University of Waterloo Waterloo, Ontario N2L 3G1 Email: [email protected]
INSTITUTE OF INSURANINSTITUTE OF INSURANCE AND CE AND PENSION RESEARCHPENSION RESEARCH
Gerber-Shiu analysis with a generalized penalty function
Eric C.K. Cheung David Landriault
Gordon E. Willmot Jae-Kyung Woo
08-14
2008 IIPR Reports 2008-01 “On the regulator-insurer-interaction in a structural model”, Carole Bernard and An Chen
2008-02 “Evidence of Cohort Effect on ESRD Patients Based on Taiwan’s NHIRD”, Juang Shing-Her, Chiao Chih-
Hua, Chen Pin-Hsun, and Lin Yin-Ju
2008-03 “Measurement and Transfer of Catastrophic Risks. A Simulation Analysis”, Enrique de Alba, Jesús Zúñiga,
and Marco A. Ramírez Corzo
2008-04 “An Accelerating Quasi-Monte Carlo Method for Option Pricing Under the Generalized Hyperbolic Lévy
Process”, Junichi Imai and Ken Seng Tan
2008-05 “Conditional tail moments of the Exponential Family and its transformed distributions”, Joseph H.T. Kim
2008-06 “Fast Simulation of Equity-Linked Life Insurance Contracts with a Surrender Option”, Carole Bernard and
Christiane Lemieux
2008-07 “The Munich Chain-Ladder Method: A Bayesian Approach”, Enrique de Alba
2008-08 “Projecting Immigration: A Survey of Current Practices”, Ivana Djukic
2008-09 “Basic Living Expenses for the Canadian Elderly”, Bonnie-Jeanne MacDonald, Doug Andrews, and Robert
L. Brown
2008-10 “Competing Views of Social Security Pension Design and Its Impact on Financing”, Robert L. Brown
2008-11 “Perturbed MAP risk models with dividend barrier strategies”, Eric C.K. Cheung and David Landriault
2008-12 “Dependent risk models with bivariate phase-type distributions”, Andrei L. Badescu, Eric C.K. Cheung,
and David Landriault
2008-13 “Analysis of a generalized penalty function in a semi-Markovian risk model”, Eric C.K. Cheung and David
Landriault
2008-14 “Gerber-Shiu analysis with a generalized penalty function”, Eric C.K. Cheung, David Landriault, Gordon
E. Willmot, and Jae-Kyung Woo
Gerber-Shiu analysis with a generalized penalty function
Eric C.K. Cheung, David Landriault, Gordon E. Willmot and Jae-Kyung Woo!
November 16, 2008
Abstract
A generalization of the usual penalty function is proposed, and a defective re-newal equation is derived for the Gerber-Shiu discounted penalty function in theclassical risk model. This is used to derive the trivariate distribution of the deficitat ruin, the surplus prior to ruin, and the surplus immediately following the secondlast claim before ruin. The marginal distribution of the last interclaim time beforeruin is derived and studied, and its joint distribution with the claim causing ruin isderived.
Keywords: defective renewal equation, Dickson-Hipp transform, compound ge-ometric distribution, exponential distribution, mixed Erlang distribution, phase-typedistribution, NBU, NWU, DFR.
Acknowledgement: Support for David Landriault and Gordon E. Willmot fromgrants from the Natural Sciences and Engineering Research Council of Canada is grate-fully acknowleged. Support from the Munich Reinsurance Company is also gratefullyacknowledged by Gordon E. Willmot as is support for Eric C.K. Cheung and Jae-KyungWoo from the Institute for Quantitative Finance and Insurance at the University of Wa-terloo.
!Department of Statistics and Actuarial Science, University of Waterloo, 200 University AvenueWest, Waterloo, Canada. Corresponding Author: Jae-Kyung Woo ([email protected])
1
1 Introduction and background
In the classical compound Poisson risk model, the insurer’s surplus process {U (t) , t ! 0}is defined as
U (t) = u + ct" S (t) ,
where u is the initial surplus, c is the premium rate and S (t) is the aggregate claimamount up to time t. The aggregate claims process {S (t) , t ! 0} is assumed to be acompound Poisson process with
S (t) =
!"
