Weak Approximation of SDEs by Discrete-Time Processes

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.

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):𝑑𝑋_𝑡=𝜎|||𝑋_𝑡|||𝑑𝑊_𝑡,(1.1)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,𝑑𝑋_𝑡=𝛿𝑡,𝑋_𝑡𝑑𝑡+𝜎𝑡,𝑋^𝑡|||𝑋_𝑡|||𝑑𝑊_𝑡,(1.2)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:𝑑𝑋_𝑡=𝑏𝑡,𝑋_𝑡𝑑𝑡+𝑎𝑡,𝑋^𝑡𝑑𝑊^𝑡,𝑋₀=𝑥₀,(1.3)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,(1.4)for some finite constant 𝐾>0. We assume that SDE (1.3) 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=(𝑋_𝑡∶𝑡≥0) is a real-valued continuous (ℱ_𝑡)-adapted process such that ℙ-almost surely,𝑋_𝑡=𝑥₀+^𝑡0𝑏𝑟,𝑋_𝑟𝑑𝑟+^𝑡0𝑎𝑟,𝑋_𝑟𝑑𝑊^𝑟∀𝑡≥0.(1.5)Here the latter is an Itô-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|||𝑎(𝑡,𝑥)−𝑎𝑡,𝑥_|||≤ℎ|||𝑥−𝑥^|||∀𝑡∈ℝ₊,𝑥,𝑥^∈ℝ,(1.6)for some strictly increasing ℎ∶ℝ₊→ℝ₊ with ∫⁰⁰⁺ℎ⁻²(𝑢)𝑑𝑢=∞. Note that (1.6) and Lipschitz continuity of 𝑏 even imply the existence of a strongly unique strong solution (Yamada-Watanabe criterion [7]). But the notion of strong solutions and strong uniqueness is beyond our interest.

Our starting point is the fact that any weak solution of (1.3) is a solution of the following martingale problem and vice versa (cf. [8, Section 5.4.B], or [9, Theorem 1.27]).

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, and 𝑋=(𝑋_𝑡∶𝑡≥0) is a real-valued continuous (ℱ_𝑡)-adapted process such that𝑀_𝑡=𝑋_𝑡−𝑥₀−^𝑡0𝑏𝑟,𝑋_𝑟𝑑𝑟(1.7)provides a (continuous, mean-zero) square-integrable (ℱ_𝑡)-martingale with compensator⟨𝑀⟩_𝑡=^𝑡0𝑎²𝑟,𝑋^𝑟𝑑𝑟.(1.8)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 (𝑌_𝛼,𝑍_𝛼)→(𝑌,𝑍) in the cádlàg space implies convergence in distribution ∫𝑌_𝛼𝑑𝑍_𝛼→∫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 𝑡^𝜖𝑛=𝑛𝜖 for every 𝑛∈ℕ₀ and 𝜖>0.

For every 𝛼∈ℕ we fix some 𝜖_𝛼>0 such that 𝜖_𝛼→0. For the sake of clarity, we also set 𝑡^𝛼𝑛=𝑡^𝜖𝛼_𝑛(=𝑛𝜖_𝛼) for all 𝑛∈ℕ₀. 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→𝑥₀, 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∈ℕ₀ with 𝑡^𝛼𝑛≤𝑡.

Definition 2.1. Suppose that 𝑋^𝛼=(𝑋^𝛼𝑡∶𝑡≥0) is a real-valued process on some probability space (Ω,ℱ,ℙ) whose trajectories are constant on the intervals [𝑡^𝛼𝑛,𝑡^𝛼𝑛+1), 𝑛∈ℕ₀. Then 𝑋^𝛼 is called a solution of the (𝜖_𝛼,𝑎_𝛼,𝑏_𝛼,𝑥_𝛼)-martingale problem if𝑀^𝛼𝑡=𝑋^𝛼𝑡−𝑥_𝛼−^{𝑛𝛼(𝑡)−1}_𝑖=0𝑏_𝛼𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖𝜖^𝛼(2.1)provides a (zero-mean) square-integrable martingale (with respect to the natural filtration) with compensator𝑀^𝛼_𝑡=^{𝑛𝛼(𝑡)−1}_𝑖=0𝑎^2𝛼𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖𝜖^𝛼.(2.2)

