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 of .
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.[2]
The earliest formalization of Exponential Tilting is often attributed to Esscher[3] with its use in importance sampling being attributed to David Siegmund.[4]
^H.U. Gerber & E.S.W. Shiu (1994). "Option pricing by Esscher transforms". Transactions of the Society of Actuaries. 46: 99–191.
^Cruz, Marcelo (2015). Fundamental Aspects of Operational Risk and Insurance Analytics. Wiley. pp. 784–796. ISBN 978-1-118-11839-9.
^Butler, Ronald (2007). Saddlepoint Approximations with Applications. Cambridge University Press. pp. 156. ISBN 9780521872508.
^Siegmund, D. (1976). "Importance Sampling in the Monte Carlo Study of Sequential Tests". The Annals of Statistics. 4 (4): 673–684. doi:10.1214/aos/1176343541.
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