On the Tail Probabilities of Aggregated Lognormal Random Fields with Small Noise

Motivation. The integral of lognormal random fields is the central quantity of many probabilistic models in portfolio risk analysis, spatial point processes, etc. (see, e.g., Liu and Xu [12, 14]). The current analysis is of interest particularly for risk analysis of short-term behavior of a large size portfolio under high correlations. We elaborate more on this application. Consider a portfolio consisting of n assets denoted by S1, ..., Sn, each of which is associated to a weight, denoted by w1, ..., wn. The total value is S = ∑n i=1wiSi. Of interest is the tail behavior of S. A stylized model assumes that Si’s are lognormal random variables. Then, the total value is the sum of n correlated lognormal random variables (Ahsan [1], Duffie and Pan [6], Glasserman et al. [10], Basak and Shapiro [3], Deutsch [5], Foss and Richards [8]). Under such a setting, one may employ a latent space approach by embedding S1, ..., Sn in a Gaussian process. More precisely, we construct a