converges to Feller’s branching diffusion ([4]), i.e., to the solution of SDE (1). Lindvall [14] showed this ap- proximation via the convergence of the finite-dimensional distributions, for which the shape

of the Laplace transforms of the transition probabilities is essential. Here we shall exploit the martingale property of the Galton-Watson process (with offspring varianceσ²). The lat- ter 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 Z_n = PZn−1

i=1 Nn−1,i for all n ≥ 1, where {N_n,i :n ≥0, i≥1} is a family of i.i.d.N0-valued random variables with mean 1 and vari- ance σ². In addition we require that the 4-th moment of N1,1 is finite. Thereby Zn has a finite 4-th moment for everyn∈N0. Actually, in [14] the finiteness of the 4-th 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 that cases (cf. [21]).

SettingZ_t

n =Z_nwe obtain a re-scaled version,Z, ofZ. Recallt_n=n, henceZis a process havingN0 ={0, ,2, . . .}as both its index set and its state space. Now pick (_α)⊂R+such that α → 0, and recall our convention t^α_n = t_n^α and that btc denotes the largest element s of N0 with s ≤ t. Regard the process Z^α as continuous-time process, X^α, by setting X_t^α = Z_btc^α

α, and suppose X₀^α = bx₀c_α. The latter requires that Z₀ actually depends on α. The domain of X^α is denoted by (Ωα,F_α,Pα). It is easy to see that M^α defined in (8) provides a (zero-mean) square-integrable martingale. Moreover the compensator of M^α is given byhM^αi_t=σ²Pnα(t)−1

i=0 X_t^αα

i α since in this case, Eα

h (M_t^αα

n)²− hM^αi_t^α_n

− (M_t^αα

n−1)²− hM^αi_t^α

n−1

F_t^Xα^α

n−1

i

= 0 can be checked easily with help of

Eα

h X_t^αα

n|X_t^αα n−1

i

=X_t^αα

n−1 and Varα

h X_t^αα

n|X_t^αα n−1

i

=σ²X_t^αα

n−1 α. (11) The formulae in (11) are immediate consequences of the well-known moment formulae forZ (see [7] p.6) and (F_t^Xα^α

n ) denotes the natural filtration induced by X^α. Hence X^α solves the (_α, a, b, x_α)-martingale problem of Definition 2.1 witha(t, x) =p

|x|,b≡0 andx_α=bx₀c_α. It remains to show (10). To this end we state the following lemma.

Lemma 3.1 Assume ξ1, ξ2, . . . are independent random variables on some probability space (Ω,F,P)withE[ξ_i] = 0andsup_i∈NE[ξ⁴_i]<∞. Letν be a further random variable on(Ω,F,P) being independent of (ξ_i), 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 thatE[(Pν

i=1ξ_i)⁴]≤CE[ν²].

Proof By the finiteness of the fourth moments the law of total expectation yields E

X^ν

i=1

ξi

4

=X

n∈N n

X

i1=1 n

X

i2=1 n

X

i3=1 n

X

i4=1

E h

ξi1ξi2ξi3ξi4

i

P[ν =n].

Since theξ_i are independent and centered, the summand on the right-hand side might differ from 0 only if eitheri₁=i₂ =i₃=i₄, ori₁ =i₂ and i₃=i₄ 6=i₁, ori₁=i₃ andi₂ =i₄6=i₁,

ori₁ =i₄ and i₂=i₃ 6=i₁. Hence, E

X^ν

i=1

ξ_i4

≤ X

n∈N

(n+ 3n(n−1)) sup

i,j∈N

E h

ξ_i²ξ_j²i

P[ν =n] ≤ 4 sup

i,j∈N

E h

ξ_i²ξ_j²i E

h ν²i

.

This yields the claim of the lemma withC= 4 sup_i,j∈NE[ξ²_iξ²_j]. 2 With help of Lemma 3.1 we obtain

Eα

h|X_t^αα n −X_t^αα

n−1|⁴i

= Eα

⁻¹α X_tα^α

n−1

X

i=1

(_αNn−1,i−_α)

4

= Eα

⁻¹α X_tα^α

n−1

X

i=1

(Nn−1,i−1)

4 ⁴_α

≤ C Eα

h

(⁻¹_α X_t^αα

n−1)²i

⁴_α ≤ C 1 +Eα

h (X_t^αα

n−1)⁴i ²_α

for some suitable constantC >0. This shows that (10) 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).

