International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 275747 |

Henryk Zähle, "Weak Approximation of SDEs by Discrete-Time Processes", International Journal of Stochastic Analysis, vol. 2008, Article ID 275747, 15 pages, 2008.

Weak Approximation of SDEs by Discrete-Time Processes

Academic Editor: Nikolai Leonenko
Received29 Oct 2007
Revised25 Jan 2008
Accepted20 Feb 2008
Published23 Mar 2008


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,(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,𝑑𝑋𝑡=𝛿𝑡,𝑋𝑡𝑑𝑡+𝜎𝑡,𝑋𝑡|||𝑋𝑡|||𝑑𝑊𝑡,(1.2)is weakly approximated by two different types of population-size-dependent GWPs (in the sense of [36]) with immigration, where 𝛿 and 𝜎 are suitable nonnegative continuous functions on +×. 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=𝑏𝑡,𝑋𝑡𝑑𝑡+𝑎𝑡,𝑋𝑡𝑑𝑊𝑡,𝑋0=𝑥0,(1.3)where 𝑥0 and 𝑊 is a one-dimensional Brownian motion. The coefficients 𝑎 and 𝑏 are continuous functions on +× satisfying||||||+||||||||𝑥||𝑎(𝑡,𝑥)𝑏(𝑡,𝑥)𝐾1+𝑡+,𝑥,(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,,(𝑡),)} where (Ω,,(𝑡),) is a filtered probability space with (𝑡) satisfying the usual conditions, 𝑊=(𝑊𝑡𝑡0) is an (𝑡)-Brownian motion, and 𝑋=(𝑋𝑡𝑡0) is a real-valued continuous (𝑡)-adapted process such that -almost surely,𝑋𝑡=𝑥0+𝑡0𝑏𝑟,𝑋𝑟𝑑𝑟+𝑡0𝑎𝑟,𝑋𝑟𝑑𝑊𝑟𝑡0.(1.5)Here the latter is an It̂o-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)𝑎𝑡,𝑥||||||𝑥𝑥|||𝑡+,𝑥,𝑥,(1.6)for some strictly increasing ++ with 00+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,,(𝑡),)} is said to be a solution of the (𝑎, 𝑏, 𝑥0)-martingale problem if (Ω,,(𝑡),) is a filtered probability space with (𝑡) satisfying the usual conditions, and 𝑋=(𝑋𝑡𝑡0) is a real-valued continuous (𝑡)-adapted process such that𝑀𝑡=𝑋𝑡𝑥0𝑡0𝑏𝑟,𝑋𝑟𝑑𝑟(1.7)provides a (continuous, mean-zero) square-integrable (𝑡)-martingale with compensator𝑀𝑡=𝑡0𝑎2𝑟,𝑋𝑟𝑑𝑟.(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., [1012]). 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 𝑌𝛼𝑑𝑍𝛼𝑌𝑑𝑍 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() the space of cádlàg functions from + to . We equip 𝐷() 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 𝑡𝜖𝑛=𝑛𝜖 for every 𝑛0 and 𝜖>0.

For every 𝛼 we fix some 𝜖𝛼>0 such that 𝜖𝛼0. For the sake of clarity, we also set 𝑡𝛼𝑛=𝑡𝜖𝛼𝑛(=𝑛𝜖𝛼) for all 𝑛0. Now suppose that 𝑎𝛼 and 𝑏𝛼 are measurable functions on +× such that 𝑎𝑎𝛼 and 𝑏𝑏𝛼 converge to 0 as 𝛼, where is the usual supremum norm. Let (𝑥𝛼) satisfy 𝑥𝛼𝑥0, and suppose that 𝑋𝛼 is a solution of the following (𝜖𝛼,𝑎𝛼,𝑏𝛼,𝑥𝛼)-martingale problem for every 𝛼1. Here we write 𝑛𝛼(𝑡) for the largest 𝑛0 with 𝑡𝛼𝑛𝑡.

Definition 2.1. Suppose that 𝑋𝛼=(𝑋𝛼𝑡𝑡0) is a real-valued process on some probability space (Ω,,) whose trajectories are constant on the intervals [𝑡𝛼𝑛,𝑡𝛼𝑛+1), 𝑛0. Then 𝑋𝛼 is called a solution of the (𝜖𝛼,𝑎𝛼,𝑏𝛼,𝑥𝛼)-martingale problem if𝑀𝛼𝑡=𝑋𝛼𝑡𝑥𝛼𝑛𝛼(𝑡)1𝑖=0𝑏𝛼𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖𝜖𝛼(2.1)provides a (zero-mean) square-integrable martingale (with respect to the natural filtration) with compensator𝑀𝛼𝑡=𝑛𝛼(𝑡)1𝑖=0𝑎2𝛼𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖𝜖𝛼.(2.2)