The 𝑋^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 that𝔼_𝛼|||𝑋^{𝛼𝑡𝛼𝑛}−𝑋^𝛼𝑡_𝛼𝑛−1|||^𝑞≤𝐶^𝑇1+𝔼^𝛼|||𝑋^{𝛼𝑡𝛼𝑛−1}|||^𝑞𝜖^𝛿𝛼(2.3)for every 𝛼≥1 and 𝑛∈ℕ with 𝑡^𝛼𝑛≤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^𝛼𝑡_𝛼𝑛|^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 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=(𝑍_𝑛∶𝑛∈ℕ₀) that can be defined recursively as follows. Choose an initial state 𝑍₀∈ℕ and set 𝑍_𝑛=∑^𝑍𝑛−1_𝑖=1𝑁_{𝑛−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 variance 𝜎². 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∈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 𝑍^𝜖𝑡_𝜖𝑛=𝜖𝑍_𝑛 we obtain a rescaled version, 𝑍^𝜖, of 𝑍. Recall 𝑡^𝜖𝑛=𝑛𝜖, hence 𝑍^𝜖 is a process having 𝜖ℕ₀={0,𝜖,2𝜖,…} as both its index set and its state space. Now pick (𝜖_𝛼)⊂ℝ₊ such that 𝜖_𝛼→0, and recall our convention 𝑡^𝛼𝑛=𝑡^𝜖𝛼_𝑛 and that ⌊𝑡⌋_𝜖 denotes the largest element 𝑠 of 𝜖ℕ₀ with 𝑠≤𝑡. Regard the process 𝑍^𝜖𝛼 as continuous-time process, 𝑋^𝛼, by setting 𝑋^𝛼𝑡=𝑍^𝜖𝛼_{⌊𝑡⌋𝜖𝛼}, and suppose that 𝑋^𝛼0=⌊𝑥₀⌋_𝜖𝛼. The latter requires that 𝑍₀ 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=𝜎²∑^{𝑛𝛼(𝑡)−1𝑖=0}𝑋^𝛼𝑡_𝛼𝑖𝜖_𝛼 since, in this case,𝔼_𝛼𝑀^{𝛼𝑡𝛼𝑛}²−𝑀^𝛼_𝑡𝛼𝑛−𝑀^{𝛼𝑡𝛼𝑛−1}²−𝑀^𝛼_{𝑡𝛼𝑛−1}∣ℱ^{𝑋𝛼𝑡𝛼𝑛−1}=0(3.1)can be checked easily with help of𝔼_𝛼𝑋^{𝛼𝑡𝛼𝑛}∣𝑋^𝛼𝑡_𝛼𝑛−1=𝑋^{𝛼𝑡𝛼𝑛−1},𝕍𝑎𝑟_𝛼𝑋^{𝛼𝑡𝛼𝑛}∣𝑋^𝛼𝑡_𝛼𝑛−1=𝜎²𝑋^𝛼𝑡_𝛼𝑛−1𝜖_𝛼.(3.2)The formulae in (3.2) are immediate consequences of the well-known moment formulae for 𝑍 (see [17, page 6]) and (ℱ^𝑋𝛼_𝑡𝛼𝑛) 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=⌊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[𝜉_𝑖]=0 and sup_𝑖∈ℕ𝔼[𝜉^4𝑖]<∞. Let 𝜈 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 𝜉_𝑖, such that 𝔼[(∑^𝜈𝑖=1𝜉_𝑖)⁴]≤𝐶𝔼[𝜈²].

Proof. By the finiteness of the fourth moments the law of total expectation yields𝔼[(^𝜈_𝑖=1𝜉_𝑖)⁴]=_{𝑛𝑛∈ℕ𝑖}_1𝑛=1_𝑖2𝑛=1_𝑖3𝑛=1_𝑖4=1𝔼𝜉_𝑖1𝜉_𝑖2𝜉_𝑖3𝜉_𝑖4ℙ[𝜈=𝑛].(3.3)Since the 𝜉_𝑖 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[(^𝜈_𝑖=1𝜉_𝑖)⁴]≤_𝑛∈ℕ𝑛+3𝑛𝑛−1)sup_{𝑖,𝑗∈ℕ}𝔼𝜉^2𝑖𝜉^2𝑗ℙ[𝜈=𝑛]≤4sup_{𝑖,𝑗∈ℕ}𝔼𝜉^2𝑖𝜉^2𝑗𝔼𝜈².(3.4)This yields the claim of the lemma with 𝐶=4sup_{𝑖,𝑗∈ℕ}𝔼[𝜉^2𝑖𝜉^2𝑗].

With help of Lemma 3.1 we obtain𝔼_𝛼|||𝑋^{𝛼𝑡𝛼𝑛}−𝑋^𝛼𝑡_𝛼𝑛−1|||⁴==𝔼𝔼_𝛼𝛼[|^{𝜖𝛼−1𝑋𝛼𝑡𝛼𝑛−1}_𝑖=1𝜖^𝛼𝑁_{𝑛−1,𝑖}−𝜖_𝛼|⁴][|^{𝜖𝛼−1𝑋𝛼𝑡𝛼𝑛−1}_𝑖=1𝑁^{𝑛−1,𝑖}−1|⁴]𝜖^4𝛼≤𝐶𝔼_𝛼𝜖^𝛼−1𝑋^𝛼𝑡_𝛼𝑛−1²𝜖^4𝛼≤𝐶1+𝔼_𝛼𝑋^{𝛼𝑡𝛼𝑛−1}⁴𝜖^2𝛼(3.5)for 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 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):𝑋^𝜖𝑡_𝜖𝑛−𝑋^𝜖𝑡_𝜖𝑛−1=𝑏𝑡^𝜖𝑛−1,𝑋^𝜖𝑡_𝜖𝑛−1𝜖+𝑎𝑡^𝜖𝑛−1,𝑋^𝜖𝑡_𝜖𝑛−1𝑉^{𝜖𝑡𝜖𝑛},𝑋^𝜖𝑡_𝜖0=𝑥_𝜖.(4.1)Here, (𝑥_𝜖) is a sequence in ℝ satisfying 𝑥_𝜖→𝑥₀ as 𝜖→0, and 𝑉^𝜖={𝑉^𝜖𝑡_𝜖𝑛∶𝑛∈ℕ} is a family of independent centered random variables with variance 𝜖 and 𝔼_𝜖[|𝑉^𝜖𝑡_𝜖𝑛|^𝑞]≤𝐶𝜖^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^𝜖𝑡_𝜖𝑛=√𝜖𝜉_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^𝜖𝑡_𝜖𝑛∶𝑛∈ℕ} are identically distributed. Below we will see that the independence is necessary neither.

