Hello 


I am trying to derive the second equation displayed in section 7.1 (or p. 41) of this [article](http://ta.twi.tudelft.nl/users/vuik/numanal/kort_afst.pdf) or equation (6.3) of this [book](http://books.google.co.uk/books?id=0dyagVg20XQC&pg=PA182&dq=6.3+conditional+sampling+copula&hl=en&ei=AvrOTayfGMuBhQfV3-HyDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDcQ6AEwAA#v=onepage&q=6.3%2520conditional%2520sampling%2520copula&f=false). I've seen this in many documents discussing conditional sampling with copulae. 


Suppose $C(u\_1,...u\_d)$ is a d dimensional copula and $C\_j(u\_1,...u\_j)=C(u\_1,...u\_j,1,...,1)$, $j \in {2,...,d-1}$ is the $j$ dimensional marginal copula. 


I am trying to prove the following relationship as in page 420 of this [book](http://books.google.co.uk/books?id=I_Ua05TV1M0C&printsec=frontcover&dq=statistics+of+financial+markets&hl=en&ei=dwPPTZ3uHsK6hAfGyOD9DA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CEIQ6AEwAA#v=snippet&q=conditional%2520inverse%2520method&f=false) (search inside the book for "conditional inverse method" in google books to see this page), 


$\mathbb{P}\left(U\_{j}\leq u\_{j},U\_1=u\_{1},\ldots,U\_{j-1}=u\_{j-1}\right)=\frac{\partial^{j-1}C\_{j}\left(u\_{1},\ldots,u\_{j}\right)}{\partial u\_{1},\ldots,\partial u\_{j-1}}$


Maybe something along the lines of, 


$\mathbb{P}\left(U\_{j}\leq u\_{j},U\_1=u\_{1},\ldots,U\_{j-1}=u\_{j-1}\right)$


$=\lim\_{\triangle u\_{1},\ldots,\triangle u\_{j-1}\rightarrow0}\mathbb{P}\left(U\_{j}\leq u\_{j},u\_{1}\leq U\_{1}\leq u\_{1}+\triangle u\_{1},\ldots,u\_{j-1}\leq U\_{j-1}\leq u\_{j-1}+\triangle u\_{j-1}\right) $


then the paper i'm reading tells me if this is right it should equal


$=\lim\_{\triangle u\_{1},\ldots,\triangle u\_{j-1}\rightarrow0}\frac{C\_{j}\left(u\_{1}+\triangle u\_{1},\ldots,u\_{j-1}+\triangle u\_{j-1},u\_{j}\right)-C\left(u\_{1},\ldots,u\_{j}\right)}{\triangle u\_{1}\cdots\triangle u\_{j-1}}$ ... (\*)


Where the fact the copula $C\_j$ is the joint distribution of $j$ uniform random variables, $U\_1,...,U\_j$ is used. But I dont quite understand where the denominator in (\*) $\triangle u\_{1}\cdots\triangle u\_{j-1}$ comes from??


In fact the derivation in the paper goes against my understanding of probability. 


For example how does the author divide by $\mathbb{P}\left(U\_{j}\leq u\_{j},U=u\_{1},\ldots,U\_{j-1}=u\_{j-1}\right)$ in the derivation. I would have imagined since the $u\_i$'s are continuous this quantity would be 0... but how come its not??



Proof of conditional copula relation to the marginal copulas 
Have a look at:
<http://publications.rwth-aachen.de/record/59254/files/04_198.pdf>


in Section 2.2, the derivation is done for you.