The 𝑋𝛼 could be defined on different probability spaces (Ω𝛼,𝛼,𝛼). However, we assume without loss of generality that Ω𝛼=𝐷(), 𝛼=(𝐷()), and 𝑋𝛼 is the coordinate process of 𝛼 (each cádlàg process induces a corresponding law on 𝐷()). We further assume that there are some 𝑞>2 and 𝛿>1 such that𝔼𝛼|||𝑋𝛼𝑡𝛼𝑛𝑋𝛼𝑡𝛼𝑛1|||𝑞𝐶𝑇1+𝔼𝛼|||𝑋𝛼𝑡𝛼𝑛1|||𝑞𝜖𝛿𝛼(2.3)for every 𝛼1 and 𝑛 with 𝑡𝛼𝑛𝑇, where 𝐶𝑇>0 is some finite constant that may depend on 𝑇. (By an induction on 𝑛, (2.3) implies immediately that 𝔼𝛼[|𝑋𝛼𝑡𝛼𝑛|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 the corresponding law on 𝐷(). Moreover, let 𝛼 be the law (on 𝐷()) of 𝑋𝛼 subject to (2.1)–(2.3). Then 𝛼 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 0-valued Markov process 𝑍=(𝑍𝑛𝑛0) that can be defined recursively as follows. Choose an initial state 𝑍0 and set 𝑍𝑛=𝑍𝑛1𝑖=1𝑁𝑛1,𝑖 for all 𝑛1, where {𝑁𝑛,𝑖𝑛0,𝑖1} is a family of i.i.d. 0-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 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 𝑍𝜖𝑡𝜖𝑛=𝜖𝑍𝑛 we obtain a rescaled version, 𝑍𝜖, of 𝑍. Recall 𝑡𝜖𝑛=𝑛𝜖, hence 𝑍𝜖 is a process having 𝜖0={0,𝜖,2𝜖,} as both its index set and its state space. Now pick (𝜖𝛼)+ such that 𝜖𝛼0, and recall our convention 𝑡𝛼𝑛=𝑡𝜖𝛼𝑛 and that 𝑡𝜖 denotes the largest element 𝑠 of 𝜖0 with 𝑠𝑡. Regard the process 𝑍𝜖𝛼 as continuous-time process, 𝑋𝛼, by setting 𝑋𝛼𝑡=𝑍𝜖𝛼𝑡𝜖𝛼, and suppose that 𝑋𝛼0=𝑥0𝜖𝛼. The latter requires that 𝑍0 actually depends on 𝛼. The domain of 𝑋𝛼 is denoted by (Ω𝛼,𝛼,𝛼). 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 𝑀𝛼𝑡=𝜎2𝑛𝛼(𝑡)1𝑖=0𝑋𝛼𝑡𝛼𝑖𝜖𝛼 since, in this case,𝔼𝛼𝑀𝛼𝑡𝛼𝑛2𝑀𝛼𝑡𝛼𝑛𝑀𝛼𝑡𝛼𝑛12𝑀𝛼𝑡𝛼𝑛1𝑋𝛼𝑡𝛼𝑛1=0(3.1)can be checked easily with help of𝔼𝛼𝑋𝛼𝑡𝛼𝑛𝑋𝛼𝑡𝛼𝑛1=𝑋𝛼𝑡𝛼𝑛1,𝕍𝑎𝑟𝛼𝑋𝛼𝑡𝛼𝑛𝑋𝛼𝑡𝛼𝑛1=𝜎2𝑋𝛼𝑡𝛼𝑛1𝜖𝛼.(3.2)The formulae in (3.2) are immediate consequences of the well-known moment formulae for 𝑍 (see [17, page 6]) and (𝑋𝛼𝑡𝛼𝑛) 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|, 𝑏0 and 𝑥𝛼=𝑥0𝜖𝛼. 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,,) with 𝔼[𝜉𝑖]=0 and sup𝑖𝔼[𝜉4𝑖]<. Let 𝜈 be a further random variable on (Ω,,) being independent of (𝜉𝑖), taking values in 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[(𝜈𝑖=1𝜉𝑖)4]𝐶𝔼[𝜈2].

Proof. By the finiteness of the fourth moments the law of total expectation yields𝔼[(𝜈𝑖=1𝜉𝑖)4]=𝑛𝑛𝑖1𝑛=1𝑖2𝑛=1𝑖3𝑛=1𝑖4=1𝔼𝜉𝑖1𝜉𝑖2𝜉𝑖3𝜉𝑖4[𝜈=𝑛].(3.3)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=𝑖4𝑖1, or 𝑖1=𝑖3 and 𝑖2=𝑖4𝑖1, or 𝑖1=𝑖4 and 𝑖2=𝑖3𝑖1. Hence,𝔼[(𝜈𝑖=1𝜉𝑖)4]𝑛𝑛+3𝑛𝑛1)sup𝑖,𝑗𝔼𝜉2𝑖𝜉2𝑗[𝜈=𝑛]4sup𝑖,𝑗𝔼𝜉2𝑖𝜉2𝑗𝔼𝜈2.(3.4)This yields the claim of the lemma with 𝐶=4sup𝑖,𝑗𝔼[𝜉2𝑖𝜉2𝑗].

