Parameter Estimation for Stochastic McKean-Vlasov Equations

• Activity Number: 245 - SLDS CSpeed 4
• Type: Contributed
• Date/Time: Wednesday, August 11, 2021 : 10:00 AM to 11:50 AM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Louis Sharrock* and Nikolas Kantas and Grigorios Pavliotis and Panos Parpas
• Companies: Imperial College London and Imperial College London and Imperial College London and Imperial College London
• Keywords: stochastic McKean-Vlasov equation; parameter estimation; maximum likelihood; consistency; asymptotic normality; stochastic gradient descent

Abstract

In this presentation, we consider the problem of parameter estimation for a fully observed McKean-Vlasov stochastic differential equation (MVSDE), and the associated system of weakly interacting particles. We begin by establishing consistency and asymptotic normality of the offline maximum likelihood estimator (MLE) of the interacting particle system (IPS) in the limit as the number of particles (N) tends to infinity. We then propose a recursive MLE for the MVSDE, which evolves according to a stochastic gradient ascent algorithm on the asymptotic log-likelihood of the IPS. Under suitable assumptions, we prove that this estimator converges in L1 to the stationary points of the asymptotic log-likelihood of the MVSDE in the joint limit as N and t tend to infinity. Under the additional assumption of global strong concavity, we also demonstrate that our estimator converges in L2 to the unique maximiser of the asymptotic log-likelihood of the MVSDE, and establish an L2 convergence rate. Our results are demonstrated via several numerical examples of practical interest, including a linear mean field model, and a stochastic opinion dynamics model.