[1] Alexandros Beskos, Omiros Papaspiliopoulos, and Gareth O. Roberts. Retrospective exact simulation of diffusion sample paths with applications. Bernoulli, 12(6):1077-1098, 2006. [ bib | http ]
We present an algorithm for exact simulation of a class of Itô's diffusions. We demonstrate that when the algorithm is applicable, it is also straightforward to simulate diffusions conditioned to hit specific values at predetermined time instances. We also describe a method that exploits the properties of the algorithm to carry out inference on discretely observed diffusions without resorting to any kind of approximation apart from the Monte Carlo error.

Keywords: conditioned diffusion processes; discretely observed diffusions; exact simulation; Monte Carlo maximum likelihood; rejection sampling
[2] Alexandros Beskos, Omiros Papaspiliopoulos, and Gareth O. Roberts. A factorisation of diffusion measure and finite sample path constructions. Methodol. Comput. Appl. Probab., 10(1):85-104, 2008. [ bib | http ]
In this paper we introduce decompositions of diffusion measure which are used to construct an algorithm for the exact simulation of diffusion sample paths and of diffusion hitting times of smooth boundaries. We consider general classes of scalar time-inhomogeneous diffusions and certain classes of multivariate diffusions. The methodology presented in this paper is based on a novel construction of the Brownian bridge with known range for its extrema, which is of interest on its own right.

Keywords: Rejection sampling; Exact simulation; Conditioned Brownian motion; Boundary hitting times
[3] Krzysztof Latuszyński, Ioannis Kosmidis, Omiros Papaspiliopoulos, and Gareth O. Roberts. Simulating events of unknown probabilities via reverse time martingales. Random Structures Algorithms, 38(4):441-452, 2011. [ bib | http ]
Let s in(0,1) be uniquely determined but only its approximations can be obtained with a finite computational effort. Assume one aims to simulate an event of probability s. Such settings are often encountered in statistical simulations. We consider two specific examples. First, the exact simulation of non-linear diffusions ([3]). Second, the celebrated Bernoulli factory problem ([10, 13]) of generating an f(p)-coin given a sequence X1,X2,... of independent tosses of a p-coin (with known f and unknown p). We describe a general framework and provide algorithms where this kind of problems can be fitted and solved. The algorithms are straightforward to implement and thus allow for effective simulation of desired events of probability s. Our methodology links the simulation problem to existence and construction of unbiased estimators.

Keywords: Keywords: perfect simulation; Bernoulli factory; retrospective sampling; unbiased simulation

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