Weak quantitative propagation of chaos via differential calculus on the space of measures
Jean-Fran\c{c}ois Chassagneux, Lukasz Szpruch, Alvin Tse
TL;DR
This paper establishes quantitative bounds on the difference between a functional of a probability measure and its empirical approximation, using differential calculus on the space of measures, with applications to propagation of chaos in particle systems.
Contribution
It introduces regularity conditions and techniques for deriving precise asymptotic expansions for functionals on measure spaces, applicable to McKean-Vlasov SDEs and mean-field models.
Findings
Quantitative estimates of propagation of chaos.
Asymptotic expansion of functionals with respect to sample size.
Weak propagation of chaos properties for interacting particle systems.
Abstract
Consider the metric space of square integrable laws on with the topology induced by the 2-Wasserstein distance . Let be a function and be the empirical measure of a sample of random variables distributed as . The main result of this paper is to show that under suitable regularity conditions, we have \[ |\Phi(\mu) - \mathbb{E}\Phi(\mu_N)|= \sum_{j=1}^{k-1}\frac{C_j}{N^j} + O(\frac{1}{N^k}), \] for some positive constants that do not depend on , where corresponds to the degree of smoothness. We distinguish two cases: a) is the empirical measure of -samples from ; b) is a marginal law of McKean-Vlasov stochastic differential equation in which case is an empirical law of marginal laws of the corresponding…
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Weak quantitative propagation of chaos via differential calculus on the space of measures
Jean-François Chassagneux
Lukasz Szpruch
School of Mathematics, University of Edinburgh
Alvin Tse
School of Mathematics, University of Edinburgh
Abstract
Consider the metric space of square integrable laws on with the topology induced by the 2-Wasserstein distance . Let be a function and be the empirical measure of a sample of random variables distributed as . The main result of this paper is to show that under suitable regularity conditions, we have
[TABLE]
for some positive constants that do not depend on , where corresponds to the degree of smoothness. We distinguish two cases: a) is the empirical measure of -samples from ; b) is a marginal law of McKean-Vlasov stochastic differential equation in which case is an empirical law of marginal laws of the corresponding particle system. The first case is studied using functional derivatives on the space of measures. The second case relies on an Itô-type formula for the flow of probability measures and is intimately connected to PDEs on the space of measures, called the master equation in the literature of mean-field games. We state the general regularity conditions required for each case and analyse the regularity in the case of functionals of the laws of McKean-Vlasov SDEs. Ultimately, this work reveals quantitative estimates of propagation of chaos for interacting particle systems. Furthermore, we are able to provide weak propagation of chaos estimates for ensembles of interacting particles and show that these may have some remarkable properties.
1 Introduction
The aim of this work is to provide an exact weak error expansion between a (nonlinear) functional of the empirical measure and its deterministic limit , . We distinguish two cases: a) is the empirical measure of -samples from ; b) is the marginal law of a process described by a McKean-Vlasov stochastic differential equation (McKV-SDE), in which case is the empirical measure of the marginal laws of the corresponding particle system.
In the first case where is the empirical measure of -samples from , the only interesting case is when the functional is non-linear. To provide some context to our results, one may, for example, assume that is Lipschitz continuous with respect to the Wasserstein distance, i.e, there exists a constant such that
[TABLE]
one could bound by . Consequently, following [15] or [14], the rate of convergence in the number of samples deteriorates as the dimension increases. On the other hand, recently, authors [13, Lem. 5.10] made a remarkable observation that if the functional is twice-differentiable with respect to the functional derivative (see Section 2.1.1), then one can obtain a dimension-independent bound for the strong error , , which is of order (as expected by CLT). Here, we study a weak error and show that, (see Theorem 2.17) if is -times differentiable with respect to the functional derivative, then indeed we have
[TABLE]
for some positive and explicit constants that do not depend on . The result is of independent interest, but is also needed to obtain a complete expansion for the error in particle approximations of McKV-SDEs that we discuss next.
The second situation we treat in this work concerns estimates of propagation-of-chaos type 111We would like to remark that our results also cover a situation where the law is induced by a system of stochastic differential equations with random initial conditions that are not of McKean-Vlasov type in which case the samples are i.i.d.. Consider a probability space with a -dimensional Brownian motion . We are interested in the McKean-Vlasov process with interacting kernels and , starting from a random variable , defined by the SDE222We assume without loss of generality that the dimensions of and are the same because we will not make any non-degeneracy assumption on the diffusion coefficient in our work. In particular, one dimension of could be time itself.