#
N(t)$i=1
Yi, N (t) > 0
0, N (t) = 0,
where the claim number process {N (t) , t ! 0} is a Poisson process with rate ! > 0.That is, let V1 be the time of the first claim, and Vi for i = 2, 3, ..., be the timebetween the (i" 1)-th claim and the i-th claim. Then, {V1, V2, ...} is an independent andidentically distributed sequence of exponential random variables, each with mean 1/!.The claim sizes {Yi}"i=1 are independent and identically distributed random variableswith common density p, cumulative distribution function (cdf) P (·) = 1 " P (·), andLaplace transform %p (s) =
&"0 e#syp (y) dy. The claim number process {N (t) , t ! 0} is
assumed to be independent of the claim size random variables {Yi}"i=1. The positiveloading condition
&"0 yp(y) dy < c/! is also assumed to hold.
Let T be the time to ruin for the surplus process {U (t) , t ! 0} defined by T =inf {t ! 0 : U (t) < 0} with T = # if ruin does not occur (i.e. U (t) ! 0 for all t ! 0).Of interest in ruin theory is the analysis of the Gerber-Shiu discounted penalty function,namely
m (u) = E'e#!T w1
(U
(T#)
, |U (T )|)1 (T <#) |U (0) = u
*, (1)
where " (" ! 0) is a force of interest, w1 is the so-called penalty function and 1 (A)holds for the indicator function of the event A. An operator known as the Dickson-Hippoperator Tr (with Re(r) ! 0) defined as
Trf (y) =+ "
ye#r(x#y)f (x) dx,
for an integrable function f (see e.g. Li and Garrido (2004)) plays a role in the analysisof the expected discounted penalty function m (u). Also, let # = # (") ! 0 be definedby
! + "
c" # =
!
c%p (#) . (2)
Gerber and Shiu (1998) showed that {m (u) , u ! 0} satisfies a defective renewal equationof the form
m (u) = $!
+ u
0m (u" y) f! (y) dy +
!
cT"%1 (u) , (3)
where$! =
!
c
+ "
0e#"yP̄ (y) dy < 1, (4)
2
f! (y) =T"p (y)
T0T"p (0), y ! 0, (5)
and%1 (u) =
+ "
uw1 (u, y " u) p (y) dy. (6)
In this paper, we propose to generalize (1) as follows. Let Rn = u +$n
i=1 (cVi " Yi)for n = 1, 2, ..., and define R0 = u. Then, Rn represents the surplus immediately afterthe n-th claim if n ! 1. Then, RN(T )#1 is the surplus immediately after the penultimate(second last) claim before ruin occurs, with the understanding that RN(T )#1 = u if ruinoccurs on the first claim. Note that RN(T )#1 is not necessarily the minimum value ofthe surplus U (t) before ruin. Therefore, we replace (1) by
m! (u) = E'e#!T w
(U
(T#)
, |U (T )| , RN(T )#1
)1 (T <#) |U (0) = u
*, (7)
and our aim is to study (7) in this paper. In particular, this generalization allows us toanalyze the distribution of the final interclaim time before ruin and its relationship toother ruin-related quantities of interest.
In Section 2, we show that m! (u) satisfies a defective renewal equation of the form(3), i.e. with the same values of $! and f! (y) given by (4) and (5) respectively. Wealso give an expression for the solution to this equation. In Section 3, we obtain thetrivariate distribution of U (T#), |U (T )|, and RN(T )#1 (see also Badescu et al. (2008)),while in Section 4 we study the joint and marginal distributions of the last interclaimtime before ruin occurs and the claim causing ruin (see Dufresne and Gerber (1988)).
