A general method for debiasing a Monte Carlo estimator

Abstract

Consider a stochastic process Xn, n = 0, 1, 2, ...such that EXn → x∞ as n → ∞. The sequence {Xn} may be a deterministic one, obtained by using a numerical integration scheme, or obtained from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation but we will assume that we can sample from the distribution of X1, X2, ...Xm for finite m. We propose a scheme for unbiased estimation of the limiting value x∞, together with estimates of standard error and apply this to examples including numerical integrals, root-finding and option pricing in a Heston Stochastic Volatility model.

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

“…As mentioned in the Introduction, the construction of our estimator exploits an idea that was been developed previously in Glynn (2011) andMcLeish (2011). The idea is to introduce an integervalued random variable N, independent of the X (k)'s and let p (k) = P (N = k) for k ≥ 0, where p (k) > 0 for each k ≥ 0.…”

“…replications can be easily implemented in parallel computations. Our estimator builds on the multilevel Monte Carlo method introduced in Giles (2008), and the debiasing idea studied recently in Glynn (2011) andMcLeish (2011); related ideas are discussed in Bujok, Hambly, and Reisinger (2013) and Section 9 of Giles (2015). In particular, we note that if X (n) = (X 1 (n) , ...,X d (n)) is the vector of empirical means of i.i.d.…”

“…The approach in Glynn (2011) andMcLeish (2011) basically involves introducing independent randomization in order to represent the infinite sum in the previous display as an expectation with respect to an integer valued random variable, N, after dividing the n-th term by P (N = n) = p (n). Then, the effort in the construction of the estimator is focused on finding a suitable random variable ∆ n such that E (∆ n ) = Eg (X (n + 1)) − Eg (X (n)) and also satisfying that ∑ E |∆ n | 2 /p (n) < ∞, (implying that the tails of N should be sufficiently heavy).…”

Exact estimation for Markov chain equilibrium expectations

“…We start by briefly describing and (slightly) generalizing the algorithms and analysis presented in Glynn (2012, 2013) and McLeish (2011), following related earlier work by Rychlik (1990Rychlik ( , 1995.…”

General multilevel adaptations for stochastic approximation algorithms of Robbins–Monro and Polyak–Ruppert type

“…The remaining fast regime in which errors decay like N −1/2 is omitted in our analysis. In this setting unbiased multilevel estimates of [8] and [12] can be used in which case a central limit theorem immediately follows from classical theory. We will use Landau notation and write a n = O(b n ) for two sequences (a n ) and (b n ) if b n is eventually strictly positive and lim sup |a n |/b n < ∞.…”