22 X.S. Lin, G.E. Willmot Insurance: Mathematics and Economics 27 2000 19–44
Define now the associated compound geometric DF Ku = 1 − ¯ Ku by
¯ Ku =
∞
X
n=1
β 1 + β
1 1 + β
n
¯ G
∗n
u, u ≥ 0,
1.11 where ¯
G
∗n
u is the tail of the n-fold convolution of Gu. It is easy to see that ¯ Ku is the solution to the following
renewal equation: ¯
Ku = 1
1 + β Z
u
¯ Ku − x dGx +
1 1 + β
¯ Gu,
u ≥ 0. 1.12
¯ Ku may also be viewed as the Laplace transform at the time of ruin T . To see this, we let wx
1
, x
2
= 1. It follows immediately from 1.8 and 1.9 that H u = ¯
Gu. Thus, with wx
1
, x
2
= 1, φu is the Laplace transform of T and ¯
Ku = φ u. We now look at the special case when δ = 0. It is clear that
Gx = 1
p
1
Z
x
¯ P y dy.
1.13 Thus, in this case,
¯ Ku = ψu.
1.14 We now state a theorem which shows that the solution to 1.5 can be expressed in terms of ¯
Ku.
Theorem 1.1 Lin and Willmot 1999. The solution φu to 1.5 may be expressed as
φ u = 1
β Z
u
H u − x dKx + 1
1 + β H u,
1.15 φ u = −
1 β
Z
u
¯ Ku − x dH x −
H 0 β
¯ Ku +
1 β
H u. 1.16
If H u is differentiable, then φu may be expressed as φ u = −
1 β
Z
u
¯ Ku − xH
′
x dx − H 0
β ¯
Ku + 1
β H u,
u ≥ 0. 1.17
The proof of Theorem 1.1 and a detailed discussion of Gx, H u and Ku are given in Lin and Willmot 1999.
2. Equilibrium distributions and reliability classifications
Equilibrium distributions associated with a given distribution function and reliability classifications play an important role in our analysis. Many results in the following sections heavily depend on equilibrium distributions.
Reliability classifications enable us to derive upper and lower bounds for functions of the deficit. In what follows, we briefly discuss equilibrium distributions and some reliability classes. Let P x, x 0, be a
distribution function. The equilibrium DF P
1
x = 1 − ¯ P
1
x of P x is defined by P
1
x = R
x
¯ P y dyp
1
. In the present context P
1
x may be interpreted as the DF of the amount of the drop in the surplus level, given that there is a drop e.g., Bowers et al. 1997, Chapter 12. The DF P x is said to be decreasing increasing failure rate or
DFR IFR if ¯ P x + y ¯
P x is nondecreasing nonincreasing in x for fixed y ≥ 0. A larger class of distributions is the new worse better than used or NWU NBU class with ¯
P x + y ≥ ≤ ¯ P x ¯
P y for all x ≥ 0 and y ≥ 0.
X.S. Lin, G.E. Willmot Insurance: Mathematics and Economics 27 2000 19–44 23
The mean residual lifetime of P x is defined by r
P
x = R
∞ x
¯ P y dy
¯ P x
= r
P
¯ P
1
x ¯
P x ,
x ≥ 0. 2.1
Another class larger than the DFR IFR class is the increasing decreasing mean residual lifetime or IMRL DMRL class for which r
P
x is nondecreasing nonincreasing for x ≥ 0. Similarly, P x is new worse better than used in expectation or NWUE NBUE if r
P
x ≥ ≤r
P
0, or equivalently ¯ P
1
x ≥ ≤ ¯ P x, x ≥ 0. See Fagiuoli
and Pellerey 1994 and Lin and Willmot 1999 and references therein for further details on these classifications. The notion of higher order equilibrium DFs is also needed in what follows. Let P
2
x = 1 − ¯ P
2
x be the equilibrium DF of P
1
x. The DF P
2
x is called the second-order equilibrium DF of P x. Similarly, we define the nth order equilibrium DF of P x by
P
n
x = 1 − ¯ P
n
x = R
x
¯ P
n−1
y dy R
∞
¯ P
n−1
y dy ,
n = 1, 2, . . . , 2.2
where ¯ P
x = 1 − P x = ¯
P x. It can be shown that the mean of the DF P
n
x, n = 1, 2, . . . , is given by Z
∞
¯ P
n
x dx = p
n+1
n + 1p
n
, 2.3
and that the relation between the DFs P
n
x and P x is given by ¯
P
n
x = 1
p
n
Z
∞ x
y − x
n
dP y. 2.4
See Hesselager et al. 1998 and references therein for further details. The connection of the equilibrium DFs P
n
x of P x and the equilibrium DFs G
n
x of G x = Gx, where
P x is the individual claim size DF and Gx is the associated “claim size” DF to the renewal equation 1.5 can now be established.
It is easy to see from 1.9 that in the special case when δ = 0, Gx = P
1
x since ρ = 0. Thus, G
n
x = P
n+1
x in this case. Let µ
n
ρ be the nth moment of Gx, i.e. µ
n
ρ = Z
∞
x
n
dGx. 2.5
Thus, µ
n
0 = Z
∞
x
n
dP
1
x = 1
p
1
Z
∞
x
n
¯ P x dx =
p
n+1
n + 1p
1
. 2.6
For ρ 0, we have µ
n
ρ = µ
n
R
∞
e
−ρx
dP
n+1
x R
∞
e
−ρx
dP
1
x .
2.7 The equilibrium DF G
n
x of Gx can be expressed in terms of the equilibrium DF P
n
x. In fact we have, analogous to 1.9,
¯ G
n
x = R
∞
e
−ρy
¯ P
n
x + y dy R
∞
e
−ρy
¯ P
n
y dy .
2.8 The derivation of 2.7 and 2.8 can be found in Section 3 of Lin and Willmot 1999.
24 X.S. Lin, G.E. Willmot Insurance: Mathematics and Economics 27 2000 19–44
3. Convolution of equilibrium distributions and compound geometric tails
