University of Waterloo Waterloo, Ontario N2L 3G1 Email: iipr@uwaterloo.ca

INSTITUTE OF INSURANINSTITUTE OF INSURANCE AND CE AND PENSION RESEARCHPENSION RESEARCH

Generalized penalty function analysis with a class of Coxian interclaim time distributions

Gordon E. Willmot Jae-Kyung Woo

08-16

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

2008-15 “Generalized penalty functions in dependent Sparre Andersen models”, Eric C.K. Cheung, David

Landriault, Gordon E. Willmot, and Jae-Kyung Woo

2008-16 “Generalized penalty function analysis with a class of Coxian interclaim time distributions”, Gordon E.

Willmot and Jae-Kyung Woo

Generalized penalty function analysiswith a class of Coxian interclaim time distributions

Gordon E. Willmot and Jae-Kyung Woo!

November 16, 2008

Abstract

Gerber-Shiu analysis with the generalized penalty function proposed by Cheung et al. (2008a) isconsidered in the Sparre Andersen risk model with a Kn family distribution for the interclaim time.A defective renewal equation and its solution for the present Gerber-Shiu function are derived,and their forms are natural for analysis which jointly involves the time of ruin and the surplusimmediately prior to ruin. The results are then used to find explicit expressions for various defectivejoint and marginal densities, including those involving the claim causing ruin and the last interclaimtime before ruin. The case with mixed Erlang claim amounts is considered in some detail.

Keywords: Sparre Andersen risk process, Kn family of distributions, combinations and mixturesof Erlangs, defective renewal equation, Lagrange polynomials

Acknowledgement: Support for Gordon E. Willmot from the Natural Sciences and EngineeringResearch Council of Canada and the Munich Reinsurance Company is gratefully acknowleged, as issupport for Jae-Kyung Woo from the Institute for Quantitative Finance and Insurance at the Universityof Waterloo.

!Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo,Canada. Corresponding Author: Jae-Kyung Woo (jkwoo@math.uwaterloo.ca)

1

1 Introduction and preliminaries

Consider the insurer’s surplus process at time t defined as {Ut; t ! 0} with Ut = u + ct "!Nt

i=1 Yi,and u ! 0 is the initial surplus. The number of claims process {Nt; t ! 0} is assumed to be a renewalprocess, with V1 the time of the first claim and Vi the time between the (i" 1)th and the ith claim fori = 2, 3, 4, . . .. It is assumed that {Vi}"i=1 is an independent and identically distributed (iid) sequenceof positive random variables with common probability density function (pdf) k(t) and distributionfunction (df) K(t) = 1"K(t).

In the present paper, we consider the model of Li and Garrido (2005), whereby k(t) is a pdf fromthe Kn class of densities, and has Laplace transform "k(s) =

#"0 e#stk(t)dt given by

"k(s) =!(s)$m

i=1("i + s)ni(1)

where "i > 0 for i = 1, 2, . . . , m with "i #= "j for i #= j. Also, ni is a nonnegative integer fori = 1, 2, . . . , m, and n = n1 + · · · + nm > 0, while !(s) is a polynomial of degree n " 1 or less (thedenominator of (1) is a polynomial of degree n). In this paper we adopt the notational conventionthat the empty product is 1, and the empty sum is 0. The classical compound Poisson risk model (e.g.Gerber and Shiu, 1998) is recovered in the exponential case with m = n = 1, the Erlang(n) renewalrisk model (Li and Garrido, 2004) with m = 1, and nm = n, and the generalized Erlang renewal riskmodel (Gerber and Shiu, 2005) with ni = 1 for i = 1, 2, . . . , n, and !(s) =

$mi=1 "ni

i in all these cases.As pointed out by Li and Garrido (2005), a partial fraction expression of (1) results in

"k(s) =m%

i=1

ni%

j=1

ai,j

("i + s)j(2)

where

ai,j =1

(ni " j)!dni#j

dsni#j

&'

(

m)

k=1,k $=i

!(s)("k + s)nk

*+

,

-----s=#!i

.

Inversion of (2) results in

k(t) =m%

i=1

ni%

j=1

ai,jtj#1e#!it

(j " 1)!, (3)

and the Kn class may be viewed in terms of finite combinations of Erlangs. Also, it is assumed thatthe claim sizes {Yi}"i=1 with Yi the size of the ith claim are iid positive random variables with pdf p(y),df P (y) = 1"P (y), and Laplace transform "p(s) =

#"0 e#syp(y)dy. Premiums are paid continuously at

rate c, and the positive security loading condition E[Y1] < cE[V1] is assumed to hold.

Let T be the time to ruin defined by T = inf{t ! 0 : U(t) < 0} with T =$ if Ut ! 0 for all t ! 0,and # ! 0 may be viewed as a discount factor. The classical Gerber-Shiu discounted penalty functionis defined (Gerber and Shiu, 1998) by

m",12 (u) = E.e#"T w12 (UT" , |UT |) I (T <$)

---U0 = u/, u ! 0, (4)

where w12(x, y) is the so-called penalty function for x > 0, y > 0, and I(.) is the indicator function. Ifruin occurs, the surplus prior to ruin is UT" and the deficit at ruin is |UT |.

2

In this paper, we study a generalized form of Gerber-Shiu discounted penalty function in (4) whichincludes a new quantity in the penalty function introduced by Cheung et al. (2008a), i.e.

m" (u) = E.e#"T w (UT" , |UT | , RNT#1) I (T <$)

---U0 = u/, u ! 0, (5)

where Rn = u +!n

i=1(cVi"Yi) for n = 1, 2, . . . , and R0 = u. Thus, RNT#1 is the surplus immediatelyafter the second last claim before ruin occurs if NT > 1, and is equal to the initial surplus u if ruinoccurs on the first claim. We may study quantities involving RNT#1 such as the last interclaim timebefore ruin VNT = (UT" " RNT#1)/c, discussed by Cheung et al. (2008a) in the classical compoundPoisson risk model.