With help of Lemma 3.1 we obtain𝔼𝛼|||𝑋𝛼𝑡𝛼𝑛𝑋𝛼𝑡𝛼𝑛1|||4=𝔼𝛼[|𝜖𝛼1𝑋𝛼𝑡𝛼𝑛1𝑖=1𝜖𝛼𝑁𝑛1,𝑖𝜖𝛼|4]=𝔼𝛼[|𝜖𝛼1𝑋𝛼𝑡𝛼𝑛1𝑖=1𝑁𝑛1,𝑖|14]𝜖4𝛼𝐶𝔼𝛼𝜖𝛼1𝑋𝛼𝑡𝛼𝑛12𝜖4𝛼𝐶1+𝔼𝛼𝑋𝛼𝑡𝛼𝑛14𝜖2𝛼(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 𝑋𝛼 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 +× 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):𝑋𝜖𝑡𝜖𝑛𝑋𝜖𝑡𝜖𝑛1𝑡=𝑏𝜖𝑛1,𝑋𝜖𝑡𝜖𝑛1𝑡𝜖+𝑎𝜖𝑛1,𝑋𝜖𝑡𝜖𝑛1𝑉𝜖𝑡𝜖𝑛,𝑋𝜖𝑡𝜖0=𝑥𝜖.(4.1)Here, (𝑥𝜖) is a sequence in satisfying 𝑥𝜖𝑥0 as 𝜖0, and 𝑉𝜖={𝑉𝜖𝑡𝜖𝑛𝑛} is a family of independent centered random variables with variance 𝜖 and 𝔼𝜖[|𝑉𝜖𝑡𝜖𝑛|𝑞]𝐶𝜖𝑞/2 for all 𝑛, 𝜖(0,1], some 𝑞>2, and some finite constant 𝐶>0, where (Ω𝜖,𝜖,𝜖) denotes the domain of 𝑉𝜖. For instance, one may set 𝑉𝜖𝑡𝜖𝑛=𝜖𝜉𝑛 where {𝜉𝑛𝑛} 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 {𝑉𝜖𝑡𝜖𝑛𝑛} are identically distributed. Below we will see that the independence is necessary neither.

By virtue of (1.4), 𝑋𝜖𝑡𝜖𝑛 has a finite 𝑞th moment if 𝑋𝜖𝑡𝜖𝑛1 has. It follows by induction that the solution 𝑋𝜖=(𝑋𝜖𝑡𝜖𝑛𝑛0) of (4.1) is 𝑞-integrable, and hence square integrable. Equation (4.1) is obviously equivalent to the stochastic sum equation𝑋𝜖𝑡𝜖𝑛=𝑥𝜖+𝑛1𝑖=0𝑏𝑡𝜖𝑖,𝑋𝜖𝑡𝜖𝑖𝜖+𝑛1𝑖=0𝑎𝑡𝜖𝑖,𝑋𝜖𝑡𝜖𝑖𝑉𝜖𝑡𝜖𝑖+1.(4.2)Suppose that (𝜖𝛼) is an arbitrary sequence with 𝜖𝛼(0,1] and 𝜖𝛼0, set 𝑥𝛼=𝑥𝜖𝛼 and recall our convention 𝔼𝛼=𝔼𝜖𝛼, 𝑋𝛼𝑋𝜖𝛼, 𝑡𝛼𝑛=𝑡𝜖𝛼𝑛. Then it is easy to see that 𝑀𝛼 defined in (2.1) provides a (mean-zero) square-integrable (𝑋𝛼𝑡)-martingale. Moreover, 𝑀𝛼𝑡𝛼𝑛 coincides with the second sum on the right-hand side of (4.2). Therefore, we also