By virtue of (1.4), 𝑋^𝜖𝑡_𝜖𝑛 has a finite 𝑞th moment if 𝑋^𝜖𝑡_𝜖𝑛−1 has. It follows by induction that the solution 𝑋^𝜖=(𝑋^𝜖𝑡_𝜖𝑛∶𝑛∈ℕ₀) of (4.1) is 𝑞-integrable, and hence square integrable. Equation (4.1) is obviously equivalent to the stochastic sum equation𝑋^𝜖𝑡_𝜖𝑛=𝑥_𝜖+^𝑛−1_𝑖=0𝑏𝑡^𝜖𝑖,𝑋^𝜖𝑡_𝜖𝑖𝜖+^𝑛−1_𝑖=0𝑎𝑡^𝜖𝑖,𝑋^𝜖𝑡_𝜖𝑖𝑉^{𝜖𝑡𝜖𝑖+1}.(4.2)Suppose that (𝜖_𝛼) is an arbitrary sequence with 𝜖_𝛼∈(0,1] and 𝜖_𝛼→0, set 𝑥_𝛼=𝑥_𝜖𝛼 and recall our convention 𝔼_𝛼=𝔼_𝜖𝛼, 𝑋^𝛼≡𝑋^𝜖𝛼, 𝑡^𝛼𝑛=𝑡^𝜖𝛼_𝑛. Then it is easy to see that 𝑀^𝛼 defined in (2.1) provides a (mean-zero) square-integrable (ℱ^𝑋𝛼_𝑡)-martingale. Moreover, 𝑀^𝛼𝑡_𝛼𝑛 coincides with the second sum on the right-hand side of (4.2). Therefore, we also obtain𝑀^𝛼_𝑡𝛼𝑛==^𝑛_{𝑖=1𝑛−1}𝔼_𝛼𝑎𝑡^𝛼𝑖−1,𝑋^𝛼𝑡_𝛼𝑖−1𝑉^{𝜖𝛼𝑡𝛼𝑖}²∣ℱ^𝑋𝛼_{𝑡𝛼𝑛−1}_𝑖=0=_𝑛−1𝑎²𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖𝔼^𝛼𝑉^{𝜖𝛼𝑡𝛼𝑖+1}²_𝑖=0𝑎²𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖𝜖^𝛼(4.3)which 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 from𝔼_𝛼|||𝑋^{𝛼𝑡𝛼𝑛}−𝑋^𝛼𝑡_𝛼𝑛−1|||^𝑞≤2_𝑞−1𝔼^𝛼|||𝑏𝑡^𝛼𝑛−1,𝑋^𝛼𝑡_𝛼𝑛−1𝜖^𝛼|||^𝑞+𝔼^𝛼|||𝑎𝑡^𝛼𝑛−1,𝑋^𝛼𝑡_𝛼𝑛−1|||^𝑞𝔼_𝛼|||𝑉^{𝜖𝛼𝑡𝛼𝑛}|||^𝑞≤2_𝑞−1𝐾2^𝑞−11+𝔼^𝛼|||𝑋^{𝛼𝑡𝛼𝑛−1}|||^𝑞𝜖^𝑞𝛼+𝐾2^𝑞−11+𝔼^𝛼|||𝑋^{𝛼𝑡𝛼𝑛−1}|||^𝑞𝐶𝜖^𝛼𝑞/2(4.4)for which we used (4.1), the independence of 𝑋^𝛼𝑡_𝛼𝑛−1 of 𝑉^𝜖𝛼, (1.4), and 𝔼_𝛼[|𝑉^𝜖𝛼_𝑡𝛼𝑛|^𝑞]≤𝐶𝜖^𝛼𝑞/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^𝜖𝑡_𝜖𝑛∶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^𝜖𝑡_𝜖𝑛 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^𝜖𝑡_𝜖1=√𝜖𝜉₁(1) and 𝑉^𝜖𝑡_𝜖𝑛=√𝜖𝜉_𝑛(𝑓_𝑛(𝑉^𝜖𝑡_𝜖1,…,𝑉^𝜖𝑡_𝜖𝑛−1)), 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 conditioning𝔼_𝛼𝑉^{𝜖𝛼𝑡𝛼𝑛}∣ℱ^𝑋𝛼_{𝑡𝛼𝑛−1}=𝔼0,𝔼^𝛼𝑉^{𝜖𝛼𝑡𝛼𝑖+1}²∣ℱ^𝑋𝛼_{𝑡𝛼𝑛−1}=𝜖^𝛼,1≤𝑖≤𝑛−1,_𝛼|||𝑎𝑡^𝛼𝑛−1,𝑋^𝛼𝑡_𝛼𝑛−1𝑉^{𝜖𝛼𝑡𝛼𝑛}|||^𝑞≤𝐶𝜖^𝛼𝑞/2.(4.5)If the 𝜉_𝑛(𝑖) are not identically distributed, then the 𝑉^𝜖𝑡_𝜖𝑛 are typically not independent. In particular, the approximating process 𝑋^𝜖 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 exist 𝐾^>0 and 𝛼₀≥1 such that for all 𝛼≥𝛼₀, 𝑡≥0, and 𝑥∈ℝ,|||𝑎_𝛼(𝑡,𝑥)|||+|||𝑏_𝛼(𝑡,𝑥)|||≤𝐾^1+||𝑥||.(5.1)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 |∑^𝑚𝑖=1𝑦_𝑖|^𝑝≤𝑚^𝑝−1∑^𝑚𝑖=1|𝑦_𝑖|^𝑝 for all 𝑚∈ℕ, 𝑝≥1 and 𝑦₁,…,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 𝜖ℕ₀={0,𝜖,2𝜖,…} which is smaller than or equal to 𝑥. Moreover, we assume without loss of generality that 𝜖_𝛼≤1.

