A Deep Learning Approach for Rare Event Simulation in Diffusion Processes

We address the challenge of estimating rare events associated with stochastic differential equations using importance sampling. The importance sampling zero variance measure in these settings can be inferred from a solution to the Hamilton-Jacobi-Bellman partial differential equation (HJB-PDE) associated with a value function for the underlying process. Guided by this equation, we use a neural network to learn the zero variance change of measure. To improve performance of our estimation, we pursue two new ideas. First, we adopt a loss function that combines three objectives which collectively contribute to improving the performance of our estimator. Second, we embed our rare event problem into a sequence of problems with increasing rarity. We find that a well chosen schedule of rarity increase substantially speeds up rare event simulation. Our approach is illustrated on Brownian motion, Orstein-Uhlenbeck (OU) process, Cox–Ingersoll–Ross (CIR) process as well as Langevin double-well diffusion.