2 A defective renewal equation
In order to solve for m! (u), we condition on the time (t) and the amount (y) of the firstclaim. This results in the integral equation
m! (u) =+ "
0e#!t {&! (u + ct) + % (u + ct, u)}!e##tdt, (8)
where
&! (t) =+ t
0m! (t" y) p (y) dy, (9)
and% (t, u) =
+ "
tw (t, y " t, u) p (y) dy (10)
generalizes (6). A change of variables from t to x = u + ct in (8) results in
m! (u) =!
c
+ "
ue#(!+"
c )(x#u) {&! (x) + % (x, u)} dx,
3
which may be expressed as
m! (u) =!
c
,T!+"
c&! (u)
-+ '! (u) , (11)
where'! (u) =
!
c
+ "
ue#
!+"c (x#u)% (x, u) dx. (12)
To solve (11), we will use Laplace transforms (denoted by a “ % ” over the function).Thus, from (9) and (10), (11) may be expressed as
%m! (s) =!
c
%&!
(#+!
c
)" %m! (s) %p (s)
s" #+!c
+ %'! (s) .
Solving for %m! (s) results in
%m! (s).
s" ! + "
c+
!
c%p (s)
/=
!
c%&!
0! + "
c
1+
0s" ! + "
c
1%'! (s) .
Assuming %m!(#) <# and using (2), replacement of s by # results in
!
c%&!
0! + "
c
1=
0! + "
c" #
1%'! (#) ,
from which it follows that
%m! (s).
s" ! + "
c+
!
c%p (s)
/= (s" #) %'! (s) +
0! + "
c" #
1 ,%'! (#)" %'! (s)
-.
But from Eq. (4.4) of Willmot and Dickson (2003), the Laplace transform of (5) satisfies.
s" ! + "
c+
!
c%p (s)
/= (s" #)
,1" $!
%f! (s)-
,
which implies that
%m! (s),
1" $!%f! (s)
-= %'! (s) +
0! + "
c" #
1 %'! (#)" %'! (s)s" #
.
Inversion of this transform relationship results in the defective renewal equation
m! (u) = $!
+ u
0m! (u" y) f! (y) dy + v! (u) , (13)
wherev! (u) = '! (u) +
0! + "
c" #
1T"'! (u) . (14)
The equation (13) reduces to (3) if w (x, y, z) = w1 (x, y). To see this, first note thatf (y) = r {Trf (y)}" d
dyTrf (y). Thus, from (14),
v! (u) =! + "
c{T"'! (u)}" d
duT"'! (u) .
4
But, from (6) and (10), % (x, u) = %1 (x), in turn implying from (12) that '! (u) =(!/c)T!+"
c%1 (u). Therefore,
v! (u) =! + "
c
.T"
2!
cT!+"
c%1 (u)
3/" d
duT"
.!
cT!+"
c%1 (u)
/
=!
c
! + "
c
,T!+"
cT"%1 (u)
-" !
c
d
du
,T!+"
cT"%1 (u)
-
=!
cT"%1 (u) .
In order to avoid di!erentiation of v! (u) and hence w (x, y, z) by virtue of (10) and(12), we introduce the compound geometric density
g! (u) ="4
n=1
(1" $!) ($!)n f!n! (u) , u > 0, (15)
where f!n! (u) is the n-fold convolution of the density f! (u) with itself. We remark(e.g. Willmot and Lin, 2001, Chapter 9) that g! (u) = "G
$! (u) where G! (u) =
E5e#!T 1 (T <#) |U (0) = u
6is a compound geometric tail. The Laplace transform
of (15) is
%g! (s) =+ "
0e#sug! (u) du =
"4
n=1
(1" $!)7$!
%f! (s)8n
=1" $!
1" $!%f! (s)
" (1" $!) ,
and, from (13),
%m! (s) =%v! (s)
1" $!%f! (s)
=%v! (s)1" $!
{%g! (s) + 1" $!} ,
from which it follows that the solution to (13) may be expressed in the well-known form(see e.g. Resnick (1992, Section 3.5))
m! (u) = v! (u) +1
1" $!
+ u
0g! (u" t) v! (t) dt. (16)
For the present analysis, it is convenient to express (16) directly in terms of '! (u) using(14). Thus, (16) may be expressed as
m! (u) = '! (u) +0
! + "
c" #
1+ "
ue#"(z#u)'! (z) dz +
11" $!
+ u
0g! (u" z) '! (z) dz
+#+!
c " #
1" $!
