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Retrospective exact simulation of diffusion sample paths with applications. (English) Zbl 1129.60073

Let \(X=\{ X_t,\;0\leq t\leq T\} \) be a scalar random process determined as the solution of the stochastic differential equation (SDE) of the type \[ dX_t=a(X_t)dt+\sigma (X_t)\,d\beta_t, \tag{1} \] \(X_0=x\in \mathbb{R},\) \(t\in [0,T]\) driven by the Brownian motion \(\{\beta_t,\;t\in [0,T]\}.\) Under some regularity conditions on \(a(\cdot)\) and \( \sigma (\cdot)\) there exists a weakly unique global solution of SDE(1). Actually, it suffices to consider \(\sigma (\cdot) =1.\) Simulation and inference of such SDEs require some discrete approximation for \(X_{t+\Delta}\) as \(\Delta\to \infty\). The exact algorithm returns skeletons of exact paths of \(X\).
In the paper, an easier and more flexible method is proposed applying the rejection sampling algorithm, presented by A. P. Beskos and G. O. Roberts [Ann. Appl. Probab. 15, 2422–2444 (2005; Zbl 1101.60060)]. Now the authors relax the boundedness condition on the drift and require that the functional \(a^2+a'\) is bounded from below and \(\lim \sup_{u\to \infty}(a^2+a')(u)\) and \(\lim \sup_{u\to -\infty}(a^2+a')(u)\) are not both \(+\infty .\)
In sections 2 and 3 the exact algorithm is presented and its extension, discussed above, is analytically described. The efficiency of the algorithm, its applications to simulation exactly from the otherwise intractable logistic growth model, the use for simulation of Monte-Carlo maximum likelihood for discretely observed diffusions and for exact simulation of diffusions conditioned to hit specific values at predeterminated time instances are considered in other sections. Finally, some general remarks about the exact algorithm and possible extensions in future research are presented. The appendix contains the proof of several propositions employed above.

MSC:

60J60 Diffusion processes
60G17 Sample path properties

Citations:

Zbl 1101.60060
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References:

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