[TABLE]
where denotes the law of and functions and satisfy suitable conditions so that there exists a unique weak solution (see e.g. [34] or, for more up-to-date panorama on research on existence and uniqueness, see [32, 20, 1, 12]). McKean-Vlasov SDE (1.1) can be derived as a limit of interacting diffusions. Indeed, one can approximate the law by the empirical measure generated by particles defined as
[TABLE]
where , are independent -dimensional Brownian motions and , are i.i.d. random variables with the same distribution as . It is well known, [34, Prop. 2.1], that the property of propagation of chaos is equivalent to weak convergence of measure-valued random variables to . A common strategy is to establish tightness of and to identify the limit by showing that converges weakly to . This approach does not reveal quantitative bounds we seek in this paper, but is a very active area of research. We refer the reader to [18, 34, 29] for the classical results in this direction and to [22, 3, 16, 30, 26] for an account (non-exhaustive) of recent results. On the other hand, the results on quantitative propagation of chaos are few and far in between. In the case when coefficients of (1.1) depend on the measure component linearly, i.e., are of the form
[TABLE]
with being Lipschitz continuous in both variables, it follows from a simple calculation [34] to see that . We refer to Sznitman’s result as strong propagation of chaos. Note that in this work we treat the case of McKean-Vlasov SDEs with coefficients with general measure dependence. In that case, as explicitly demonstrated in [7, Ch. 1], the rate of strong propagation of chaos deteriorates with the dimension . This is due to the fact that one needs to estimate the difference between the empirical law of i.i.d. samples from and itself using results such as [15] or [14]. In the special case when the diffusion coefficient is constant, with linear measure dependence on the drift (which lies in some negative Sobolev space), the rate of convergence in the total variation norm has been shown to be in [21]. Of course, in a strong setting, is widely considered to be optimal as it corresponds to the size of stochastic fluctuations as predicted by the CLT. In this work, we are interested in weak quantitative estimates of propagation of chaos. Indeed, this new direction of research has been put forward very recently by two independent works [24, Ch. 9] and [31, Th. 2.1]. The authors presented novel weak estimates of propagation of chaos for linear functions in measure, i.e. with being smooth. This gives the rate of convergence , plus the error due to approximation of the functional of the initial law (see [31, Lem. 4.6] for a discussion of a dimensional-dependent case). While the aim of [31] is to establish quantitative propagation of chaos for the Boltzmann’s equation, in a spirit of Kac’s programme [23, 28], Theorem 6.1 in [31, Th. 6.1] specialises their result to McKV-SDEs studied here, but only for elliptic diffusion coefficients that do not depend on measure and symmetric Lipschitz drifts with linear measure dependence. The key idea behind both results is to work with the semigroup that acts on the space of functions of measure, sometimes called the lifted semigroup, which can be viewed a dual to the space of probability measures on as presented in [30]. A similar research programme, but in the context of mean-field games with a common noise, has been successfully undertaken in [6]. In this work, the authors study the master equation, which is a PDE driven by the Markov generator of a lifted semi-group. They show that existence of classical solution to that PDE is the key to obtain quantitative bounds between an -player Nash system and its mean field limit. Indeed, perturbation analysis of the PDE on the space of measures leads to the weak error being of the order .
In this work we build on these observations, and identify minimal assumptions for the expansion in number of particles to hold. Next, we verify these assumptions for McKV-SDEs with a general drift and general (and possibly non-elliptic) diffusion coefficients. We also consider non-linear functionals of measure. The main theorem in this paper, Theorem 2.17, states that given sufficient regularity we have
[TABLE]
where are constants that do not depend on .
As mentioned above, the method of expansion relies heavily on the calculus on and we follow the approach presented by P. Lions in his course at Collège de France [27] (redacted by Cardaliaguet [5]). To obtain such an expansion, one needs to rely on some smoothness property for the solution of (1.1) as it is always the case when one wants to gets error expansion for some approximating procedure (see [36] for a similar expansion on the weak error expansion of SDE approximation with time-discretisation). The important object in our study, similarly to [6], is the PDE written on the space , which corresponds to the lifted semigroup and comes from the Itô’s formula of functionals of measures established in [4] and [9]. Smoothness properties on the functions (see Definition 2.6) required for expansion (2.17) to hold are formulated in Theorem 2.9. A natural question is then to identify some sufficient conditions on the SDE coefficients to guarantee the smoothness property of the functions . We give one possible answer to this question in Theorem 2.17. In that theorem, we show that if the coefficients of the SDE are smooth, then the functions are also smooth enough provided that is itself smooth. This result, which is expected, comes from an extension of Theorem 7.2 in [4] (see Theorem 2.15).
While in the current paper, we assume high order of smoothness of the coefficients of McKV-SDEs and , we anticipate this general approach to be valid under a less regular setting. Indeed, when working with a strictly elliptic setting with some structural conditions, the lifted semigroup may be smooth even in the case when drift and diffusion coefficients are irregular. This has been demonstrated in [11, 12]. Similarly, does not need to be smooth for the lifted semigroup to be differentiable in the measure direction. This has been shown using techniques of Malliavin calculus in [10]. Finally, when the underlying equation has some special structure, the more classical approach can be deployed to study weak propagation of chaos property [2]. The analysis of irregular cases goes beyond the scope of this paper.
To sum up, there are three main contributions in this paper. Firstly, the main result (Theorem 2.17) allows us to use Romberg extrapolation to obtain an estimator of with weak error being in the order of , for each . (See Section 1.1 for details.) Thus, effectively, a higher-order particle system (in terms of the weak error) can be constructed up to a desired order of approximation. Secondly, the analysis in this paper makes use of the notions of measure derivatives and linear functional derivatives by generalising them to an arbitrary order of differentiation. This is in line with the approach in [10]. Some properties (e.g. Lemma 2.5) relate the regularity of the two notions of derivatives in measure and might be of an independent interest. In particular, the generalisation of Theorem 7.2 in [4] from second order derivatives in measure to higher order derivatives is proven to be useful in the analysis of McKean-Vlasov SDEs in general. Finally, as a by-product of the weak error expansion, a version of the law of large numbers in terms of functionals of measures is developed in Theorem 2.12.
1.1 Romberg extrapolation and ensembles of particles
In this section we construct an ensemble particle system in the spirit of Richardson’s extrapolation method [33] that has been studied in the context of time-discretisation of SDEs in [36] and in the context of discretisation of SPDEs in [19].