obtain𝑀𝛼𝑡𝛼𝑛=𝑛𝑖=1𝔼𝛼𝑎𝑡𝛼𝑖1,𝑋𝛼𝑡𝛼𝑖1𝑉𝜖𝛼𝑡𝛼𝑖2𝑋𝛼𝑡𝛼𝑛1=𝑛1𝑖=0𝑎2𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖𝔼𝛼𝑉𝜖𝛼𝑡𝛼𝑖+12=𝑛1𝑖=0𝑎2𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖𝜖𝛼(4.3)which 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|||𝑋𝛼𝑡𝛼𝑛𝑋𝛼𝑡𝛼𝑛1|||𝑞2𝑞1𝔼𝛼|||𝑏𝑡𝛼𝑛1,𝑋𝛼𝑡𝛼𝑛1𝜖𝛼|||𝑞+𝔼𝛼|||𝑎𝑡𝛼𝑛1,𝑋𝛼𝑡𝛼𝑛1|||𝑞𝔼𝛼|||𝑉𝜖𝛼𝑡𝛼𝑛|||𝑞2𝑞1𝐾2𝑞11+𝔼𝛼|||𝑋𝛼𝑡𝛼𝑛1|||𝑞𝜖𝑞𝛼+𝐾2𝑞11+𝔼𝛼|||𝑋𝛼𝑡𝛼𝑛1|||𝑞𝐶𝜖𝛼𝑞/2(4.4)for which we used (4.1), the independence of 𝑋𝛼𝑡𝛼𝑛1 of 𝑉𝜖𝛼, (1.4), and 𝔼𝛼[|𝑉𝜖𝛼𝑡𝛼𝑛|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 {𝑉𝜖𝑡𝜖𝑛𝑛} 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 𝑉𝜖𝑡𝜖𝑛 are not independent. For instance, let {𝜉𝑛(𝑖)𝑛,𝑖} 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 𝑉𝜖𝑡𝜖1=𝜖𝜉1(1) and 𝑉𝜖𝑡𝜖𝑛=𝜖𝜉𝑛(𝑓𝑛(𝑉𝜖𝑡𝜖1,,𝑉𝜖𝑡𝜖𝑛1)), 𝑛2, where 𝑓𝑛 is any measurable mapping from 𝑛1 to . This follows from the following relations which can be shown easily with help of the functional representation theorem for conditional expectations respectively by conditioning𝔼𝛼𝑉𝜖𝛼𝑡𝛼𝑛𝑋𝛼𝑡𝛼𝑛1=0,𝔼𝛼𝑉𝜖𝛼𝑡𝛼𝑖+12𝑋𝛼𝑡𝛼𝑛1=𝜖𝛼𝔼,1𝑖𝑛1,𝛼|||𝑎𝑡𝛼𝑛1,𝑋𝛼𝑡𝛼𝑛1𝑉𝜖𝛼𝑡𝛼𝑛|||𝑞𝐶𝜖𝛼𝑞/2.(4.5)If the 𝜉𝑛(𝑖) are not identically distributed, then the 𝑉𝜖𝑡𝜖𝑛 are typically not independent. In particular, the approximating process 𝑋𝜖 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 𝛼01 such that for all 𝛼𝛼0, 𝑡0, and 𝑥,|||𝑎𝛼|||+|||𝑏(𝑡,𝑥)𝛼|||(𝑡,𝑥)𝐾||𝑥||1+.(5.1)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 |𝑚𝑖=1𝑦𝑖|𝑝𝑚𝑝1𝑚𝑖=1|𝑦𝑖|𝑝 for all 𝑚, 𝑝1 and 𝑦1,,𝑦𝑚. As a first consequence of (5.1) we obtain Lemma 5.1. For every 𝑥+ we write 𝑥𝜖 for the largest element of 𝜖0={0,𝜖,2𝜖,} which is smaller than or equal to 𝑥. Moreover, we assume without loss of generality that 𝜖𝛼1.

