A direct approach to the discounted penalty function

Abstract

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closed-form expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step ap-proach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al.

Full text

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420
A DIRECT APPROACH TO THE DISCOUNTED
PENALTY FUNCTION
Hansjo ¨rg Albrecher*, Hans U. Gerber†, and Hailiang Yang‡
ABSTRACT
This paper provides a new and accessible approach to establishing certain results concerning the
discounted penalty function. The direct approach consists of two steps. In the first step, closed-
form expressions are obtained in the special case in which the claim amount distribution is a
combination of exponential distributions. A rational function is useful in this context. For the
second step, one observes that the family of combinations of exponential distributions is dense.
Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus
process has downward and upward jumps, modeled by two independent compound Poisson
processes. If the distribution of the upward jumps is exponential, a series of new results can be
obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W.
Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step ap-
proach is also applied when an independent Wiener process is added to the surplus process.
Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational
and Applied Mathematics 233: 1773–1784 (2010)], which uses different methods.
1. INTRODUCTION
This paper provides a new and accessible approach to establishing certain results concerning the dis-
counted penalty function. The method consists of two steps. In the first step, results are derived for
the case where the claim amount distribution is a combination of exponential distributions. A rational
function is a handy tool in this context. The second step is based on the observation that any claim
amount distribution can be obtained as a limit of a sequence of combinations of exponential distri-
butions. Thus it suffices to translate in general terms the results of the first step, in order to obtain
results for an arbitrary claim amount distribution. The approach of this paper is partly inspired by
Dufresne and Gerber (1989), which features the first step in the special case of the probability of ruin
function.
The paper considers a model for the surplus process with downward and upward jumps, given by two
independent compound Poisson processes. It is noted that the classical model with deterministic pre-
miums can be retrieved as a limit. The two-step approach is particularly fruitful in the case in which
the upward jumps are exponentially distributed. The corresponding results are believed to be new, and
several of the results in Gerber and Shiu (1998) can be found as limits. However, it should be mentioned
that in Gerber and Shiu, the penalty may also depend on the surplus prior to ruin, which is not the
case in this paper. In the last section, the two-step approach is applied to the model, where an inde-
* Hansjo ¨rg Albrecher, PhD, is a Professor of Actuarial Science, Faculty of Business and Economics, University of Lausanne, CH-1015 Lausanne,
Switzerland, hansjoerg.albrecher@unil.ch.
†Hans U. Gerber, ASA, PhD, Distinguished Visiting Professor at the University of Hong Kong, is an Honorary Professor of Actuarial Science,
Faculty of Business and Economics, University of Lausanne, CH-1015 Lausanne, Switzerland, hgerber@unil.ch.
‡Hailiang Yang, ASA, PhD, is a Professor in the Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong
Kong, hlyang@hkusua.hku.hk.

