Unbiased Estimation with Square Root Convergence for SDE Models

Abstract

In many settings in which Monte Carlo methods are applied, there may be no known algorithm for exactly generating the random object for which an expectation is to be computed. Frequently, however, one can generate arbitrarily close approximations to the random object. We introduce a simple randomization idea for creating unbiased estimators in such a setting based on a sequence of approximations. Applying this idea to computing expectations of path functionals associated with stochastic differential equations (SDEs), we construct finite variance unbiased estimators with a "square root convergence rate" for a general class of multidimensional SDEs. We then identify the optimal randomization distribution. Numerical experiments with various path functionals of continuous-time processes that often arise in finance illustrate the effectiveness of our new approach.

Exact estimation for Markov chain equilibrium expectations

“…2 As for N , we note that the same proof technique as for Proposition 1 of Rhee and Glynn (2013) establishes that the optimal choice for the distribution of N is to choose P(N ≥ k) proportional to…”

Finite variance unbiased estimation of stochastic differential equations

“…As a result, it is not straightforward to obtain estimators with finite variance unless we impose strong conditions on b, σ and g as done in Rhee and Glynn (2015). Most of the recent research has been focused to obtain schemes with finite variance without imposing severe regularity conditions.…”

“…However, due to the time discretization step, these methods inherently contain a bias which vanishes asymptotically. A way to address this issue is through bias reduction schemes such as the randomization method (random number of discretization steps) of Rhee and Glynn (2015) which can be seen as a randomized version of the multilevel Monte Carlo method of Giles (2008). However, the randomization method has a finite cost but infinite variance in full generality (when b, σ , g are Lipschitz continuous).…”

Unbiased Monte Carlo for optimization and functions of expectations via multi-level randomization

“…So such estimator is slightly more desirable than the one we consider here and in fact Rhee and Glynn (2015) use this flexibility on N in order to optimize its selection. However, such estimator requires the introduction of an auxiliary sequence in order to facility variance analysis.…”