Obviously, with w(x, y, v) = 1 in (5),

G" (u) = E.e#"T I (T <$)

---U0 = u/, u ! 0, (6)

and for # = 0 (6) is the ruin probability $(u) = Pr(T <$|U0 = u). We remark that G"(u) = 1"G"(u)is a compound geometric tail and satisfies the defective renewal equation

G"(u) = %"

0 u

0G"(u" y)f"(y)dy + %"

0 "

uf"(y)dy, (7)

and Li and Garrido (2005) have identified %" and the ladder height pdf f"(y).

As was the case in Li and Garrido (2005), the analysis of (5) depends heavily on the Dickson-Hippoperator (e.g. Li and Garrido, 2004) defined for a function h(x) by Trh(x) = erx

#"x e#ryh(y)dy. If

"h(s) =#"0 e#syh(y)dy is the Laplace transform of h(x), the Laplace transform of Trh(x) is given by#"

0 e#sx{Trh(x)}dx = {"h(r)" "h(s)}/(s" r).Also, Lundberg’s (generalized) fundamental equation

"p(s)"k(# " cs) = 1 (8)

is of central importance in the ensuing analysis, and Li and Garrido (2005) showed that (8) has exactlyn roots &1, &2, · · · , &n with nonnegative real part Re(&j) ! 0 in the complex plane. We shall henceforthassume (as did Li and Garrido, 2005) that these roots are distinct, i.e. &i #= &j for i #= j.

It follows from (1) and (2) that

!(# " cs) =

1m)

k=1

("k + # " cs)nk

2m%

i=1

ni%

j=1

ai,j

("i + # " cs)j ,

is still a polynomial in s of degree n" 1 or less. More generally, if 'i,j are constants, then as pointedout by Li and Garrido (2005),

q(s) =

1m)

k=1

("k + # " cs)nk

2m%

i=1

ni%

j=1

'i,j

("i + # " cs)j , (9)

is a polynomial in s of degree n " 1 or less. Therefore, from the theory of Lagrange polynomials, (9)may be reexpressed as

q(s) =n%

i=1

q(&i)

&'

(

n)

j=1,j $=i

s" &j

&i " &j

*+

, . (10)

3

Next, we note that functions of the form

h(u) =0 "

0e#"tr(u + ct)k(t)dt, (11)

where r(x) is a function and k(t) is given by (3), are of interest in later sections. The Laplace transformof (11) is

"h(s) =0 "

0e#su

0 "

0e#"tr(u + ct)k(t)dtdu

=0 "

0e#("#cs)t

30 "

0e#s(u+ct)r(u + ct)du

4k(t)dt

=0 "

0e#("#cs)t

30 "

0e#sxr(x)dx"

0 ct

0e#sxr(x)dx

4k(t)dt.

Thus if the Laplace transform of r(x) is "r(s) =#"0 e#sxr(x)dx, then

"h(s) = "r(s)"k(# " cs)"0 "

0e#sx

10 "

x/ce#("#cs)tk(t)dt

2r(x)dx.

It is straightforward but tedious to show using (3) that the Laplace transform of (11) may then beexpressed as

"h(s) = "r(s)"k(# " cs)"m%

i=1

ni%

j=1

'i,j

("i + # " cs)j , (12)

where

'i,j =ni%

k=j

("1)k#jai,k"r(k#j)5

!i+"c

6

(k " j)!ck#j,

and "r(j)(s) =#"0 ("x)je#sxr(x)dx. Clearly, (12) is completely specified by "k(s) and "r(s). This Laplace

transform relationship is used in Section 2 to derive a defective renewal equation for (5), and to showthat this is a generalization of that obtained by Li and Garrido (2005) for its special case (4). Thisgeneralization, in addition to allowing for analysis involving RNT#1 and related quantities, is in factalso more natural for use in joint analysis involving T and UT" , as is also discussed.

In Section 3, the results of Section 2 are used to obtained the trivariate “discounted” defectivedistribution of UT" , |UT |, and RNT#1. It is instructive to note that this distribution is su!cient todetermine that of UT" , |UT | , RNT#1, and XT where XT = inf0%t<T Ut is the minimum surplus beforeruin (and thus the last ladder height before ruin is XT + |UT |). See Cheung et al. (2008b) for furtherdetails. Joint and marginal distributions of the claim causing ruin (e.g. Dufresne and Gerber, 1988)given by YNT = UT" + |UT |, and the last interclaim time before ruin are also obtained. The specialcase with mixed Erlang claim sizes is considered in Section 4.

2 Defective renewal equations

In this section we demonstrate that (5) satisfies the defective renewal equation

m"(u) = %"

0 u

0m"(u" y)f"(y)dy + v"(u), (13)

4

where %" and f"(y) are the same as in the special case (7) of (13). The function v"(u) will alsobe identified. Much information can be obtained from (13), including solutions for m"(u) which arediscussed further in subsequent sections of the paper.

