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Optimal control of diffusion processes is intimately connected to the problemof solving certain Hamilton-Jacobi-Bellman equations. Building on recentmachine learning inspired approaches towards high-dimensional PDEs, weinvestigate the potential of $\textit{iterative diffusion optimisation}$techniques, in particular considering applications in importance sampling andrare event simulation, and focusing on problems without diffusion control, withlinearly controlled drift and running costs that depend quadratically on thecontrol. More generally, our methods apply to nonlinear parabolic PDEs with acertain shift invariance. The choice of an appropriate loss function being acentral element in the algorithmic design, we develop a principled frameworkbased on divergences between path measures, encompassing various existingmethods. Motivated by connections to forward-backward SDEs, we propose andstudy the novel $\textit{log-variance}$ divergence, showing favourableproperties of corresponding Monte Carlo estimators. The promise of thedeveloped approach is exemplified by a range of high-dimensional and metastablenumerical examples.

Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural  networks: perspectives from the theory of controlled diffusions and measures  on path space