Lemma 5.1. For 𝑞>2 and 𝛿>1 satisfying (2.3) and every 𝑇>0, sup𝛼𝛼0𝔼𝛼sup𝑡𝑇|||𝑋𝛼𝑡|||𝑞<.(5.2)

Proof. First of all, note that for the proof it actually suffices to require 𝑞2 and 𝛿1. Set 𝑆=sup𝛼𝛼0|𝑥𝛼|𝑞 and 𝑆𝛼𝑡=𝔼𝛼[max1𝑖𝑛𝛼(𝑡)|𝑀𝛼𝑡𝛼𝑖𝑀𝛼𝑡𝛼𝑖1|𝑞]. Using Proposition A.1 in the appendix and (5.1) we obtain, for all 𝑡>0 and 𝛼𝛼0,𝔼𝛼sup𝑖𝑛𝛼(𝑡)|||𝑋𝛼𝑡𝑖|||𝑞3𝑞1{𝔼𝛼sup𝑖𝑛𝛼(𝑡)|||𝑀𝛼𝑡𝑖|||𝑞+𝑆+𝔼𝛼[(𝑛𝛼(𝑡)1𝑖=0|||𝑏𝛼𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖|||𝜖𝛼)𝑞]}3𝑞1𝐶𝑞{𝔼𝛼[|𝑛𝛼(𝑡)1𝑖=0𝑎2𝛼𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖𝜖𝛼|𝑞/2]+𝑆𝛼𝑡+𝑆+𝔼𝛼[[𝑛𝛼(𝑡)1𝑖=0|𝑏𝛼𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖|𝜖𝛼]𝑞]}𝑘𝑞{𝔼𝛼[[𝑛𝛼(𝑡)1𝑖=0𝐾|||𝑋1+𝛼𝑡𝛼𝑖|||2𝜖𝛼]𝑞/2]+𝑆𝛼𝑡+𝑆+𝔼𝛼[[𝑛𝛼(𝑡)1𝑖=0𝐾|||𝑋1+𝛼𝑡𝛼𝑖|||𝜖𝛼]𝑞]},(5.3)where 𝐶𝑞 is independent of 𝑡 and 𝛼, and 𝑘𝑞=3𝑞1𝐶𝑞. By Hölder's inequality we get𝔼𝛼[[𝑛𝛼(𝑡)1𝑖=0𝐾|||𝑋1+𝛼𝑡𝛼𝑖|||2𝜖𝛼]𝑞/2]𝔼𝛼[(𝑛𝛼(𝑡)1𝑖=02𝐾2|||𝑋1+𝛼𝑡𝛼𝑖|||2𝑞/2)(𝑛𝛼(𝑡)1𝑖=0𝜖𝛼(𝑞/2)/(𝑞/21))𝑞/21]𝔼𝛼[(𝑛𝛼(𝑡)1𝑖=02𝑞/212𝐾2𝑞/2|||𝑋1+𝛼𝑡𝛼𝑖|||𝑞)𝑛𝛼(𝑡)𝑞/21𝜖𝛼𝑞/2]𝑐𝑞𝑡𝑞/2+𝑐𝑞𝑡𝑛𝑞/21𝛼(𝑡)1𝑖=0𝔼𝛼sup𝑗𝑖|||𝑋𝛼𝑡𝛼𝑗|||𝑞𝜖𝛼,(5.4)where 𝑐𝑞=2𝑞/21(2𝐾2)𝑞/2. Analogously, with 𝑐𝑞=2𝑞1𝐾𝑞,𝔼𝛼[[𝑛𝛼(𝑡)1𝑖=0𝐾|||𝑋1+𝛼𝑡𝛼𝑖|||𝜖𝛼]𝑞]𝑐𝑞𝑡𝑞+𝑐𝑞𝑡𝑛𝑞1𝛼(𝑡)1𝑖=0𝔼𝛼sup𝑗𝑖|||𝑋𝛼𝑡𝛼𝑗|||𝑞𝜖𝛼.(5.5)Moreover, by (2.3) and (5.1) we obtain, for all 𝑡𝑇 and 𝛼𝛼0,𝑆𝛼𝑡𝑛𝛼(𝑡)𝑖=1𝔼𝛼|||𝑀𝛼𝑡𝛼𝑖𝑀𝛼𝑡𝛼𝑖1|||𝑞2𝑛𝑞1𝛼(𝑡)𝑖=1𝔼𝛼|||𝑋𝛼𝑡𝛼𝑖𝑋𝛼𝑡𝛼𝑖1|||𝑞+|||𝑏𝛼𝑡𝛼𝑖1,𝑋𝛼𝑡𝛼𝑖1|||𝑞𝜖𝑞𝛼2𝑛𝑞1𝛼(𝑡)1𝑖=0𝐶𝑇1+𝔼𝛼|||𝑋𝛼𝑡𝛼𝑖|||𝑞𝜖𝛿𝛼+𝔼𝛼𝐾|||𝑋1+𝛼𝑡𝛼𝑖|||𝑞𝜖𝑞𝛼𝑐𝑞,𝑇𝑡+𝑐𝑛𝑞,𝑇𝛼(𝑡)1𝑖=0𝔼𝛼sup𝑗𝑖|||𝑋𝛼𝑡𝛼𝑗|||𝑞𝜖𝛼,(5.6)where 𝑐𝑞,𝑇=2𝑞1(𝐶𝑇+𝐾). By all account we have, for all 𝑡𝑇 and 𝛼𝛼0,𝔼𝛼sup𝑖𝑛𝛼(𝑡)|||𝑋𝛼𝑡𝑖|||𝑞𝑘𝑞𝑐{𝑆+𝑞+𝑐𝑞+𝑐𝑞,𝑇𝑡𝑞11(1+𝜖𝛼𝑛𝛼(𝑡)1𝑖=0𝔼𝛼sup𝑗𝑖||𝑋𝛼𝑡𝛼𝑗||𝑞𝑘)}𝑞𝑆+𝐶𝑞,𝑇+𝐶𝑞,𝑇𝜖𝛼𝑛𝛼(𝑡)1𝑖=0𝔼𝛼sup𝑗𝑖||𝑋𝛼𝑡𝛼𝑗||𝑞,(5.7)where 𝐶𝑞,𝑇=𝑘𝑞(𝑐𝑞+𝑐𝑞+𝑐𝑞,𝑇)(𝑇𝑞11). An application of Lemma A.2 yields𝔼𝛼sup𝑠𝑡||𝑋𝛼𝑠||𝑞=𝔼𝛼sup𝑖𝑛𝛼(𝑡)||𝑋𝛼𝑡𝑖||𝑞𝑘𝑞𝑆+𝐶𝑞,𝑇1+𝐶𝑞,𝑇𝜖𝛼𝑛𝛼(𝑡)+𝐶𝑞,𝑇𝜖𝛼𝑛𝛼(𝑡)𝑆,(5.8)where we emphasize that the constants 𝑘𝑞, 𝑆, and 𝐶𝑞,𝑇 are independent of 𝑡𝑇 and 𝛼𝛼0. This proves Lemma 5.1 since limsup𝛼(1+𝐶𝑞,𝑇𝜖𝛼)𝑛𝛼(𝑡) is bounded by exp(𝑡𝐶𝑞,𝑇) (note that 𝑛𝛼(𝑡)=𝑡/𝜖𝛼1𝑡/𝜖𝛼).

