Abstract
We consider a system of N interacting particles, governed by transport and diffusion, that converges in a mean-field limit to the solution of a McKean–Vlasov equation. From the observation of a trajectory of the system over a fixed time horizon, we investigate nonparametric estimation of the solution of the associated nonlinear Fokker–Planck equation, together with the drift term that controls the interactions, in a large population limit \(N \rightarrow \infty \). We build data-driven kernel estimators and establish oracle inequalities, following Lepski’s principle. Our results are based on a new Bernstein concentration inequality in McKean–Vlasov models for the empirical measure around its mean, possibly of independent interest. We obtain adaptive estimators over anisotropic Hölder smoothness classes built upon the solution map of the Fokker–Planck equation, and prove their optimality in a minimax sense. In the specific case of the Vlasov model, we derive an estimator of the interaction potential and establish its consistency.
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Notes
Usually required to control the model for convergence to equilibrium when T is large.
In particular, it is a first step toward the interesting problem of testing the hypothesis \(F=0\) against a set of local alternatives that quantify how far F is from being constant.
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Acknowledgements
Informal discussions with colleagues at CEREMADE are gratefully acknowledged; we thank in particular, Pierre Cardaliaguet, Djalil Chafaï and Stéphane Mischler. We also thank Denis Belomestny and Nicolas Fournier for insightful comments. This work partially answers a problem that was posed to us by Sylvie Méléard almost two decades ago (at a time we did not have the proper tools to address it!).
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Appendix
Appendix
1.1 Characterisation of sub-Gaussian random variables
We recall a classical definition of a sub-Gaussian random variable. Recommended reference is [10].
Definition 30
A real-valued random variable Z such that \(\mathbb {E}[Z]=0\) is \(\lambda ^2\) sub-Gaussian if one of the following conditions is satisfied, each statement implying the next:
-
(i)
Laplace transform condition
$$\begin{aligned} \mathbb {E}\big [\exp (zZ)\big ] \le \exp \big (\tfrac{1}{2}\lambda ^2 z^2\big )\;\;\text {for every}\;\;z \in \mathbb {R}. \end{aligned}$$ -
(ii)
Moment condition
$$\begin{aligned} \mathbb {E}\big [Z^{2p}\big ] \le p! (4\lambda ^2)^p\;\;\text {for every integer}\;\;p \ge 1. \end{aligned}$$ -
(iii)
Orlicz condition
$$\begin{aligned} \mathbb {E}\big [\exp \big (\tfrac{1}{8\lambda ^2}Z^2\big )\big ] \le 2. \end{aligned}$$ -
(iv)
Laplace transform condition (bis)
$$\begin{aligned} \mathbb {E}\big [\exp (zZ)\big ] \le \exp \big (\tfrac{24}{2}\lambda ^2 z^2\big )\;\;\text {for every}\;\;z \in \mathbb {R}. \end{aligned}$$
We will also use the following additive property of sub-Gaussian random variables: if the random variables \(Z_i\) are independent and \(\lambda _i^2\) sub-Gaussian, then \(\rho (Z_1+Z_2)\) is \(|\rho |^2(\lambda _1^2+\lambda _2^2) \) sub-Gaussian for every \(\rho \in \mathbb {R}\).
1.2 Proof of Lemma 20
By Assumption 4, the estimate
holds for every \((t,x)\in [0,T] \times \mathbb {R}^d\), where \( b_0 = \sup _{t \in [0,T]}|b(t,0,\delta _0)|\). Remember that
are independent d-dimensional \(\overline{\mathbb P}^{N}\)-Brownian motions. By Minkowski’s and Jensen’s inequality, we have
where \(\zeta _{T}^{i} = \sup _{0 \le t \le T}\big |\int _0^t \sigma (s,X_s^i)d\overline{B}_{s}^i\big | \). Integrating w.r.t. \(\overline{\mathbb P}^N\), we also have
We infer by Grönwall’s lemma
and plugging this estimate in (72) we infer
Applying Grönwall’s lemma again, we derive
Taking the exponent 2p and expectation w.r.t. \(\overline{\mathbb P}^N\), we further obtain
with \(C_6=50\,\mathrm {e}^{3|b|_{\mathrm {Lip}}T}\). By Assumption 1, the initial condition \(|X_0^i|\) satisfies
hence for every \(p \ge 1\), we obtain
since \(\gamma _1 \ge 1\). By Burkholder–Davis–Gundy’s inequality with constant \((C^\star )^{p/2} p^{p/2}\) for some numerical constant \(C^\star \), see e.g. Barlow and Yor [4], we also have
Putting these estimates together, we conclude
and Lemma 20 is established with \(C_2 = C_6\big (\tfrac{\gamma _1}{\gamma _0}+T(b_0+8C^\star \mathrm {e}\big |\mathrm {Tr}(c)\big |_\infty )\big )\).
