Parameter estimation for macroscopic pedestrian dynamics models from microscopic data

In this paper we develop a framework for parameter estimation in macroscopic pedestrian models using individual trajectories -- microscopic data. We consider a unidirectional flow of pedestrians in a corridor and assume that the velocity decreases with the average density according to the fundamental diagram. Our model is formed from a coupling between a density dependent stochastic differential equation and a nonlinear partial differential equation for the density, and is hence of McKean--Vlasov type. We discuss identifiability of the parameters appearing in the fundamental diagram from trajectories of individuals, and we introduce optimization and Bayesian methods to perform the identification. We analyze the performance of the developed methodologies in various situations, such as for different in- and outflow conditions, for varying numbers of individual trajectories and for differing channel geometries

Paramater estimation for the McKean-Vlasov stochastic differential equation

We consider the problem of parameter estimation for a stochastic McKean-Vlasov equation, and the associated system of weakly interacting particles. We first establish consistency and asymptotic normality of the offline maximum likelihood estimator for the interacting particle system in the limit as the number of particles $N\rightarrow\infty$. We then propose an online estimator for the parameters of the McKean-Vlasov SDE, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. We prove that this estimator converges in $\mathbb{L}^1$ to the stationary points of the asymptotic log-likelihood of the McKean-Vlasov SDE in the joint limit as $N\rightarrow\infty$ and $t\rightarrow\infty$, under suitable assumptions which guarantee ergodicity and uniform-in-time propagation of chaos. We then demonstrate, under the additional assumption of global strong concavity, that our estimator converges in $\mathbb{L}^2$ to the unique maximiser of this asymptotic log-likelihood function, and establish an $\mathbb{L}^2$ convergence rate. We also obtain analogous results under the assumption that, rather than observing multiple trajectories of the interacting particle system, we instead observe multiple independent replicates of the McKean-Vlasov SDE itself or, less realistically, a single sample path of the McKean-Vlasov SDE and its law. Our theoretical results are demonstrated via two numerical examples, a linear mean field model and a stochastic opinion dynamics model

Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

In this paper we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in~\cite{DuncanPavliotis2016}. We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean-Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions

Parameter estimation of discretely observed interacting particle systems

In this paper, we consider the problem of joint parameter estimation for drift and diffusion coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution of the process and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval $[0, T]$, we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step ($\Delta_n$) and the number of particles ($N$) satisfy $\Delta_n \rightarrow 0$ and $N \rightarrow \infty$, and asymptotically normal when additionally the condition $\Delta_n N \rightarrow 0$ holds

Multi-index Importance Sampling for McKean-Vlasov Stochastic Differential Equation

This work introduces a novel approach that combines the multi-index Monte Carlo (MC) method with importance sampling (IS) to estimate rare event quantities expressed as an expectation of a smooth observable of solutions to a broad class of McKean-Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator, previously introduced in our works (Ben Rached et al., 2022a,b), to the multi-index setting. We formulate a new multi-index DLMC estimator and conduct a comprehensive cost-error analysis, leading to improved complexity results. To address rare events, an importance sampling scheme is applied using stochastic optimal control of the single level DLMC estimator. This combination of IS and multi-index DLMC not only reduces computational complexity by two orders but also significantly decreases the associated constant compared to vanilla MC. The effectiveness of the proposed multi-index DLMC estimator is demonstrated using the Kuramoto model from statistical physics. The results confirm a reduced complexity from $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the single level DLMC estimator (Ben Rached et al., 2022a) to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$ for the considered example, while ensuring accurate estimation of rare event quantities within the prescribed relative error tolerance $\mathrm{TOL}_\mathrm{r}$.Comment: Extension to works 2207.06926 and 2208.0322

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework. Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts

Directed Chain Stochastic Differential Equations

We propose a particle system of diffusion processes coupled through a chain-like network structure described by an infinite-dimensional, nonlinear stochastic differential equation of McKean-Vlasov type. It has both (i) a local chain interaction and (ii) a mean-field interaction. It can be approximated by a limit of finite particle systems, as the number of particles goes to infinity. Due to the local chain interaction, propagation of chaos does not necessarily hold. Furthermore, we exhibit a dichotomy of presence or absence of mean-field interaction, and we discuss the problem of detecting its presence from the observation of a single component process.Comment: 32 page

Systemic Risk and Default Clustering for Large Financial Systems

As it is known in the finance risk and macroeconomics literature, risk-sharing in large portfolios may increase the probability of creation of default clusters and of systemic risk. We review recent developments on mathematical and computational tools for the quantification of such phenomena. Limiting analysis such as law of large numbers and central limit theorems allow to approximate the distribution in large systems and study quantities such as the loss distribution in large portfolios. Large deviations analysis allow us to study the tail of the loss distribution and to identify pathways to default clustering. Sensitivity analysis allows to understand the most likely ways in which different effects, such as contagion and systematic risks, combine to lead to large default rates. Such results could give useful insights into how to optimally safeguard against such events.Comment: in Large Deviations and Asymptotic Methods in Finance, (Editors: P. Friz, J. Gatheral, A. Gulisashvili, A. Jacqier, J. Teichmann) , Springer Proceedings in Mathematics and Statistics, Vol. 110 2015