Discussion of Hans U. Gerber and Elias S.W. Shiu’s
”The Time Value of Ruin in a Sparre Andersen Model”
Cary Chi-Liang Tsai∗
Professors Gerber and Shiu use an innovative approach based on a Sparre Andersen model
(more general than the ones used in the study of ruin theory before) for the interclaim time
assumption to derive the renewal equation for the expected discounted function φ which is
usually used to study a variety of interesting and important quantities in the problems of
classical ruin theory by appropriate choices of the penalty function w. The purpose of this
discussion is to show that the discounted joint distribution function of U (T −) and |U (T )|
can be obtained from the expected discounted function.
Gerber and Shiu express g(y) by (eq. (9.3))
Qn
i=1
cn
g(y) =
n
λi Y
[ Tρj ]p(y),
j=1
and further simplify it to a very elegant and simple form (eq. (9.10)) as
Qn
g(y) =
i=1
cn
n
n
Y
λi X
[ (
j=1 k=1, k6=j
1
)Tρ p](y),
ρk − ρj j
a linear function of Tρj p(y)’s.
Define the distribution function Γj (y) by (see also Lin and Willmot (1999))
Z y
Γj (y) =
Z 0∞
0
Z yZ ∞
Tρj p(t)dt
=
Tρj p(t)dt
Z 0∞ Zt ∞
0
e−ρj (x−t) dP (x)dt
t
e−ρj (x−t) dP (x)dt
,
R R
R
j = 1, 2, · · · , n. Since y∞ t∞ e−ρj (x−t) dP (x)dt = y∞ e−ρj (x−y) P (x)dx, we have
Z ∞
Z ∞
Γj (y) = 1 − Γj (y) = Zy∞
0
Tρj p(t)dt
=
Tρj p(t)dt
y
e−ρj (x−y) P (x)dx
Ej
(1)
R∞
R ∞ R ∞ −ρ (x−t)
R
j
dP (x)dt = 0∞ e−ρj x P (x)dx = [Tρj P ](0), j =
0 Tρj p(t)dt = 0
t e
where Ej =
1, 2, · · · , n. Then
Qn
E=
R∞
i=1
cn
n
n
Y
λi X
[ (
Ry
j=1 k=1, k6=j
1
)Ej ]
ρk − ρj
where E = 0 g(x)dx. Let G(y) = 0 g(x)dx/E; then the associated ”claim size” distribution G can be expressed as a weighted average of Γj ’s (each Γj plays the same role as G if the
∗
Cary Chi-Liang Tsai, ASA, PhD., is Assistant Professor of Department of Statistics and Actuarial Science, Simon Fraser
University, Burnaby, BC V5A 1S6, Canada, Email: cltsai@sfu.ca
interclaim time random variables are independent with a common exponential distribution
of mean 1/λj for fixed j), that is,
Qn
n
n
Y
λi X
[ (
i=1
cn
G(y) =
j=1 k=1, k6=j
where
n
Y
(
k=1, k6=j
n
Y
wj = X
n
n
X
1
wj Γj (y),
)Ej Γj (y)]/E =
ρk − ρj
j=1
1
)Ej
ρk − ρj
n
Y
1
)Ei
(
i=1 h=1, h6=i ρh − ρi
with
Pn
j=1
wj = 1, and G(y) = 1 − G(y) =
i=1
cn
=
Pn
j=1
n
Y
λi (
k=1, k6=j
1
)Ej
ρk − ρj
E
(2)
wj Γj (y).
R
Since E = 0∞ g(y)dy < 1 (see eq. (11.5)), let E = 1/(1 + β), where β = 1/E − 1 > 0.
Then the defective renewal equation (9.2) for φ(u) can be rewritten as
1 Zu
1
φ(u) =
φ(u − y)dG(y) +
B(u)
1+β 0
1+β
(3)
where
Qn
B(u) = (1 + β)
i=1
cn
n
λi Y
[ Tρj ]ω(u) = (1 + β)
j=1
Qn
i=1
cn
n
n
Y
λi X
[ (
j=1 k=1, k6=j
1
)Tρ ω](u).
ρk − ρj j
(4)
By Lin and Willmot (1999),
φ(u) = −
1
1
1Zu
K(u − x)dB(x) + B(u) − B(0)K(u),
β 0
β
β
(5)
∞
X
β
1 n ∗n
(
) G (u), u ≥ 0, is a compound geometric
n=1 1 + β 1 + β
distribution function with K(0) = 1/(1+β), and K(u) satisfies the defective renewal equation
where K(u) = 1 − K(u) =
1 Zu
1
K(u) =
K(u − x)dG(x) +
G(u),
1+β 0
1+β
u ≥ 0.
(6)
When w(x,
y) = 1, φ(u) reduces to E[e−δT I(T < ∞)|U (0) = u]. Then B(u) in (4) becomes
Qn
Z ∞
n
n
n
X
Y
X
1
i=1 λi
−ρj (x−u)
(1+β)
[
(
)
e
P
(x)dx]
=
wj Γj (u) = G(u). Compare (6)
cn j=1 k=1, k6=j ρk − ρj u
j=1
with (3), we conclude that K(u) = φ(u)|w=1 = E[e−δT I(T < ∞)|U (0) = u].
Moreover, the discounted joint distribution function of U (T −) and |U (T )| can be produced from φ(u) by an appropriate choice of the penalty function w(x, y). Then an explicit
expression for the discounted joint distribution function of U (T −) and |U (T )| can be obtained from (5). To see this, for any fixed x and y, let
(
w(x1 , x2 ) =
1, if x1 ≤ x, x2 ≤ y,
0, otherwise.
2
R R R
By (2.1), φ(u) becomes 0x 0y 0∞ e−δt f (x1 , x2 , t|u)dtdx1 dx2 = F (x, y|u), the discounted joint
distribution function of U (T −) and |U (T )|.
For j = 1, 2, · · · , n, if 0 < x ≤ u, Tρj ω(u) = 0, and if 0 ≤ u < x,
Tρj ω(u) =
=
Z x
u
−ρj (x1 −u)
e
Z ∞
u
−
Z x1 +y
x1
−ρj (x1 −u)
e
Z ∞
u+y
dP (x2 )dx1
−ρj (x−u)
P (x1 )dx1 − e
−ρj (x1 −u−y)
e
Z ∞
x
P (x1 )dx1 + e
e−ρj (x1 −x) P (x1 )dx1
−ρj (x−u)
Z ∞
x+y
−ρj (x−u)
= Ej [Γj (u) − Γj (u + y)] − Ej e
e−ρj (x1 −x−y) P (x1 )dx1
[Γj (x) − Γj (x + y)]
by (1). Therefore,
B(u) = [G(u) − G(u + y)] −
n
X
wj e−ρj (x−u) [Γj (x) − Γj (x + y)],
(7)
j=1
and for u > 0
dB(u) = dG(u + y) − dG(u) −
n
X
wj ρj e−ρj (x−u) [Γj (x) − Γj (x + y)]du.
(8)
j=1
R
u
Equation (5) becomes F (x, y|u) = −(1/β)
K(u − t)dB(t) + (1/β)B(u) − (1/β)B(0)K(u)
Rx 0
for 0 ≤ u < x, and F (x, y|u) = −(1/β) 0 K(u − t)dB(t) − (1/β)B(0)K(u) for 0 < x ≤ u.
Replacing B(u) and dB(t) with (7) and (8), respectively, and using (6) gives the discounted
joint distribution function of U (T −) and |U (T )| as follows (setting u = 0 yields F (x, y|0)
from K(0) = 1/(1 + β) and (6)):