1.3 Proof of Lemma 22
Fix \(\mathcal I_k = \{i_1,\ldots , i_k\} \subset \{1,\ldots , N\}\). For \(g: [0,T] \times (\mathbb {R}^d)^{k} \times (\mathbb {R}^d)^\ell \rightarrow \mathbb {R}^d\), we define
For technical convenience, we establish a slightly stronger, replacing \(\mathcal V_{2p}^N\big (f(t,\cdot )\big )\) in (46) by
for every \(\mathcal I_{k-\ell +1} \subset \{1,\ldots , N\}\) with cardinality \(k-\ell +1\) and every function \(g: [0,T] \times (\mathbb {R}^d)^{k-\ell +1} \times (\mathbb {R}^d)^\ell \rightarrow \mathbb {R}^d\), Lipschitz continuous in the space variables, that defines in turn a class \(\mathcal G_{k-\ell +1,\ell }\). In particular \(\mathcal V_{2p,\ell }^N\big (f(t,\cdot )\big )\) and \({\mathcal V}_{2p, \ell }^N\big (g_{\mathcal I_{k-\ell +1}}(t,\cdot )\big )\) agree for \(\ell = k\) in which case the class \(\mathcal G_{1,k}\) coincide with \(\mathcal G_k\) and we obtain Lemma 22. We prove the result by induction.
Step 1: The case \(\ell = 1\). For \(g\in \mathcal G_{k,1}\), \(x^{k}\in (\mathbb {R}^d)^{k}\) and \(\mathcal I \subset \{1,\ldots ,N\}\), let
where we write \(\mu ^{\mathcal J}_t(dx) = |\mathcal J|^{-1}\sum _{i \in \mathcal J}\delta _{X_t ^i}(dx)\) for the empirical measure in restriction to \(\mathcal I\). Observe that \(\Lambda _t^{\mathcal I}(g,x^{k})\) is a sum of independent and centred random variables under \(\overline{\mathbb P}^{N}\). We write
since \(|\mathcal I_k|=k\). We obtain the decomposition
with
The term I is controlled by the smoothness of g:
where the last estimate stems from Lemma 20. For the term II, writing \(g = (g^1,\ldots , g^d)\) where the functions \(g^j\) are real-valued, we further have
Moreover, for every \(x\in \mathbb {R}^{d}\), the term
is the sum of independent centred random variables that are independent of \((X_t^{i_1}, \ldots , X_t^{i_k})\) and
is \(\lambda ^2\) sub-Gaussian with \(\lambda ^2 = 24C_2|g^j(t,\cdot )|_{\mathrm {Lip}}^{2}\) via the same estimate as for I and the fact that (ii) implies (iv) in Definition 30. Thanks to the additivity property of independent sub-Gaussian random variables, we further infer that \(\Lambda _t^{\mathcal I_k^c}(g^j,x^{k})\) is \(\widetilde{\lambda }^2\) sub-Gaussian with
Conditioning on \((X_t^{i_1}, \ldots , X_t^{i_k})\), we derive
by (ii) of Definition 30. Plugging this estimate in (73), we obtain
and putting together our estimates for I and II, we conclude
with \(K_1 = 16(k^2+24d^2)C_2\). This establishes Lemma 22 for g in the case \(\ell = 1\).
Step 2: We assume that (46) holds for \(\mathcal V_{2p,\ell }^N\big (g_{{\mathcal I}_{k-\ell +1}}(t,\cdot )\big )\), for every \(\mathcal I_{k-\ell +1} \subset \{1,\ldots , N\}\) with cardinality \(k-\ell +1\) and every \(g \in \mathcal G_{k-\ell +1,\ell }\) with \(\ell < k\). Let \(g \in {\mathcal G}_{k-\ell ,\ell +1}\) and \({\mathcal I}_{k-\ell } \subset \{1,\ldots , N\}\). We have:
with
Let \(i_0 \in \mathcal I_{k-\ell }^c\) and put \(\mathcal I_{k-\ell +1} = \mathcal I_{k-\ell } \cup \{i_0\}\). The term IV can be rewritten as
where, for fixed \(y \in \mathbb {R}^d\), the function \(g'(t,x_{i_1}, \ldots x_{i_{k-\ell }}, x_{i_0},y^\ell )(y) = g(t,x_{i_1}, \ldots x_{i_{k-\ell }},(y,y^\ell ))\) with the artificial variable \(x_{i_0}\) belongs to \(\mathcal G_{k-\ell +1,\ell }\). By the induction hypothesis and noting that \(\sup _{y \in \mathbb {R}^d}|g'(t,\cdot ,y)|_{\mathrm {Lip}} \le |g(t,\cdot )|_{\mathrm {Lip}}\), we infer
We split the sum in III over indices in \(\mathcal I_{k-\ell }\) and \(\mathcal I_{k-\ell }^c\). If \(i \in \mathcal I_{k-\ell }\), in the same way as for IV, we write
with \(\mathcal I_{k-\ell +1} = \mathcal I_{k-\ell } \cup \{i_0\}\) for some arbitrary \(i_0 \in \mathcal I_{k-\ell }^c\) and with \(g''(t,x_{i_1},\ldots , x_{i_{k-\ell }},x_{i_0},y^\ell ) =g\big (t,x_{i_1},\ldots , x_{i_{k-\ell }},(x_i,y^\ell )\big )\), where i coincides with one of the \(i_j \in \mathcal I_{k-\ell }\). Also, \(g''\) belongs to \(\mathcal G_{k-\ell +1,\ell }\). If \(i \in \mathcal I_{k-\ell }^c\), we write
with \(g'''(t,x_{i_1},\ldots , x_{i_{k-\ell }}, x_i,y^\ell ) =g\big (t,x_{i_1},\ldots , x_{i_{k-\ell }},(x_i,y^\ell )\big )\) and \(g'''\) belongs to \(\mathcal G_{k-\ell +1,\ell }\) as well. We infer
by the induction hypothesis and noting again that \(|g''(t,\cdot )|_{\mathrm {Lip}}\) and \(|g'''(t,\cdot )|_{\mathrm {Lip}}\) are controlled by \(|g(t,\cdot )|_{\mathrm {Lip}}\). We conclude
with \(K_{\ell +1} = 4 K_\ell \). The proof of Lemma 22 is complete.