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A DIRECT APPROACH TO THE DISCOUNTED PENALTY FUNCTION
421
pendent Wiener process is added. Again, some of the results are new, and some are somewhat related
to results of Zhang et al. (2010), which uses different methods.
In the literature, sophisticated methods have been developed and applied, such as renewal theory,
Laplace transforms, and deep analytical tools. In contrast, the two-step approach of this paper appears
simple.
In the following, we give an incomplete account of the literature of two-sided jump models. Ruin
theory for a compound Poisson risk model with two-sided jumps is a classical object of study; see, for
instance, Segerdahl (1939) and Crame ´r (1955). Boucherie and Boxma (1996) noticed in a queueing
context that upward jumps can be interpreted as an increase of interarrival times (during which the
premiums are collected with constant intensity) in a risk model with negative jumps only. Hence,
quantities that are invariant with respect to scaling of the time axis can equivalently be obtained from
the corresponding renewal model with appropriately adjusted interarrival times and negative jumps. In
particular, a Pollaczek-Khintchine formula for the ruin probability for the model with two-sided jumps
is obtained in Boucherie et al. (1997). If one is interested in time-dependent quantities such as the
time of ruin, the analysis is usually more delicate. Based on martingale techniques, some first-exit
problems for compound Poisson processes and, more generally, for Le ´vy processes with two-sided jumps
were studied recently by, among others, Perry et al. (2002), Kou and Wang (2003), Jacobsen (2005),
and Xing et al. (2008) under certain types of assumptions on the jump distributions. For related results
on explicit Wiener-Hopf factorization for classes of Le ´vy models with two-sided jumps and applications
to mathematical finance, see, for example, Levendorskii (2004), Lewis and Mordecki (2008), and As-
mussen et al. (2008). Cai et al. (2009) apply results on two-sided exponential jumps to the pricing of
perpetual American put options.
Risk models without a deterministic premium component, where both the premium income and the
aggregate claim process are modeled by (independent) compound Poisson processes, recently were
investigated in several papers. Temnov (2004) compares the resulting ruin probabilities with the ones
of the classical risk model. Defective renewal equations for the discounted penalty function in such
models are studied in Bao (2006), Labbe ´ and Sendova (2009), and Zhang et al. (2010); see also
Schmidli (2010) for a general approach using a change of measure. For an extension to renewal models
see Zhang and Yang (2010).
2. THE TWO-SIDED JUMPS MODEL
Let U(t) denote the surplus of a company at time t. We model the surplus process as
U(t) ? u ? S(t),t ? 0.
Here u ? 0 is the initial surplus, and {S(t)} is a compound Poisson process with positive and negative
jumps. We prefer the alternative but mathematically equivalent formulation
U(t) ? u ? S (t) ? S (t),t ? 0.(2.1)
12
Here {S1(t)} and {S2(t)} are independent compound Poisson processes, each with positive jumps only.
The first represents the aggregate claims; it is given by the Poisson parameter ? and the probability
density function p(x), x ? 0. The second is given by the Poisson parameter ? and the probability density
function q(x), x ? 0. We interpret S2(t) as the aggregate income, which includes premiums and gains.
We prefer not to have a term with deterministic premium income ct in (2.1). This facilitates some of
the calculations, and the case with deterministic premiums always can be retrieved as a limiting case.
We introduce the discounted penalty function ?(u) in this model. The penalty at ruin is given by a
penalty function w(x), x ? 0: If the deficit at ruin is x, the penalty is w(x). In the following, ? ? 0 is
a constant force for discounting the penalty, and T is the time of ruin. Then
??T
?(u) ? E[ew(?U(T))I(T ? ?)?U(0) ? u]

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422NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 4
is the expectation of the discounted penalty at ruin, considered as a function of the initial surplus u.
Let h ? 0. By distinguishing according to the time and the amount of the first jump before time h
(if there is such a jump), we see by conditioning that
?(u) ? ?? e
? ?(u ? x)p(x) dx dt ? ?? e
00
? ?? e
? ?(u ? x)q(x) dx dt ? e
00
huh
?
?(?????)t
?(?????)t
? w(x ? u)p(x) dx dt
u0
h
?
?(?????)t
?(?????)h
?(u). (2.2)
We differentiate this equation with respect to h and set h ? 0 in the resulting equation. This yields
the equation
?? ?(u ? x)p(x) dx ? ?? w(x ? u)p(x) dx ? ?? ?(u ? x)q(x) dx ? (? ? ? ? ?)?(u) ? 0
0u
u
??
0
(2.3)
(see also eq. [4.2] of Labbe ´ and Sendova 2009). The function ?(u) can be characterized as the unique
solution of this integral equation. To see this, consider the mapping
? ?(u ? x)p(x) dx ?
0
? ? ? ? ?? ? ? ? ?
u
?
??
?(u) →
? w(x ? u)p(x) dx
u
?
?
?
? ?(u ? x)q(x) dx,
0
? ? ? ? ?
which is a contraction and has a unique fixed point. This is based on the Contraction Mapping Theorem,
which can be found in many textbooks, for example, Burden and Faires (1989).
3. COMBINATION OF EXPONENTIALS
We make the additional assumption that the claim size distribution is a combination of n exponential
distributions,
?
i?1
n
?? x
i
p(x) ?
A? e
i
,x ? 0,(3.1)
i
where ?1? ?2? ? ? ? ? ?nand A1? ? ? ? ? An? 1. This differs from a mixture, because some of the
Aivalues can be negative as long as p(x) ? 0. For the time being, no restriction is imposed on q(x).
Then the discounted penalty function is of the form
?
k?1
n
?r u
k
?(u) ?
C e
k
,u ? 0.(3.2)
Indeed, the function (3.2) satisfies equation (2.3), if r1, . . . , rn, C1, . . . , Cnare properly chosen. To
obtain the conditions for these 2n coefficients, we substitute (3.1) and (3.2) in (2.3). In the resulting
equation, we compare the coefficients ofto see that r1, . . . , rnmust be solutions of the equatione
? ?? e
?
0
? ? r
i?1
i
?r u
k
n
?
?i
?rx
?
Aq(x) dx ? (? ? ? ? ?) ? 0.(3.3)
i
This is a generalized Lundberg’s equation. The proof in Zhang et al. (2010, Section 2) can be adapted
to show that (3.3) has exactly n solutions with a positive real part (the latter property is neces-
sary in (3.2) because ?(u) → 0 for u → ?). One of these solutions is real valued, say, r1? R, with
0 ? R ? ?1. The other n ? 1 solutions r2, . . . , rnhave a real part exceeding R. Hence, the first

