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On Higher Derivatives of Expectations

Listed author(s):
  • Robert de Rozario

    (University of NSW, Sydney, Australia)

It is understood that derivatives of an expectation $E [\phi(S(T)) | S(0) = x]$ with respect to $x$ can be expressed as $E [\phi(S(T)) \pi | S(0) = x]$, where $S(T)$ is a stochastic variable at time $T$ and $\pi$ is a stochastic weighting function (weight) independent of the form of $\phi$. Derivatives of expectations of this form are encountered in various fields of knowledge. We establish two results for weights of higher order derivatives under the dynamics given by (\ref{dynamics}). Specifically, we derive and solve a recursive relationship for generating weights. This results in a tractable formula for weights of any order.

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Paper provided by EconWPA in its series Risk and Insurance with number 0308001.

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Length: 6 pages
Date of creation: 19 Aug 2003
Handle: RePEc:wpa:wuwpri:0308001
Note: Type of Document - LaTex; prepared on IBM PC ; to print on PostScript; pages: 6 ; figures: included. In the process of being submitted
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  1. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
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