Let be a Borel-measurable function and define . Observe that
[TABLE]
Hence, the weak error reads . By the result of Theorem 2.17, we can apply the technique of Romberg extrapolation to construct an estimator which approximates such that the weak error is of the order of . More precisely, for , since is independent of ,
[TABLE]
and
[TABLE]
Hence,
[TABLE]
For general , we can use a similar method to show that
[TABLE]
where
[TABLE]
To motivate the study of the weak error expansion we will analyse an estimator that uses ensembles of particles. Fix . The ensembles are indexed by . For , consider
[TABLE]
where are independent ensembles each consisting of -dimensional Brownian motions; and are independent ensembles each consisting of i.i.d. random variables with the same distribution as . We consider the following estimator
[TABLE]
Next we analyse mean-square error333We look at the mean-square error for simplicity, but a similar computation could be done to verify the Lindeberg condition and produce CLT with an appropriate scaling. of this estimator
[TABLE]
The first term on the right-hand side is studied in Theorem 2.17 and, provided that the coefficients of (1.1) are sufficiently smooth, it converges with order . Control of the second term follows from the qualitative strong propagation of chaos. Indeed, we write
[TABLE]
where denotes the solution of (1.1) driven by with initial data . Hence, independence implies that
[TABLE]
On the other hand,
[TABLE]
where Jensen’s inequality is used. Using the fact is Lipschitz continuous and the result on a dimension-free bound for strong propagation of chaos, established in [35], there exists a constant with no dependence on such that
[TABLE]
Consequently, we have
[TABLE]
Since there are ensembles corresponding to the estimator and each ensemble has sub-particle systems with particles each, , the total number of interactions is . When we take and the mean-square error is of the order (since is a constant). The corresponding number of interactions is of the order . The message here is that as the smoothness increases, less interactions among particles are needed when approximating the law of McKean-Vlasov SDE (1.1). We would like to stress out again that the dimension of the system does not deteriorate the rate of convergence, in contrast to results presented in the literature [8, 15, 30]. It is instructive to compare the above computation with a usual mean-square analysis of a single particle system
[TABLE]
As above, invoking strong propagation of chaos, one can show that the second term is of order O\Big{(}N^{-1}\Big{)}. That means that there would be no gain to go beyond what we can obtain from the strong propagation of chaos analysis to control the first term. Taking results in mean-square error being of the order and number of interactions . That clearly demonstrates that working with ensembles of particles leads to an improvement in quantitative properties of propagation of chaos, which is interesting on its own but can also be explored when simulating particle systems on the computer.
Notations.
- •
The Wasserstein metric is defined by
[TABLE]
where denotes the set of couplings between and i.e. all measures on such that and for every .
- •
Uniqueness in law of (1.1) implies that for any random variables such that , we have . Therefore, we adopt the notation if only the law of the process is concerned.
- •
When the total number of particles is clear from context, we will often simply write for .
- •
For any , we denote their inner product by . Since different measure derivatives lie in different tensor product spaces, we use to denote the Euclidean norm for any tensor product space in the form .
- •
The law of any random variable is denoted by . For any function , its lift is defined by .
- •
Also, stands for a copy of , which is useful to represent the Lions’ derivative of a function of a probability measure. Any random variable defined on is represented by as a pointwise copy on . In the section on regularity, we shall introduce a sequence of copies of , denoted by . As before, any random variable defined on is represented by as a pointwise copy on .
- •
For , we define the following subsets of , :
[TABLE]
and
[TABLE]
We often denote and . We shall also sometimes use the convention for simplicity of notation.
- •
With the above definition, we denote
[TABLE]
- •
For any function , we always denote by the partial derivatives of in the variable at whenever they exist.
- •
denotes the set of square integrable random variables, the set of square-integrable progressively measurable processes such that .
2 Method of weak error expansion
2.1 Calculus on the space of measures
Our method of proof is based on expansion of an auxiliary map satisfying a PDE on the Wasserstein space. One of the most important tools of the paper is thus the theory of differentiation in measure.
We make an intensive use of the so-called “L-derivatives” and “linear functional derivatives” that we recall now, following essentially [6]. We also introduce a higher-order version of this derivative as this is needed in the proofs of our expansion.
2.1.1 Linear functional derivatives
A continuous function is said to be the linear functional derivative of , if
- •
for any bounded set , has at most quadratic growth in uniformly in ,
- •
for any ,
[TABLE]
For the purpose of our work, we need to introduce derivatives at any order .
Definition 2.1**.**
For any , the -th order linear functional of the function is a continuous function from satisfying
- •
for any bounded set , has at most quadratic growth in uniformly in ,
- •
for any ,
[TABLE]
provided that the -th order derivative is well defined.
The above derivatives are defined up to an additive constant via (2.1). They are normalised by
[TABLE]
We make the following easy observation, which will be useful in the latest parts.
Lemma 2.2**.**
If admits linear functional derivatives up to order , then the following expansion holds
[TABLE]
Proof.
We define
[TABLE]
and apply Taylor-Lagrange formula to up to order , namely
[TABLE]
It remains to show that
[TABLE]
by induction. Since (2.4) holds trivially for , we suppose that (2.4) holds for Then
[TABLE]
Taking gives (2.4) for . This completes the proof. ∎
2.1.2 L-derivatives
The above notion of linear functional derivatives is not enough for our work. We shall need to consider further derivatives in the non-measure argument of the derivative function.
If the function is of class , we consider the intrinsic derivative of that we denote
[TABLE]
The notation is borrowed from the literature on mean field games and corresponds to the notion of “L-derivative” introduced by P.-L. Lions in his lectures at Coll?ge de France [27]. Traditionally, it is introduced by considering a lift on an space of the function and using the Fr?chet differentiability of this lift on this Hilbert space. The equivalence between the two notions is proved in [8, Tome I, Chapter 5], where the link with the notion of derivatives used in optimal transport theory is also made.
