In the setting of a weakly interacting particle system, the authors detail a control-based importance sampling scheme for estimation of statistics of the form \[ \mathbb{E}\left[\exp\left\{-NG\left(\mu_T\right)\right\}\right]\,, \] where \(\mu_T\) is the empirical measure of the particles at the terminal time \(T\), there are \(N\) particles in total, and \(G\) is an appropriate function. This involves defining a suitably controlled version of the underlying system, and using statistics from this controlled system to estimate quantities of interest for the original system with reduced variance for large \(N\). Suitable controls are given using a zero-viscosity Hamilton-Jacobi-Bellman (HJB) equation on a Wasserstein space.
One of the main results of the present paper is that, under certain assumptions (including boundedness and continuity of coefficients, and existence and uniqueness of solutions) for a bounded and continuous function \(G\), this procedure requires only a subexponential number of samples to achieve a given relative error as \(N\to\infty\). Under stronger assumptions, the number of samples required vanishes in the limit, so that a single sample suffices for large \(N\) in some situations.
These results are complemented by explicit consideration of some linear-quadratic examples in which the HJB equation can be solved analytically, and by numerical results.