Research Article | Open Access
Henryk Zähle, "Weak Approximation of SDEs by Discrete-Time Processes", International Journal of Stochastic Analysis, vol. 2008, Article ID 275747, 15 pages, 2008. https://doi.org/10.1155/2008/275747
Weak Approximation of SDEs by Discrete-Time Processes
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.
It is well known that a rescaled version of the classical Galton-Watson process (GWP) with offspring variance weakly converges to the unique solution of the following one-dimensional stochastic differential equation (SDE):where is a one-dimensional Brownian motion (cf. ). One might ask whether it is possible to approximate more general SDEs, driven by a Brownian motion, by generalized GWPs. In  it will be shown that this is actually possible. In fact, in  the solution of the SDE,is weakly approximated by two different types of population-size-dependent GWPs (in the sense of [3–6]) with immigration, where and are suitable nonnegative continuous functions on . Here the methods of  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 .
To be exact, we focus on the following one-dimensional SDE:where and is a one-dimensional Brownian motion. The coefficients and are continuous functions on satisfyingfor some finite constant . We assume that SDE (1.3) has a weak solution. It means that there exists a triplet where is a filtered probability space with satisfying the usual conditions, is an -Brownian motion, and is a real-valued continuous -adapted process such that -almost surely,Here the latter is an It-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 in (uniformly in ) andfor some strictly increasing with . Note that (1.6) and Lipschitz continuity of 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.
Definition 1.1. A tuple is said to be a solution of the (, , )-martingale problem if is a filtered probability space with satisfying the usual conditions, and is a real-valued continuous -adapted process such thatprovides a (continuous, mean-zero) square-integrable -martingale with compensatorThe 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 ()-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 , see also references therein. In  general conditions are given, under which the convergence in distribution in the cádlàg space implies convergence in distribution 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 the space of cádlàg functions from to . We equip with the topology generated by the Skohorod convergence on compacts and consider it as a measurable space with respect to its Borel -algebra. Moreover, we set for every and .
For every we fix some such that . For the sake of clarity, we also set for all . Now suppose that and are measurable functions on such that and converge to as , where is the usual supremum norm. Let satisfy , and suppose that is a solution of the following -martingale problem for every . Here we write for the largest with .
Definition 2.1. Suppose that is a real-valued process on some probability space whose trajectories are constant on the intervals , . Then is called a solution of the -martingale problem ifprovides a (zero-mean) square-integrable martingale (with respect to the natural filtration) with compensator
The could be defined on different probability spaces . However, we assume without loss of generality that , and is the coordinate process of (each cádlàg process induces a corresponding law on ). We further assume that there are some and such thatfor every and with , where is some finite constant that may depend on . (By an induction on , (2.3) implies immediately that for all and . Lemma 5.1 will provide an even stronger statement.) The following theorem shows that converges in distribution to the unique solution of (1.3).
Here, symbolizes weak convergence. The proof of Theorem 2.2 will be carried out in Section 5. The finiteness of the th moments for some 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 to is hardly possible. The assumption 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 , that is, to the solution of SDE (1.1). Lindvall  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 ). The latter is an -valued Markov process that can be defined recursively as follows. Choose an initial state and set for all , where is a family of i.i.d. -valued random variables with mean and variance . In addition, we require that the fourth moment of is finite. Thereby has a finite fourth moment for every . Actually, in  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. ).
Setting we obtain a rescaled version, , of . Recall , hence is a process having as both its index set and its state space. Now pick such that , and recall our convention and that denotes the largest element of with . Regard the process as continuous-time process, , by setting , and suppose that . The latter requires that actually depends on . The domain of is denoted by . It is easy to see that defined in (2.1) provides a (zero-mean) square-integrable martingale. Moreover, the compensator of is given by since, in this case,can be checked easily with help ofThe formulae in (3.2) are immediate consequences of the well-known moment formulae for (see [17, page 6]) and denotes the natural filtration induced by . Hence, solves the -martingale problem of Definition 2.1 with , and . It remains to show (2.3). To this end we state the following lemma.
Lemma 3.1. Assume that are independent random variables on some probability space with and . Let be a further random variable on being independent of , taking values in and satisfying . Then there is some finite constant , depending only on the second and the fourth moments of the , such that .
Proof. By the finiteness of the fourth moments the law of total expectation yieldsSince the are independent and centered, the summand on the right-hand side might differ from only if either , or and , or and , or and . Hence,This yields the claim of the lemma with .
With help of Lemma 3.1 we obtainfor some suitable constant . This shows that (2.3) holds too. Hence, the assumptions of Theorem 2.2 are fulfilled, and the theorem implies that 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 ; see also references therein. As before we suppose that and are continuous functions on satisfying (1.4), and that SDE (1.3) possesses a unique weak solution. Now let , recall the notation introduced in Section 2, and consider the following stochastic difference equation (weak Euler scheme):Here, is a sequence in satisfying as , and is a family of independent centered random variables with variance and for all , , some , and some finite constant , where denotes the domain of . For instance, one may set where is a family of independent centered random variables with variance and the th moment being bounded uniformly in . Note that we do not require that the random variables are identically distributed. Below we will see that the independence is necessary neither.
By virtue of (1.4), has a finite th moment if has. It follows by induction that the solution of (4.1) is -integrable, and hence square integrable. Equation (4.1) is obviously equivalent to the stochastic sum equationSuppose that is an arbitrary sequence with and , set and recall our convention , , . Then it is easy to see that defined in (2.1) provides a (mean-zero) square-integrable -martingale. Moreover, coincides with the second sum on the right-hand side of (4.2). Therefore, we also obtainwhich shows that solves the -martingale problem of Definition 2.1. For an application of Theorem 2.2 it thus remains to show (2.3). But (2.3) follows fromfor which we used (4.1), the independence of of , (1.4), and . Hence, Theorem 2.2 ensures that converges in distribution to the unique solution of SDE (1.3).
