Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
4.4. Focus on the dependence structure
Lemma 4.4.2. Let f be a continuous concave (resp. convex) function on a set S. Then the
sequence (f(xi) − f(xi−1))i with xi = x0 + iδ is decreasing (resp. increasing).
Proof. Let ˜ Fi : [xi−1, xi+1] ↦→ R be defined as
˜Fi(x) =
x − xi−1
xi+1 − xi−1
f(xi−1) + xi+1 − x
f(xi+1).
xi+1 − xi−1
Since f is concave, we have for all x ∈ [xi−1, xi+1], f(x) ≥ ˜ Fi(x). In particular for x = xi, we
get
f(xi) ≥ (f(xi−1) + f(xi+1))/2 ⇔ f(xi) − f(xi−1) ≥ f(xi+1) − f(xi).
Lemma 4.4.3. Let f be a completely monotone function and c > 0. The function fc : x ↦→
f(x) − f(x + c) is also completely monotone.
Proof. As f being completely monotone, (−1) n f (n) is decreasing. Furthermore, we have
(−1) n f (n)
c (x) = (−1) n f (n) (x) − (−1) n f (n) (x + c) ≥ 0
as x < x + c. Thus fc is completely monotone.
In particular, if f is convex decreasing, then x ↦→ f(x) − f(x + c) is also convex decreasing,
and x ↦→ f(x + c) − f(x) is concave increasing.
Let ∆i,j be the height of stairstep in the graphical representation of the distribution
function DW1,W2 . We have ∆i,j = DW1,W2 (xi, yj) − DW1,W2 (xi−1, yj−1), where xi, yj ∈
Im(FW1 ) × Im(FW2 ), i.e. xi = FW1 (i) and yj = FW2 (j).
Proposition 4.4.4. The maximal stairstep, representing the highest difference between distribution
functions DW1,W2 and CY1,Y2 , is ∆∞,0 = ∆0,∞.
Proof. We are looking for the maximum of ∆i,j. By algebraic manipulations, we have
DW1,W2 (xi, yj) = P (W1 ≤ i, W2 ≤ j) = 1 − LΘ(i + 1) − LΘ(j + 1) + LΘ(i + j + 2).
Since LΘ is completely monotone, the function x ↦→ LΘ(x + c) − LΘ(x) is concave increasing,
cf. Lemma 4.4.3. Similarly, for a constant c > 0, the function gc : x ↦→ 1 − LΘ(c + 1) − LΘ(x +
1) + LΘ(x + 2 + c) is concave increasing.
Furthermore, we have
∆i,j = DW1,W2 (xi, yj) − DW1,W2 (xi, yj−1) + DW1,W2 (xi, yj−1) − DW1,W2 (xi−1, yj−1)
Using the function f defined above, we have DW1,W2 (xi, yj)−DW1,W2 (xi, yj−1) = gi(j)−gi(j −
1). Hence, (DW1,W2 (xi, yj) − DW1,W2 (xi, yj−1))j is a decreasing sequence, using Lemma 4.4.2
and gi is concave. DW1,W2 (xi, yj−1) − DW1,W2 (xi−1, yj−1) = gi+1(j + 1) − gi+1(j) + LΘ(i) −
LΘ(i + 1). By Lemma 4.4.3, the function gi+1 is concave and by Lemma 4.4.2 the sequence
(H(xi, yj−1) − H(xi−1, yj−1))j is decreasing.
Therefore for a fixed i, (∆i,j)j is a sum of two decreasing sums of j. Since Archimedean
copulas are symmetric, we deduce
max
i,j≥0 ∆i,j = max
i≥j ∆i,j = max
i≥0 max
i≥j≥0 ∆i,j = max
i
∆i,0.
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