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May 10th, 2017, 01:45 PM   #1
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Lebesgue-Stieltjes integrals, lower partial moments, probability distributions

Working through a lecture in performance management course (topic about downside-oriented risk measures and lower partial moments), I have encountered several expressions of the following form:

LPM(2,t)(R)=integral((t-R)^2dF(R))=E[(max{t-R;0})^2]

where F(R) is the cumulative distribution function, E is the expectation, LPM is the lower partial moment

or, for example,

LPM(2,t)(R)=t*(2*LPM(1,t)(R) - t*LPM(0,t)(R)) + integral(R^2dF(R))

It was also mentioned in one article that the integrals are of Lebesgue-Stieltjes type.

I was not taught anything like this either at university or at school, so I would ask you to provide any links to books/chapters from books/articles from where I can fill this gap in knowledge. I want to be able to understand the above transformations and conduct them on my own, I also would like to cover the whole theory behind this. Also, how large is the volume of theory I have to cover, given that I only made some simple integral calculus at school? Thank you very much.

Last edited by skipjack; May 10th, 2017 at 10:08 PM.
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distributions, integrals, lebesgue, lebesguestieltjes, lebesguestiljets, lower, moments, partial, probability, stiljets



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