In
actuarial science, the Esscher transform is a transform that takes a
probability density
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
''f''(''x'') and transforms it to a new probability density ''f''(''x''; ''h'') with a parameter ''h''. It was introduced by F. Esscher in 1932 .
Definition
Let ''f''(''x'') be a probability density. Its Esscher transform is defined as
:
More generally, if ''μ'' is a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
, the Esscher transform of ''μ'' is a new probability measure ''E
h''(''μ'') which has
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 ...
:
with respect to ''μ''.
Basic properties
; Combination
: The Esscher transform of an Esscher transform is again an Esscher transform: ''E
h''
1 ''E
h''
2 = ''E
h''
1 + ''h''2.
; Inverse
: The inverse of the Esscher transform is the Esscher transform with negative parameter: ''E'' = ''E''
−''h''
; Mean move
: The effect of the Esscher transform on the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
is moving the mean:
::
Examples
See also
*
Esscher principle
*
Exponential tilting
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 X is known as the natural exponen ...
References
*
* {{cite journal
, title=On the Probability Function in the Collective Theory of Risk
, journal=Skandinavisk Aktuarietidskrift
, volume=15
, issue = 3
, year=1932
, pages=175–195
, last=Esscher
, first=F.
, doi = 10.1080/03461238.1932.10405883
Actuarial science
Transforms