4 Example 2: Weak Euler scheme approximation of (3)

As a second application of Theorem 2.2, we establish a weak Euler scheme approximation of SDE (3). Our assumptions are partially weaker then the assumptions of classical results on weak functional Euler scheme approximations. A standard reference for Euler schemes is the monograph [12]; see also references therein. As before we suppose thataandbare continuous functions onR+×Rsatisfying (4), and that SDE (3) possesses a unique weak solution. Now let > 0, recall the notation introduced in Section 2 and consider the following stochastic difference equation (weak Euler scheme)

X_t n−X_t

n−1 = b(t_n−1, X_t

n−1) + a(t_n−1, X_t n−1) V_t

n, X_t

0 =x. (12)

Here (x) is a sequence in R satisfying x → x₀ as → 0, and V = {V_t

n : n ∈ N} is a family of independent centered random variables with variance and E[|V_t

n|^q] ≤ C^q/2 for all n∈ N, ∈ (0,1], some q > 2 and some finite constant C > 0, where (Ω,F,P) denotes the domain of V. For instance, one may set V_t

n = √

ξ_n where {ξ_n : n ∈ N} is a family of independent centered random variables with variance 1 and q-th moment being bounded uniformly in n. Note that we do not require that the random variables {V_t

n : n ∈ N} are identically distributed. Below we will see that the independence is necessary neither.

By virtue of (4),X_t

n has a finiteq-th moment ifX_t

n−1 has. It follows by induction that the solution X = (X_t

n :n∈ N0) of (12) is q-integrable, and hence square-integrable. Equation (12) is obviously equivalent to the stochastic sum equation

X_t

n = x +

n−1

X

i=0

b(t_i, X_t

i) +

n−1

X

i=0

a(t_i, X_t

i) V_t

i+1. (13)

Suppose (_α) is an arbitrary sequence with _α ∈ (0,1] and _α → 0, set x_α =x_α and recall our convention Eα = Eα, X^α ≡ X^α, t^α_n = t_n^α. Then it is easy to see that M^α defined in (8) provides a (mean-zero) square-integrable (F_t^Xα)-martingale. MoreoverM_t^αα

n coincides with the second sum on the right-hand side of (13). Therefore we also obtain

hM^αi_t^α_n =

n

X

i=1

Eα

a(t^α_i−1, X_t^αα

i−1) V_tα^α i

2 F_t^Xα^α

n−1

=

n−1

X

i=0

a²(t^α_i, X_t^αα

i) Eα

h (V_tα^α

i+1)² i

=

n−1

X

i=0

a²(t^α_i, X_t^αα

i) α (14)

which shows that X^α solves the (α, a, b, xα)-martingale problem of Definition 2.1. For an application of Theorem 2.2 it thus remains to show (10). But (10) follows from

Eα

h|X_t^αα n −X_t^αα

n−1|^qi

≤ 2^q−1 n

Eα

h

|b(t^α_n−1, X_t^αα

n−1)α|^qi +Eα

h

|a(t^α_n−1, X_t^αα

n−1)|^qi Eα

h

|V_tα^α n|^qi o

≤ 2^q−1n

K2^q−1

1 +Eα[|X_t^αα n−1|^q]

^q_α+K2^q−1

1 +Eα[|X_t^αα n−1|^q]

C^q/2_α o

(15) for which we used (12), the independence ofX_t^αα

n−1 ofV^α, (4) andEα[|V_tα^α

n |^q]≤C^q/2_α . Hence Theorem 2.2 ensures thatX^α converges in distribution to the unique solution of SDE (3).

As mentioned above, the independence of the random variables{V_t

n:n∈N}is not necessary.

The independence was used for (14), (15) and the martingale property of M^α. But these relations may be valid even if theV_t

n are not independent. For instance, let{ξ_n(i) :n, i∈N} be an array of independent centered random variables with variance 1 and q-th moments being bounded above by someC > 0 uniformly in n, i, for some q > 2. Then the martingale property of M^α and the main statements of (14) and (15) remain true for V_t

1 = √

ξ₁(1) and V_t

n =√

ξn(fn(V_t

1, . . . , V_t

n−1)), n≥2, where fn is any measurable mapping fromRⁿ⁻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:

Eα

h V_tα^α

n |F_t^Xα^α n−1

i

= 0, Eα

h (V_t^αα

i+1)²|F_t^Xα^α n−1

i

=α (1≤i≤n−1) and

Eα

h|a(t^α_n−1, X_t^αα

n−1)V_tα^α n |^qi

≤C^q/2_α . If the ξ_n(i) are not identically distributed, then the V_t

n are typically not independent. In particular, the approximating processX may be non-Markovian.