Proposition 5.2. If (𝛼) is tight, then the coordinate process of any weak limit point, that has no mass outside of 𝐶(), is a solution of the (𝑎,𝑏,𝑥0)-martingale problem of Definition 1.1.

Proof. We consider a weakly convergent subsequence whose limit, , has no mass outside of 𝐶(). By an abuse of notation, we denote this subsequence by (𝛼) either. We further write 𝑋 for the coordinate process of . Since 𝑋 is -almost surely continuous, we know [19, Theorem 3.7.8] that𝛼𝜋𝑡11,,𝑡𝑘𝜋𝑡11,,𝑡𝑘(5.9)for all 𝑡1,,𝑡𝑘+, where 𝜋𝑡1,,𝑡𝑘𝐷()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 (𝑋𝑡)-martingale and has 𝑀 defined in (1.8) as compensator. Here, (𝑋𝑡) denotes the natural augmentation of the filtration (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,sup𝑡𝑇𝔼|||𝑋𝑡|||𝑞sup𝑡𝑇liminf𝑁lim𝛼𝔼𝛼|||𝑋𝛼𝑡|||𝑞𝑁sup𝑡𝑇sup𝛼𝛼0𝔼𝛼|||𝑋𝛼𝑡|||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 (𝑋𝑡)-martingale. It suffices to show that 𝑀 is an (𝑋𝑡)-martingale; see [20, page 75]. The latter is true if and only if𝔼𝑋𝑡+𝑠𝑋𝑡𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟𝑑𝑟𝑙𝑖=1𝑖𝑋𝑡𝑖=0(5.11)holds for all 0𝑡1<𝑡𝑙𝑡, 𝑠0, 𝑙1 and bounded 1,,𝑙𝐶() (do not confuse 𝑡𝑖 and 𝑡𝛼𝑖). Since 𝑋𝛼 solves the (𝜖𝛼,𝑎𝛼,𝑏𝛼,𝑥𝛼)-martingale problem, we have𝔼𝛼[(𝑋𝛼𝑡+𝑠𝑋𝛼𝑡𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝛼𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖𝜖𝛼)𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖]=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 provinglim𝛼𝔼𝛼𝑋𝛼𝑢𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖𝑋=𝔼𝑢𝑙𝑖=1𝑖𝑋𝑡𝑖(5.13)for every 𝑢0, which together with (5.19) below implies the required convergence. To this end we set 𝑥(𝑁)𝑁=(𝑁𝑥) for all 𝑥 and 𝑁>0. The right-hand side of|||𝔼𝛼𝑋𝑢𝑙𝛼,(𝑁)𝑖=1𝑖𝑋𝛼𝑡𝑖𝔼𝛼𝑋𝛼𝑢𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖|||𝔼𝛼|||𝑋𝑢𝛼,(𝑁)𝑋𝛼𝑢|||𝑙𝑖=1𝑖(5.14)can be estimated, for every 𝑇𝑢, bysup𝑟𝑇sup𝛼𝛼0𝔼𝛼|||𝑋𝛼𝑟|||1||𝑋𝛼𝑟||>𝑁𝑙𝑖=1𝑖(5.15)which tends to 0 as 𝑁 since {𝑋𝛼𝑟𝑟𝑇,𝛼1} is uniformly integrable by (5.2). Therefore, we havelim𝑁𝔼𝛼𝑋𝑢𝑙𝛼,(𝑁)𝑖=1𝑖𝑋𝛼𝑡𝑖=𝔼𝛼𝑋𝛼𝑢𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖uniformlyin𝛼𝛼0(5.16)(and uniformly in 𝑢𝑇, for every 𝑇>0). By (5.9) we further obtain for every 𝑁>0,lim𝛼𝔼𝛼𝑋𝑢𝑙𝛼,(𝑁)𝑖=1𝑖𝑋𝛼𝑡𝑖𝑋=𝔼𝑢𝑙(𝑁)𝑖=1𝑖𝑋𝑡𝑖(5.17)since the mapping (𝑥1,,𝑥𝑙+1)𝑥(𝑁)𝑙+1𝑙𝑖=1𝑖(𝑥𝑖) from 𝑙+1 to 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 giveslim𝑁𝔼𝑋𝑢𝑙(𝑁)𝑖=1𝑖𝑋𝑡𝑖𝑋=𝔼𝑢𝑙𝑖=1𝑖𝑋𝑡𝑖(5.18)which along with (5.16) and (5.17) implies (5.13). It remains to showlim𝛼𝔼𝛼[𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝛼𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖𝜖𝛼𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖]=𝔼𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟𝑑𝑟𝑙𝑖=1𝑖𝑋𝑡𝑖.