+ u
0g! (u" t)
+ "
te#"(z#t)'! (z) dzdt.
Interchanging the order of integration results in+ u
0g! (u" t)
+ "
te#"(z#t)'! (z) dzdt
=+ u
0'! (z)
+ z
0e#"(z#t)g! (u" t) dtdz +
+ "
u'! (z)
+ u
0e#"(z#t)g! (u" t) dtdz.
5
Therefore,
m! (u) = '! (u) +1
1" $!
+ u
0'! (z)
.g! (u" z) +
0! + "
c" #
1+ z
0e#"(z#t)g! (u" t) dt
/dz
+0
! + "
c" #
1+ "
u'! (z)
.e#"(z#u) +
11" $!
+ u
0e#"(z#t)g! (u" t) dt
/dz.
In other words, the solution to (13) may be expressed as
m! (u) = '! (u) ++ "
0'! (z) (! (u, z) dz, (17)
where
(! (u, z) =
9 11#$"
:g! (u" z) +
(#+!
c " #) & z
0 e#"(z#t)g! (u" t) dt;, z < u
(#+!
c " #) ,
e#"(z#u) + 11#$"
& u0 e#"(z#t)g! (u" t) dt
-, z > u.
(18)
A probabilistic interpretation of (17) is given in the next section.
Note that Equation (22) of Landriault and Willmot (2008) expresses the joint dis-counted density of U(T#) and |U(T )| as p(x + y))!(u, x) where
)! (u, x) =
9#
c(1#$")
& x0 e#"(x#t)g! (u" t) dt, x < u
#c e#"(x#u) + #
c(1#$")
& u0 e#"(x#t)g! (u" t) dt, x > u.
Integrating out y, the marginal discounted density of U(T#) is h!!(x|u) = P̄ (x))!(u, x).Using the above and (2), it is clear that (18) may be expressed as
(! (u, z) =%p(#)h!!(z|u)
P̄ (z)+
g!(u" z)1" $!
1(z < u) .
We remark that when " = 0, it follows that # = 0, $0 = * (0) and g0 (u) = "*$ (u) with* (u) the ruin probability. Thus, (18) may be expressed as
(0 (u, z) =
91
1#%(0)
:#c [* (u" z)" * (u)]" *$ (u" z)
;, z < u
#c
1#%(u)1#%(0) , z > u.
(19)
The function (! (u, z) and its special case (0 (u, z) defined by (18) and (19) are of centralimportance in what follows.
3 The trivariate distribution
In this section, we consider the joint distribution of U (T#) , |U (T )|, and RN(T )#1. Thesame joint distribution has also been studied using a completely di!erent approach byBadescu et al. (2008) in a model with dependency by connecting the risk process to anequivalent fluid flow process. For ruin occuring on the first claim, the joint density ofthe surplus (x) and the deficit (y) was shown by Landriault and Willmot (2008) to be
h1 (x, y |u) =!
ce#
!c (x#u)p (x + y) , x > u, (20)
6
and in this case the time of ruin is (x" u) /c and RN(T )#1 equals u.
For ruin on subsequent claims, there is no such simple functional relationship be-tween these random variables, and we denote the joint density of the time of ruin (t),the surplus before ruin (x), the deficit at ruin (y), and the surplus after the secondlast claim before ruin (z) to be h2 (t, x, y, z |u). See Landriault and Willmot (2008) forfurther discussion of this issue. Let h2,! (x, y, z |u) =
&"0 e#!th2 (t, x, y, z |u) dt be the
discounted density of U (T#) , |U (T )|, and RN(T )#1 for ruin on claims subsequent to thefirst. Using (20), it follows that the Laplace transform of the (defective) distribution ofT , U (T#), |U (T )|, and RN(T )#1 is given by
E'e#!T#sU(T")#v|U(T )|#rRN(T )"11 (T <#) |U (0) = u
*
=!
c
+ "
u
+ "
0e#
!+"c (x#u)#sx#vy#rup (x + y) dydx
++ "
0
+ "
0
+ "
0e#sx#vy#rzh2,! (x, y, z |u) dydxdz.
