Exponential Tilting is used in Monte Carlo Estimation for rare-event simulation, and rejection and importance sampling in particular.
In mathematical finance [1] 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.
is the cumulant generating function (CGF) defined as We call the
The exponentially tilted measure in many cases has the same parametric form as that of
In such examples, the random variable generation may not always be straightforward.
[7] In statistical mechanics, the energy of a system in equilibrium with a heat bath has the Boltzmann distribution:
Exponential tilting then corresponds to changing the temperature:
Exponential tilting then corresponds to changing both the temperature and the chemical potential.
In many cases, the tilted distribution belongs to the same parametric family as the original.
This is particularly true when the original density belongs to the exponential family of distribution.
This simplifies random variable generation during Monte-Carlo simulations.
Exponential tilting may still be useful if this is not the case, though normalization must be possible and additional sampling algorithms may be needed.
In addition, there exists a simple relationship between the original and tilted CGF, We can see this by observing that Thus, Clearly, this relationship allows for easy calculation of the CGF of the tilted distribution and thus the distributions moments.
Moreover, it results in a simple form of the likelihood ratio.
One common application is sampling from a distribution conditional on a sub-region of the domain, i.e.
can meaningfully reduce the required amount of sampling or the variance of an estimator.
The saddlepoint approximation method is a density approximation methodology often used for the distribution of sums and averages of independent, identically distributed random variables that employs Edgeworth series, but which generally performs better at extreme values.
From the definition of the natural exponential family, it follows that Applying the Edgeworth expansion for
is referred to as the saddle-point, and the above expansion is always evaluated at the expectation of the tilted distribution.
leads to the final representation of the approximation given by Using the tilted distribution
and accepting with probability where That is, a uniformly distributed random variable
[10][11] The above leads to the alternate representation of the stochastic differential equation
Girsanov's Formula states the likelihood ratio
Therefore, Girsanov's Formula can be used to implement importance sampling for certain SDEs.
small, the algorithm uses exponential tilting to derive the importance distribution.
The algorithm is used in many aspects, such as sequential tests,[12] G/G/1 queue waiting times, and
is the only tilting parameter producing bounded relative error (
[13] We can only see the input and output of a black box, without knowing its structure.
When we generate random numbers, the output may not be within the same common parametric class, such as normal or exponential distributions.
In other words, the rules of the game are that the algorithm may use simulated values from