Nested simulation concerns estimating functionals of a conditional expectation via simulation. In this paper, we propose a new method based on kernel ridge regression to exploit the smoothness of the conditional expectation as a function of the multidimensional conditioning variable. Asymptotic analysis shows that the proposed method can effectively alleviate the curse of dimensionality on the convergence rate as the simulation budget increases, provided that the conditional expectation is sufficiently smooth. The smoothness bridges the gap between the cubic root convergence rate (that is, the optimal rate for the standard nested simulation) and the square root convergence rate (that is, the canonical rate for the standard Monte Carlo simulation). We demonstrate the performance of the proposed method via numerical examples from portfolio risk management and input uncertainty quantification.

Last week, Elon Musk, the billionaire founder of Tesla Motors, SpaceX, and other cutting-edge companies, took a surprising question at the Code Conference, a technology event in California. What, a man in the audience asked, did Musk make of the idea that we are living not in the real world, but in an elaborate computer simulation? Musk exhibited a surprising familiarity with this concept. "I've had so many simulation discussions it's crazy," Musk said. Citing the speed with which video games are improving, he suggested that the development of simulations "indistinguishable from reality" was inevitable.