Stated another way, the trivariate Laplace transform of h2,! (x, y, z |u) is given by+ "
0
+ "
0
+ "
0e#sx#vy#rzh2,! (x, y, z |u) dydxdz
= m! (u)" !
c
+ "
u
+ "
0e#
!+"c (x#u)#sx#vy#rup (x + y) dydx, (21)
where m! (u) is chosen with the penalty function w (x, y, z) = e#sx#vy#rz. With thischoice of w, (10) becomes
% (t, u) = e#st#ru+ "
te#v(y#t)p (y) dy = e#st#ru
+ "
0e#vyp (y + t) dy.
Thus, from (12),
'! (u) =!
c
+ "
ue#
!+"c (x#u)#sx#ru
+ "
0e#vyp (x + y) dydx
=!
c
+ "
u
+ "
0e#
!+"c (x#u)#sx#vy#rup (x + y) dydx. (22)
Therefore, using (17) and (22), (21) becomes+ "
0
+ "
0
+ "
0e#sx#vy#rzh2,! (x, y, z |u) dydxdz
= m! (u)" '! (u)
=+ "
0'! (z) (! (u, z) dz
=!
c
+ "
0(! (u, z)
+ "
z
+ "
0e#
!+"c (x#z)#sx#vy#rzp (x + y) dydxdz
=+ "
0
+ "
0
+ x
0e#sx#vy#rz
.!
ce#
!+"c (x#z)p (x + y) (! (u, z)
/dzdydx,
7
after interchanging the order of integration of x and z. Equating coe"cients of e#sx#vy#rz
results inh2,! (x, y, z |u) =
!
ce#
!+"c (x#z)p (x + y) (! (u, z) , x > z. (23)
The marginal trivariate defective density of U (T#), |U (T )|, and RN(T )#1 for ruin onclaims subsequent to the first is given by (23) with " = 0.
Note that, using (20), an equivalent representation for the discounted densityh2,! (x, y, z |u) is
h2,! (x, y, z |u) = e#"c (x#z)h1 (x, y |z ) (! (u, z) .
One concludes that (! (u, z) is the Laplace transform of the time at which the surplusprocess reaches the level z at the second last claim prior to ruin given an initial surplusof u. Therefore, if we now return to (17), a probabilistic interpretation can be given asfollows. One easily observes from (11) that the term '!(u) from m!(u) is contributed bythe case where ruin occurs upon the first claim. The same '!(u) also appears as the firstterm on the right-hand side of (17), and therefore the second term
&"0 '! (z) (! (u, z) dz
in (17) must be due to ruin occurring as a result of at least two claims. The latterevent can only happen with the surplus process reaching level z from u at the secondlast claim prior to ruin, and this random time has Laplace transform (! (u, z). Afterreaching z from u, the next claim has to cause ruin, and this contributes '!(z) to thefunction m!(u). Since the level z is arbitrary and can go from 0 to #, this explains theintegral
&"0 '! (z) (! (u, z) dz.
The functional form of (23) with respect to both x and y is quite simple. In par-ticular, the marginal bivariate discounted defective density of U (T#) and RN(T )#1 isgiven by
h3,! (x, z |u) =+ "
0h2,! (x, y, z |u) dy
=!
ce#
!+"c (x#z)P (x) (! (u, z) , x > z.
Similarly, the marginal discounted defective density of RN(T )#1 is given by
h4,! (z |u) =+ "
zh3,! (x, z |u) dx
=!
c
,T!+"
cP (z)
-(! (u, z) .
Note also that RN(T )#1 has a discrete mass point at u resulting from ruin on the firstclaim given by
Pr(RN(T )#1 = u
)=
+ "
u
+ "
0h1 (x, y |u) dydx
=!
c
+ "
ue#
!c (x#u)
+ "
0p (x + y) dydx
=!
c
+ "
ue#
!c (x#u)P (x) dx
=!
cT!
cP (u) .
8
4 The last interclaim time and the claim causing ruin
In this section, we demonstrate how the results may be used to identify the distributionof the last interclaim time before ruin, namely VN(T ), as well as its joint distributionwith the claim causing ruin YN(T ). The marginal distribution of YN(T ) was obtained byDufresne and Gerber (1988) and will not be reproduced here.