To begin the analysis, we condition on the time t and the amount y of the first claim, resulting inthe integral equation for m"(u) given by

m" (u) = (" (u) +0 "

0e#"t)" (u + ct) k(t)dt, (14)

where("(u) =

0 "

0e#"t*(u + ct, u)k(t)dt (15)

with*(x, u) =

0 "

xw(x, y " x, u)p(y)dy, (16)

and)" (x) =

0 x

0m" (x" y) p(y)dy. (17)

In order to express (14) in the form (13), we will use Laplace transforms, denoted by ‘%’ above theletter. Note that the right hand term of (14) is of the form (11) with r(x) = )"(x), and thus from (12)and (14),

"m"(s) = "("(s) + ")"(s)"k(# " cs)"m%

i=1

ni%

j=1

'!i,j

("i + # " cs)j (18)

where '!i,j are constants. But ")"(s) = "m"(s)"p(s) from (17), and thus we may rewrite (18) as

"m"(s)7

1" "p(s)"k(# " cs)8

= "("(s)"q!(s)$m

k=1("k + # " cs)nk, (19)

where

q!(s) =

1m)

k=1

("k + # " cs)nk

2m%

i=1

ni%

j=1

'!i,j

("i + # " cs)j

is of the form (9) and is a polynomial of degree n" 1 or less.Therefore, from (8) and (10), q!(s) may be expressed as

q!(s) =n%

i=1

1"("(&i)

m)

k=1

("k + # " c&i)nk

2n)

j=1,j $=i

9s" &j

&i " &j

:, (20)

where it is tacitly assumed that "m"(&i) < $ for i = 1, 2, . . . , n, to ensure that the left side of (19)vanishes when s = &i.

Multiplication of (19) by$m

k=1("k + # " cs)nk and using (20) results in

"m"(s)

1m)

k=1

("k + # " cs)nk

271" "p(s)"k(# " cs)

8

= "("(s)

1m)

k=1

("k + # " cs)nk

2"

n%

i=1

1"("(&i)

m)

k=1

("k + # " c&i)nk

2n)

j=1,j $=i

9s" &j

&i " &j

:.

(21)

5

But from Li and Garrido (2005, equations 19 and 22), the defective density %"f"(y) has Laplacetransform

%""f"(s) = 1"

$mk=1("k + # " cs)nk

cn$n

j=1(&j " s)

71" "p(s)"k(# " cs)

8,

which may be rearranged as1

m)

k=1

("k + # " cs)nk

271" "p(s)"k(# " cs)

8=

71" %"

"f"(s)8

("c)nn)

j=1

(s" &j).

Substitution of this expression into the left side of (21) followed by division of both sides of (21) by("c)n $n

j=1(s" &j) results in

"m"(s)7

1" %""f"(s)

8= "v"(s) (22)

where

"v"(s) = "("(s)$m

k=1(s"!k+"

c )nk

$nj=1(s" &j)

"n%

i=1

"("(&i)$m

k=1(&i " !k+"c )nk

(s" &i)$n

j=1,j $=i(&i " &j). (23)

Inversion of (22) yields (13), and it remains to identify v"(u) by inverting (23).

As$m

k=1(s " ("k + #)/c)nk and$n

j=1(s " &j) are polynomials of degree n with leading coe!cientequal to unity, the polynomial

q0(s) =

1m)

k=1

9s" "k + #

c

:nk2"

n)

j=1

(s" &j)

is of degree n" 1. Because q0(&i) =$m

k=1(&i " ("k + #)/c)nk , (10) yields

q0(s) =n%

i=1

1m)

k=1

9&i "

"k + #

c

:nk2

n)

j=1,j $=i

9s" &j

&i " &j

:.

Therefore, equating these two expressions for q0(s) and rearranging yields the identity$m

k=1

5s" !k+"

c

6nk

$nj=1(s" &j)

= 1 +n%

i=1

$mk=1

5&i " !k+"

c

6nk

(s" &i)$n

j=1,j $=i(&i " &j).

Replacement of the coe!cient of "("(s) in (23) by the right hand side of this expression yields

"v"(s) = "("(s) +n%

i=1

$mk=1

5&i " !k+"

c

6nk

$nj=1,j $=i(&i " &j)

1"("(s)" "("(&i)

s" &i

2,

i.e.,

"v"(s) = "("(s) +n%

i=1

a!i

1"("(&i)" "("(s)

s" &i

2(24)

where

a!i =

$mk=1

5!k+"

c " &i

6nk

$nj=1,j $=i(&j " &i)

. (25)

6

Inversion of (24) yields

v"(u) = ("(u) +n%

i=1

a!i T#i("(u). (26)

The defective renewal equation (13) with v"(u) given by (26) will be used in the next section toanalyze quantities of interest for this model.

The defective renewal equation derived by Li and Garrido (2005) for m",12(u) in (4), while a specialcase of the above result, is expressed directly in terms of the function

*12(x) =0 "

xw12(x, y " x)p(y)dy, (27)

and (27) obviously cannot appear directly in this generalization. It is instructive to note that thepresent generalization is needed for analysis beyond that which involves the surplus prior to ruin UT"

and the deficit at ruin |UT |, including in particular that involving RNT#1 and/or XT as defined inSection 1. Such analysis is in fact the subject matter of Section 3. Moreover, this generalization isactually more natural for use with analysis which includes the surplus prior to ruin UT" and the timeof ruin T , and in particular that which also involves the deficit at ruin |UT |. The reason for this isthat such distributions are functionally di"erent if ruin occurs on the first claim than if ruin occurs onsubsequent claims, due to the fact that UT" = u + cT if ruin occurs on the first claim. As discussedin the next section, the contribution due to ruin on the first claim is captured by the function ("(u),which appears separately in (26). In fact, in Landriault and Willmot (2008), the trivariate distributionof T, UT" , and |UT | in the classical compound Poisson model was derived by analytic inversion of theassociated trivariate Laplace transform, and this e"ectively required that v",12(u) be reexpressed in thegeneralized form (26) in order to identify the separate contribution to the distribution of ruin on thefirst claim. For further discussion of these issues, see Cheung et al. (2008a, 2008b).

