| Risk is the core of risk theory. The nature of the risk is uncertain and random loss. The insurance industry itself faces the risk of claims, especially the "large claims", which has a significant impact on the company’s operating condition, we use heavy-tailed distribution to describe such claims in mathematics. If the event occurrence probability is greater than the tail of the normal distribution, then the event follows a heavy-tailed distribution.PH distribution is an important branch of heavy-tailed distribution. Univariate PH distributions have the advantage of closure, namely their distribution, densities, Laplace transforms and all their moments can be written in a closed form, and thus, they become more and more popular in the study of correlation analysis of stochastic models.This article starts from the PH distribution of continuous time Markov chain. According to the basic concept of Log Phase-type distribution of continuous time Markov chain, we study Log-PH distribution of its basic properties, such as moments, Laplace transform, Kronecker product. Taking reference of bivariate PH distribution of continuous time Markov chain theory, we derive the corresponding distribution function, density function, moment and Laplace conversion of bivariate Log-PH distribution.The conditional tail expectation is about heavy-tailed distribution for consider average loss above quantile. This article is an extension of the conditional tail expectation from Log phase-type distribution and multivariate PH distribution to multivariate Log phase-type distribution. Using the order statistic method and Markov chain properties of multivar-iate Log-PH distribution, we construct the conditional tail expectation of multivariate Log-PH distribution.Finally, the paper puts forward some problems about the study of financial risk measurement of the class of heavy-tailed distributions and we discuss the dynamic development and prospect about Log phase-type distribution. |
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