In terms of U (T#) , |U (T )| , and RN(T )#1, it is clear that VN(T ) = (U (T#) "RN(T )#1)/c and YN(T ) = U (T#) + |U (T )|. As
e#vVN(T )#sYN(T ) = e#(s+ vc )U(T")#s|U(T )|+ v
c RN(T )"1 ,
it follows that the bivariate Laplace transform may be obtained with " = 0 and thepenalty function w (x, y, z) = e#(s+ v
c )x#sy+ vc z. In this case, (10) yields
% (t, u) = e#(s+ vc )t+ v
c u+ "
te#s(y#t)p (y) dy = e#
vc (t#u)
+ "
te#syp (y) dy,
and (12) becomes
'0 (u) =!
c
+ "
ue#
!+vc (x#u)
+ "
xe#syp (y) dydx.
A change of variables from x to t = (x" u)/c results in
'0 (u) =+ "
0e#vt
.!e##t
+ "
u+cte#syp (y) dy
/dt. (24)
The marginal Laplace transform of VN(T ) is obtained with s = 0, in which case (24)reduces to
'0 (u) =+ "
0e#vt
,!e##tP (u + ct)
-dt. (25)
Then, from (17) and (25)
E5e#vVN(T )1 (T <#) |U (0) = u
6
=+ "
0e#vt
,!e##tP (u + ct)
-dt +
+ "
0(0 (u, z)
+ "
0e#vt
,!e##tP (z + ct)
-dtdz
=+ "
0e#vt
.!e##t
2P (u + ct) +
+ "
0P (z + ct) (0 (u, z) dz
3/dt, (26)
from which it follows that the marginal defective density of the last interclaim timeVN(T ) is
h5 (t |u) = !e##t
2P (u + ct) +
+ "
0P (z + ct) (0 (u, z) dz
3, t > 0. (27)
We now consider the proper cdf
H5,u (t) = 1"H5,u (t) = Pr(VN(T ) $ t |T <# , U (0) = u
)
9
of VN(T ) |T <# . Clearly, (26) with v = 0 yields * (u) =&"0 h5 (t |u) dt, and so
H5,u (t) =1
* (u)
+ "
th5 (y |u) dy. (28)
Using (27) followed by some manipulations, (28) becomes
H5,u (t) =&"t !e##y
(P (u + cy) +
&"0 P (z + cy) (0 (u, z) dz
)dy
* (u)
=&"ct
#c e#
!c y
(P (u + y) +
&"0 P (z + y) (0 (u, z) dz
)dy
* (u)
= e##t
&"0
#c e#
!c y
(P (u + y + ct) +
&"0 P (z + y + ct) (0 (u, z) dz
)dy
* (u). (29)
But (10) with w (x, y, z) = 1 results in % (t, u) = P (t), and (12) reduces to
'0 (u) =!
c
+ "
0e#
!c yP (u + y) dy.
With this choice of '0 (u), (17) becomes the ruin probability, i.e.
* (u) =!
c
+ "
0e#
!c yP (u + y) dy +
!
c
+ "
0
.+ "
0e#
!c yP (z + y) dy
/(0 (u, z) dz
=!
c
+ "
0e#
!c y
.P (u + y) +
+ "
0P (z + y) (0 (u, z) dz
/dy. (30)
For any x ! 0, let P x (t) = 1 " Px (t) = P (x + t) /P (x), so that Px (t) is a residuallifetime (or excess loss) cdf. Then combining (29) and (30) results in
H5,u (t) = e##t+ "
0
.au (y) P u+y (ct) +
+ "
0bu (y, z) P z+y (ct) dz
/dy , (31)
where au (y) = #c e#
!c yP (u + y) /* (u) and bu (y, z) = #
c e#!c yP (z + y) (0 (u, z) /* (u)
and so (31) expresses the distribution of VN(T ) |T <# as the minimum of an exponential(with mean 1/!) random variable and an independent random variable whose cdf is amixture over x of the cdf Px (z).