In this context, higher order derivatives are introduced by iterating the operator and the derivation in the non-measure arguments. Namely, at order , one considers
[TABLE]
This leads in particular to the notion of a fully function that will be of great interest for us (see [9]).
Definition 2.3** (Fully ).**
A function is fully if the following mappings
[TABLE]
are well-defined and continuous for the product topologies.
Let us observe for later use that if the function is fully and moreover satisfies, for any compact subset ,
[TABLE]
then it follows from Theorem 3.3 in [9] that can be expanded along the flow of marginals of an It? process. Namely, let where
[TABLE]
with and , then
[TABLE]
In order to prove our expansion, we need to iterate the application of the previous chain rule and in order to proceed, we need to use higher order derivatives of the measure functional.
Inspired by the work [10], for any , we formally define the higher order derivatives in measures through the following iteration (provided that they actually exist): for any , and , the function is defined by
[TABLE]
and its corresponding mixed derivatives in space are defined by
[TABLE]
Since this notation for higher order derivatives in measure is quite cumbersome, we introduce the following multi-index notation for brevity. This notation was first proposed in [10].
Definition 2.4** (Multi-index notation).**
Let be non-negative integers. Also, let be an -dimensional vector of non-negative integers. Then we call any ordered tuple of the form or a multi-index. For a function , the derivative is defined as
[TABLE]
if this derivative is well-defined. For any function , we define
[TABLE]
if this derivative is well-defined. Finally, we also define the order 444 We do not consider ‘zeroth’ order derivatives in our definition, i.e. at least one of , and must be non-zero, for every multi-index \big{(}n,\ell,(\beta_{1},\ldots,\beta_{n})\big{)}. (resp. ) by
[TABLE]
As for the first order case, we can establish the following relationship with linear functional derivatives, see e.g. [6] for the correspondence up to order 2,
[TABLE]
provided one of the two derivatives is well-defined.
Next, we deduce the following lemma that will be useful later on.
Lemma 2.5**.**
Let and assume that . Then
[TABLE]
for some constant .
Proof.
We sketch the proof by induction in dimension one, for ease of notation. Let .
First, we compute that
[TABLE]
Let denote the derivative w.r.t. the th component of the spatial variables. From the convention of normalisation (2.2), we simply obtain that
[TABLE]
Let and assume that
[TABLE]
Then, observing that
[TABLE]
we recover
[TABLE]
Setting in (2.10), we then obtain
[TABLE]
The proof is concluded by invoking the boundedness assumption of along with Young’s inequality.
∎
2.2 Weak error expansion along dynamics
To state our expansion for the dynamic case, we will need some notion of smoothness given in the following definition.
Definition 2.6**.**
Let be a positive integer. A function is of class if the following conditions hold:
- i)
is jointly continuous on . 2. ii)
For all , is fully . 3. iii)
Let be a positive constant. For all and ,
[TABLE] 4. iv)
- •
: is continuously differentiable on .
- •
: for all and all , the function
[TABLE]
is continuously differentiable on . 5. v)
The functions
[TABLE]
are continuous.
We define recursively the functions , , , that are used to prove the expansion.
Definition 2.7**.**
- i)
For , we set and define by
[TABLE]
Assuming that belongs to the class , we set as
[TABLE] 2. ii)
For , we define by
[TABLE]
Assuming that belongs to the class , we set as
[TABLE]
A key point in our work is to show that the previous definition is licit under some assumptions on the coefficient functions and (Theorem 2.16 and Theorem 2.17).
Before we proceed we state the following assumptions
- (Lip)
and are Lipschitz continuous with respect to the Euclidean norm and the norm.
- (UB)
There exists such that , for every and .
It will become apparent from the proofs that when working only with (Lip), higher order integrability conditions would need to be stated in Definition (2.6). We refrain from this extension and assume (UB) to improve readability of the paper, but encourage a curious reader to perform this simple extension.
We begin with the following technical lemma.
Lemma 2.8**.**
Assume (Lip) and (UB). Let be a positive integer and be a continuous function. Consider given by and set , where . If is of class , then the following statements hold:
- (i)
* satisfies on the following PDE*
[TABLE]
with terminal condition , where denotes the diffusion operator
[TABLE] 2. (ii)
* can be expanded along the flow of random measure associated to the particle system (1.2) as follows, for all ,*
[TABLE]
*where is a square integrable martingale with . *
Proof.
(i) By the flow property, we observe that the function is constant. Indeed, Applying the chain rule in both time and measure arguments between and , we get
[TABLE]
Dividing by and letting allows to recover the first claim.
(ii) To recover the expansion, we use the known strategy of considering finite dimensional projection of . Namely, for a fixed number of particles , we define
[TABLE]
From Definition 2.6(ii), (iv) and (v), we have that is (see Proposition 3.1 in [9]. Recalling the link between the derivatives of and (again see Proposition 3.1 in [9]), we can apply the classical Ito’s formula to to get
[TABLE]
We first note that the term in (2.15) is precisely (2.12) evaluated at and is thus equal to zero. We now study the local martingale term in (2.14). We simply compute
[TABLE]
where we have used (UB). Using Definition 2.6(iii), we have
[TABLE]
which concludes that is a square integrable martingale.
