Asymptotic Analysis of the Loss Given Default in the Presence of Multivariate Regular Variation

If {Xj, } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

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Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation

Advances in Applied Probability ◽

10.1017/s0001867800006224 ◽

2013 ◽

Vol 45 (01) ◽

pp. 139-163 ◽

Cited By ~ 5

Author(s):

Bikramjit Das ◽

Abhimanyu Mitra ◽

Sidney Resnick

Keyword(s):

Classical Theory ◽

Regular Variation ◽

Environmental Science ◽

Asymptotic Model ◽

Multivariate Regular Variation ◽

Tail Risk ◽

Risk Estimates ◽

Hidden Regular Variation ◽

Definition Of

Multivariate regular variation plays a role in assessing tail risk in diverse applications such as finance, telecommunications, insurance, and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to inaccurate and useless estimates of probabilities of joint tail regions. This problem can be partly ameliorated by using hidden regular variation (see Resnick (2002) and Mitra and Resnick (2011)). We offer a more flexible definition of hidden regular variation that provides improved risk estimates for a larger class of tail risk regions.

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Positive decreasing solutions of second order quasilinear ordinary differential equations in the framework of regular variation

Filomat ◽

10.2298/fil1509995m ◽

2015 ◽

Vol 29 (9) ◽

pp. 1995-2010 ◽

Cited By ~ 1

Author(s):

Jelena Milosevic ◽

Jelena Manojlovic

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Asymptotic Behavior ◽

Differential Equations ◽

Asymptotic Analysis ◽

Ordinary Differential Equations ◽

Regular Variation ◽

Second Order ◽

Precise Information ◽

Varying Coefficients

This paper is concerned with asymptotic analysis of positive decreasing solutions of the secondorder quasilinear ordinary differential equation (E) (p(t)?(|x'(t)|))'=q(t)?(x(t)), with the regularly varying coefficients p, q, ?, ?. An application of the theory of regular variation gives the possibility of determining the precise information about asymptotic behavior at infinity of solutions of equation (E) such that lim t?? x(t)=0, lim t?? p(t)?(-x'(t))=?.

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