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A DIRECT APPROACH TO THE DISCOUNTED PENALTY FUNCTION
423
term of the sum in (3.2) is dominating for u → ?. We exclude the unlikely case where some of the rk
coincide. A comparison of the coefficients of yields the conditione
?? ? r
?
k?1
iki
?? u
i
n
C
?
ki
?
,i ? 1, . . . , n,(3.4)
with the notation
?
?? x
i
? ? ? ? w(x)edx. (3.5)
ii
0
This is a system of n linear equations to determine C1, . . . , Cn. Its coefficient matrix is the Cauchy
matrix, which has a known inverse. However, we prefer to solve (3.4) by a more direct method. We
note that the parameter ? and the function q(x) do not appear explicitly in (3.4). This explains why
(3.4) is formally the same as that in the classical case. The formula corresponding to (3.4) in the
classical case can be found, for example, in Gerber et al. (2006; see formula (42)).
4. THE TRICK WITH THE RATIONAL FUNCTION
We define a rational function Q(r) that is associated with ?(u) in (3.2):
n
Ck
Q(r) ?
.(4.1)
?r ? r
k?1
k
If we know the function Q(r), information concerning ?(u) can readily be obtained according to the
formulas
C ? lim (r ? r )Q(r),
h
r→rh
?(0) ? lim rQ(r).
r→??
?k?1
ˆ ?(?) ?? e
0
h ? 1, . . . , n,(4.2)
h
(4.3)
To verify the latter, observe that ?(0) ?
Ck. Moreover, if
n
n
?
Ck
??u
?(u) du ? ?
,
? ? 0,(4.4)
?
k?1
? ? r
k
denotes the Laplace transform of the function ?(u), we have
ˆ ?(?) ? ?Q(??),
? ? 0.(4.5)
The trick is now to find equivalent and more useful expressions for the function Q(r), and to apply
(4.2), (4.3), and (4.5) to these expressions. For this purpose, we note that the function Q(r) is com-
pletely determined by the following three properties:
P1. It is a rational function of the type with a polynomial of degree at most n ? 1 divided by one of
degree n.
P2. Its poles are r1, . . . , rn.
P3. Q(?i) ? ?i/?i, i ? 1, . . . , n, according to (3.4).
As a first application of this idea, we define the rational function
nnn
?
?
r ??
? ? ?
j
j
i
(? ? r )
j
?
k?1
? ?
j?1
?
k
k?1i?1,i?j
ji
Q (r) ?
1
.(4.6)
n
(r ? r )
k