5 Proof of Theorem 2.2

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 existK⁰ >0 and α₀ ≥1 such that for all α≥α₀,t≥0 and x∈R,

|a_α(t, x)|+|b_α(t, x)| ≤K⁰(1 +|x|). (16)

This is true since we assumed (4) and uniform convergence ofa_α and b_α to the coefficientsa andb, respectively. Throughout this section we will frequently use the well-known inequality

|Pm

i=1yi|^p≤m^p−1Pm

i=1|y_i|^p for all m∈N,p≥1 andy1, . . . , ym ∈R. As a first consequence of (16) we obtain Lemma 5.1. For every x ∈ R+ we write bxc for the largest element of N0 ={0, ,2, . . .} which is smaller than or equal tox. Moreover we assume without loss of generalityα≤1.

Lemma 5.1 For q >2 and δ >1 satisfying (10) and every T >0, sup

α≥α₀Eα

sup

t≤T

|X_t^α|^q

<∞. (17)

Proof First of all note that for the proof it actually suffices to require q ≥ 2 and δ ≥ 1.

SetS = sup_α≥α0|x_α|^q and S_t^α=Eα[max_{1≤i≤nα(t)}|M_t^αα i −M_t^αα

i−1|^q]. Using Proposition A.1 and (16) we obtain for allt >0 andα≥α0,

Eα

sup

i≤nα(t)

|X_t^α_i|^q

(18)

≤ 3^q−1

Eα

sup

i≤nα(t)

|M_t^α_i|^q

+S+Eα

^nα

(t)−1

X

i=0

|b_α(t^α_i, X_t^αα

i)|_αq

≤ 3^q−1C_q

Eα

nα(t)−1

X

i=0

a²_α(t^α_i, X_t^αα

i)_α

q 2

+S_t^α+S+Eα

h

nα(t)−1

X

i=0

|b_α(t^α_i, X_t^αα

i)|_αiq

≤ k_q

Eα

h

nα(t)−1

X

i=0

(K⁰(1 +|X_t^αα

i|))²_αi^q₂

+S_t^α+S+Eα

h

nα(t)−1

X

i=0

K⁰(1 +|X_t^αα

i|)_αiq

whereC_q is independent of tand α, and k_q= 3^q−1C_q. By H¨older’s inequality we get

Eα

h

nα(t)−1

X

i=0

(K⁰(1 +|X_t^αα

i|))²_αiq/2

≤ Eα

nα(t)−1

X

i=0

(2K⁰²(1 +|X_t^αα

i|²))^q/2

nα(t)−1

X

i=0

(q/2)/(q/2−1) α

q/2−1

≤ Eα

nα(t)−1

X

i=0

2^q/2−1(2K⁰²)^q/2(1 +|X_t^αα i|^q)

n_α(t)^q/2−1q/2_α

≤ cq t^q/2 + cq t^q/2−1

nα(t)−1

X

i=0

Eα

sup

j≤i

|X_t^αα j|^q

α

wherecq= 2^q/2−1(2K⁰²)^q/2. Analogously, with ¯cq= 2^q−1K^0q,

Eα

h

nα(t)−1

X

i=0

K⁰(1 +|X_t^αα

i|)_αiq

≤ c¯_q t^q + c_q t^q−1

nα(t)−1

X

i=0

Eα

sup

j≤i

|X_t^αα j|^q

_α.

Moreover, by (10) and (16) we obtain for allt≤T and α≥α₀, S_t^α ≤

nα(t)

X

i=1

Eα

h

|M_t^αα i −M_t^αα

i−1|^qi

≤ 2^q−1

nα(t)

X

i=1

Eα

h

|X_t^αα i −X_t^αα

i−1|^q+|b_α(t^α_i−1, X_t^αα

i−1)|^qq_α i

≤ 2^q−1

nα(t)−1

X

i=0

C_T

1 +Eα

h|X_t^αα i|^qi

^δ_α+Eα

h

K⁰(1 +|X_t^αα i|^q)i

^q_α

≤ c_q,T t + c_q,T

nα(t)−1

X

i=0

Eα

sup

j≤i

|X_t^αα j|^q

_α

wherec_q,T = 2^q−1(C_T +K⁰). By all account we have for allt≤T and α≥α₀,

Eα

sup

i≤nα(t)

|X_t^α_i|^q

≤ k_q

S+ (c_q+ ¯c_q+c_q,T)(t^q−1∨1)

