Biased Monte Carlo Simulation, Multilevel Paradigm

We introduce in this chapter the paradigm of multilevel simulation, whose aim is to dramatically reduce the bias in a Monte Carlo simulation when the (probability distribution of the) random variable under consideration cannot be simulated at a reasonable cost but can be approximated by simulable random variables with a controlled complexity. As typical examples let us cite the discretization scheme of a stochastic process or nested Monte Carlo simulations. The paradigm relies on the existence of both a strong rate and an expansion of the weak error convergence of approximating random variables. We propose an in-depth analysis of both weighted and regular multilevel methods in an abstract framework, in presence of a higher or first order expansion of the weak error. Various applications are detailed like the pricing of path-dependent or forward start options, quantile computations in actuarial sciences (SCR). When the strong convergence rate is fast enough (like the Milstein scheme for diffusion), multilevel simulation behaves like an unbiased simulation. We conclude with a section about randomized multilevel quantization.