Efficient simulation of tail probabilities of sums of correlated lognormals

Asmussen, Søren; Blanchet, José; Juneja, Sandeep; Rojas-Nandayapa, Leonardo

doi:10.1007/s10479-009-0658-5

Cited by 45 publications

(55 citation statements)

References 19 publications

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“…This will facilitate the understanding of the approach in the Generalized Gamma case. From the expression of ℓ in (28), the idea of the conditional MC estimator is to use the fact that exponential RV Y i is equal in distribution to G × S i where G is a Gamma distribution with shape N and scale 1 and S = (S 1 , · · · , S N ) are uniformly distributed over the simplex {s i > 0, N i=1 s i = 1} and independent of G, see [21]. Then, using this representation, the probability ℓ can be expressed as…”

Section: A Generalized Gamma Casementioning

confidence: 99%

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On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances

Rached

Botev

Kammoun

et al. 2018

IEEE Trans. Wireless Commun.

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We consider the problem of evaluating the cumulative distribution function (CDF) of the sum of order statistics, which serves to compute outage probability (OP) values at the output of generalized selection combining receivers. Generally, closed-form expressions of the CDF of the sum of order statistics are unavailable for many practical distributions. Moreover, the naive Monte Carlo (MC) method requires a substantial computational effort when the probability of interest is sufficiently small. In the region of small OP values, we propose instead two effective variance reduction techniques that yield a reliable estimate of the CDF with small computing cost. The first estimator, which can be viewed as an importance sampling estimator, has bounded relative error under a certain assumption that is shown to hold for most of the challenging distributions. An improvement of this estimator is then proposed for the Pareto and the Weibull cases. The second is a conditional MC estimator that achieves the bounded relative error property for the Generalized Gamma case and the logarithmic efficiency in the Log-normal case. Finally, the efficiency of these estimators is compared via various numerical experiments.[13] distributions. On the other hand, efficient simulation methods have been also developed for the estimation of the CDF of the sum of RVs such as the Log-normal [3], [14]-[17] and the Generalied Gamma [3].In the general case where L < N and apart from the exponential and Gamma RVs, closed-form expressions of the CDF of partial sums of ordered RVs are out of reach for many challenging distributions and are still open problems. This is for instance the case of the Log-normal RV which models shadowing [18] and weak-to-moderate turbulence channels in free space optical communication systems [19]. The Weibull variate, which has also received an increasing interest and has been shown to fit realistic propagation channels [20], is another example where the CDF of sums of order statistics is not known to possess a closed-form expression. Thus, it is important to propose alternative approaches to compute the CDF of sums of ordered RVs with arbitrary distributions.

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Section: A Generalized Gamma Casementioning

confidence: 99%

“…Using the asymptotic behavior of the right hand side term given in [28] P (X 1 ≤ γ th /L) ∼ 1 √ 2π log (L/γ th ) exp − (log(γ th /L)) 2 2 (77) and following the same steps as for the case L = N, we get…”

Section: B Log-normal Casementioning

confidence: 99%

On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances

Rached

Botev

Kammoun

et al. 2018

IEEE Trans. Wireless Commun.

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“…Despite the tractability of multivariate log-normal distribution, the first result which derives the asymptotic tail behaviour of the sum of log-normal risks appeared recently in Asmussen and Rojas-Nandayapa (2008), see also Albrecher et al (2006). The recent contribution Asmussen et al (2011) derives an explicit asymptotic expansion (u → ∞) of to model the aggregated risk utilizing a log-elliptical framework, which has been recently discussed in RojasNandayapa (2008), Kortschak and Hashorva (2013) and Hashorva (2013). The latter two papers derived (under different conditions) the following asymptotic expansion…”

Section: Introductionmentioning

confidence: 99%

Second Order Asymptotics of Aggregated Log-Elliptical Risk

Kortschak

Hashorva

2013

Methodol Comput Appl Probab

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In this paper we establish the error rate of first order asymptotic approximation for the tail probability of sums of log-elliptical risks. Our approach is motivated by extreme value theory which allows us to impose only some weak asymptotic conditions satisfied in particular by log-normal risks. Given the wide range of applications of the log-normal model in finance and insurance our result is of interest for both rare-event simulations and numerical calculations. We present numerical examples which illustrate that the second order approximation derived in this paper significantly improves over the first order approximation.

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“…Asmussen et al [11] considered the case in which X is a multidimensional Gaussian vector. This setting is motivated by considering d correlated asset prices, each following a Black-Scholes dynamic in which individual stock prices are lognormal.…”

Section: Introductionmentioning

confidence: 99%

Efficient simulation of tail probabilities of sums of dependent random variables

Blanchet¹,

Rojas-Nandayapa²

2011

J. Appl. Probab.

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We study asymptotically optimal simulation algorithms for approximating the tail probability of P(e X 1 + · · · + e X d > u) as u → ∞. The first algorithm proposed is based on conditional Monte Carlo and assumes that (X 1 , . . . , X d ) has an elliptical distribution with very mild assumptions on the radial component. This algorithm is applicable to a large class of models in finance, as we demonstrate with examples. In addition, we propose an importance sampling algorithm for an arbitrary dependence structure that is shown to be asymptotically optimal under mild assumptions on the marginal distributions and, basically, that we can simulate efficientlyExtensions that allow us to handle portfolios of financial options are also discussed.

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Efficient simulation of tail probabilities of sums of correlated lognormals

Cited by 45 publications

References 19 publications

On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances

On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances

Second Order Asymptotics of Aggregated Log-Elliptical Risk

Efficient simulation of tail probabilities of sums of dependent random variables

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