## Zack Bezemek : Large Deviations and Importance Sampling for Weakly Interacting Diffusions

- Uploaded by schrett ( 5 Views )We consider an ensemble of N interacting particles modeled by a system of N stochastic differential equations (SDEs). The coefficients of the SDEs are taken to be such that as N approaches infinity, the system undergoes Kac’s propagation of chaos, and is well-approximated by the solution to a McKean-Vlasov Equation. Rare but possible deviations of the behavior of the particles from this limit may reflect a catastrophe, and computing the probability of such rare events is of high interest in many applications. In this talk, we design an importance sampling scheme which allows us to numerically compute statistics related to these rare events with high accuracy and efficiency for any N. Standard Monte Carlo methods behave exponentially poorly as N increases for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field control. This HJB Equation is seen to be connected to the large deviations rate function for the empirical measure on the ensemble of particles. We identify conditions under which our scheme is provably asymptotically optimal in N in the sense of log-efficiency. We also provide evidence, both analytical and numerical, that with sufficient regularity of the solution to the HJB Equation, our scheme can have vanishingly small relative error as N increases.

## Jake Madrid : Stochastic Extinction events in Large Populations Prior to Entering the Metastable State

- Uploaded by schrett ( 12 Views )We will explore the role of demographic stochasticity in triggering extinction events in models of large finite populations. While prior works have focused on large fluctuations from quasi-stationary distributions, we instead consider extinction events occurring before entering a metastable state. Since such extinction events require only slight deviations from the mean-field trajectories, we can derive the approximating extinction probability PDE with a modified Robin-type boundary condition. We then investigate the utility of this approximation by comparing to the Lotka-Volterra model as well as the Lotka-Volterra model with logistic growth.