For the special case with the penalty function given by w12(x, y) we will now recover the defectiverenewal equation

m",12(u) = %"

0 u

0m",12(u" y)f"(y)dy + v",12(u) (28)

of Li and Garrido (2005) for (4).It is clear from (16) that in this special case ("(u) in (15) reduces to

(",12(u) =0 "

0e#"t*12(u + ct)k(t)dt (29)

with *12(x) given by (27). But (29) is again of the form (11), and then by (12) its Laplace transformmay be expressed as

"(",12(s) = "*12(s)"k(# " cs)"m%

i=1

ni%

j=1

'!!i,j

("i + # " cs)j ,

where '!!i,j are constants. Therefore, using (1) and (9), it follows that

"(",12(s)

1m)

k=1

9s" "k + #

c

:nk2

= "*12(s)q1(s) + q2(s) (30)

whereq1(s) =

!(# " cs)("c)n

(31)

7

and q2(s) are both polynomials of degree n" 1 or less. Hence, using (10), we may write for k = 1, 2,

qk(s) =n%

i=1

qk(&i)n)

j=1,j $=i

9s" &j

&i " &j

:. (32)

Therefore, using (30), (23) becomes

"v",12(s) ="(",12(s)

$mk=1(s"

!k+"c )nk

$nj=1(s" &j)

"n%

i=1

"(",12(&i)$m

k=1(&i " !k+"c )nk

(s" &i)$n

j=1,j $=i(&i " &j)

="*12(s)q1(s) + q2(s)$n

j=1(s" &j)"

n%

i=1

"*12(&i)q1(&i) + q2(&i)(s" &i)

$nj=1,j $=i(&i " &j)

.

But from (32) with k = 2, the terms involving q2(s) cancel in this expression, yielding

"v",12(s) ="*12(s)q1(s)$nj=1(s" &j)

"n%

i=1

"*12(&i)q1(&i)(s" &i)

$nj=1,j $=i(&i " &j)

,

and replacing q1(s) in this expression by the right side of (32) with k = 1 results in

"v",12(s) =n%

i=1

q1(&i)$nj=1,j $=i(&i " &j)

3"*12(s)" "*12(&i)

s" &i

4.

That is, using (31)

"v",12(s) =n%

i=1

bi

3"*12(&i)" "*12(s)

s" &i

4, (33)

wherebi =

!(# " c&i)cn

$nj=1,j $=i(&j " &i)

. (34)

Inversion of (33) yields Li and Garrido’s result (2005), namely (28) with

v",12(u) =n%

i=1

biT#i*12(u). (35)

We remark that in the special case of (1) with !(s) =$m

k=1 "nkk , (34) simplifies to

bi =$m

k=1("k/c)nk

$nj=1,j $=i(&j " &i)

,

and using Equation 4 of Li and Garrido (2004), (35) may be expressed as

v",12(u) =

1m)

k=1

9"k

c

:nk2

T#1T#2 · · ·T#n*12(u).

Also, when w12(x, y) = 1, (27) becomes *12(x) = P (x), and (28) reduces to (7). But this impliesthat v",12(y) = %"

#"y f"(x)dx, which from (35) yields immediately that

%"

0 "

yf"(x)dx =

n%

i=1

biT#iP (y) =n%

i=1

bi

0 "

0e##ixP (x + y)dx.

8

This implies in turn for y = 0 that

%" =n%

i=1

bi

0 "

0e##ixP (x)dx, (36)

and also (by di"erentiating with respect to y) that

f"(y) =1%"

n%

i=1

bi

0 "

0e##ixp(x + y)dx =

1%"

n%

i=1

biT#ip(y). (37)

Furthermore using (1), (34) may be expressed as

bi ="k(# " c&i)

$mk=1

5!k+"

c " &i

6nk

$nj=1,j $=i(&j " &i)

= "k(# " c&i)a!i

where a!i is given by (25). Also, if w(x, y, v) = e#rvw12(x, y), a similar (but more tedious) argumentmay be used to show that v"(u) with Laplace transform (23) may be expressed as

v",12,r(u) = e#run%

i=1

{bi,r[T#i+r*12(u)] + ei,r[T#i+r(",12(u)]} (38)

wherebi,r =

!(# " c&i " cr)cn

$nj=1,j $=i(&j " &i)

and

ei,r =

7$mk=1

5!k+"

c " &i

6nk8"

7$mk=1

5!k+"

c " &i " r6nk

8

$nj=1,j $=i(&j " &i)

,

but the details are omitted. Clearly, (38) reduces to (35) when r = 0.

3 Associated densities

In this section, we derive various joint and marginal densities involving UT" , |UT | and RNT#1 by usingsolutions to m"(u) in (13) with v"(u) given by (26). First, the general solution to (13) is given by (e.g.Resnick (1992, Section 3.5))

m"(u) = v"(u) +1

1" %"

0 u

0g"(u" y)v"(y)dy, (39)

where g"(u) = "G&"(u) is the compound geometric density given by

g" (u) ="%

n=1

(1" %") (%")n f!n" (u) , u ! 0, (40)

9

and f!n" (u) is the density of the n-fold convolution of f"(u). Then from (13) and (26), (39) yields

m"(u) = ("(u) +n%

i=1

a!i

0 "

ue##i(v#u)("(v)dv

+1

1" %"

10 u

0g"(u" v)("(v)dv +

n%

i=1

a!i

0 u

0g"(u" t)

0 "

te##i(v#t)("(v)dvdt

2,

and interchanging the order of integration of v and t gives0 u

0g"(u" t)

0 "

te##i(v#t)("(v)dvdt

=0 u

0("(v)

0 v

0e##i(v#t)g"(u" t)dtdv +

0 "

u("(v)

0 u

0e##i(v#t)g"(u" t)dtdv.

Therefore,

m"(u) = ("(u) +1

1" %"

0 u

0("(v)

1g"(u" v) +

n%

i=1

a!i

0 v

0e##i(v#t)g"(u" t)dt

2dv

+n%

i=1

a!i

0 "

u("(v)

3e##i(v#u) +

11" %"

0 u

0e##i(v#t)g"(u" t)dt

4dv,

which may be written as

m"(u) = ("(u) +0 "

0("(v)+"(u, v)dv, (41)

where

+" (u, v) =

1 11#$!