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424NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 4
This function satisfies properties P1–P3 above, which shows that Q1(r) ? Q(r). Applying (4.2) to Q1(r),
we find that
? ??
?
j?1k?1i?1,i?j
j
C ?
n
h
(r ? r )
?
k?1,k?h
nnn
?
r ? ?
h
? ? ?
j
j
i
(? ? r )
jk
i
,h ? 1, . . . , n,(4.7)
hk
which is the solution of the system of linear equations (3.4). In the following we shall consider the
special case where q(x) is exponential, in order to obtain more attractive results.
5. EXPONENTIAL GAINS
Here and in the following sections we consider the special case where q(x) ? ?e??xfor some ? ? 0.
Then (3.3) becomes the equation
?
? ? r
i?1
i
n
??
i
?
A
? ?
? (? ? ? ? ?) ? 0. (5.1)
i? ? r
This equation has exactly n ? 1 solutions, namely, r1, . . . , rn(introduced earlier) and a unique negative
solution ??. Consider now the rational function
?
?? ? r
i?1
Q (r) ?
n
2
? ? r
?
i
?
A
?
i?1
i
nn
??
ii
(? ? r)A
? (? ? ?)A
?
i?1
ii
? ? ?
ii
.(5.2)
?
? ?
? (? ? ? ? ?)
i? ? r
? ? r
We note that r ? ?? is a common zero of the numerator and the denominator. With this, one verifies
that conditions P1–P3 are satisfied, from which it follows that Q2(r) ? Q(r). From (4.2) and
rule we getl’Hopital’s
ˆ
?
?? ? r
i?1
i
C ?
n
h
? ? r
h
?
A
?
(? ? r )
i?1
nn
??
ii
(? ? r )A
? (? ? ?)A
?
i?1
?
hii
? ? ?
ih
,(5.3)
?
i
? ?
i
22
(? ? r )
ihh
which is an alternative to (4.7). Furthermore, an application of (4.3) leads to
nn
??i
?(0) ?
A? ? (? ? ?)
ii
A,(5.4)
?
i?1
?
i?1
??
i
? ? ? ? ?? ? ?
i
which simplifies to
n
?? ? ?
i
? ? ?
i
?(0) ?
A?
i
.(5.5)
?
i?1
i
? ? ? ? ?
6. GENERAL CLAIM AMOUNT DISTRIBUTIONS
Because any claim amount distribution can be obtained as a limit from an appropriate sequence of
combinations of exponential distributions (see, e.g., Dufresne 2007), certain results of Section 5 lead
to results for an arbitrary p(x). The recipe is simple: Rewrite the results in terms of p(x) and w(x),
instead of Ai, ?i, ?i. For example, by observing that for r ? ?1

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A DIRECT APPROACH TO THE DISCOUNTED PENALTY FUNCTION
425
Figure 1
The Zeros of the Lundberg Function
nn
??
?i
?? x
?(? ?r)y
ii
A
?
A ? ?w(x)e
0
? ? w(x)A? ? e e
?
0
i?1
?? w(x)? e p(x ? y) dy dx,
00
dx? edy
?
i?1
?
i?1
iii
0
? ? r
i
n
??
ry ?? (x?y)
i
dy dx
ii
0
??
ry
we can rewrite (5.4) as
???
?
??y
?(0) ?
? w(x)p(x) dx ? (? ? ?)? w(x)? e
?
p(x ? y) dy dx .(6.1)
?
000
? ? ? ? ?
Here ?? is now the negative zero of the Lundberg function
?
L(r) ? ?M(r) ? ?
? (? ? ? ? ?),(6.2)
? ? r
where M(r) ?
The function L(r) is defined for r in the interval where M(r) exists. Note that ?/(? ? ? ? ?) is the
discounted probability that the surplus process has a downward jump before the first upward jump;
this explains the first integral in (6.1). Let g(x) denote the discounted probability density function of
the deficit at ruin for initial surplus zero. Because (6.1) holds for arbitrary w(x), it follows that
p(x) ? (? ? ?)? e
?
? ? ? ? ?
Formula (5.2) can be rewritten as
erxp(x) dx is the moment generating function of p(x); see Figure 1.
?
?0
?
?
??y
g(x) ?
p(x ? y) dy ,x ? 0.(6.3)
?
0
1(? ? r)N(r) ? (? ? ?)N(??)
L(r)
Q (r) ?
2
,(6.4)
? ? r