Lemma 5.1. For 𝑞>2 and 𝛿>1 satisfying (2.3) and every 𝑇>0, sup_𝛼≥𝛼0𝔼_𝛼sup_𝑡≤𝑇|||𝑋_𝛼𝑡|||^𝑞<∞.(5.2)

Proof. First of all, note that for the proof it actually suffices to require 𝑞≥2 and 𝛿≥1. Set 𝑆=sup_𝛼≥𝛼0|𝑥_𝛼|^𝑞 and 𝑆^𝛼𝑡=𝔼_𝛼[max_{1≤𝑖≤𝑛𝛼(𝑡)}|𝑀^𝛼𝑡_𝛼𝑖−𝑀^𝛼𝑡_𝛼𝑖−1|^𝑞]. Using Proposition A.1 in the appendix and (5.1) we obtain, for all 𝑡>0 and 𝛼≥𝛼₀,𝔼_𝛼sup_{𝑖≤𝑛𝛼(𝑡)}|||𝑋_𝛼𝑡𝑖|||^𝑞≤3_𝑞−1{𝔼_𝛼sup_{𝑖≤𝑛𝛼(𝑡)}|||𝑀_𝛼𝑡𝑖|||^𝑞+𝑆+𝔼^𝛼[(∑^{𝑛𝛼(𝑡)−1}_𝑖=0|||𝑏_𝛼≤3_𝑞−1𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖|||𝜖^𝛼)^𝑞]}≤𝑘_𝑞𝐶_𝑞{𝔼_𝛼[|∑^{𝑛𝛼(𝑡)−1}_𝑖=0𝑎^2𝛼𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖𝜖^𝛼|^𝑞/2]+𝑆^𝛼𝑡+𝑆+𝔼_𝛼[[∑^{𝑛𝛼(𝑡)−1}_𝑖=0|𝑏_𝛼𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖|𝜖^𝛼]^𝑞]}{𝔼_𝛼[[∑^{𝑛𝛼(𝑡)−1}_𝑖=0𝐾^1+|||𝑋^{𝛼𝑡𝛼𝑖}|||²𝜖_𝛼]^𝑞/2]+𝑆^𝛼𝑡+𝑆+𝔼_𝛼[[∑^{𝑛𝛼(𝑡)−1}_𝑖=0𝐾^1+|||𝑋^{𝛼𝑡𝛼𝑖}|||𝜖_𝛼]^𝑞]},(5.3)where 𝐶_𝑞 is independent of 𝑡 and 𝛼, and 𝑘_𝑞=3^𝑞−1𝐶_𝑞. By Hölder's inequality we get𝔼_𝛼[[∑^{𝑛𝛼(𝑡)−1}_𝑖=0𝐾^1+|||𝑋^{𝛼𝑡𝛼𝑖}|||²≤𝔼_𝛼𝜖_𝛼]^𝑞/2][(∑^{𝑛𝛼(𝑡)−1}_𝑖=02𝐾^21+|||𝑋^{𝛼𝑡𝛼𝑖}|||²≤𝔼_𝛼^𝑞/2)(∑^{𝑛𝛼(𝑡)−1}_𝑖=0𝜖^{𝛼(𝑞/2)/(𝑞/2−1)})^𝑞/2−1][(∑^{𝑛𝛼(𝑡)−1}_𝑖=02^𝑞/2−12𝐾^2^𝑞/21+|||𝑋^{𝛼𝑡𝛼𝑖}|||^𝑞≤𝑐_𝑞)𝑛^𝛼(𝑡)^𝑞/2−1𝜖^𝛼𝑞/2]𝑡^𝑞/2+𝑐_𝑞𝑡∑^{𝑛𝑞/2−1𝛼(𝑡)−1}_𝑖=0𝔼_𝛼sup_𝑗≤𝑖|||𝑋_{𝛼𝑡𝛼𝑗}|||^𝑞𝜖^𝛼,(5.4)where 𝑐_𝑞=2^𝑞/2−1(2𝐾^2)^𝑞/2. Analogously, with 𝑐_𝑞=2^𝑞−1𝐾^𝑞,𝔼_𝛼[[^{𝑛𝛼(𝑡)−1}_𝑖=0𝐾^1+|||𝑋^{𝛼𝑡𝛼𝑖}|||𝜖_𝛼]^𝑞]≤𝑐_𝑞𝑡^𝑞+𝑐_𝑞𝑡^{𝑛𝑞−1𝛼(𝑡)−1}_𝑖=0𝔼_𝛼sup_𝑗≤𝑖|||𝑋_{𝛼𝑡𝛼𝑗}|||^𝑞𝜖^𝛼.(5.5)Moreover, by (2.3) and (5.1) we obtain, for all 𝑡≤𝑇 and 𝛼≥𝛼₀,𝑆^𝛼𝑡≤^{𝑛𝛼(𝑡)}_𝑖=1𝔼_𝛼|||𝑀^{𝛼𝑡𝛼𝑖}−𝑀^𝛼𝑡_𝛼𝑖−1|||^𝑞≤2_{𝑛𝑞−1𝛼(𝑡)}_𝑖=1𝔼_𝛼|||𝑋^{𝛼𝑡𝛼𝑖}−𝑋^𝛼𝑡_𝛼𝑖−1|||^𝑞≤2_{𝑛𝑞−1𝛼(𝑡)−1}+|||𝑏_𝛼𝑡^𝛼𝑖−1,𝑋^𝛼𝑡_𝛼𝑖−1|||^𝑞𝜖^𝑞𝛼_𝑖=0𝐶^𝑇1+𝔼^𝛼|||𝑋^{𝛼𝑡𝛼𝑖}|||^𝑞𝜖^𝛿𝛼+𝔼_𝛼𝐾^1+|||𝑋^{𝛼𝑡𝛼𝑖}|||^𝑞≤𝑐_𝑞,𝑇𝜖^𝑞𝛼𝑡+𝑐_{𝑛𝑞,𝑇}^{𝛼(𝑡)−1}_𝑖=0𝔼_𝛼sup_𝑗≤𝑖|||𝑋_{𝛼𝑡𝛼𝑗}|||^𝑞𝜖^𝛼,(5.6)where 𝑐_𝑞,𝑇=2^𝑞−1(𝐶_𝑇+𝐾^). By all account we have, for all 𝑡≤𝑇 and 𝛼≥𝛼₀,𝔼_𝛼sup_{𝑖≤𝑛𝛼(𝑡)}|||𝑋_𝛼𝑡𝑖|||^𝑞≤𝑘^𝑞{𝑆+𝑐_𝑞+𝑐_𝑞+𝑐_𝑞,𝑇𝑡^𝑞−1∨1(1+𝜖^{𝛼𝑛𝛼(𝑡)−1}_𝑖=0𝔼_𝛼sup_𝑗≤𝑖||𝑋^𝛼𝑡_𝛼𝑗||^𝑞≤𝑘)}_𝑞𝑆+𝐶_𝑞,𝑇+𝐶^𝑞,𝑇𝜖_𝛼𝑛^{𝛼(𝑡)−1}_𝑖=0𝔼_𝛼sup_𝑗≤𝑖||𝑋^𝛼𝑡_𝛼𝑗||^𝑞,(5.7)where 𝐶_𝑞,𝑇=𝑘_𝑞(𝑐_𝑞+𝑐_𝑞+𝑐_𝑞,𝑇)(𝑇^𝑞−1∨1). An application of Lemma A.2 yields𝔼_𝛼sup_𝑠≤𝑡||𝑋^𝛼𝑠||^𝑞=𝔼^𝛼sup_{𝑖≤𝑛𝛼(𝑡)}||𝑋^𝛼𝑡_𝑖||^𝑞≤𝑘^𝑞𝑆+𝐶_𝑞,𝑇1+𝐶^𝑞,𝑇𝜖_𝛼^{𝑛𝛼(𝑡)}+𝐶_𝑞,𝑇𝜖_𝛼^{𝑛𝛼(𝑡)}𝑆,(5.8)where we emphasize that the constants 𝑘_𝑞, 𝑆, and 𝐶_𝑞,𝑇 are independent of 𝑡≤𝑇 and 𝛼≥𝛼₀. This proves Lemma 5.1 since limsup_𝛼→∞(1+𝐶_𝑞,𝑇𝜖_𝛼)^{𝑛𝛼(𝑡)} is bounded by exp(𝑡𝐶_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,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 (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] thatℙ_𝛼∘𝜋^𝑡−1_{1,…,𝑡𝑘}⇒ℙ∘𝜋^𝑡−1_{1,…,𝑡𝑘}(5.9)for all 𝑡₁,…,𝑡_𝑘∈ℝ₊, where 𝜋_{𝑡1,…,𝑡𝑘}∶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^𝑋𝑡)-martingale and has ⟨𝑀⟩ defined in (1.8) as compensator. Here, (ℱ^𝑋𝑡) denotes the natural augmentation of the filtration (ℱ^𝑋𝑡) 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,sup_𝑡≤𝑇𝔼|||𝑋_𝑡|||^𝑞≤sup_𝑡≤𝑇liminf_𝑁→∞lim_𝛼→∞𝔼_𝛼|||𝑋^𝛼𝑡|||^𝑞𝑁≤sup_𝑡≤𝑇sup_𝛼≥𝛼0𝔼_𝛼|||𝑋^𝛼𝑡|||^𝑞<∞.(5.10)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^𝑋𝑡)-martingale. It suffices to show that 𝑀 is an (ℱ^𝑋𝑡)-martingale; see [20, page 75]. The latter is true if and only if𝔼𝑋_𝑡+𝑠−𝑋_𝑡−^{𝑡𝑡+𝑠}𝑏𝑟,𝑋_𝑟𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖=0(5.11)holds for all 0≤𝑡₁<⋯≤𝑡_𝑙≤𝑡, 𝑠≥0, 𝑙≥1 and bounded ℎ₁,…,ℎ_𝑙∈𝐶(ℝ) (do not confuse 𝑡_𝑖 and 𝑡^𝛼𝑖). Since 𝑋^𝛼 solves the (𝜖_𝛼,𝑎_𝛼,𝑏_𝛼,𝑥_𝛼)-martingale problem, we have𝔼_𝛼[(𝑋^{𝛼𝑡+𝑠}−𝑋^𝛼𝑡−^{𝑛𝛼(𝑡+𝑠)−1}_𝑖=_{𝑛𝛼(𝑡)}𝑏_𝛼𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖𝜖^𝛼)^𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖]=0.(5.12)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 provinglim_𝛼→∞𝔼_𝛼𝑋^𝛼𝑢𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖=𝔼𝑋^𝑢𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖(5.13)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|||𝔼_𝛼𝑋^{𝑢𝑙𝛼,(𝑁)}_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖−𝔼^𝛼𝑋^𝛼𝑢𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖|||≤𝔼^𝛼|||𝑋^{𝑢𝛼,(𝑁)}−𝑋^𝛼𝑢|||^𝑙_𝑖=1‖‖ℎ_𝑖‖‖_∞(5.14)can be estimated, for every 𝑇≥𝑢, bysup_𝑟≤𝑇sup_{𝛼′≥𝛼0}𝔼_𝛼′|||𝑋^𝛼′𝑟|||1||_{𝑋𝛼′𝑟}||_>𝑁^𝑙_𝑖=1‖‖ℎ_𝑖‖‖_∞(5.15)which tends to 0 as 𝑁→∞ since {𝑋^𝛼′_𝑟∶𝑟≤𝑇,𝛼^≥1} is uniformly integrable by (5.2). Therefore, we havelim_𝑁→∞𝔼_𝛼𝑋^{𝑢𝑙𝛼,(𝑁)}_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖=𝔼^𝛼𝑋^𝛼𝑢𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖uniformlyin𝛼≥𝛼⁰(5.16)(and uniformly in 𝑢≤𝑇, for every 𝑇>0). By (5.9) we further obtain for every 𝑁>0,lim_𝛼→∞𝔼_𝛼𝑋^{𝑢𝑙𝛼,(𝑁)}_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖=𝔼𝑋^{𝑢𝑙(𝑁)}_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖(5.17)since the mapping (𝑥₁,…,𝑥_𝑙+1)↦𝑥^(𝑁)𝑙+1∏^𝑙𝑖=1ℎ_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 giveslim_𝑁→∞𝔼𝑋^{𝑢𝑙(𝑁)}_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖=𝔼𝑋^𝑢𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖(5.18)which along with (5.16) and (5.17) implies (5.13). It remains to showlim_𝛼→∞𝔼_𝛼[^{𝑛𝛼(𝑡+𝑠)−1}_𝑖=_{𝑛𝛼(𝑡)}𝑏_𝛼𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖𝜖^𝛼𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖]=𝔼^{𝑡𝑡+𝑠}𝑏𝑟,𝑋_𝑟𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖.(5.19)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),lim_𝑁→∞=𝔼𝔼^𝛼_𝛼[∑^{𝑛𝛼(𝑡+𝑠)−1}_{𝑖=𝑛𝛼(𝑡)}𝑏_𝛼𝑡^𝛼𝑖,𝑋^{𝑡𝛼,(𝑁)}_𝛼𝑖𝜖∏_{𝛼𝑙𝑖=1}ℎ_𝑖𝑋^𝛼𝑡_𝑖][∑^{𝑛𝛼(𝑡+𝑠)−1}_{𝑖=𝑛𝛼(𝑡)}𝑏_𝛼𝑡^𝛼𝑖,𝑋^𝛼𝑡_𝛼𝑖𝜖∏_{𝛼𝑙𝑖=1}ℎ_𝑖𝑋^𝛼𝑡_𝑖]uniformlyin𝛼≥𝛼⁰,(5.20)respectively,lim_𝑁→∞𝔼^{𝑡𝑡+𝑠}𝑏𝑟,𝑋^𝑟(𝑁)𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖=𝔼^{𝑡𝑡+𝑠}𝑏𝑟,𝑋_𝑟𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖.(5.21)By the uniform convergence of 𝑏_𝛼 to 𝑏 and (𝑛_𝛼(𝑡+𝑠)−𝑛_𝛼(𝑡))𝜖_𝛼≤𝑠+𝜖_𝛼, we also have𝔼_𝛼[^{𝑛𝛼(𝑡+𝑠)−1}_𝑖=_{𝑛𝛼(𝑡)}𝑏_𝛼𝑡^𝛼𝑖,𝑋^{𝑡𝛼,(𝑁)}_𝛼𝑖𝜖^𝛼𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖=𝔼^𝛼[^{𝑛𝛼(𝑡+𝑠)−1}_𝑖=_{𝑛𝛼(𝑡)}𝑏𝑡^𝛼𝑖,𝑋^{𝑡𝛼,(𝑁)}_𝛼𝑖𝜖^𝛼𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖+𝑜^𝛼(1).(5.22)Moreover, we have𝔼_𝛼[^{𝑛𝛼(𝑡+𝑠)−1}_𝑖=_{𝑛𝛼(𝑡)}𝑏𝑡^𝛼𝑖,𝑋^{𝑡𝛼,(𝑁)}_𝛼𝑖𝜖^𝛼𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖=𝔼^𝛼^{𝑡𝑡+𝑠}𝑏𝑟,𝑋^{𝑟𝛼,(𝑁)}𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖+𝑜^𝛼(1)(5.23)which is a consequence of the dominated convergence theorem and|∑≤_{𝑖=𝑛𝛼(𝑡)}∫^{𝑛𝛼(𝑡+𝑠)−1}𝑏⌊𝑡^𝛼𝑖,𝑋^{𝑡𝛼,(𝑁)}_𝛼𝑖𝜖^𝛼−∫^{𝑡𝑡+𝑠}𝑏𝑟,𝑋^{𝑟𝛼,(𝑁)}𝑑𝑟|_𝑡+𝑠⌋_{𝜖𝛼−𝜖𝛼}⌊_𝑡⌋_𝜖𝛼|||𝑏⌊𝑟⌋_𝜖𝛼,𝑋^{𝑟𝛼,(𝑁)}−𝑏𝑟,𝑋^{𝑟𝛼,(𝑁)}|||𝑑𝑟+𝑜^𝛼(1)(5.24)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) lim_𝛼→∞𝔼_𝛼^{𝑡𝑡+𝑠}𝑏𝑟,𝑋^{𝑟𝛼,(𝑁)}𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖=^{𝑡𝑡+𝑠}lim_𝛼→∞𝔼_𝛼𝑏𝑟,𝑋^{𝑟𝛼,(𝑁)}^𝑙_𝑖=1=ℎ_𝑖𝑋^𝛼𝑡_𝑖𝑑𝑟_{𝑡𝑡+𝑠}𝔼𝑏𝑟,𝑋^𝑟(𝑁)^𝑙_𝑖=1=𝔼_{𝑡𝑡+𝑠}ℎ_𝑖𝑋^𝑡_𝑖𝑑𝑟𝑏𝑟,𝑋^𝑟(𝑁)𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖(5.25)which along with (5.22) and (5.23) implieslim_𝛼→∞𝔼_𝛼[^{𝑛𝛼(𝑡+𝑠)−1}_𝑖=_{𝑛𝛼(𝑡)}𝑏_𝛼𝑡^𝛼𝑖,𝑋^{𝑡𝛼,(𝑁)}_𝛼𝑖𝜖^𝛼𝑙_𝑖=1ℎ_𝑖𝑋^𝛼𝑡_𝑖]=𝔼^{𝑡𝑡+𝑠}𝑏𝑟,𝑋^𝑟(𝑁)𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖.(5.26)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𝑀^2𝑡+𝑠−𝑀^2𝑡−^{𝑡𝑡+𝑠}𝑎²𝑟,𝑋^𝑟𝑑𝑟^𝑙_𝑖=1ℎ_𝑖𝑋^𝑡_𝑖=0(5.27)holds for all 0≤𝑡₁<⋯≤𝑡_𝑙≤𝑡, 𝑠≥0, 𝑙≥1 and bounded ℎ₁,…,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^𝛼𝑟)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 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^𝛼𝑡) 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, ℚ_𝛼|||𝑌^{𝛼𝑡−ℎ}−𝑌^𝛼𝑡|||^𝑚/2|||𝑌_𝛼𝑡−𝑌^𝛼𝑡+ℎ|||^𝑚/2≤𝐶^𝑇ℎ^𝛾.(5.28)Then (ℚ_𝛼) is tight.

