Bounds for the probability generating functional of a Gibbs point process

We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics such as the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity, and higher-order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.

Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

This article provides importance sampling algorithms for computing the probabilities of various types ruin of spectrally negative Lévy risk processes, which are ruin over the infinite time horizon, ruin within a finite time horizon and ruin past a finite time horizon. For the special case of the compound Poisson process perturbed by diffusion, algorithms for computing probabilities of ruins by creeping (i.e. induced by the diffusion term) and by jumping (i.e. by a claim amount) are provided. It is shown that these algorithms have either bounded relative error or logarithmic efficiency, as t,x→∞t,x→∞, where t>0t>0 is the time horizon and x>0x>0 is the starting point of the risk process, with y=t/xy=t/x held constant and assumed either below or above a certain constant.

Diffusion-weighted magnetic resonance imaging detects significant prostate cancer with high probability

PURPOSE We prospectively assessed the diagnostic accuracy of diffusion-weighted magnetic resonance imaging for detecting significant prostate cancer. MATERIALS AND METHODS We performed a prospective study of 111 consecutive men with prostate and/or bladder cancer who underwent 3 Tesla diffusion-weighted magnetic resonance imaging of the pelvis without an endorectal coil before radical prostatectomy (78) or cystoprostatectomy (33). Three independent readers blinded to clinical and pathological data assigned a prostate cancer suspicion grade based on qualitative imaging analysis. Final pathology results of prostates with and without cancer served as the reference standard. Primary outcomes were the sensitivity and specificity of diffusion-weighted magnetic resonance imaging for detecting significant prostate cancer with significance defined as a largest diameter of the index lesion of 1 cm or greater, extraprostatic extension, or Gleason score 7 or greater on final pathology assessment. Secondary outcomes were interreader agreement assessed by the Fleiss κ coefficient and image reading time. RESULTS Of the 111 patients 93 had prostate cancer, which was significant in 80 and insignificant in 13, and 18 had no prostate cancer on final pathology results. The sensitivity and specificity of diffusion-weighted magnetic resonance imaging for detecting significant PCa was 89% to 91% and 77% to 81%, respectively, for the 3 readers. Interreader agreement was good (Fleiss κ 0.65 to 0.74). Median reading time was between 13 and 18 minutes. CONCLUSIONS Diffusion-weighted magnetic resonance imaging (3 Tesla) is a noninvasive technique that allows for the detection of significant prostate cancer with high probability without contrast medium or an endorectal coil, and with good interreader agreement and a short reading time. This technique should be further evaluated as a tool to stratify patients with prostate cancer for individualized treatment options.

Voting Power and Probability

Voting power is commonly measured using a probability. But what kind of probability is this? Is it a degree of belief or an objective chance or some other sort of probability? The aim of this paper is to answer this question. The answer depends on the use to which a measure of voting power is put. Some objectivist interpretations of probabilities are appropriate when we employ such a measure for descriptive purposes. By contrast, when voting power is used to normatively assess voting rules, the probabilities are best understood as classical probabilities, which count possibilities. This is so because, from a normative stance, voting power is most plausibly taken to concern rights and thus possibilities. The classical interpretation also underwrites the use of the Bernoulli model upon which the Penrose/Banzhaf measure is based.

Stochastic simulation of rare events

Stochastic simulation is an important and practical technique for computing probabilities of rare events, like the payoff probability of a financial option, the probability that a queue exceeds a certain level or the probability of ruin of the insurer's risk process. Rare events occur so infrequently, that they cannot be reasonably recorded during a standard simulation procedure: specifc simulation algorithms which thwart the rarity of the event to simulate are required. An important algorithm in this context is based on changing the sampling distribution and it is called importance sampling. Optimal Monte Carlo algorithms for computing rare event probabilities are either logarithmic eficient or possess bounded relative error.