∎
Theorem 2.9** (Weak error expansion: dynamic case).**
Assume (Lip) and (UB). Suppose that Definition 2.7 is well-posed for . Then the weak error in the particle approximation can be expressed as
[TABLE]
where and
[TABLE]
and and
[TABLE]
Proof.
Part 1: We first check that the constants are well defined.
For , we first show that the function
[TABLE]
is continuous. Indeed, let be a sequence converging to in the product topology. Then there exists a sequence of random variable such that converging to with law in . By continuity of , Definition 2.7 and Definition 2.6(v),
[TABLE]
in probability. Next, since is bounded,
[TABLE]
where the last inequality follows from the fact that is of class , by Definition 2.6(iii). By de La Vallée Poussin Theorem, the previous computation shows that is uniformly integrable and thus . Observing that is continuous, we conclude that is also continuous (hence measurable) and therefore is well-defined.
Hence, by the definition of , for each , the function
[TABLE]
is continuous. Also, by the previous argument along with Definition 2.6(iii), we can see that is uniformly bounded. Therefore, the function
[TABLE]
is also uniformly bounded. By the dominated convergence theorem, the function
[TABLE]
is continuous. This shows that is well-defined.
Part 2: We now proceed with the proof of the expansion, which is done by induction on .
Base step: We decompose the weak error as
[TABLE]
Applying Lemma 2.8(ii) for the first term in the right-hand side and taking expectation on both side, we obtain that
[TABLE]
Recalling the definition of in (2.11), we get
[TABLE]
From Part 1, we know that is uniformly bounded and thus , where does not depend on . This proves the induction for the base step.
Induction step: Assume that for ,
[TABLE]
Then, we observe that
[TABLE]
which leads to
[TABLE]
Applying Lemma 2.8(ii) to , we obtain that
[TABLE]
Inserting this back into (2.19), we get
[TABLE]
The proof is concluded by observing that , due to the uniform boundedness of given in Part 1.
∎
2.3 Weak error expansion for the initial condition
Assuming enough smoothness of the functions , we can take care of the terms appearing in the previous theorem, which are error made at time [math]. The following weak error analysis relies on the notion of linear functional derivatives. We first start by studying the weak error generated between the evaluation of the function at a measure and its empirical measure counterparts. We prove two results: one dealing mainly with low order expansion and the order one, available at any order.
The main assumption we work with relates to the couple , where is a function with domain .
- (p-LFD)
The th order linear functional derivative of exists and is continuous and that for any family of random variable identically distributed with law the following holds
[TABLE]
for some positive constant .
We first make the following observation regarding assumption (p-LFD), that will be of later use.
Remark 2.10**.**
(i) Lemma 2.5 states that
[TABLE]
for every , for every , and for some . This means that for any , the couple satisfies (p-LFD). This polynomial growth condition is motivated by our example of application, stated in Section 2.4, that relies on the smoothness of the coefficients.
(ii) The following simple example of measure functional shows that the above condition is reasonable to consider: For any bounded smooth function , we set The linear derivative functional of order can then be computed by induction, using the normalisation convention (2.2), to obtain that
[TABLE]
which easily relates to (2.20) .
Theorem 2.11**.**
Let be i.i.d. random variables with law . The following statements hold:
- (i)
Let (p-LFD) hold with for . Then
[TABLE] 2. (ii)
Let (p-LFD) hold with for . Suppose that . Then
[TABLE]
where and is independent of .
Proof.
Let and , We also consider i.i.d. random variables with law that are also independent of .
- (i)
By the definition of linear functional derivatives, we have
[TABLE]
We introduce measures
[TABLE]
and notice that
[TABLE]
Therefore,
[TABLE]
To conclude part (i), we observe that
[TABLE]
by assumption (p-LFD) with . 2. (ii)
We continue the expansion of (2.21). To avoid a further interpolation in measure between and , we proceed via integration by parts. Let
[TABLE]
and note that . Then, by a similar method as the derivation of (2.4),
[TABLE]
Therefore, by integration by parts,
[TABLE]
For the final term in (LABEL:eq:exp_second_step), by exchangeability, we rewrite
[TABLE]
As before, we introduce measures
[TABLE]
Then
[TABLE]
Combining (2.21), (LABEL:eq:exp_second_step), (2.23) and (2.24) gives
[TABLE]
Using the fact that satisfies assumption (p-LFD) with , the statement for part (ii) is established.
∎
In principle, we can continue the above expansion to higher orders. However, in the next theorem we present a simplified argument that allows for complete weak error expansion. The simplification is at the cost of requiring one extra order of regularity in the assumption. However, we believe the argument is of independent interest.
Theorem 2.12** (Weak error expansion: static case).**
Let be a positive integer and . Suppose that assumption (p-LFD) holds for , for each . Then, for i.i.d. random variables with law ,
[TABLE]
where
[TABLE]
for some i.i.d. random variables with law that are also independent of .
Proof.
Let . By Lemma 2.2, we have
[TABLE]
with
[TABLE]
where . Observe that by assumption (p-LFD) all the terms in the expansion are well defined. We study them now. For , we have
[TABLE]
Now let and observe that
[TABLE]
Suppose that at least one of the is different from the other , . Without loss of generality, we assume that this is the case for . We then observe that
[TABLE]
by conditioning on . Therefore, when ,
[TABLE]
It remains to study the remainder term above. We rewrite
[TABLE]
Let be a subset of . We denote and introduce
[TABLE]
Then
[TABLE]
For , we simply observe that
[TABLE]
For , we consider defined above and work with the special choice , which implies, by exchangeability, that
[TABLE]
For later use, we denote
[TABLE]
We will now work iteratively from to .
