Remarks on style only:If you write "Theorem" it should be followed by "Proof" or preceded by the exact citation of the reference where that exact same stated theorem appears. For your first "Theorem" there is no "Proof" given, that's fine but then it's better to make it a "Remark".Generally, use more delimiters like "Notation" and "Example" etc: it makes reading easier when notations are referenced in one place and ideas explained in another.For a good example on p.12 you have a "Remark" where it seems you explain how what you are saying relates to what others have said before: that's very good. Do that more.For a bad example, on the bottom of p.12 you have "Denote a...b= min ( a , b ) and a ... b = max ( a , b ) and we call the k-tranche asthe k-tranche"well this should be in a separate "Notation" environment, and also the second half of the statement needs rereading.In general, try to formalize your ideas into Theorems, this is a good exercise for you, because it helps you to really make precise statements.The reader has less time to read your paper than you had to write it,and most likely he is not as clever as you (), therefore you must spoon feed him with very concise truths.Because he is not inside your head, there is a risk that he will misinterpret EVERY sign you write,yet your goal is to let that person understand ONE , maybe two, ideas that are in your head, and not in other people's papers.Actually, refining your ideas into theorems will make obvious and apparent how your contribution differs from others'.Maybe you should split this into two documents: one giving a review of the litterature, and one presenting separately your framework, with minimal reference to the rest of the litterature.Right now, it seems there is a lot of material that you are referencing, to present your own idea clearly, it might help to make these references shorter, assuming the reader is familiar with the field. But since you have done a lot of work on reviewing all the approaches you mention, definitely put that in a separate document used as litterature review.For example, this part about copula functions - I don't think it's the central idea in your text - well it's really not obvious to me, and it is distracting me from the important things in the rest of the paper. Otherwise, the function defined by the equality C ( F 1 ( x 1 ) , ? F n ( x n ) )should be verified whether or not it represents a multivariate distribution function. Fromthe theorem this fact does not follow though such situation is typical in copula financialapplications. The exceptions might be the case when marginal distributions are uniquelydefine the joint distributionMaybe give an example where C ( F 1 ( x 1 ) , ? F n ( x n ) ) doesn't represent a df?Are you related to I.I Gikhman?