Exponential Familia Techniques for the Lognormal Left Tail

ABSTRACT:

Sums of lognormals random variables arise in a wide variety of disciplines
such as engineering, economics, insurance or nance, and are often employed
in modeling across the sciences.

Since the lognormal is subexponential and heavy-tailed, then the asymp-
totic properties of the probability that the sum goes to innity (right tail) are
typically analyzed using subexponential techniques. In contrast, the study
of the asymptotics of the probability that the sum becomes very small (left
tail) is a light-tailed problem and the typical tools would be saddlepoint
or large deviations techniques. This faces, however, the problem that the
Laplace transform is not explicit.

In this talk we discuss a decomposition of the Laplace transform of the
Lognormal distribution which can be employed to provide sharp approxima-
tions for left tail probabilities of a sum of lognormals. The Laplace transform
is decomposed into a closed-form expression given in terms of the Lambert
W function corresponding to the asymptotic approximation provided by
Laplace’s method, and the term in the decomposition is the expected value
of a function which is close to 1, hence negligible.

The proposed decomposition is used as basis for studying rst the ex-
ponentially tilted density which can asymptotically become concentrated
around the mode point of the integrand dening the Laplace transform and
has a shape that can be described either as Lognormal, Gamma or Normal.
Next a saddlepoint-type-approximation for the left tail is presented, and ef-
cient simulation algorithms for estimating both the Laplace transform and
left tail probabilities are suggested.

Numerical examples are presented in a range of parameters that we con-
sider realistic for nancial and engineering applications.

Posted on May 23, 2014 in Seminars, Stochastic Modeling