Lemma 5.4. 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 0≤𝑡^≤𝑡^≤𝑇, limsup_𝛼→∞ℚ_𝛼|||𝑌^𝛼𝑡′−𝑌^𝛼𝑡_′′|||^𝑚≤𝐶^𝑇𝑡^−𝑡^^𝛾.(5.29)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. 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 𝐶_𝑞>0 such that for every 0≤𝑡^≤𝑡^ and 𝛼≥𝛼₀,𝔼_𝛼|||𝑋^𝛼𝑡′−𝑋^𝛼𝑡_′′|||^𝑞≤𝐶^𝑞{𝜖^𝛼𝑞/2𝔼_𝛼[(^{𝑛𝛼(𝑡′′)−1}_{𝑖=𝑛𝛼(𝑡′)}1+|||𝑋^{𝛼𝑡𝛼𝑖}|||²+𝜖_𝑞𝛼)^𝑞/2]𝔼_𝛼[(^{𝑛𝛼(𝑡′′)−1}_𝑖=_{𝑛𝛼(𝑡′)}1+|||𝑋^{𝛼𝑡𝛼𝑖}|||)_𝑞]+^{𝑛𝛼(𝑡′′)−1}_𝑖=_{𝑛𝛼(𝑡′)}1+𝔼^𝛼|||𝑋^{𝛼𝑡𝛼𝑖}|||^𝑞𝜖^{𝛿⋀𝑞𝛼}}.(5.30)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≤𝐶_𝑞{𝜖^𝛼𝑞/2(∑^{𝑛𝛼(𝑡′′)−1}_{𝑖=𝑛𝛼(𝑡′)}𝔼_𝛼1+|||𝑋^{𝛼𝑡𝛼𝑖}|||^2𝑞/2)(∑^{𝑛𝛼(𝑡′′)−1}_{𝑖=𝑛𝛼(𝑡′)}1^{𝑞/(𝑞−2)})^𝑞/2−1+𝜖^𝑞𝛼(∑^{𝑛𝛼(𝑡′′)−1}_{𝑖=𝑛𝛼(𝑡′)}𝔼_𝛼1+|||𝑋^{𝛼𝑡𝛼𝑖}|||^𝑞)(∑^{𝑛𝛼(𝑡′′)−1}_{𝑖=𝑛𝛼(𝑡′)}1^{𝑞/(𝑞−1)})^𝑞−1+∑^{𝑛𝛼(𝑡′′)−1}_{𝑖=𝑛𝛼(𝑡′)}(1+𝔼_𝛼|||𝑋^{𝛼𝑡𝛼𝑖}|||^𝑞≤𝐶_𝑞,𝑇)𝜖^{𝛿⋀𝑞𝛼}}≤𝜖^𝛼𝑞/2𝑛^𝛼𝑡^−𝑛^𝛼𝑡^^𝑞/2+𝜖^𝑞𝛼𝑛^𝛼𝑡^−𝑛^𝛼𝑡^^𝑞+𝜖^{𝛿⋀𝑞𝛼}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,(5.31)where 𝐶_𝑞,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 (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 get𝔼_𝛼|||𝑋^{𝛼𝑡−ℎ}−𝑋^𝛼𝑡|||^𝑞/2|||𝑋_𝛼𝑡−𝑋^𝛼𝑡+ℎ|||^𝑞/2≤𝔼^𝛼|||𝑋^{𝛼𝑡−ℎ}−𝑋^𝛼𝑡|||^𝑞^1/2𝔼_𝛼|||𝑋^𝛼𝑡−𝑋^𝛼𝑡+ℎ|||^𝑞^1/2.(5.32)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(3ℎ)^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,𝔼_𝛼|||𝑋^{𝛼𝑡−ℎ}−𝑋^𝛼𝑡|||^𝑞/2|||𝑋_𝛼𝑡−𝑋^𝛼𝑡+ℎ|||^𝑞/2≤𝐶^2𝑞,𝑇3^𝛾ℎ^𝛾(5.33)for 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 (ℙ_𝛼).

