Exact simulation of diffusions

By Alexandros Beskos, Gareth O. Roberts. 2005

321 citations recorded by the source corpus.

We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without further reference to the dynamics of the target process.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4f6ebe59a070b8f72138c266019a4ccc85428010

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In Artificial Intelligence

In Statistics and Probability

Citations in the corpus

On nonnegative unbiased estimators (2013, background): “Various articles (Beskos and Roberts, 2005; Beskos et al., 2006; Beskos, Papaspiliopoulos and Roberts, 2006; Jourdain and Sbai, 2007; Fearnhead, Papaspiliopoulos and Roberts, 2008; Olsson and Ströjby, 2011; Sermaidis et al., 2012) describe the casef : x 7→ exp(x) and its applications to perfect…”
Penalized nonparametric mean square estimation of the coefficients of diffusion processes (2007, methodology): “Instead, we use the exact retrospective simulation method described in Beskos et al. (2004) and Beskos and Roberts (2005).”
Diffusion-Based Voice Conversion with Fast Maximum Likelihood Sampling Scheme (2021, background): “At the same time, it is possible to design unbiased (Henry-Labord ` ere et al., 2017) or even exact (Beskos & Roberts, 2005) numerical solvers for some particular SDE types.”
Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach (2011, unknown): “2) The unbiased estimation of the diffusion transition density (Beskos, Papaspiliopoulos, Roberts and Fearnhead, 2006; Beskos, Papaspiliopoulos and Roberts, 2009); this approach builds on the exact simulation approach (Beskos and Roberts, 2005; Beskos et al., 2006).”
Retrospective exact simulation of diffusion sample paths with applications (2006, methodology): “An initial version of the exact algorithm was given in Beskos and Roberts (2005).”
Maximum-likelihood estimation for diffusion processes via closed-form density expansions (2013, methodology): “In particular, among various numerical methods, Lo [46] proposed to employ a numerical solution of Kolmogorov equation for transition density; Pedersen [55], Brandt and SantaClara [21], Durham and Gallant [25], Stramer and Yan [60], Beskos and Roberts [17], Beskos et al.”
On Simulated Likelihood of Discretely Observed Diffusion Processes and Comparison to Closed-Form Approximation (2007, unknown): “The exact simulation approach is also a data augmentation method but is built on exact simulation of diffusion sample paths (Beskos and Roberts 2005; Beskos et al. 2006a).”
Statistical inference for dynamical systems: A review (2012, background): “In [13] the authors develop an algorithm to “perfectly” sample the paths of Xt at any finite number of points, i.”
Monte Carlo Maximum Likelihood Estimation for Discretely Observed Diffusion Processes (2009, methodology): “EA was introduced in [9] and [7]; it returns a draw from any finite-dimensional distribution of the target SDE by rejection sampling with proposals from the Wiener measure.”
A probabilistic interpretation of the parametrix method. (2015, methodology): “Before discussing the nature of the above probabilistic representation formula, let us remark that a similar formula is available for one-dimensional diffusion processes (see [5]) which is strongly based on explicit formulas that one may obtain using the Lamperti formula.”
On the exact and ε-strong simulation of (jump) diffusions (2013, methodology): “Reconsidering our initial Implementable Exact Algorithm (Algorithm 2), recall that after simulating the end point from biased Brownian motion (Algorithm 2 Step 1), the remainder of the proposal sample path can be simulated exactly from the law of a Brownian bridge (see Theorem 1).”
An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process (2012, methodology): “strong dependence of the computational cost of the algorithm on the parameters of the equation, arises also for the exact sampling algorithm introduced in [5], which can be also applied to equation (1), see [6].”
Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion) (2006, methodology): “For 1 i n, let ωÅi = .ωÅi,s, s∈ [ti−1, ti]/ and ΦÅi ={ΨÅi , ΥÅi } be the accepted elements of the conditional EA1 algorithm applied on the time interval [ti−1, ti]; S.ωÅi / will denote the corresponding output skeleton.”
A Factorisation of Diffusion Measure and Finite Sample Path Constructions (2008, methodology): “In comparison with our previous work in Beskos and Roberts (2005) and Beskos et al. (2006), the methodology is now much more advanced and includes: probabilistic results on the absolute maximum of Brownian paths and the maximum of Bessel process paths; a novel rejection sampling algorithm for the…”
Particle filters for partially observed diffusions (2007, methodology): “Our approach does not rely on time discretization but builds on recent work on the exact algorithm (EA) for the simulation of diffusions (Beskos and Roberts, 2005; Beskos et al., 2006a, 2005a) and on the unbiased estimation of the diffusion transition density (Beskos et al., 2006b, 2005b).”
Applied Stochastic Differential Equations (2019, unknown): “8 Exact Algorithm The exact algorithm (Beskos and Roberts, 2005; Beskos et al., 2006a,b, 2008; Iacus, 2008) is an algorithm that can be used for simulating exact samples (without a discretization error) from SDEs of the form”