Gonçalo dos Reis: Numerical methods for McKean-Vlasov equations. Particles, no particles, and the super-measure case
We present a fully probabilistic Euler scheme for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth in the measure component (also in the spatial component, and random initial condition). We develop a split-step scheme attaining an almost \(1/2\) root-mean-square error rate in stepsize and addresses a gap in the literature regarding efficient numerical methods and their convergence rate for this class of McKean Vlasov SDEs. Time allowing, we discuss if existing methods are fit-for-purpose and a new method for the numerical approximation of MV-SDE without resorting to interacting particles systems.
Valentine Genon-Catalot: Probabilistic properties and parametric inference of small variance nonlinear self-stabilizing stochastic differential equations
Let \((X_t)\) be solution of a one-dimensional McKean-Vlasov stochastic differential equation with classical drift term \(V (\alpha, x)\), self-stabilizing term \(\Phi(\beta, x)\) and small noise amplitude \(\varepsilon\). Our aim is to study the estimation of the unknown parameters \(\alpha\), \(\beta\) from a continuous observation of \((X_t, \, t \in [0, T])\) under the double asymptotic framework \(\varepsilon\) tends to \(0\) and \(T\) tends to infinity. After centering and normalization of the process, uniform bounds with respect to \(t \ge 0\) and \(\varepsilon\) are derived. We then build an explicit approximate log-likelihood leading to consistent and asymptotically Gaussian estimators with original rates of convergence: the rate for the estimation of \(\alpha\) is either \(\varepsilon^{-1}\) or \(\sqrt{T} \varepsilon^{-1}\), the rate for the estimation of \(\beta\) is \(\sqrt{T}\). This is joint work with Catherine Larédo (INRAE, Université Paris-Saclay).
References:
[1] V. Genon-Catalot, C. Larédo. (2021). Probabilistic properties and statistical inference of small variance nonlinear self-stabilizing stochastic differential equations Stoch. Proc. and Appl. 142, 513-548.
[2] V. Genon-Catalot, C. Larédo. (2021). Parametric inference for small variance and long time horizon McKean-Vlasov diffusion models. To appear in Electronic Journal of Statistics.
Marc Hoffmann: Some statistical inference results for interacting particle models in a mean-field limit
We propose a theoretical statistical analysis for systems of interacting diffusions, possibly with common noise and/or degenerate diffusion components, in a mean-field regime. These models are more or less widely used in finance, MFG, systemic risk analysis, behaviourial sociology or ecology. We consider several inference issues such as: i) nonparametric estimation of the solution of the underlying Fokker-Planck type equation or the drift of the system ii) testing for the interaction between components iii) estimation of the interaction range between particles. This talk is based on joint results with C. Fonte and L. Della Maestra.
Nikolas Kantas: Parameter estimation for the McKean-Vlasov stochastic differential equation
We consider the problem of parameter estimation for a McKean stochastic differential equation, and the associated system of weakly interacting particles. The problem is motivated by many applications in areas such as neuroscience, social sciences (opinion dynamics, cooperative behaviours), financial mathematics, statistical physics. We will first survey some model properties related to propagation of chaos and ergodicity and then move on to discuss the problem of parameter estimation both in offline and on-line settings. In the on-line case, we propose an online estimator, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. The talk will present our convergence results and then show some numerical results for two examples, a linear mean field model and a stochastic opinion dynamics model. This is joint work with Louis Sharrock, Panos Parpas and Greg Pavliotis. Preprint: https://arxiv.org/abs/2106.13751
Catherine Larédo: Parametric inference for small variance McKean-Vlasov stochastic differential equations (I): Fixed time horizon
We consider a process \((X_t)\) solution of a one-dimensional McKean-Vlasov stochastic differential equation with classical drift term \(V (\alpha, x)\), interaction term \(\Phi(\beta, x)\) depending on unknown parameters \(\alpha\), \(\beta\) and small diffusion coefficient \(\varepsilon\). Adding this interaction term leads to Markov processes with transitions both depending on \(X_t\) and on its distribution. We study the probabilistic properties of \((X_t)\) as \(\varepsilon \to 0\) and exhibit an approximating Gaussian process for \((X_t)\). Then, based on a continuous observation of \((X_t)\) on a fixed time interval \([0, T]\), we study the estimation of \((\alpha, \beta)\). We build explicit estimators using an approximate likelihood obtained from the likelihood of a proxi-model. For fixed \(T\), we prove that, as \(\varepsilon \to 0\), \(\alpha\) can be consistently estimated with rate \(\varepsilon^{-1}\), but not \(\beta\). Next, we consider \(n\) i.i.d. paths \((X^i_t,\, i=1,\dots,n)\) continuously observed on \([0, T]\). We derive consistent and asymptotically Gaussian estimators of \((\alpha, \beta)\) with rates \(\sqrt{n} \varepsilon^{-1}\) for \(\alpha\) and \(\sqrt{n}\) for \(\beta\). Finally, we prove that the statistical experiments generated by \((X_t, \, t \in [0, T])\) and the proxi-model are asymptotically equivalent in the sense of the Le Cam \(\Delta\)-distance both for the continuous observation of one path and for \(n\) i.i.d. paths under the condition \(\sqrt{n}\varepsilon \to 0\). This result justifies the statistical method proposed here. This is joint work with Valentine Genon-Catalot (Université de Paris).