1 +_α

nα(t)−1

X

i=0

Eα

sup

j≤i

|X_t^αα j|^q

≤ (k_qS+C_q,T) +C_{q,T α}

nα(t)−1

X

i=0

Eα

sup

j≤i

|X_t^αα j|^q

whereC_q,T =k_q(c_q+ ¯c_q+c_q,T)(T^q−1∨1). An application of Lemma A.2 yields Eα

sup

s≤t

|X_s^α|^q

= Eα

sup

i≤nα(t)

|X_t^α

i|^q

≤ (kqS+Cq,T)(1+Cq,T α)^nα(t)+(Cq,T α)^nα(t)S, (19) where we emphasize that the constantsk_q,S andC_q,T are independent oft≤T andα≥α₀. This proves Lemma 5.1 since lim sup_α→∞(1 +Cq,Tα)^nα(t) is bounded by exp(tCq,T) (note

thatn_α(t) =bt/_αc₁ ≤t/_α). 2

Proposition 5.2 If (Pα) is tight then the coordinate process of any weak limit point, that has no mass outside ofC(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 writeX for the coordinate process of P. SinceX is P-almost surely continuous, we know ([3]

Theorem 3.7.8) that

Pα◦π_t⁻¹

1,...,tk ⇒P◦π_t⁻¹

1,...,tk (20)

for allt1, . . . , tk ∈R+, where πt1,...,tk :D(R)→ R^k is the usual coordinate projection. In the remainder of the proof we will show in three steps thatM defined in (6) is square-integrable, provides an ( ¯F_t^X)-martingale and hashMidefined in (7) 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 (20) and (17) we obtain for everyT >0, sup

t≤T E[|X_t|^q] ≤ sup

t≤T

lim inf

N→∞ lim

α→∞Eα[|X_t^α|^q∧N] ≤ sup

t≤T

sup

α≥α₀Eα[|X_t^α|^q] < ∞. (21) Taking (4) into account we conclude thatM defined in (6) is square-integrable.

Step 2. We next show that M is an ( ¯F_t^X)-martingale. It suffices to show that M is an (F_t^X)-martingale, see [2] p. 75. The latter is true if and only if

E

X_t+s−X_t− Z t+s

t

b(r, X_r)drY^l

i=1

h_i(X_ti)

= 0 (22)

holds for all 0≤t1<· · · ≤t_l≤t,s≥0,l≥1 and boundedh1, . . . , h_l∈C(R) (do not confuse t_i and t^α_i). SinceX^α solves the (_α, a_α, b_α, x_α)-martingale problem, we have

Eα

X_t+s^α −X_t^α−

nα(t+s)−1

X

i=nα(t)

b_α(t^α_i, X_t^αα

i)_αY^l

i=1

h_i(X_t^α_i)

= 0. (23)

We are going to verify (22) by showing that the left-hand side of (23) converges to the left-hand side of (22) asα→ ∞. We begin with proving

α→∞lim Eα

X_u^α

l

Y

i=1

hi(X_t^α_i)

= E

Xu l

Y

i=1

hi(Xti)

(24) for every u ≥ 0, which together with (28) below implies the required convergence. To this end we setx^(N)= (−N ∨x)∧N for allx∈Rand N >0. The right-hand side of

Eα

X_u^α,(N)

l

Y

i=1

hi(X_t^α_i)

− Eα

X_u^α

l

Y

i=1

hi(X_t^α_i)

≤ Eα

X_u^α,(N)−X_u^α

l

Y

i=1

kh_ik_∞

can be estimated, for everyT ≥u, by sup

r≤T

sup

α⁰≥α0

Eα⁰

h

|X_r^α0|1_|Xr^α0_|>Ni Y^l

i=1

kh_ik_∞

which tends to 0 as N → ∞ since {X_r^α0 : r ≤ T, α⁰ ≥ 1} is uniformly integrable by (17).

Therefore we have

N→∞lim Eα

X_u^α,(N)

l

Y

i=1

hi(X_t^α_i)

= Eα

X_u^α

l

Y

i=1

hi(X_t^α_i)

uniformly in α≥α0 (25) (and uniformly inu≤T, for every T >0). By (20) we further obtain for everyN >0,

α→∞lim Eα

X_u^α,(N)

l

Y

i=1

hi(X_t^α_i)

= E

X_u^(N)

l

Y

i=1

hi(Xti)

(26)

since the mapping (x₁, . . . , x_l+1) 7→ x^(N_l+1⁾Ql

i=1h_i(x_i) from R^l+1 to R is bounded and con- tinuous. This is the reason why we introduced the truncation x^(N). By virtue of (21), an application of the dominated convergence theorem gives

Nlim→∞E

X_u^(N)

l

Y

i=1

h_i(X_ti)

= E

X_u

l

Y

i=1

h_i(X_ti)