Various conclusions follow in a straightforward manner from (31). First, P x (t) $ 1,implying that H5,u (t) $ e##t, which is to say that VN(T ) |T <# is stochasticallydominated by a generic interclaim time random variable V (i.e.
(VN(T ) |T <#
)$st V ).
This is in agreement with intuition, as (all else equal) one would expect ruin to occurafter a relatively short interclaim time rather than a long one.
Other bounds on H5,u (t) follow from reliability properties of P (y). The cdf P (y) isnew worse (better) than used or NWU (NBU) if P (x + y) ! ($) P (x) P (y) for all x ! 0and y ! 0. The NWU (NBU) class of distributions includes the decreasing (increasing)failure rate or DFR (IFR) class for which P x (y) is nondecreasing (nonincreasing) in x
10
for fixed y. See Barlow and Proschan (1975) for further details. If P (y) is NWU thenP x (t) ! P (t), implying that
e##tP (ct) $ H5,u (t) $ e##t , (32)
whereas if P (y) is NBU then (32) is replaced by
H5,u (t) $ e##tP (ct) . (33)
Also, if P (y) is DFR, then P x (t) is also DFR, and since the DFR property is preservedunder mixing, H5,u (t) is also DFR, as is clear from (31).
In the exponential claim case with P (y) = 1" e#&y, the left-hand side of (32) and(33) are equalities, so that H5,u (t) = 1 " e#(#+c&)t is again an exponential cdf. Moregenerally, if P (y) has the mixed Erlang density (see e.g. Willmot and Woo (2007))
p (y) ="4
j=1
qj' ('y)j#1 e#&y
(j " 1)!, y > 0 ,
then Px (y) also has the mixed Erlang density (see e.g. Willmot and Lin (2001, pp.20-21))
px (t) ="4
j=1
qj (x)' ('t)j#1 e#&t
(j " 1)!,
where
qj (x) =
$"k=j qk
(&x)k"j
(k#j)!$"
i=0
($"k=i+1 qk
) (&x)i
i!
.
We remark that * (u) and *$ (u) are expressible as easily computable damped exponen-tial series (see e.g. Willmot and Woo (2007)) so that evaluation of bu (y, z) is straight-forward (and omitted here). Then the density associated with the cdf 1" e#tH5,u (t) isagain of mixed Erlang form, i.e.
+ "
0
.au (y) pu+y (t) +
+ "
0bu (y, z) pz+y (t) dz
/dy =
"4
j=1
qj (u)' ('t)j#1 e#&t
(j " 1)!,
whereqj (u) =
+ "
0
.au (y) qj (u + y) +
+ "
0bu (y, z) qj (z + y) dz
/dy .
Thus, with Qj (u) =$"
k=j+1 qk (u), it follows (see e.g. Willmot and Lin (2001, p.12))that
+ "
0
.au (y) P u+y (t) +
+ "
0bu (y, z) P z+y (t) dz
/dy = e#&t
"4
j=0
Qj (u)('t)j
j!,
which implies from (31) that
H5,u (t) = e#(#+c&)t"4
j=0
Qj (u)(c't)j
j!.
11
We remark that the mixed Erlang class of distributions is dense in the class of pos-itive continuous distributions (see e.g. Tijms (1994, p.163)). Another generalizationof the exponential distribution is the phase-type class of distributions (see e.g. As-mussen (2000, Chapter VIII)), which is also dense (see e.g. Latouche and Ramaswami(1999, p.55)). When the claim size follows a phase-type distribution with representationPH (a,Q), its density and cdf are given by p (y) = aeQyq and 1 " P (y) = 1 " aeQy1respectively, where 1 is a column vector of ones and q = "Q1. In such case, (27) canbe expressed as
h5 (t |u) = !e##t
2aeQ(u+ct)1 +
+ "
0(0 (u, z)aeQ(z+ct)1dz
3
= !a2eQu +
+ "
0(0 (u, z) eQzdz
3e(cQ##I)t1,
where I is the identity matrix. Therefore, use of (28) leads to
H5,u (t) = a (u) e(cQ##I)t1 , (34)
wherea (u) =
1* (u)
a2eQu +
+ "
0(0 (u, z) eQzdz
37I" c
!Q
8#1.
