Modelling Dependence with Copulas - IFOR

2 Copulas
2.1 Mathematical Introduction
Definition 1. Let S 1 , S 2 ,...S n be nonempty subsets of R, whereR denotes the
extended real line [−∞, ∞]. Let H be a real function of n variables such that
DomH = S 1 × S 2 × ...× S n and let B =[a, b] beann-box all of whose vertices
are in DomH. Then the H-volume of B is given by
V H (B) = ∑ sgn(c)H(c), (2.1.1)
where the sum is taken over all vertices c of B, and sgn(c) isgivenby
sgn(c) =
{
1, if ck = a k foranevennumberofk’s,
−1, if c k = a k for an odd number of k’s.
(2.1.2)
Equivalently, the H-volume of an n-box B =[a, b] isthenth order difference of
H on B
V H (B) =△ b aH(t) =△bn a n
△ bn−1
a n−1
...△ b1
a 1
H(t),
wherewedefinethen first order differences as
△ b k
ak
H(t) =H(t 1 ,... ,t k−1 ,b k ,t k+1 ,... ,t n ) − H(t 1 ,... ,t k−1 ,a k ,t k+1 ,... ,t n ).
Definition 2. A real function H of n variables is n-increasing if V H (B) ≥ 0 for all
n-boxes B whose vertices lies in DomH.
Suppose that the domain of a real function H of n variables is given by DomH =
S 1 × S 2 × ...× S n where each S k has a least element a k .WesaythatH is grounded
if H(t) = 0 for all t in DomH such that t k = a k for at least one k. If each S k is
nonempty and has a greatest element b k ,thenwesaythatH has margins, and the
one-dimensional margins of H are the functions H k given by DomH k = S k and
H k (x) =H(b 1 ,... ,b k−1 ,x,b k+1 ,... ,b n ) for all x in S k . (2.1.3)
Higher dimensional margins are defined by fixing fewer places in H. One-dimensional
margins will be called simply “margins” and for k ≥ 2 k-dimensional margins will
be called “k-margins”.
Lemma 2.1. Let S 1 ,S 2 ,... ,S n be nonempty subsets of R, andletH be a grounded
n-increasing function with domain S 1 × S 2 × ...× S n . Then H is nondecreasing in
each argument, that is, if (t 1 ,... ,t k−1 ,x,t k+1 ,... ,t n ) and
(t 1 ,... ,t k−1 ,y,t k+1 ,... ,t n ) are in DomH and x