1.4 (Sketch of) proof of Proposition 13
Step 1: Thanks to Chapters 6 and 9 of [7], since F, G are bounded and \(\mu _0\) satisfies Assumption 1, it can be shown that (3) admits a unique probability solution \(\mu \) in the sense of [7], absolutely continuous w.r.t. the Lebesgue measure, that we still denote \(\mu (t,x) = \mu _t(x)\). Moreover \(\mu \in \mathcal {H}_{\text {loc}}^{\delta /2,\delta } = \cap _{(t_0,x,_0)\in (0,T)\times \mathbb {R}^d} \mathcal H^{\delta /2,\delta }(t_0,x_0)\) for every \(0< \delta < 1\). The main arguments of these properties rely on the existence of a suitable Lyapunov function associated to (3), following the terminology of [51] and [7] (for instance \(x \mapsto 1+ |x|^2\)) together with Sobolev embeddings.
Step 2: Define
and
which are well defined since \(\beta ,\beta ' >1\). Consider next the Cauchy problem associated to (3) in its strong form:
Taking \(\delta =\beta -\lfloor \beta \rfloor \) we obtain \(\widetilde{a}_i, \widetilde{a} \in \mathcal {C}^{(\beta -\lfloor \beta \rfloor )/2,\beta -\lfloor \beta \rfloor }_{\text {loc}}\) by Step 1.
Step 3: Using \(\text {inf} \; \widetilde{a} > -\infty \) and the existence of a Lyapunov function associated to the problem, by Theorem 2.3 of [2], there exists a unique solution \(\widetilde{\mu }\) of (74). Moreover, \(\widetilde{\mu }\) is continuous on \((0,T) \times \mathbb {R}^d\) and
It is also the unique solution defined in Theorem 12 of Chapter 1 of [27], therefore the unique integrable solution of the problem (3). By uniqueness, \(\mu = \widetilde{\mu }\).
Step 4: If \(\lfloor \beta \rfloor = 1 \), we obtain \(\mu \in \mathcal {H}^{(1+\beta )/2, 1+\beta }(t_0,x_0)\) for every \((t_0,x_0)\in (0,T)\times \mathbb {R}^d\). Otherwise, we can iterate the process thanks to results of Section 8.12 in [42]: successively:
-
Since \(\partial _{x_{k'}}\widetilde{a}_k\) and \(\partial _{x_k} \widetilde{a}\) are in \(\mathcal {C}^{(\beta -\lfloor \beta \rfloor )/2,\beta -\lfloor \beta \rfloor }_{\text {loc}}\), we have
$$\begin{aligned} \partial _{x_k} \mu \in \mathcal {C}^{1+(\beta -\lfloor \beta \rfloor )/2,2+ \beta -\lfloor \beta \rfloor }_{\text {loc}}. \end{aligned}$$ -
Since \(\partial _{t}\widetilde{a}_k\) and \(\partial _{t} \widetilde{a}\) are now in \(\mathcal {C}^{(\beta -\lfloor \beta \rfloor )/2,\beta -\lfloor \beta \rfloor }_{\text {loc}}\), we have
$$\begin{aligned} \partial _{t} \mu \in \mathcal {C}^{1+(\beta -\lfloor \beta \rfloor )/2,2+ \beta -\lfloor \beta \rfloor }_{\text {loc}}. \end{aligned}$$
Therefore, if \(\lfloor \beta \rfloor = 2 \), we obtain \(\mu \in \mathcal {H}^{(1+\beta )/2, 1+\beta }(t_0,x_0)\) for every \((t_0,x_0)\in (0,T)\times \mathbb {R}^d\). Otherwise, we can iterate again the process and so on. The result follows.
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Della Maestra, L., Hoffmann, M. Nonparametric estimation for interacting particle systems: McKean–Vlasov models. Probab. Theory Relat. Fields 182, 551–613 (2022). https://doi.org/10.1007/s00440-021-01044-6
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DOI: https://doi.org/10.1007/s00440-021-01044-6
Keywords
- Nonparametric estimation
- Statistics and PDE
- Interacting particle systems
- McKean–Vlasov models
- Oracle inequalities
- Goldenshluger–Lepski method
- Anisotropic estimation