Search

Continuous-time modeling of random searches is designed to be robust to the sampling rate while the spatial model is required to be of rotation-invariant type, which is often computationally prohibitive. Such computational difficulty may be circumvented by employing a model with independent components. We demonstrate that its disadvantages in statistical properties are blurred under lower frequency. We propose a quantitative criterion for choice of the sampling rate at which a spatial model with independent components resembles a rotation-invariant model. Our findings have the potential to assist the observer to employ simpler models in the continuous-time framework to avoid expensive computation required for statistical inference., (© 2012 American Physical Society)

Exact yet simple simulation algorithms are developed for a wide class of Ornstein-Uhlenbeck processes with tempered stable stationary distribution of finite variation with the help of their exact transition probability between consecutive time points. Random elements involved can be divided into independent tempered stable and compound Poisson distributions, each of which can be simulated in the exact sense through acceptance-rejection sampling, respectively, with stable and gamma proposal distributions. We discuss various alternative simulation methods within our algorithms on the basis of acceptance rate in acceptance-rejection sampling for both high- and low-frequency sampling. Numerical results illustrate their advantage relative to the existing approximative simulation method based on infinite shot noise series representation. [ABSTRACT FROM AUTHOR]

To enhance efficiency in Monte Carlo simulations, we develop an adaptive stratified sampling algorithm for allocation of sampling effort within each stratum, in which an adaptive variance reduction technique is applied. Given the number of replications in each batch, our algorithm updates allocation fractions to minimize the work-normalized variance of the stratified estimator of the mean. We establish the asymptotic normality of the stratified estimator of the mean as the number of batches tends to infinity. Although implementation of the proposed algorithm requires a small amount of initial work, the algorithm has the potential to yield substantial improvements in estimator efficiency. Equally important is that the adaptive framework avoids the need for frequent recalibration of the parameters of the variance reduction methods applied within each stratum when changes occur in the experimental conditions governing system performance. To illustrate the applicability and effectiveness of our algorithm, we provide numerical results for a Black-Scholes option pricing, where we stratify the underlying Brownian motion with respect to its terminal value and apply an importance sampling method to normal random variables filling in the Brownian path. Relative to the estimator variance with proportional allocation, the proposed algorithm achieved a fourfold reduction in estimator variance with a negligible increase in computing time. [ABSTRACT FROM AUTHOR]

The author proposes stochastic approximation methods of finding the optimal measure change by the exponential tilting for Lévy processes in Monte Carlo importance sampling variance reduction. In accordance with the structure of the underlying Lévy measure, either a constrained or unconstrained algorithm of the stochastic approximation is chosen. For both cases, the almost sure convergence to a unique stationary point is proved. Numerical examples are presented to illustrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

We propose an approach to a twofold optimal parameter search for a combined variance reduction technique of the control variates and the important sampling in a suitable pure-jump Lévy process framework. The parameter search procedure is based on the two-time-scale stochastic approximation algorithm with equilibrated control variates component and with quasi-static importance sampling one. We prove the almost sure convergence of the algorithm to a unique optimum. The parameter search algorithm is further embedded in adaptive Monte Carlo simulations in the case of the gamma distribution and process. Numerical examples of the CDO tranche pricing with the Gamma copula model and the intensity Gamma model are provided to illustrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Combined control variates and importance sampling variance reduction and its two-fold optimality are investigated. Two-time-scale stochastic approximation algorithm is applied in parameter search for the combination and almost sure convergence of the algorithm to the unique optimum is proved. The parameter search procedure is further incorporated into adaptive Monte Carlo simulation, and its law of large numbers and central limit theorem are proved to hold. An numerical example is provided to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]