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426NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 4
with the auxiliary function
??
ry
N(r) ? ?? w(x)? e p(x ? y) dy dx,
00
(6.5)
and L(r) as given in (6.2). In general, Q2(r) is not a rational function, and its use should be restricted,
for example, to r ? 0. According to (4.5), the Laplace transform of ?(u) is
1(? ? ?)N(??) ? (? ? ?)N(??)
L(??)
ˆ ?(?) ?
,
? ? 0,(6.6)
? ? ?
for an arbitrary p(x).
We note that (6.1) can be recovered from (6.4) by means of (4.3). For this, it is important to take
the limit as r → ??.
Now we turn to formula (5.3). In general, it is not useful unless h ? 1. Assuming the existence of R
(the positive zero of L(r)), we have the asymptotic formula
?Ru
?(u) ? Cefor u → ?,(6.7)
with
1(? ? R)N(R) ? (? ? ?)N(??)
L?(R)
C ?
.(6.8)
? ? R
Let us consider the special case where w(x) ? 1. Then ?(u) is the expected present value of 1
payable at the time of ruin. From (6.1) we have
1 ? (? ? ?)? ? e
?
? ? ? ? ?
??
?
??y
?(0) ?
p(x ? y) dy dx . (6.9)
?
00
To evaluate this expression, we use the formula
??
1
r
ry
? ? e p(x ? y) dy dx ?
00
[M(r) ? 1](6.10)
for r ? ?? and the fact that L(??) ? 0. After simplification we find that
??
?(0) ? 1 ?
.(6.11)
?(? ? ? ? ?)
The negative term of this surprisingly simple formula shows the effect of discounting and the possibility
of survival.
7. THE CLASSICAL MODEL
In the classical model, the surplus at time t is U(t) ? u ? ct ?S1(t). This model can be obtained as a
limit from the model of Section 6. For ? → ?, ? → ? such that ?/? ? c, we have S2(t) ? ct in the
limit. By taking this limit, we can retrieve several known results in a straightforward manner.
First, (6.2) becomes
L(r) ? ?M(r) ? cr ? (? ? ?)(7.1)
in the limit. Then ?? is the negative zero and R (if it exists) the positive zero of L(r). Formula (6.1)
leads to
?(0) ? ? w(x)? e
00
c
??
?
??y
p(x ? y) dy dx(7.2)

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A DIRECT APPROACH TO THE DISCOUNTED PENALTY FUNCTION
427
in the limit, and (6.3) yields
?
?
c
??y
g(x) ? ? ep(x ? y) dy,x ? 0.(7.3)
0
This result is formula (3.14) in Gerber and Shiu (1997) and (3.4) in Gerber and Shiu (1998). From
(6.4) we see that Q2(r) can be written as
? w(x)? (e
00
L(r)
ˆ ?(?)
Gerber and Shiu (1998). Moreover, from (6.8) we get the limiting value
? w(x)? (e
00
L?(R)
??
?
ry
??y
Q (r) ?
2
? e)p(x ? y) dy dx.(7.4)
The resulting formula for
? ? Q2(??) is contained in (2.48) together with (2.52) and (2.56) of
??
?
Ry
??y
C ?? e)p(x ? y) dy dx,(7.5)
which is formula (4.10) in Gerber and Shiu (1998).
Finally, if w(x) ? 1, formula (6.11) yields
?
?c
?(0) ? 1 ?
(7.6)
in the limit. This is formula (3.9) in Gerber and Shiu (1998). Furthermore, (7.5) simplifies to
?
1
R
1
?
C ??
. (7.7)
??
L?(R)
To verify this, use (6.10) with r ? R and ?? and L(R) ? L(??) ? 0.
8. THE PROBABILITY OF RUIN
In the limit ? ? 0 and the special case w(x) ? 1, ?(u) becomes ?(u), the probability of ruin. We
assume a positive loading, that is, that ?/? ? ?p1in Section 6 and c ? ?p1in Section 7, where p1
denotes the mean claim amount. Then ? ? 0 in the limit. From (6.1) we see that
?
?(0) ?
(1 ? ?p ),(8.1)
1
? ? ?
and (7.2) yields a classical result of ruin theory,
?p1
c
?(0) ?
.(8.2)
Of course, this formula can also be obtained as a limit from (8.1). Formula (8.1) can be reformulated
as
??
?
1 ? ?(0) ?? ?p.(8.3)
??
1
? ? ?
Note that the expression ?/? ? ?p1is the expected increase of the surplus per unit time. To obtain
(8.3) as a limit from (6.11), observe that ?/? → ?/? ? ?p1for ? → 0. Formula (8.1) can also be
reformulated as
???p
??
E[S (1)]
11
?(0) ????
,(8.4)
? ? ?? ? ? ?/?? ? ?? ? ? E[S (1)]
2