;g" (u" v) +

!ni=1 a!i

# v0 e##i(v#t)g" (u" t) dt

<, v < u

!ni=1 a!i

7e##i(v#u) + 1

1#$!

# u0 e##i(v#t)g" (u" t) dt

8, v > u.

(42)

Also, for # = 0,

+0(u, v) =

1 11#%(0)

;"$&(u" v)"

!ni=1 a!i

# v0 e##i(v#t)$&(u" t)dt

<, v < u

!ni=1 a!i

7e##i(v#u) " 1

1#%(0)

# u0 e##i(v#t)$& (u" t) dt

8, v > u,

(43)

since %0 = $(0) and g0(u) = "$&(u). In (43), the &i are the roots of (8) when # = 0. Also, (41)may be interpreted probabilistically by regarding ("(u) as the contribution from ruin occurring onthe first claim and

#"0 ("(v)+"(u, v)dv as the contribution for ruin on the other claims. See Cheung

et al. (2008a, Section 3) for further details. Clearly, the classical compound Poisson risk model withK(t) = 1 " e#!t is the special case of the present model with m = n = 1 in (1), and (42) and (43)easily simplify to the result given by Cheung et al. (2008a).

We remark that for v > u, +"(u, v) =!n

i=1 A!i (u)e##i(v#u) where

A!i (u) = a!i

31 +

11" %"

0 u

0e##itg"(t)dt

4=

a!i1" %"

0 u

0#e##itdG"(t).

Therefore, an integral of the form#"0 #(v)+"(u, v)dv (which appears in (41), for example), may be

expressed as 0 "

0#(v)+"(u, v)dv =

0 u

0#(v)+"(u, v)dv +

n%

i=1

A!i (u)T#i#(u).

10

Now, we may express (5) as (Cheung et al., 2008b)

m"(u) =0 "

0

0 "

uw(x, y, u)h1,"(x, y|u)dxdy +

0 "

0

0 "

0

0 "

vw(x, y, v)h2,"(x, y, v|u)dxdydv,

where h1,"(x, y|u) and h2,"(x, y, v|u) are the discounted densities of UT" , |UT |, and RNT#1 for ruin onthe first claim or claims subsequent to the first respectively. Using (3), we have

h1," (x, y |u) =1cp(x + y)

m%

i=1

ni%

j=1

ai,j

=x#u

c

>j#1e#(!i+")(x"u

c )

(j " 1)!, x > u. (44)

A change of variable from t to x = u + ct in (15), together with (16) yields

("(u) =0 "

0

0 "

uw(x, y, u)h1,"(x, y|u)dxdy, (45)

and thusm"(u) = ("(u) +

0 "

0

0 "

0

0 "

vw(x, y, v)h2,"(x, y, v|u)dxdydv.

Then it is clear by comparison of the right side of the above equation and (41) that0 "

0

0 "

0

0 "

vw(x, y, v)h2,"(x, y, v|u)dxdydv =

0 "

0("(v)+"(u, v)dv.

By (45), comparing coe!cients of w(x, y, v) results in the defective joint density of UT" , |UT |, andRNT#1, namely

h2,"(x, y, v|u) = h1,"(x, y|v)+"(u, v), x > v, (46)

where h1,"(x, y|u) and +"(u, v) are given by (44) and (42) respectively. Also, integrating out y in (46)yields the marginal bivariate discounted defective density of UT" and RNT#1, namely

h3,"(x, v|u) =1cP (x)+"(u, v)

m%

i=1

ni%

j=1

ai,j

=x#v

c

>j#1e#(!i+")(x"v

c )

(j " 1)!, x > v.

We remark that from these results, the joint distribution of UT" , |UT | , XT , and RNT#1 may bederived (see Cheung et al., 2008b). In fact, if we generalize m"(u) to

m!" (u) = E

.e#"T w! (UT" , |UT | , XT , RNT#1) I (T <$)

---U0 = u/,

then m!"(u) satisfies the defective renewal equation

m!"(u) = %"

0 u

0m!

"(u" y)f"(y)dy + v!" (u),

where %" and f"(y) are given by (36) and (37) respectively, and

v!" (u) =0 "

u

0 "

0w!(x + u, y " u, u, u)h1,"(x, y|0)dxdy

+0 "

u

0 "

0

0 x

0w!(x + u, y " u, u, v + u)h2,"(x, y, v|0)dvdxdy,

11

with h1,"(x, y|u) and h2,"(x, y, v|u) in turn given by (44) and (46) respectively.

We next turn our attention to the joint distribution of the last interclaim time before ruin andthe claim causing ruin, and also their marginal distributions. In the classical compound Poisson riskmodel, these results were studied by Cheung et al. (2008a) and are thus generalized here. SinceVNT = (UT" " RNT#1)/c and YNT = UT" + |UT |, we get the bivariate Laplace transform of VNT

and YNT with w(x, y, v) = e#s1((x#v)/c)#s2(x+y) in (5). In this case, from (16) we get *(u + ct, u) =e#s1t

#"u+ct e#s2yp(y)dy, and from (15),

("(u) =0 "

0

0 "

u+cte#s1t#s2y

7e#"tk(t)p(y)

8dydt, (47)

where k(t) is given by (3). Thus from (41) and (47)

E.e#"T#s1VNT

#s2YNT 1 (T <$)---U0 = u

/=

0 "

0

0 "

0e#s1t#s2yh4,"(t, y|u)dydt,

where

h4," (t, y |u) = p(y)m%

i=1

ni%

j=1

ai,jtj#1e#(!i+")t

(j " 1)!