(5.19)Taking (5.1) and (𝑛𝛼(𝑡+𝑠)𝑛𝛼(𝑡))𝜖𝛼𝑠+𝜖𝛼 into account we obtain, analogously to (5.16) and (5.18),lim𝑁𝔼𝛼[𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝛼𝑡𝛼𝑖,𝑋𝑡𝛼,(𝑁)𝛼𝑖𝜖𝛼𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖]=𝔼𝛼[𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝛼𝑡𝛼𝑖,𝑋𝛼𝑡𝛼𝑖𝜖𝛼𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖]uniformlyin𝛼𝛼0,(5.20)respectively,lim𝑁𝔼𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟(𝑁)𝑑𝑟𝑙𝑖=1𝑖𝑋𝑡𝑖=𝔼𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟𝑑𝑟𝑙𝑖=1𝑖𝑋𝑡𝑖.(5.21)By the uniform convergence of 𝑏𝛼 to 𝑏 and (𝑛𝛼(𝑡+𝑠)𝑛𝛼(𝑡))𝜖𝛼𝑠+𝜖𝛼, we also have𝔼𝛼[𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝛼𝑡𝛼𝑖,𝑋𝑡𝛼,(𝑁)𝛼𝑖𝜖𝛼𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖=𝔼𝛼[𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝑡𝛼𝑖,𝑋𝑡𝛼,(𝑁)𝛼𝑖𝜖𝛼𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖+𝑜𝛼(1).(5.22)Moreover, we have𝔼𝛼[𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝑡𝛼𝑖,𝑋𝑡𝛼,(𝑁)𝛼𝑖𝜖𝛼𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖=𝔼𝛼𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟𝛼,(𝑁)𝑑𝑟𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖+𝑜𝛼(1)(5.23)which is a consequence of the dominated convergence theorem and|𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝑡𝛼𝑖,𝑋𝑡𝛼,(𝑁)𝛼𝑖𝜖𝛼𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟𝛼,(𝑁)𝑑𝑟|𝑡+𝑠𝜖𝛼𝜖𝛼𝑡𝜖𝛼|||𝑏𝑟𝜖𝛼,𝑋𝑟𝛼,(𝑁)𝑏𝑟,𝑋𝑟𝛼,(𝑁)|||𝑑𝑟+𝑜𝛼(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) lim𝛼𝔼𝛼𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟𝛼,(𝑁)𝑑𝑟𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖=𝑡𝑡+𝑠lim𝛼𝔼𝛼𝑏𝑟,𝑋𝑟𝛼,(𝑁)𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖=𝑑𝑟𝑡𝑡+𝑠𝔼𝑏𝑟,𝑋𝑟(𝑁)𝑙𝑖=1𝑖𝑋𝑡𝑖𝑑𝑟=𝔼𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟(𝑁)𝑑𝑟𝑙𝑖=1𝑖𝑋𝑡𝑖(5.25)which along with (5.22) and (5.23) implieslim𝛼𝔼𝛼[𝑛𝛼(𝑡+𝑠)1𝑖=𝑛𝛼(𝑡)𝑏𝛼𝑡𝛼𝑖,𝑋𝑡𝛼,(𝑁)𝛼𝑖𝜖𝛼𝑙𝑖=1𝑖𝑋𝛼𝑡𝑖]=𝔼𝑡𝑡+𝑠𝑏𝑟,𝑋𝑟(𝑁)𝑑𝑟𝑙𝑖=1𝑖𝑋𝑡𝑖.(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 if𝔼𝑀2𝑡+𝑠𝑀2𝑡𝑡𝑡+𝑠𝑎2𝑟,𝑋𝑟𝑑𝑟𝑙𝑖=1𝑖𝑋𝑡𝑖=0(5.27)holds for all 0𝑡1<𝑡𝑙𝑡, 𝑠0, 𝑙1 and bounded 1,,𝑙𝐶(). 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 {(𝑋𝛼𝑟)2𝑟𝑡+𝑠,𝛼1}. 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 𝛼 and refer to any laws on 𝐷(), and 𝑌𝛼 and 𝑌 are the respective coordinate processes. By an abuse of notation, we denote the corresponding expectations by 𝛼 and 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 (𝑌𝛼𝑡) is tight in for every rational 𝑡0. Let 𝑚>0, 𝛾>1, and assume for every 𝑇>0 that there is some finite constant 𝐶𝑇>0 such that for all 𝛼1 and 𝑡,0 with 0𝑡 and 𝑡+𝑇, 𝛼|||𝑌𝛼𝑡𝑌𝛼𝑡|||𝑚/2|||𝑌𝛼𝑡𝑌𝛼𝑡+|||𝑚/2𝐶𝑇𝛾.(5.28)Then (𝛼) 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 𝛼1 and 0𝑡𝑡𝑇, limsup𝛼𝛼|||𝑌𝛼𝑡𝑌𝛼𝑡|||𝑚𝐶𝑇𝑡𝑡𝛾.(5.29)Then if 𝛼, the limit has no mass outside of 𝐶().

