On non-polynomial lower error bounds for adaptive strong approximation of SDEs
Recently, it has been shown in  that there exists a system of stochastic differential equations (SDE) on the time interval with infinitely often differentiable and bounded coefficients such that the Euler scheme with equidistant time steps converges to the solution of this SDE at the final time in the strong sense but with no polynomial rate. Even worse, in  it has been shown that for any sequence , which may converge to zero arbitrary slowly, there exists an SDE on with infinitely often differentiable and bounded coefficients such that no approximation of the solution of this SDE at the final time based on evaluations of the driving Brownian motion at fixed time points can achieve a smaller absolute mean error than the given number . In the present article we generalize the latter result to the case when the approximations may choose the location as well as the number of the evaluation sites of the driving Brownian motion in an adaptive way dependent on the values of the Brownian motion observed so far.
Let , , consider a -dimensional system of autonomous stochastic differential equations (SDE)
with a deterministic initial value , a drift coefficient , a diffusion coefficient and an -dimensional driving Brownian motion , and assume that (1) has a unique strong solution . Our computational task is to approximate by means of methods that use finitely many evaluations of the driving Brownian motion . In particular we are interested in the following question: under which assumptions on the coefficients and exists a method of the latter type, which converges to in absolute mean with a polynomial rate?
It is well-known that if the coefficients and are globally Lipschitz continuous then the classical Euler scheme achieves the rate of convergence , see . Moreover, the recent literature on numerical approximation of SDEs contains a number of results on approximation schemes that are specifically designed for non-Lipschitz coefficients and achieve polynomial convergence rates for suitable classes of such SDEs, see e.g. [16, 12, 18, 25, 38, 35, 37, 3, 21, 4] for SDEs with globally monotone coefficients and see e.g. [2, 8, 5, 1, 32, 17, 19, 23, 24, 33, 11] for SDEs with possibly non-monotone coefficients.
On the other hand, it has recently been shown in  that for any sequence , which may converge to zero arbitrary slowly, there exists an SDE (1) with and and with infinitely often differentiable and bounded coefficients and such that no approximation of based on finitely many evaluations of the driving Brownian motion converges in absolute mean faster than the given sequence . More formally,
In particular, there exists an SDE (1) with infinitely often differentiable and bounded coefficients and such that its solution at the final time can not be approximated with a polynomial rate of convergence based on finitely many evaluations of the driving Brownian motion . We add that the latter statement in the special case when the approximation is given by the Euler scheme with equidistant time steps has first been shown in .
Note that the time points that are used by an approximation in (2) are fixed, and therefore this negative result does not cover approximations that may choose the number as well as the location of the evaluation sites of the driving Brownian motion in an adaptive way, e.g. numerical schemes that adjust the actual step size according to a criterion that is based on the values of the driving Brownian motion observed so far, see e.g. [6, 29, 30, 27, 34, 22, 13, 14] and the references therein. See Section 4 for the formal definition of that type of approximations. It is well-known that for SDEs (1) with (essentially) globally Lipschitz continuous coefficients and adaptive approximations can not achieve a better rate of convergence compared to what is best possible for non-adaptive ones, which at the same time coincides with the best possible rate of convergence that can be achieved by any approximation based on , see [29, 30]. However, as has recently turned out, this is not necessarily the case anymore if the coefficients and are not both globally Lipschitz continuous. In  it has been shown that for the one-dimensional squared Bessel process, which is the solution of the SDE (1) with and for the following holds: the best possible rate of convergence that can be achieved by any approximation based on equals , i.e. there exist such that
while the best possible rate of convergence that can be achieved by approximations based on adaptively chosen evaluations of the driving Brownian motion equals infinity. More formally, for every there exists and a sequence of approximations based on adaptively chosen evaluations of such that
In view of the latter result one might hope that a non-polynomial lower error bound in (2) could be overcome by using adaptive approximations, see also the discussion in [7, p. 2]. In the present article we prove that the pessimistic alternative is true. We show that for any sequence , which may converge to zero arbitrary slowly, there exists an SDE (1) with and and with infinitely often differentiable and bounded coefficients and such that no approximation based on adaptively chosen evaluations of the driving Brownian motion on average can achieve a smaller absolute mean error than the given number , i.e.
for any approximation of the latter type. This fact is an immediate consequence of Corollary 2 in Section 5 together with an appropriate scaling argument. For the proof of the latter result we employ the same class of SDEs as in . Thus, roughly speaking, these SDEs can not be solved approximately in the strong sense in a reasonable computational time by means of any kind of adaptive (or nonadaptive) method based on finitely many evaluations of the driving Brownian motion .
We conjecture that a similar negative result does even if one allows for adaptive approximations based on finitely many evaluations of arbitrary linear continuous functionals of the driving Brownian motion . However, in this case one can not employ the class of SDEs from  since for every such SDE its solution at the final time can be approximated with error zero based on the evaluation of only two linear continuous functionals of the driving Brownian motion , see (2)
We briefly describe the content of the paper. In Section 2 we fix some notation. In Section 3 we briefly introduce the class of SDEs from , which is studied in this article as well. In Section 4 we formally define the class of adaptive approximations, which are analysed in this article. Our lower error bounds are stated in Section 5. The proof of the main result, Theorem 1, is carried out in Section 6.