?I(y > u + ct) + I(y > ct)

0 y#ct

0+"(u, v)dv

@. (48)

The marginal Laplace transform of VNT is obtainable with s2 = 0, in which case (47) becomes

("(u) =0 "

0e#s1t#"tk(t)P (u + ct)dt,

and from (3) and (41) it follows that

E.e#"T#s1VNT I (T <$)

---U0 = u/

=0 "

0e#s1th5,"(t|u)dt

where

h5,"(t|u) =m%

i=1

ni%

j=1

ai,jtj#1e#(!i+")t

(j " 1)!

?P (u + ct) +

0 "

0P (v + ct)+"(u, v)dv

@, t > 0. (49)

Similarly, for YNT with s1 = 0, interchanging the order of integration in (47) yields

("(u) =0 "

0e#s2yK"

9y " u

c

:I(y > u)p(y)dy

where K"(t) =# t0 e#"xk(x)dx is a discounted df. Thus from (41)

E.e#"T#s2YNT I (T <$)

---U0 = u/

=0 "

0e#s2yh6,"(y|u)dy

whereh6,"(y|u) = p(y)

?K"

9y " u

c

:I(y > u) +

0 y

0K"

9y " v

c

:+"(u, v)dv

@. (50)

12

To evaluate K"(t), if follows from (1) and (3) that

K"(t) = "k(#)"0 "

te#"xk(x)dx

=!(#)$m

i=1("i + #)ni"

m%

i=1

ni%

j=1

ai,j

0 "

t

xj#1e#(!i+")x

(j " 1)!dx

=!(#)$m

i=1("i + #)ni"

m%

i=1

e#(!i+")tni%

j=1

ai,j

j#1%

k=0

tk

k!("i + #)j#k,

i.e.,

K"(t) =!(#)$m

i=1("i + #)ni"

m%

i=1

e#(!i+")tni#1%

k=0

tk

k!

ni%

j=k+1

ai,j

("i + #)j#k. (51)

Thus, (51) may be substituted into (50).

We note that the proper pdfs corresponding to (48), (49), and (50) may be obtained by appropriatenormalization.

In the next section we consider evaluation of +"(u, v) and these densities for the class of mixedErlang claim size distributions.

4 Mixed Erlang claim amounts

Let

,k(y) =-(-y)k#1e#&y

(k " 1)!, y > 0, (52)

be the Erlang-k pdf for k = 1, 2, . . . , and the claim amount pdf be that of an Erlang mixture, i.e.,

p(y) ="%

k=1

qk,k(y), (53)

where {qk; k = 1, 2, . . .} is a discrete probability distribution. The mixed Erlang class of distributionswith density (53) is very large and includes (among others) the exponential distribution with q1 = 1,the Erlang(n) distribution with qn = 1, and the generalized Erlang distribution (e.g. Gerber and Shiu(2005)). In fact, many Kn pdf’s in (3) may be expressed in the form (53), even with m > 1 (i.e.di"erent "i’s). See Willmot and Woo (2007) for further details.

Then, using the fact that

,k(x + y) =1-

k#1%

j=0

,k#j(x),j+1(y), (54)

it was shown in Willmot (2007, Example 3.2) that G"(u) = -#1 !"k=0 Qk,",k+1(u), u ! 0, where

Qk," =!"

j=k+1 qj,", and hence that the compound geometric pdf g"(u) given by (40) is a di"erent

13

mixture of the same Erlang pdf’s, that is

g"(u) ="%

k=1

qk,",k(u), u > 0, (55)

with {qk,"; k = 0, 1, 2, . . .} a discrete compound geometric distribution. For the exponential distributionwith q1 = 1 in (53), (55) holds (trivially) with qk," = (1"%")%k

" . The associated probability generatingfunction is

Q"(z) ="%

k=0

qk,"zk =

1" %"

1" %"!"

j=1 .j,"zj, (56)

where using Willmot (2007), it follows that {.j,"; j = 1, 2, 3, . . .} is a probability distribution given by

.j," =1

-%"

0 "

0

1 "%

k=1

qj+k#1,k(x)

21n%

i=1

bie##ix

2dx

=1

-%"

"%

k=1

qj+k#1

n%

i=1

bi

0 "

0e##ix,k(x)dx

=1

-%"

"%

k=1

qj+k#1

n%

i=1

bi

9-

- + &i

:k

.

Note also that because

P (x) =1-

"%

k=0

Qk,k+1(x) (57)

where Qk =!"

j=k+1 qj , it follows from (36) that

%" =1-

n%

i=1

bi

"%

k=0

Qk

0 "

0e##ix,k+1(x)dx

=1-

n%

i=1

bi

"%

k=0

Qk

9-

- + &i

:k+1

=n%

i=1

bi

- + &i

"%

k=0

Qk

9-

- + &i

:k

.

But from Feller (1968, p.265), this may also be expressed as

%" =n%

i=1

bi

&i

11"

"%

k=1

qk

9-

- + &i

:k2

.

We now consider evaluation of +"(u, v) in (42). First assume that v & u, and thus0 v

0e##i(v#t),k(u" t)dt = -k

0 v

0e##i(v#t) (u" t)k#1e#&(u#t)

(k " 1)!dt.

14

A change in the variable of integration from t to x = u" t results in0 v

0e##i(v#t),k(u" t)dt

= -ke##i(v#u)0 u

u#v

xk#1e#(&+#i)x

(k " 1)!dx

= -ke##i(v#u)k%

j=1

(- + &i)j#k#1

(j " 1)!

7(u" v)j#1e#(&+#i)(u#v) " uj#1e#(&+#i)u

8

=k%

j=1

-k#j

(- + &i)k+1#j

1-j(u" v)j#1e#&(u#v)

(j " 1)!" e##iv -juj#1e#&u

(j " 1)!

2.

In other words, using (52), it follows for v & u that

0 v

0e##i(v#t),k(u" t)dt =

k%

j=1

-k#j

(- + &i)k+1#j

;,j(u" v)" e##iv,j(u)

<.

