Exponential Tilting (ET), Exponential Twisting, or Exponential Change of Measure (ECM) is a distribution shifting technique used in many parts of mathematics.
The different exponential tiltings of a random variable
is known as the
natural exponential family In probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF).
Definition
Univariate case
The natural exponential families (NEF) are a subset of ...
of
.
Exponential Tilting is used in
Monte Carlo Estimation for rare-event simulation, and
rejection and
importance sampling
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally att ...
in particular.
In mathematical finance Exponential Tilting is also known as Esscher tilting (or the
Esscher transform), and often combined with indirect
Edgeworth approximation and is used in such contexts as insurance futures pricing.
The earliest formalization of Exponential Tilting is often attributed to
Esscher with its use in importance sampling being attributed to
David Siegmund
__NOTOC__
David Oliver Siegmund (born November 15, 1941) is an American statistician who has worked extensively on sequential analysis. .
Overview
Given a random variable
with probability distribution
, density
, and
moment generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compar ...
(MGF)
, the exponentially tilted measure
is defined as follows:
:
where
is the
cumulant generating function
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will hav ...
(CGF) defined as
:
We call
:
the
-tilted
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of
. It satisfies
.
The exponential tilting of a random vector
has an analogous definition:
:
where