Reference:
[1] Genon-Catalot, V. and Larédo, C. (2021). Probabilistic properties and statistical inference of small variance nonlinear self-stabilizing stochastic differential equations. Stochastic Processes and their Applications 142, 513-548.
Eva Löcherbach: Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling
We consider a stochastic system of interacting neurons. The system consists of \(N\) neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to \(0\) and all other neurons receive an additional amount of potential which is a centred random variable of order \(1/\sqrt{N}\). In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as \(N\) tends to infinity, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion \(W\) which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system. Conditionally on \(W\), the different neurons are independent in the limit system. This is the "conditional propagation of chaos" property. We prove the well-posedness of the limit equation by adapting the ideas of Graham (1992) to our frame. A second part of the talk is devoted to the proof of a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of the proof is the coupling introduced in Komlos-Major-Tusnady (1976) of the point process representing the small jumps of the particle system with the limit Brownian motion. Finally, if time permits, we will also discuss extensions to a spatially structured framework. The talk is based on joint work with Xavier Erny and Dasha Loukianova.
Mateusz B. Majka: Exponential ergodicity of McKean-Vlasov SDEs with Lévy noise
We discuss convergence of solutions of McKean-Vlasov stochastic differential equations with distribution-dependent drifts, where the driving noise is a Lévy process. We present a method of quantifying convergence rates of such solutions to their stationary distributions, measured in the \(L^1\)-Wasserstein distance. The method is based on the coupling technique and involves an explicit construction of a coupling of paths of a solution to a McKean-Vlasov SDE, based on finding appropriate pairings of the jumps of the underlying Lévy noise. This technique can be also used to prove a uniform in time propagation of chaos result, which shows that interacting particle systems with Lévy noise converge to the corresponding McKean-Vlasov SDE. The talk is based on joint work with Mingjie Liang and Jian Wang.
Grigorios A. Pavliotis: On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions
I will present recent results on the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We study the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained on the torus undergoes a phase transition, i.e., if it admits more than one steady state. A typical example of such a system on the torus is given by mean field plane rotator (XY, Heisenberg, \(O(2)\)) model. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature. This is joint work with Matias Delgadino (U Texas Austin) and Rishabh Gvalani (MPI Leipzig).
Mark Podolskij: Semiparametric estimation of McKean-Vlasov SDEs
In this talk we study the problem of semiparametric estimation for a class of McKean-Vlasov stochastic differential equations. Our aim is to estimate the drift coefficient of a MV-SDE based on observations of the corresponding particle system. We propose a semiparametric estimation procedure and derive the rates of convergence for the resulting estimator. We further prove that the obtained rates are essentially optimal in the minimax sense.
Shi-Yuan Zhou: A projection estimator for nonparametric drift inference of McKean-Vlasov equations
We consider real-valued solutions of the McKean-Vlasov stochastic differential equation and construct a projection least-squares estimator for the underlying drift function in the regime where the amount of particles tends to infinity and the observed time horizon is fixed. We derive estimation rates both in the natural norm arising from the problem as well as the usual \(L^2\) norm. Additionally, the estimation procedure can also be applied to the associated particle system.