Firstly, we introduce
[TABLE]
where we define independent random variables that are also independent of and , but with the same law. We then compute that
[TABLE]
As before,
[TABLE]
so the term on the right hand side of (2.28) is equal to zero. Next, for , we define inductively
[TABLE]
This procedure is then iterated from to on the remainder term in (2.29). We thus have
[TABLE]
Next, by (2.20), we estimate the integral by
[TABLE]
where we used assumption (p-LFD). Combining with (2.30) and (2.31) gives
[TABLE]
Finally, combining with (2.27) yields
[TABLE]
∎
2.4 Expansion in terms of regularity of the drift and diffusion functions
In this subsection, we explore a sufficient condition for the expansion of an arbitrary order purely in terms of regularity of the drift and diffusion functions. It turns out that proving regularity conditions for higher order expansions for class is highly non-trivial and therefore a stronger notion of regularity in differentiating measures is proposed.
Definition 2.13**.**
A function belongs to class , if the derivatives exist for every multi-index such that and satisfy
[TABLE]
[TABLE]
for any and , for some constant .
By convention, a function defined only on will be extended to naturally by , for all .
For the time-dependent case (possibly with multi-index in time), we extend the previous definition as follows.
Definition 2.14**.**
A function is said to be in 555This definition is modified accordingly to define , where ., if
- •
: is continuously differentiable on .
- •
: for all and all , the function
[TABLE]
is continuously differentiable on . 2. 2.
, for each , where the constant in (2.32) and (2.33) is uniform in . 3. 3.
All derivatives in measure (including the zeroth order derivative) of up to the th order are jointly continuous in time, measure and space.
When it is clear from context, we will just use the notation for the two definitions above.
Note that the condition automatically implies (Lip). The following is a generalisation of Theorem 7.2 in [4] from to , for any .
Theorem 2.15**.**
Suppose that and are in , where . We consider a function defined by
[TABLE]
for some function that is also in . Then and satisfies the PDE
[TABLE]
This proof of Theorem 2.15 is postponed to the next section. We now state the key result for this part which will certify that the expansion along the dynamics is licit.
Theorem 2.16**.**
Assume (UB). Suppose that and belong to the class . Moreover, suppose that also belongs to the class . Then Definition 2.7 is well-posed for .
Proof.
We prove by induction on and prove that for each , and therefore , which establishes the claim.
For simplicity of notations, we present this proof in the case of dimension one. We commence the proof by noting that and , therefore it follows from Theorem 2.15 that .
Suppose that for , We recall the definition of as
[TABLE]
Fix . We shall first establish the smoothness of . Let be a continuous function defined by
[TABLE]
Since , for each , is also differentiable in measure with its derivative given by
[TABLE]
We observe that and are both continuous and uniformly bounded in space and measure. Therefore, by Example 3 in Section 5.2.2 of [8], is differentiable in measure with its derivative given by
[TABLE]
where is given by
[TABLE]
Formulae (2.35) and (2.36) tell us that is uniformly bounded in measure and space. Furthermore, each of and is a finite sum of products of uniformly bounded Lipschitz functions in measure and space, and is hence Lipschitz continuous as well. Finally, by the duality formula for the Kantorovich-Rubinstein distance (see Remark 6.5 in [37]), we note that there exist constants such that for every and ,
[TABLE]
where denotes the -Wasserstein metric.
Subsequently, we can repeat the same procedure to prove existence and regularity properties of higher order derivatives of . In particular, we can show that and
exist, by expressing them in terms of derivatives of up to the fourth order, and derivatives of up to the second order, which also allows us to show that they are uniformly bounded and Lipschitz continuous. In general, for any multi-index such that , we can show that exists, by expressing it in terms of derivatives of up to the th order, and derivatives of up to the th order, which again allows us to show that it is uniformly bounded and Lipschitz continuous. Thus, .
Next, we note that since , is continuously differentiable in the last component of and so is . Moreover, as mentioned above, each derivative up to the th order can be expressed in terms of derivatives of up to the th order and derivatives of up to the th order, which implies that each derivative is jointly continuous in time, measure and space, since
. Therefore, by Definition 2.14, .
We now recall the definition of , given by
[TABLE]
For fixed it follows from Theorem 2.15 that is continuously differentiable in time and that , for each Finally, all derivatives in measure of up to the th order are jointly continuous in time, measure and space, since . This implies that , which concludes the proof by the principle of induction.
∎
The following theorem is the main result of this paper and is a direct consequence of Theorem 2.9, Theorem 2.12, Theorem 2.16 and Remark 2.10(i).
Theorem 2.17** (Main result on regularity: Full expansion).**
Assume (UB). Suppose that and belong to the class . Moreover, suppose that also belongs to the class . Finally, suppose that the initial condition satisfies Then
[TABLE]
where are constants that do not depend on .
Proof.
We commence the proof by noting that , and all belong to , therefore it follows from Theorem 2.15 that . As in the proof of Theorem 2.16, we prove by induction on in order to establish that for each , . By Theorem 2.16, Definition 2.7 is well-posed for . Therefore, by Theorem 2.9, we have
[TABLE]
for some constants , where
[TABLE]
Recall that . By Remark 2.10(i) and Theorem 2.12,
[TABLE]
for some constants . Similarly, for every , since , it also follows by Remark 2.10(i) and Theorem 2.12 666Note that for , the constant in Lemma 2.5 is uniform in . Therefore, the constant in the same inequality (2.20) in Theorem 2.12 is also uniform in . The fact that the constants are well-defined follows from a similar argument as the first part of the proof of Theorem 2.9. that
[TABLE]
for some constants . The result follows by combining (2.37), (2.38) and (2.39). ∎
3 Proof of Theorem 2.15
Let be a Borel-measurable function. In this section, we study the smoothness of the function defined by
[TABLE]
There are various methods of establishing smoothness of functions of this form in the literature. One way involves considering PDE (2.12) and proving regularity properties of the solution to this PDE ([6]).