As mentioned above, the independence of the random variables is not necessary. The independence was used for (4.3), (4.4), and the martingale property of . But these relations may be valid even if the are not independent. For instance, let be an array of independent centered random variables with variance and th moments being bounded above by some uniformly in , for some . Then the martingale property of and the main statements of (4.3) and (4.4) remain true for and , , where is any measurable mapping from to . This follows from the following relations which can be shown easily with help of the functional representation theorem for conditional expectations respectively by conditioningIf the are not identically distributed, then the are typically not independent. In particular, the approximating process 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 and such that for all , , and ,This is true since we assumed (1.4) and uniform convergence of and to the coefficients and , respectively. Throughout this section we will frequently use the well-known inequality for all , and . As a first consequence of (5.1) we obtain Lemma 5.1. For every we write for the largest element of which is smaller than or equal to . Moreover, we assume without loss of generality that .
Lemma 5.1. For and satisfying (2.3) and every ,
Proof. First of all, note that for the proof it actually suffices to require and . Set and . Using Proposition A.1 in the appendix and (5.1) we obtain, for all and ,where is independent of and , and . By Hölder's inequality we getwhere . Analogously, with ,Moreover, by (2.3) and (5.1) we obtain, for all and ,where . By all account we have, for all and ,where . An application of Lemma A.2 yieldswhere we emphasize that the constants , , and are independent of and . This proves Lemma 5.1 since is bounded by (note that ).
Proposition 5.2. If is tight, then the coordinate process of any weak limit point, that has no mass outside of , is a solution of the -martingale problem of Definition 1.1.
consider a weakly convergent subsequence whose limit, , has no mass outside of . By an abuse of notation, we denote this subsequence
by either. We
further write for the
coordinate process of . Since is -almost surely
continuous, we know [19, Theorem 3.7.8] thatfor all , where is the usual
coordinate projection. In the remainder of the proof we will show in three
steps that defined in (1.7)
is square-integrable, provides an -martingale and
has defined in
(1.8) as compensator. Here, denotes the
natural augmentation of the filtration induced by .
Step 1. With help of Fatou's lemma as well as (5.9) and (5.2) we obtain, for every ,Taking (1.4) into account we conclude that defined in (1.7) is square-integrable.Step 2. We next show that is an -martingale. It suffices to show that is an -martingale; see [20, page 75]. The latter is true if and only ifholds for all , , and bounded (do not confuse and ). Since solves the -martingale problem, we haveWe 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 . We begin with provingfor every , which together with (5.19) below implies the required convergence. To this end we set for all and . The right-hand side ofcan be estimated, for every , bywhich tends to as since is uniformly integrable by (5.2). Therefore, we have(and uniformly in , for every ). By (5.9) we further obtain for every ,since the mapping from to is bounded and continuous. This is the reason why we introduced the truncation . By virtue of (5.10), an application of the dominated convergence theorem giveswhich along with (5.16) and (5.17) implies (5.13). It remains to showTaking (5.1) and into account we obtain, analogously to (5.16) and (5.18),respectively,By the uniform convergence of to and , we also haveMoreover, we havewhich is a consequence of the dominated convergence theorem andtogether with the fact that is bounded and uniformly continuous on . Finally, we get by (5.9) and the dominated convergence theorem and (5.2) which along with (5.22) and (5.23) impliesThis, (5.20), and (5.21) ensure (5.19).Step 3. It remains to show (1.8). By the uniqueness of the Doob-Meyer decomposition, has the required compensator if and only ifholds for all , , and bounded . Now, the discrete analogue of (5.27) for , , and 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 . 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 . This is why we established (5.2) for being strictly larger than .
The assumptions of Proposition 5.2 can be checked with help of the following two lemmas, where and refer to any laws on , and and are the respective coordinate processes. By an abuse of notation, we denote the corresponding expectations by 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 ; we omit the details.
Lemma 5.3. Assume that is tight in for every rational . Let , , and assume for every that there is some finite constant such that for all and with and , Then is tight.
Lemma 5.4. Let , , and assume for every that there is some finite constant such that for all and , Then if , the limit has no mass outside of .
Proposition 5.5. is tight and each limit point has no mass outside of .
Proof. Let and satisfy (2.3).
Using techniques as in the proof of Lemma 5.1 we can find a finite constant such that for
every and ,Applying Hölder's inequality to
each of the first two summands on the right-hand side, using (5.2) and setting , we may continue withwhere
are some finite
constants being independent of , and . Then Lemma 5.4 ensures that any weak limit point of has no mass
outside of . At this point, it is essential that we required and to be strictly
larger than , , respectively.
Toward the verification of tightness of we use Hölder's inequality to getIf , then (5.31) implies that both factors on the right-hand side of (5.32) are bounded by . If , then at least one of these factors vanishes since is constant on intervals of length . Hence,for all and with . That is, (5.28) holds with . Therefore, Lemma 5.3 ensures tightness of .
Here we give two auxiliary results. We first recall a square function inequality for martingales. Let be an -martingale on some probability space . The corresponding compensator is given by .
Proposition A.1 (see [21, Theorem 2.11]). For every there is some finite constant depending only on such that
The second result is a Gronwall lemma for functions with discrete domain. It can be proven by means of iterating (A.2) -times. We omit the proof since it is more or less well known.
Lemma A.2. Suppose is a mapping from to with . If there are finite constants such that then
The author thanks a referee for revealing a flaw of the original manuscript.
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