Unbiased Estimation using a Class of Diffusion Processes

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) π a given, general probability measure on (ℝd,B(ℝd)) that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schr"odinger-F"ollmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process Xtt∈ [0,1] on ℝd, d∈ mathbbN0. This latter process is constructed such that, starting at X0=0, one has X1~ π. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, citesforig,jiao consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t.~π. We show that for this methodology to achieve a mean square error of O(ε2), for arbitrary ε>0, the associated cost is O(ε-5). We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t.~π, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of O(ε2), the associated cost is, with high probability, O(ε-2|log(ε)|2+δ), for any δ>0. We implement our method on several examples including Bayesian inverse problems.