The method of Malliavin calculus is adopted in [10]. This paper proves smoothness of , for being in the form
[TABLE]
where is infinitely differentiable with bounded partial derivatives.
Article [11] considers the method of parametrix. We represent in terms of the transition density of (defined below in (3.2)). This method is applied to the case in which and are of the form
[TABLE]
for some functions and . Nonetheless, it is not clear whether this method can be applied to and with more general forms.
We follow here a different route.
Framework of analysis. We adopt the ‘variational’ approach employed in [4]. The core idea is to prove smoothness of by viewing the lift of as a composition of the map and the lift of . As [4] already proves smoothness of derivatives in measure up to the second order, we generalise that result to an arbitrary order.
The analysis of variational derivatives of solutions to classical SDEs is rather well-understood in the literature ([17], [25]). As differentiation in the direction of measure leads to rather complicated expressions, we restrict ourselves to the following special case in this section. This captures the key difficulty of this approach. The general case can be handled in an analogous way.
We consider the forward system \big{(}\{X^{s,\xi}_{t}\}_{t\in[s,T]},\{X^{s,x,\mu}_{t}\}_{t\in[s,T]}\big{)}, , which takes the form
[TABLE]
for some Borel-measurable function and one-dimensional Brownian motion . is also called the decoupled process, as it no longer depends on the law of itself.
For any sub--algebra , let denote the set of all random variables in . Let (resp. ) denote the filtration generated by Brownian motion (resp. ). Let be a random variable in . For simplicity of notations, in the following calculations, we shall denote the law by . &*First order derivative of . * We start our analysis by analysing the smoothness of the map . Suppose that the lift of with values in
[TABLE]
is Fréchet differentiable with its Fréchet derivative given by
[TABLE]
for some real-valued process that is adapted to . Then we define the derivative of with respect to the measure component by
[TABLE]
The next theorem computes explicitly.
Theorem 3.1**.**
Suppose that . Then exists and is the unique solution of the SDE
[TABLE]
Proof.
The proof is done in [4], but is included for completeness. We first define the -directional derivative D_{\xi}\big{(}X^{s,x,[\xi]}_{t}\big{)}(\eta) of in direction , given by
[TABLE]
where the limit is interpreted in the sense, i.e.
[TABLE]
Similarly, the -directional derivative of in direction is given by
[TABLE]
where both the limit and the derivative are interpreted in the sense. We proceed by formal differentiation and obtain that
[TABLE]
Hence,
[TABLE]
Recall that the lift of , i.e. , is defined by By (3.4), (3.5), and (3.6), formal differentiation of (3.2) with respect to in the direction gives
[TABLE]
By the definition of derivative in measure of , we can further rewrite (3.7) as
[TABLE]
It is then verified rigorously in Lemma 4.2 of [4] that D_{\xi}\big{(}X^{s,x,[\xi]}_{t}\big{)}(\eta) is indeed the directional derivative of in direction , by using the fact that is in .
The next step involves the consideration of a process satisfying the SDE
[TABLE]
We write
[TABLE]
and notice that
[TABLE]
where the second equality uses the fact that \big{(}{X^{(1)}}\big{)}^{s,y,[{\xi}]} is -adapted and is therefore independent of , whereas and are both -measurable. The final equality uses the fact that \big{(}{X^{(1)}}\big{)}^{s,y,[{\xi}]}_{r}\big{|}_{y={\xi}^{(1)}}=\big{(}{X^{(1)}}\big{)}^{s,{\xi}^{(1)}}_{r}. We also notice by the Fubini’s theorem that
[TABLE]
Therefore, by (3.10) and (3.11), we observe that D_{\xi}\big{(}X^{s,x,[\xi]}\big{)}(\eta) and \hat{\mathbb{E}}\big{[}U^{s,x,[{\xi}]}(\hat{\xi})\hat{\eta}\big{]} satisfy the same SDE and hence
[TABLE]
We then observe that satisfies the same SDE for any . Therefore, there is no dependence on and hence (3.9) can be rewritten as
[TABLE]
Moreover, by the fact that is in , we establish that
- (i)
[TABLE] 2. (ii)
[TABLE]
for any , and , for some constant . Indeed, (3.14) follows from the boundedness of and Gronwall’s inequality. (3.15) follows from the Lipschitz property of and Gronwall’s inequality, along with the bounds
[TABLE]
for some constant . Finally, the bounds (3.14), (3.15) and connection (3.12) allow us to establish that the Gâteaux derivative
[TABLE]
is continuous (where the space is equipped with the corresponding operator norm), which proves that (3.16) is indeed the Fréchet derivative of with respect to . By (3.12), it follows from the definition of that
[TABLE]
∎
*Higher order derivatives of . * We recall that does not depend on and hence we define
[TABLE]
Subsequently, we define inductively as in (2.6) and (2.7), the th order derivative in measure of by
[TABLE]
and its corresponding mixed derivatives by
[TABLE]
provided that these derivatives actually exist, where each derivative in is interpreted in the sense. (See Lemma 4.1 in [4] for its precise meaning.)
Next, we generalise the multi-index notation and the class to include derivatives of .
