Weak Approximation of SDEs by Discrete-Time Processes
Henryk ZΓ€hle1
Academic Editor: Nikolai Leonenko
Received29 Oct 2007
Revised25 Jan 2008
Accepted20 Feb 2008
Published23 Mar 2008
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 weakly
converges to the unique solution of the following one-dimensional stochastic
differential equation (SDE):where 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,is weakly approximated by two
different types of population-size-dependent GWPs (in the sense of
[3β6]) with immigration, where and 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: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
[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 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.,
[10β12]). 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 in the cΓ‘dlΓ g
space implies convergence in distribution of the
corresponding stochastic integrals in the
cΓ‘dlΓ g 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 cΓ‘dlΓ g processes. For this reason we denote by the space
of cΓ‘dlΓ g 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 cΓ‘dlΓ g
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).
Theorem 2.2. Suppose SDE (1.3) subject to (1.4) has a unique weak
solution, and denote by the
corresponding law on . Moreover, let be the law (on ) of subject to
(2.1)β(2.3). Then as .
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 [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 ). 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 [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 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
[18]; 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.
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 .
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 HΓΆlder'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.
Proof. We
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
[19]; 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 HΓΆlder'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 HΓΆlder'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 .
Appendix
Auxiliary Results
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
Acknowledgment
The author thanks a referee for revealing a flaw of
the original manuscript.
References
T. Lindvall, βConvergence of critical Galton-Watson branching processes,β Journal of Applied Probability, vol. 9, no. 2, pp. 445β450, 1972.
T. Fujimagari, βControlled Galton-Watson process and its asymptotic behavior,β Kodai Mathematical Seminar Reports, vol. 27, no. 1-2, pp. 11β18, 1976.
R. Höpfner, βOn some classes of population-size-dependent Galton-Watson processes,β Journal of Applied Probability, vol. 22, no. 1, pp. 25β36, 1985.
F. C. Klebaner, βGeometric rate of growth in population-size-dependent branching processes,β Journal of Applied Probability, vol. 21, no. 1, pp. 40β49, 1984.
T. Yamada and S. Watanabe, βOn the uniqueness of solutions of stochastic differential equations,β Journal of Mathematics of Kyoto University, vol. 11, pp. 155β167, 1971.
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1991.
A. S. Cherny and H.-J. Engelbert, Singular Stochastic Differential Equations, vol. 1858 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2005.
S. Méléard and S. Roelly-Coppoletta, βInteracting measure branching processes. Some bounds for the support,β Stochastics and Stochastics Reports, vol. 44, no. 1-2, pp. 103β121, 1993.
S. Roelly-Coppoletta, βA criterion of convergence of measure-valued processes: application to measure branching processes,β Stochastics, vol. 17, no. 1-2, pp. 43β65, 1986.
A. Sturm, βOn convergence of population processes in random environments to the stochastic heat equation with colored noise,β Electronic Journal of Probability, vol. 8, no. 6, pp. 1β39, 2003.
D. W. Stroock and S. R. S. Varadhan, βMultidimensional Diffusion Processes,β vol. 233 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1979.
T. G. Kurtz and P. Protter, βWeak limit theorems for stochastic integrals and stochastic differential equations,β The Annals of Probability, vol. 19, no. 3, pp. 1035β1070, 1991.
R. Lorenz, Weak approximation of stochastic delay differential equations with bounded memory by discrete time series, Ph.D. thesis, Humbold University of Berlin, Berlin, Germany, 2006.
W. Feller, βDiffusion processes in genetics,β in Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and
Probability, 1950, pp. 227β246, University of California Press, Berkeley, Calif, USA.
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics, Springer, Berlin, Germany, 1992.
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical
Statistics, John Wiley & Sons, New York, NY, USA, 1986.
C. Dellacherie and P.-A. Meyer, Probabilités et potentiel, Chapitres V á VIII: Théorie des martingales, vol. 1385 of Actualités Scientifiques et Industrielles, Hermann, Paris, France, 1980.
P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application, Probability and Mathematical Statistics, Academic Press, New York, NY, USA, 1980.