Appendix

Auxiliary Results

Here we give two auxiliary results. We first recall a square function inequality for martingales. Let 𝑀=(𝑀_𝑛∶𝑛∈ℕ₀) be an (ℱ_𝑛)-martingale on some probability space (Ω,ℱ,ℙ). The corresponding compensator is given by ⟨𝑀⟩_𝑛=∑^𝑛𝑖=1𝔼[(𝑀_𝑖−𝑀_𝑖−1)²∣ℱ_𝑖−1].

Proposition A.1 (see [21, Theorem 2.11]). For every 𝑞>0 there is some finite constant 𝐶_𝑞>0 depending only on 𝑞 such that 𝔼max_{1≤𝑖≤𝑛}|||𝑀_𝑖|||^𝑞≤𝐶^𝑞𝔼𝑀^𝑛𝑞/2+𝔼max_{1≤𝑖≤𝑛}|||𝑀_𝑖−𝑀_𝑖−1|||^𝑞.(A.1)

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 that 𝑔(𝑛)≤𝑐₀+𝑐_1𝑛−1_𝑖=0𝑔(𝑖)∀𝑛≤𝑁,(A.2)then 𝑔(𝑛)≤𝑐₀1+𝑐¹^𝑛+𝑐^𝑛1𝑔(0)<∞∀𝑛≤𝑁.(A.3)

Acknowledgment

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