(27) which along with (25) and (26) implies (24). It remains to show

α→∞lim Eα

^nα(t+s)−1 X

i=nα(t)

bα(t^α_i, X_t^αα

i)α l

Y

i=1

hi(X_t^α_i)

= E Z t+s

t

b(r, Xr)dr

l

Y

i=1

hi(Xti)

. (28) Taking (16) and (n_α(t+s)−n_α(t))_α ≤ s+_α into account we obtain analogously to (25) and (27),

Nlim→∞Eα

^nα(t+s)−1 X

i=nα(t)

b_α(t^α_i, X_t^α,(N)α i )_α

l

Y

i=1

h_i(X_t^α_i)

(29)

= Eα

^nα(t+s)−1 X

i=nα(t)

bα(t^α_i, X_t^αα

i)α l

Y

i=1

hi(X_t^α_i)

uniformly in α≥α0

respectively

N→∞lim E Z t+s

t

b(r, X_r^(N))dr

l

Y

i=1

h_i(X_ti)

= E Z t+s

t

b(r, X_r)dr

l

Y

i=1

h_i(X_ti)

. (30) By the uniform convergence ofbα tob and (nα(t+s)−nα(t))α ≤s+α, we also have

Eα

^nα(t+s)−1 X

i=nα(t)

b_α(t^α_i, X_t^α,(Nα ⁾ i )_α

l

Y

i=1

h_i(X_t^α_i)

=Eα

^nα(t+s)−1 X

i=nα(t)

b(t^α_i, X_t^α,(N)α i )_α

l

Y

i=1

h_i(X_t^α_i)

+o_α(1).

(31) Moreover we have

Eα

^nα(t+s)−1 X

i=nα(t)

b(t^α_i, X_t^α,(N)α i )_α

l

Y

i=1

h_i(X_t^α_i)

=Eα

Z t+s t

b(r, X_r^α,(N))dr

l

Y

i=1

h_i(X_t^α_i)

+ o_α(1) (32) which is a consequence of the dominated convergence theorem and

nα(t+s)−1

X

i=nα(t)

b(t^α_i, X_t^α,(N)α i )_α−

Z t+s t

b(r, X_r^α,(N))dr

≤

Z bt+sc_α−α

btc_α

b(brc_α, X_r^α,(N))−b(r, X_r^α,(N))

dr + o_α(1)

together with the fact that b is bounded and uniformly continuous on [0, t+s]×[−N, N].

Finally we get by (20) and the dominated convergence theorem and (17),

α→∞lim Eα

Z t+s t

b(r, X_r^α,(N))dr

l

Y

i=1

hi(X_t^α_i)

= Z t+s

t

α→∞lim Eα

b(r, X_r^α,(N))

l

Y

i=1

hi(X_t^α_i)

dr

=

Z t+s t

E

b(r, X_r^(N))

l

Y

i=1

hi(Xti)

dr = E Z t+s

t

b(r, X_r^(N))dr

l

Y

i=1

hi(Xti)

(33) which along with (31) and (32) implies

α→∞lim Eα

^nα(t+s)−1 X

i=nα(t)

bα(t^α_i, X_t^α,(Nα ⁾ i )α

l

Y

i=1

hi(X_t^α_i)

= E Z t+s

t

b(r, X_r^(N))dr

l

Y

i=1

hi(Xti)

.

This, (29) and (30) ensure (28).

Step 3. It remains to show (7). By the uniqueness of the Doob-Meyer decomposition,M has the required compensator if and only if

E h

M_t+s² −M_t²− Z t+s

t

a²(r, Xr)dr Y^l

i=1

hi(Xti) i

= 0 (34)

holds for all 0 ≤ t₁ <· · · ≤ t_l ≤t, s ≥ 0, l ≥ 1 and bounded h₁, . . . , h_l ∈C(R). Now, the discrete analogue of equation (34) forEα,aαandX^α holds. Proceeding similarly to the proof of (22) one can show that the left-hand side of this equation converges to the left-hand side of (34) asα→ ∞. Therefore we obtain (34). 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 (17) forq being strictly larger than 2. 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 Y^α 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 [3] Theorem 3.8.8 and [3] 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 [3]; we omit the details.

Lemma 5.3 Assume (Y_t^α) is tight in R for every rational t ≥ 0. Let m > 0, γ > 1 and assume for every T > 0 there is some finite constant CT > 0 such that for all α ≥ 1 and t, h≥0 with 0≤t−h andt+h≤T,

Qα

h

|Y_t−h^α −Y_t^α|^m/2|Y_t^α−Y_t+h^α |^m/2i

≤ CT h^γ. (35) Then(Qα) is tight.