It can easily be checked from (30) that a (u)1 = 1. It is clear from (34) that VN(T ) |T <#is phase-type distributed with representation PH (a (u) , cQ" !I). Note that the ini-tial probability vector a (u) depends on the function (0 (u, z) which in turn depends onthe ruin probability * (u). An explicit formula for * (u) with phase-type claims canbe found in Asmussen (2000, Chapter VIII, Corollary 3.1) from which evaluation of(0 (u, z) is possible.
Another class of distributions which is also dense in the class of positive continuousdistributions is the class of combinations of exponentials (see e.g. Dufresne (2006)). It isclear from (31) that the mixed distribution with tail e#tH5,u (t) is a di!erent combinationof the same exponentials (with a scale change).
For the joint distribution of VN(T ) and YN(T ), it follows from (24) and (17) that
E5e#vVN(T )#sYN(T )1 (T <#) |U (0) = u
6
=+ "
0
+ "
u+cte#vt#sy
,!e##tp (y)
-dydt
++ "
0
+ "
0
+ "
z+cte#vt#sy
,!e##tp (y) (0 (u, z)
-dydzdt
=+ "
0
+ "
u+cte#vt#sy
,!e##tp (y)
-dydt
++ "
0
+ "
ct
+ y#ct
0e#vt#sy
,!e##tp (y) (0 (u, z)
-dzdydt.
Therefore, the bivariate defective density of VN(T ) and YN(T ) is
h6 (t, y |u) =
9!e##tp (y)
& y#ct0 (0 (u, z) dz, ct < y < u + ct
!e##tp (y),
1 +& y#ct0 (0 (u, z) dz
-, y > u + ct.
Note that h6 (t, y |u) = 0 for y < ct.
12
5 References
• Asmussen, S. 2000. Ruin Probabilities, World Scientific, Singapore.
• Badescu, A.L., Cheung, E.C.K. and Landriault, D. 2008. Dependent risk Modelswith bivariate phase-type distributions. Submitted.
• Barlow, R. and Proschan, F. 1975. Statistical Theory of Reliability and Life Test-ing: Probability Models, Holt, Rinehart, and Winston, New York.
• Dufresne, D. 2006. Fitting combinations of exponentials to probability distribu-tions. Applied Stochastic models in Business and Industry 23(1): 23-48.
• Dufresne, F. and Gerber, H,U. 1988. The surpluses immediately before and atruin, and the amount of the claim causing ruin. Insurance: Mathematics andEconomics 7: 193-199.
• Gerber, H.U. and Shiu, E.S.W. 1998. On the time value of ruin. North AmericanActuarial Journal 2(1): 48-78.
• Landriault, D. and Willmot, G.E. 2008. On the joint distributions of the time toruin, the surplus prior to ruin and the deficit at ruin in the classical risk model.Submitted.
• Latouche, G. and Ramaswami, V. 1999. Introduction to Matrix Analytic Methodsin Statistic Modeling, ASA-SIAM Series on Statistics and Applied Probability,SIAM, Philadelphia PA.
• Li, S. and Garrido, J. 2004. On ruin for the Erlang(n) risk process. Insurance:Mathematics and Economics 34: 391-408.
• Resnick, S. 1992. Adventures in Stochastic Processes, Birkhauser, Boston.
• Tijms, H.C. 1994. Stochastic Models: An Algorithmic Approach, John Wiley,Chichester.
• Willmot, G.E. and Dickson, D.C.M. 2003. The Gerber-Shiu discounted penaltyfunction in the stationary renewal risk model. Insurance: Mathematics and Eco-nomics 32: 403-411.
• Willmot, G.E. and Lin, X.S. 2001. Lundberg Approximations for Compound Dis-tributions with Insurance Applications, Lecture Notes in Statistics 156, Springer-Verlag, New York.
• Willmot, G.E. and Woo, J.K. 2007. On the class of Erlang mixtures with risktheoretic applications . North American Actuarial Journal 11(2): 99-115.
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