Proposition 5.5. (𝛼) is tight and each limit point has no mass outside of 𝐶().

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 0𝑡𝑡 and 𝛼𝛼0,𝔼𝛼|||𝑋𝛼𝑡𝑋𝛼𝑡|||𝑞𝐶𝑞{𝜖𝛼𝑞/2𝔼𝛼[(𝑛𝛼(𝑡)1𝑖=𝑛𝛼(𝑡)|||𝑋1+𝛼𝑡𝛼𝑖|||2)𝑞/2]+𝜖𝑞𝛼𝔼𝛼[(𝑛𝛼(𝑡)1𝑖=𝑛𝛼(𝑡)|||𝑋1+𝛼𝑡𝛼𝑖|||)𝑞]+𝑛𝛼(𝑡)1𝑖=𝑛𝛼(𝑡)1+𝔼𝛼|||𝑋𝛼𝑡𝛼𝑖|||𝑞𝜖𝛿𝑞𝛼}.(5.30)Applying Hölder's inequality to each of the first two summands on the right-hand side, using (5.2) and setting 𝛿𝛾=(𝑞/2), we may continue with𝐶𝑞{𝜖𝛼𝑞/2(𝑛𝛼(𝑡)1𝑖=𝑛𝛼(𝑡)𝔼𝛼|||𝑋1+𝛼𝑡𝛼𝑖|||2𝑞/2)(𝑛𝛼(𝑡)1𝑖=𝑛𝛼(𝑡)1𝑞/(𝑞2))𝑞/21+𝜖𝑞𝛼(𝑛𝛼(𝑡)1𝑖=𝑛𝛼(𝑡)𝔼𝛼|||𝑋1+𝛼𝑡𝛼𝑖|||𝑞)(𝑛𝛼(𝑡)1𝑖=𝑛𝛼(𝑡)1𝑞/(𝑞1))𝑞1+𝑛𝛼(𝑡)1𝑖=𝑛𝛼(𝑡)(1+𝔼𝛼|||𝑋𝛼𝑡𝛼𝑖|||𝑞)𝜖𝛿𝑞𝛼}𝐶𝑞,𝑇𝜖𝛼𝑞/2𝑛𝛼𝑡𝑛𝛼𝑡𝑞/2+𝜖𝑞𝛼𝑛𝛼𝑡𝑛𝛼𝑡𝑞+𝜖𝛿𝑞𝛼𝑛𝛼𝑡𝑛𝛼𝑡𝐶𝑞,𝑇𝜖𝛼𝑛𝛼𝑡𝑛𝛼𝑡𝛾𝐶𝑞,𝑇𝑡𝑡+𝜖𝛼𝛾,(5.31)where 𝐶𝑞,𝑇,𝐶𝑞,𝑇>0 are some finite constants being independent of 𝑡,𝑡𝑇, and 𝛼𝛼0. Then Lemma 5.4 ensures that any weak limit point of (𝛼) has no mass outside of 𝐶(). 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 (𝛼) we use Hölder's inequality to get𝔼𝛼|||𝑋𝛼𝑡𝑋𝛼𝑡|||𝑞/2|||𝑋𝛼𝑡𝑋𝛼𝑡+|||𝑞/2𝔼𝛼|||𝑋𝛼𝑡𝑋𝛼𝑡|||𝑞1/2𝔼𝛼|||𝑋𝛼𝑡𝑋𝛼𝑡+|||𝑞1/2.(5.32)If 𝜖𝛼/2, then (5.31) implies that both factors on the right-hand side of (5.32) are bounded by 𝐶𝑞,𝑇(3)𝛾/2. If <𝜖𝛼/2, then at least one of these factors vanishes since 𝑋𝛼 is constant on intervals of length 𝜖𝛼. Hence,𝔼𝛼|||𝑋𝛼𝑡𝑋𝛼𝑡|||𝑞/2|||𝑋𝛼𝑡𝑋𝛼𝑡+|||𝑞/2𝐶2𝑞,𝑇3𝛾𝛾(5.33)for all 𝛼𝛼0 and 𝑡,0 with 𝑡+𝑇. That is, (5.28) holds with 𝑚=𝑞. Therefore, Lemma 5.3 ensures tightness of (𝛼).


Auxiliary Results

Here we give two auxiliary results. We first recall a square function inequality for martingales. Let 𝑀=(𝑀𝑛𝑛0) be an (𝑛)-martingale on some probability space (Ω,,). The corresponding compensator is given by 𝑀𝑛=𝑛𝑖=1𝔼[(𝑀𝑖𝑀𝑖1)2𝑖1].

Proposition A.1 (see [21, Theorem 2.11]). For every 𝑞>0 there is some finite constant 𝐶𝑞>0 depending only on 𝑞 such that 𝔼max1𝑖𝑛|||𝑀𝑖|||𝑞𝐶𝑞𝔼𝑀𝑛𝑞/2+𝔼max1𝑖𝑛|||𝑀𝑖𝑀𝑖1|||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 0 to +=[0,] with 𝑔(0)<. If there are finite constants 𝑐0,𝑐10 such that 𝑔(𝑛)𝑐0+𝑐1𝑛1𝑖=0𝑔(𝑖)𝑛𝑁,(A.2)then 𝑔(𝑛)𝑐01+𝑐1𝑛+𝑐𝑛1𝑔(0)<𝑛𝑁.(A.3)


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


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Copyright © 2008 Henryk Zähle. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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