Journal Article

LARGE DEVIATION PROPERTIES OF WEAKLY INTERACTING PROCESSES VIA WEAK CONVERGENCE METHODS

Fischer, Markus, Budhiraja, Amarjit, Dupuis, Paul

2012

Overview

We study large deviation properties of systems of weakly interacting particles modeled by Itô stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean—Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 34K50 / 60B10 / 60F10 / 60H10 / 60K35 / 93E20 / Brownian motion / Coefficients / Convergence / delay / Differential equations / Diffusion coefficient / interacting random processes / Large deviations / martingale problem / Martingales / McKean–Vlasov equation / optimal stochastic control / Particle interactions / Probabilities / Random variables / stochastic differential equation / Stochastic models / Stochastic processes / Stopping distances / Studies / Uniqueness / weak convergence