A general method for debiasing a Monte Carlo estimator

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.