Definition 3.2** (Multi-index notation for derivatives of ).**
Let be a multi-index. Then is defined by
[TABLE]
if this derivative is well-defined.
Definition 3.3** (Class of th order differentiable functions of ).**
The process belongs to class , if exists for every multi-index such that and
- (a)
[TABLE] 2. (b)
[TABLE]
for any , and , for some constant .
The following theorem extends Theorem 3.1 to higher order derivatives. It uses the notations
[TABLE]
and
[TABLE]
For any function , denotes the corresponding partial derivative with respect to the second component of . denotes the corresponding partial derivative with respect to the third component of . denotes the corresponding partial derivative with respect to the fourth component of .
Theorem 3.4**.**
Suppose that is . Then, for any the th order derivative in measure exists and satisfies (3.19) and (3.20). In particular, it is the unique solution of an SDE given by
[TABLE]
where is defined by the recurrence relation
[TABLE]
where is defined by and for each , the function is defined such that \theta_{k+1}\big{|}_{\{1,\ldots,i\}}=\theta and . Moreover, is given by
[TABLE]
Proof.
We remark that the functions , , are well-defined, since . We proceed by strong induction on . The base step is done in Theorem 3.1. In particular, (3.13) verifies (3.23). The main arguments in the induction step are the same as the base step. Suppose that the statement holds for all , where . Then, in particular, satisfies the SDE
[TABLE]
Let be the lift of . In the following expression, denotes the partial derivative with respect to the lifted component of . As in (3.6) and (3.7), we formally differentiate (3.24) with respect to in the direction to obtain the directional derivative
[TABLE]
We then recall that the following directional derivatives can be represented as
[TABLE]
[TABLE]
and
[TABLE]
We can therefore rewrite (3) as
[TABLE]
where, on the second last line, denotes the identity function from to itself. We now define a process \big{\{}\big{(}U_{{k^{*}}+1}\big{)}^{s,[\xi]}_{t}(v_{1},\ldots,v_{{k^{*}}+1})\big{\}}_{t\in[s,T]} that satisfies the SDE
[TABLE]
Then we write
[TABLE]
As in the proof of Theorem 3.1, we deduce that D_{\xi}\big{(}\partial^{{k^{*}}}_{\mu}X^{s,[\xi]}(v_{1},\ldots,v_{k^{*}})\big{)}(\eta) satisfies the same SDE as \hat{\mathbb{E}}\Big{[}\big{(}U_{{k^{*}}+1}\big{)}^{s,[\xi]}(v_{1},\ldots,v_{k^{*}},\hat{\xi})\hat{\eta}\Big{]}. (Note that equality of the first and third terms follows from the same argument as (3.10) and equality of the other terms follows from the same argument as (3.11).) Consequently,
[TABLE]
By the induction hypothesis, we can again establish that (as in the proof of Theorem 3.1)
- (i)
[TABLE] 2. (ii)
[TABLE]
for any , and , for some constant . Subsequently, it follows from the same reasoning as in the proof of Theorem 3.1 and (3.27) that
[TABLE]
Finally, by the recurrence relation (LABEL:recurrence_Fk) and the expression of \big{(}U_{{k^{*}}+1}\big{)}^{s,[\xi]}_{t} in (3.26), it is clear that satisfies the SDE
[TABLE]
∎
Corollary 3.5**.**
Suppose that is in . Then
Proof.
For any multi-index such that \big{|}(n,\bm{\beta})\big{|}\leq k, we have an SDE representation of , by (3.21) in Theorem 3.4. By (LABEL:recurrence_Fk) and (3.23), we know that the function in (3.21) is differentiable in the spatial components for at most times. This is exactly what we need, since . Hence, we formally differentiate times with respect to each variable , , and then use a standard Gronwall argument to establish bounds (3.19) and (3.20). (See Theorem 5.5.3 in [17] or Proposition 4.10 in [25] for details.) ∎
We are now in a position to prove Theorem 2.15, via the smoothness of and .
Proof of Theorem 2.15.
By combining (3.6), (3.12), (3.17) and (3.18), we deduce that
[TABLE]
is Fréchet differentiable with Fréchet derivative given by
[TABLE]
Next, for any fixed , we define the lifts and for functions and respectively, given by
[TABLE]
Then, we notice from equation (2.34) that
[TABLE]
By the chain rule of Fréchet differentiation, we obtain that
[TABLE]
which implies that
[TABLE]
for any . Note that the first term can be rewritten as
[TABLE]
and the second term can be rewritten by the Fubini’s theorem as
[TABLE]
Consequently, by combining (3.29) and (3.30), equation (3.28) becomes
[TABLE]
which implies that
[TABLE]
By our assumption, we know that satisfies (2.32) and (2.33), and the process satisfies (3.19) and (3.20). It follows that also satisfies (2.32) and (2.33), with the constant bound uniform in time.
By iterating this procedure, we can show that for any multi-index such that , can be computed explicitly as above and can be represented in terms of derivatives in the form and , for some , \bm{\beta^{\prime}}\in\big{(}\mathbb{N}\cup\{0\}\big{)}^{n^{\prime}} and \bm{\beta^{\prime\prime}}\in\big{(}\mathbb{N}\cup\{0\}\big{)}^{n^{\prime\prime}}, such that and . The facts that and also allow us to deduce that satisfies estimates (2.32) and (2.33), with the constant bound uniform in time. Finally, we know from Theorem 7.2 in [4] (which corresponds to Theorem 2.15 with ) that for every . Therefore, we conclude that . ∎
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