Therefore, using (55) again with v & u,0 v

0e##i(v#t)g"(u" t)dt =

"%

k=1

qk,"

0 v

0e##i(v#t),k(u" t)dt

="%

k=1

qk,"

k%

j=1

-k#j

(- + &i)k+1#j

;,j(u" v)" e##iv,j(u)

<

="%

j=1

;,j(u" v)" e##iv,j(u)

< "%

k=j

qk,"-k#j

(- + &i)k+1#j.

Multiplication by a!i /(1" %") and a change in the index of summation yields

a!i1" %"

0 v

0e##i(v#t)g"(u" t)dt

=a!i

1" %"

"%

j=0

&'

(

"%

k=j+1

qk,"-k#j#1

(- + &i)k#j

*+

,;,j+1(u" v)" e##iv,j+1(u)

<.

For notational convenience, we introduce the function

/i(x) ="%

j=0

/i,j,j+1(x), (58)

where

/i,j =a!i

1" %"

(- + &i)j

-j+1

"%

k=j+1

qk,"

9-

- + &i

:k

, (59)

and we may thus write

a!i1" %"

0 v

0e##i(v#t)g"(u" t)dt = /i(u" v)" e##iv/i(u), v & u. (60)

15

If v > u, using (60) with v = u yields

a!i1" %"

0 u

0e##i(v#t)g"(u" t)dt =

a!i e##i(v#u)

1" %"

0 u

0e##i(u#t)g"(u" t)dt

= e##i(v#u);/i(0)" e##iu/i(u)

<

= /i(0)e##i(v#u) " e##iv/i(u).

But ,1(0) = - and ,j(0) = 0 for j = 2, 3, . . . , from (52). Hence from (56), (58), and (59),

/i(0) = -/i,0 =a!i

1" %"

3Q"

9-

- + &i

:" q0,"

4= a!i

&'

(Q"

5&

&+#i

6

1" %"" 1

*+

, ,

and thus for v > u,

a!i1" %"

0 u

0e##i(v#t)g"(u" t)dt = a!i

&'

(Q"

5&

&+#i

6

1" %"" 1

*+

, e##i(v#u) " e##iv/i(u).

Thus if v > u then from (42)

+"(u, v) =n%

i=1

A

Ba!i e##i(v#u) + a!i

&'

(Q"

5&

&+#i

6

1" %"" 1

*+

, e##i(v#u) " e##iv/i(u)

C

D

=n%

i=1

a!i Q"

5&

&+#i

6

1" %"e##i(v#u) "

n%

i=1

e##iv/i(u).

Therefore, using (55), (58), and (60), (42) may be expressed as

+" (u, v) =3 !"

j=0 Aj,j+1(u" v)"!n

i=1 /i(u)e##iv, v < u!ni=1 Bie##i(v#u) "

!ni=1 /i(u)e##iv, v > u,

(61)

where

Aj =qj+1,"

1" %"+

n%

i=1

ni,j , (62)

and

Bi =a!i Q"

5&

&+#i

6

1" %". (63)

We remark that (61) is a simple function of v both for v < u and for v > u, and also that integralsinvolving ,j(x) such as those in this section are straightforward. In particular, if ct < y < u + ct, then

0 y#ct

0+"(u, v)dv =

"%

j=0

Aj

0 y#ct

0,j+1(u" v)dv "

n%

i=1

/i(u)0 y#ct

0e##ivdv

="%

j=0

Aj

0 u

u+ct#y,j+1(x)dx"

n%

i=1

/i(u)

11" e##i(y#ct)

&i

2.

16

As 0 "

y,j(x)dx =

1-

j#1%

k=0

,k+1(y), y ! 0, (64)

it follows that0 y#ct

0+"(u, v)dv =

1-

"%

j=0

Aj

j%

k=0

{,k+1(u + ct" y)" ,k+1(u)}"n%

i=1

/i(u)&i

71" e##i(y#ct)

8

and interchanging the order of summation yields

0 y#ct

0+"(u, v)dv =

1-

"%

k=0

E

F"%

j=k

Aj

G

H {,k+1(u + ct" y)" ,k+1(u)}"n%

i=1

/i(u)&i

71" e##i(y#ct)

8.

But /i(u) =!"

k=0 /i,k,k+1(u) from (58), and thus if ct < y < u + ct,

0 y#ct

0+"(u, v)dv =

1-

"%

k=0

E

F"%

j=k

Aj

G

H ,k+1(u + ct" y)

""%

k=0

&'

(1-

E

F"%

j=k

Aj

G

H +n%

i=1

/i,k

&i

51" e##i(y#ct)

6*+

, ,k+1(u). (65)

If y > u + ct, then0 y#ct

0+"(u, v)dv =

0 u

0+"(u, v)dv +

0 y#ct

u+"(u, v)dv

=

&'

(

"%

j=0

Aj

0 u

0,j+1(u" v)dv

*+

, +

1n%

i=1

Bi

0 y#ct

ue##i(v#u)dv

2"

n%

i=1

/i(u)0 y#ct

0e##ivdv

=

&'

(

"%

j=0

Aj

0 u

0,j+1(x)dx

*+

, +

1n%

i=1

Bi

0 y#u#ct

0e##ixdx

2"

n%

i=1

/i(u)&i

71" e##i(y#ct)

8,

and using (64), it follows that

0 y#ct

0+"(u, v)dv =

1-

"%

j=0

Aj

11"

j%

k=0

,k+1(u)

2

+n%

i=1

Bi

&i

71" e##i(y#u#ct)

8"

n%

i=1

/i(u)&i

71" e##i(y#ct)

8.

