MLMC for nested expectations

Abstract

This paper discusses progress and future research possibilities in applying MLMC ideas to nested expectations of the form E[g(E[f(X, Y)|X])], with an outer expectation with respect to one random variable X, and an inner conditional expectation with respect to a second random variable Y. The difficulty in treating such applications is shown to depend on whether the function g is (1) smooth, (2) continuous and piecewise smooth, or (3) discontinuous.