·
¸
1
1Zy
1+β



K(u) − K(u + y) − G(y)K(u) +
K(u + y − t)dG(t)


β
β
β 0


·
¸·
¸
Z

n
u

1X

−ρ
x
ρ
u
ρ
t

j
j
j

+
wj e
Γj (x) − Γj (x + y) ρj
e K(u − t)dt + K(u) − e



β j=1
0




if 0 ≤ u < x,
·
¸
F (x, y|u) =  1 Z x
1


K(u − t) dG(t) − dG(y + t) − G(y)K(u)



β
β
0

·
¸· Z x
¸

n

X

1

ρ
t
−ρ
x
j
j
 +
e K(u − t)dt + K(u)
wj e
Γj (x) − Γj (x + y) ρj



β j=1
0




if 0 < x ≤ u,
with
½
·
¸¾
n
X
1
G(y) +
wj e−ρj x Γj (x) − Γj (x + y)
F (x, y|0) =
1+β
j=1
.
The discounted joint probability density function of U (T −) and |U (T )| can be derived
∂ 2 F (x, y)
from f (x, y|u) =
by (2), K(0) = 1/(1 + β), ρi Γi (x) + Γ0i (x) = −eρi x [e−ρi x Γi (x)]0 =
∂x∂y
∂ −ρi x 0
p(x + y)
P (x)/Ei , and
[e
Γi (x + y)] = e−ρi x
.
∂x
Ei
3
f (x, y|u) =
 Qn

i=1 λi p(x + y)




cn 1 − K(0)


½X
¶
·
¸¾

Z u
n µ Y
n


1

−ρj x ρj u
ρj t

×
e K(u − t)dt ,
e
e − K(u) − ρj



0

j=1 k=1, k6=j ρk − ρj





if 0 ≤ u < x,

Qn




i=1 λi p(x + y)




cn 1 − K(0)


·
½X
¶
¸¾

Z x
n µ Y
n


1

−ρj x ρj x
ρj t

e K(u − x) − K(u) − ρj
×
e
e K(u − t)dt ,



ρk − ρj
0

j=1
k=1,
k6
=
j



if 0 < x ≤ u,
with
Qn
f (x, y|0) =
i=1
cn
λi
p(x + y)
n ·
X
e−ρj x
j=1
n
Y
k=1, k6=j
1
ρk − ρj
¸
which agrees with equation (8.3). Note that f (x, y|u) is a weighted average of n functions
n
n
X
Y
1
with the sum of the weights,
, equal to zero for n ≥ 2.
j=1 k=1, k6=j ρk − ρj
For n=1, F (x, y|u) and f (x, y|u) simplify to equations (2.9) and (2.12) of Tsai (2001),
respectively; for n = 2 and λ1 = λ2 , F (x, y|u) and f (x, y|u) reduce to equations (2.3) and
(2.4) of Tsai and Sun (2004), respectively. With the expression for F (x, y|u), the discounted
marginal distribution functions and probability density functions of U (T −), |U (T )| and
[U (T −) + |U (T )|] can be easily obtained.
References
Lin, X. Sheldon, and Gordon E. Willmot. 1999. ”Analysis of a Defective Renewal Equation
Arising in Ruin Theory”, Insurance: Mathematics and Economics 25: 63-84.
Tsai, Cary Chi-Liang. 2001. ”On the discounted distribution functions of the surplus process
perturbed by diffusion”, Insurance: Mathematics and Economics 28: 401-419.
Tsai, Cary Chi-Liang, and Li-Juan Sun. 2004. ”On the Discounted Distribution Functions
for the Erlang(2) Risk Process”, Insurance: Mathematics and Economics 35: 5-19.
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