A Doubly Stochastic Simulator with Applications in Arrivals Modeling and Simulation

We propose a framework that integrates classic Monte Carlo simulators and Wasserstein generative adversarial networks to model, estimate, and simulate a broad class of arrival processes with general nonstationary and multidimensional random arrival rates. Classic Monte Carlo simulators have advantages in capturing the interpretable “physics” of a stochastic object, whereas neural network–based simulators have advantages in capturing less interpretable complicated dependence within a high-dimensional distribution. We propose a doubly stochastic simulator that integrates a stochastic generative neural network and a classic Monte Carlo Poisson simulator to utilize the advantages of both. Such integration brings challenges to both theoretical reliability and computational tractability for the estimation of the simulator given real data, in which the estimation is done through minimizing the Wasserstein distance between the distribution of the simulation output and of real data. Regarding theoretical properties, we prove consistency and convergence rate for the estimated simulator under a nonparametric smoothness assumption. Regarding computational efficiency and tractability for the estimation procedure, we address a challenge in gradient evaluation that arises from the discontinuity in the Monte Carlo Poisson simulator. Numerical experiments with synthetic and real data sets are implemented to illustrate the performance of the proposed framework.