Again, using (58), it follows that for y > u + ct,0 y#ct

0+"(u, v)dv =

1-

"%

j=0

Aj +n%

i=1

Bi

&i

71" e##i(y#u#ct)

8

""%

k=0

&'

(1-

E

F"%

j=k

Aj

G

H +n%

i=1

/i,k

&i

51" e##i(y#ct)

6*+

, ,k+1(u). (66)

17

The joint density h4,"(t, y|u) may be evaluated by substitution of (65) or (66), as appropriate, into(48).

Next, we turn to the density h5,"(t|u) in (49). It follows from (54) and (57) that

P (x + y) =1-2

"%

k=0

Qk

k%

j=0

,k+1#j(y),j+1(x)

=1-2

"%

j=0

"%

k=j

Qk,j+1(x),k#j+1(y),

i.e.,

P (x + y) =1-2

"%

j=0

"%

k=0

Qj+k,j+1(x),k+1(y). (67)

Using (67), the integral in (49) may be expressed as

0 "

0P (v + ct)+"(u, v)dv =

1-2

"%

j=0

,j+1(ct)"%

k=0

Qj+k

0 "

0,k+1(v)+"(u, v)dv. (68)

Nothing that# u0 ,i(v),j(u"v)dv = ,i+j(u) because the Erlang distribution is closed under convolution,

and that

T#i,k+1(u) =0 "

ue##i(v#u),k+1(v)dv =

1-

k%

j=0

9-

- + &i

:k+1#j

,j+1(u),

it follows from (61) that0 "

0,k+1(v)+"(u, v)dv

="%

j=0

Aj

0 u

0,k+1(v),j+1(u" v)dv +

n%

i=1

BiT#i,k+1(u)"n%

i=1

/i(u)0 "

0e##iv,k+1(v)dv

="%

j=0

Aj,j+k+2(u) +1-

n%

i=1

Bi

k%

j=0

9-

- + &i

:k+1#j

,j+1(u)"n%

i=1

/i(u)9

-

- + &i

:k+1

.

But /i(u) =!"

j=0 /i,j,j+1(u) from (58), and hence

0 "

0,k+1(v)+"(u, v)dv =

"%

j=k+1

Aj#k#1,j+1(u)

+k%

j=0

11-

n%

i=1

Bi

9-

- + &i

:k+1#j2

,j+1(u)""%

j=0

1n%

i=1

/i,j

9-

- + &i

:k+12

,j+1(u).

That is, 0 "

0,k+1(v)+"(u, v)dv =

"%

j=0

Cj,k,j+1(u), (69)

18

where for j = 0, 1, 2, . . . , k, using (59) and (63),

Cj,k =1-

n%

i=1

Bi

9-

- + &i

:k+1#j

"n%

i=1

/i,j

9-

- + &i

:k+1

=n%

i=1

a!i1" %"

9-

- + &i

:k+1&'

((- + &i)j

-j+1Q"

9-

- + &i

:" (- + &i)j

-j+1

"%

l=j+1

ql,"

9-

- + &i

:l*+

,

=n%

i=1

a!i1" %"

-k#j

(- + &i)k+1#j

j%

l=0

ql,"

9-

- + &i

:l

.

For j = k + 1, k + 2, . . . , using (59) and (62),

Cj,k = Aj#k#1 "n%

i=1

/i,j

9-

- + &i

:k+1

=qj#k,"

1" %"+

n%

i=1

1/i,j#k#1 "

9-

- + &i

:k+1

/i,j

2

=qj#k,"

1" %"+

n%

i=1

a!i1" %"

(- + &i)j#k#1

-j#k

j%

l=j#k

ql,"

9-

- + &i

:l

.

Therefore the integral in (68) may be evaluated using (69). The density (49) may then in turn evaluatedusing (67) with x = ct and y = u, and by substitution of (68).

Finally, we note that evaluation of the integral in (50) and hence the density h6,"(y|u) is straight-forward but tedious using (51) and (61). The same is true of evaluation of each of the defective df’scorresponding to integrals of each (48), (49), and (50). The details are omitted.

References

[1] Cheung, Eric C.K., David Landriault, Gordon E. Willmot and Jae-Kyung Woo.2008a. Gerber-Shiu analysis with a generalized penalty function. Submitted.

[2] Cheung, Eric C.K., David Landriault, Gordon E. Willmot and Jae-Kyung Woo.2008b. Generalized penalty functions in dependent Sparre Andersen models. Submitted.

[3] Dufresne, Francois and Hans U. Gerber. 1988. The surpluses immediately before and atruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics 7(3):193-199.

[4] Feller, William. 1968. An Introduction to Probability Theory and Its Applications, Vol 1, 3rdedition, John Wiley, New York.

[5] Gerber, Hans U. and Elias S.W. Shiu. 1998. On the time value of ruin. North AmericanActuarial Journal 2(1): 48-78.

[6] Gerber, Hans U. and Elias S.W. Shiu. 2005. The time value of ruin in a Sparre Andersenmodel. North American Actuarial Journal 9(2): 49-84.

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[7] Landriault, David and Gordon E. Willmot. 2008. On the joint distributions of the timeto ruin, the surplus prior to ruin and the deficit at ruin in the classical risk model. Submitted.

[8] Li, Shuanming and Jose Garrido. 2004. On ruin for the Erlang(n) risk process. Insurance:Mathematics and Economics 34(3): 391-408.

[9] Li, Shuanming and Jose Garrido. 2005. On a general class of renewal risk process: analysisof the Gerber-Shiu function. Advances in Applied Probability 37(3): 836-856.

[10] Resnick, Sidney 1992. Adventures in Stochastic Processes, Birkhauser, Boston.

[11] Willmot, Gordon E. 2007. On the discounted penalty function in the renewal risk model withgeneral interclaim times. Insurance: Mathematics and Economics 41(1): 17-31.

[12] Willmot, Gordon E. and Jae-Kyung Woo. 2007. On the class of Erlang mixtures with risktheoretic applications . North American Actuarial Journal 11(2): 99-115.

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