Throughout this article the following notation is used. For a set , a vector space , a set , and a function we put . For sets , , a function and a subset we denote by the restriction of to . Moreover, for and we write for the Euclidean norm of . For and we denote by and the Borel -fields on and on , respectively, where the latter space is equipped with the supremum norm. For being a finite product of the latter two spaces we denote by the Borel -field on generated by the respective product topology.
3. A family of SDEs with smooth and bounded coefficients
Throughout this article we study SDEs provided by the following setting.
Let , let be a probability space with a normal filtration , and let be a standard -Brownian motion on .
Let and let be bounded and satisfy , , , , , and .
For every let and be given by
and consider the following -dimensional system of SDEs
Note that for every the functions and are infinitely often differentiable and bounded.
It is easy to see that for every the SDE (3) has a unique strong solution given by
for all .
4. Adaptive strong approximations
Let . We study general strong approximations of based on and on finitely many sequential evaluations of in the interval . Every such approximation is defined by three sequences
of measurable mappings
The sequence determines the evaluation sites of a trajectory of in the interval . The total number of evaluations is determined by the sequence of stopping rules. Finally, the sequence is used to obtain the approximation to from the observed data.
More precisely, let , let be the corresponding trajectory of and put . The sequential observation of starts at the knot . After steps the available information is then given by , where , …, , and we decide whether we stop or further evaluate according to the value of . The total number of observations of in the interval is thus given by
If , then the data is used to construct the estimate .
For obvious reasons we require that -a.s. Then the resulting approximation is given by
Without loss of generality we assume that
for all , all with and all . We put
that is the expected number of evaluations of the driving Brownian motion in the interval . We denote by the class of all methods of the above form and for we put
Clearly, for all and all .
Let us stress that the class contains in particular all methods from the literature, which use a step size control based on sequential evaluations of on average, see e.g. [6, 29, 30, 27, 34, 22, 13, 14] and the references therein. Moreover, of course contains all nonadaptive approximations based on evaluations of at fixed time points and on , as studied in . In the latter case one can take any sequences and satisfying
5. Main results
Assume the setting in Section 3 and put
as well as
Our main result is stated in Theorem 1. It provides a uniform lower bound for the mean absolute error of any strong approximation of that is based on and on sequential evaluations of in the interval on average in the case that is positive, strictly increasing and satisfies as well as . See Section 6 for the proof.
Let and let be positive, strictly increasing with and . Then for all and all we have
As a consequence of Theorem 1 we obtain a non-polynomial decay of the smallest possible mean absolute error of strong approximation of based on and on sequential evaluations of in the interval on average if additionally satisfies an exponential growth condition.
Let and let be positive, strictly increasing with and . Moreover assume that for all
Then for all we have
The following result shows that the smallest possible mean absolute error of strong approximation of based and on sequential evaluations of in the interval on average may converge to zero arbitrarily slow even then when the sequence tends to zero with any given speed.
Let and satisfy and . Then there exists and such that for all we have
We proceed similar to the proof of Corollary 4.3 in . Without loss of generality we may assume that the sequences and are strictly decreasing. Let
and for put
Note that the sequences and are strictly increasing and satisfy
Define a function by
Then is positive, strictly increasing, infinitely often
differentiable and satisfies
as well as .
Theorem 1 implies that for all
Since the sequence is decreasing hence for all
Using the assumption that the sequence is strictly decreasing we therefore conclude that for all
which completes the proof of the corollary with . ∎
For , , and define functions
as follows. If put , and and let
as well as
for . Otherwise put
where is the permutation of such that .
For , , and put
Note that due to the assumption (7) we have . Let denote the Gaussian measure on with mean and covariance function . Consider the measurable space
It is easy to see that is - measurable. Define the mapping
for all and .
is a version of the regular conditional distribution .
In the case of the statement of Lemma 1 seems to be well-known, see, e.g., [15, 28, 29, 30], but a proof of it seems not to be available in the literature. If, additionally, is constant then Lemma 1 follows from Lemma 2.9.7 in [36, p. 474], but measurability issues have not been fully addressed in the proof of the latter result. For convenience of the reader we therefore provide a proof of Lemma 1 here.
Clearly, for all the mapping
is a probability measure on .
Next, let . We show that the mapping
is - measurable. For , and define a function
for . It is easy to see that for all
and therefore for all
for . The measurability of the functions , , imply that for every the mapping
is - measurable. Thus, observing that for every the mappings
are continuous we conclude that for every the mapping
is - measurable. Hence for every
which implies that the mapping
Next, assume that , let , and . Recall the definition (11) of the time points and put
Let be the permutation of such that . Let and put