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428NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 4
which has the following interpretation: The first term is the probability that S1(t) has a jump before
S2(t), in which case ruin occurs at that time. Given that S2(t) has a jump before S1(t), the probability
of which is ?/(? ? ?), the conditional probability of ruin is E[S1(1)]/E[S2(1)]. Formula (8.4) appears
to be a natural extension of the classical result (8.2).
From (6.3) we get
?
g(x) ?
[p(x) ? ?(1 ? P(x))],x ? 0,(8.5)
? ? ?
which reduces to
?
c
g(x) ?
[1 ? P(x)],x ? 0(8.6)
in the classical model of Section 7. The latter formula can be found, for example, in Theorem 13.5.1
of Bowers et al. (1997). The determination of the Laplace transform is straightforward. In the classical
model of Section 7, we get from (7.1)
p ?? ? e
?
L(??)
with L(r) ? ?M(r) ? cr ? ?. To reconcile (8.7) with formula (2.60) in Gerber and Shiu (1998), use
(6.10) with r ? ??.
Finally, we turn to the asymptotic formula, ?(u) ? Ce?Rufor u → ?. The coefficient C is obtained
as a limit from (6.8). We have N(0) ? ?p1. By using (6.10) with r ? R and the fact that L(R) ? 0, we
find that N(R) ? ?/(? ? R). Hence,
??
?
??y
ˆ ?(?) ?
p(x ? y) dy dx ,
? ? 0,(8.7)
?
1
00
? ? ??p1
(? ? R)L?(R)
C ?
(8.8)
by (6.8). In the classical model of Section 7, this yields the formula
c ? ?p1
L?(R)
C ?
, (8.9)
which is a famous result of the Scandinavian school. To obtain (8.9) from (7.7), recall that
?/? → c ? ?p1for ? → 0.
9. PERTURBATION BY DIFFUSION
The methodology can also be applied if the surplus process (2.1) is perturbed by an independent
diffusion process. The steps are very similar. For this reason we summarize them in a condensed form
in a single section and highlight only the necessary modifications. We assume that
U(t) ? u ? S (t) ? S (t) ? ?W(t),
1
t ? 0,(9.1)
2
where {W(t)} is an independent standard Wiener process. There are two kinds of ruin in this model,
by a claim (which results in a deficit at ruin) or by oscillation. The penalty for the first is given by the
function w(x). For the second, it is given by constant w0. Instead of (2.3), the discounted penalty
function now satisfies the functional equation
?? ?(u ? x)p(x) dx ? ?? w(x ? u)p(x) dx
0u
? ?? ?(u ? x)q(x) dx ? (? ? ? ? ?)?(u) ? D??(u) ? 0
0
u
?
?
(9.2)
with the notation D ? ?2/2.

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A DIRECT APPROACH TO THE DISCOUNTED PENALTY FUNCTION
429
As in (3.1), we assume that the claim size distribution is a combination of n exponential distributions.
The discounted penalty function ?(u) is then of the form
?
k?1
n?1
?r u
k
?(u) ?
C e
k
,u ? 0.(9.3)
To determine r1, . . . ,
the n ? 1 solutions with positive real part of the equation
?
? ? r
i?1
C1, . . . ,we substitute (9.3) in (9.2) and find that r1, . . . ,arer,C,r
n?1n?1n?1
n
?
?i
?rx 2
?
A
? ?? eq(x) dx ? (? ? ? ? ?) ? Dr ? 0,(9.4)
i
0
i
and C1, . . . ,are solutions of the following system of n ? 1 equations:
?? ? r
k?1
ik
Cn?1
n?1
C
?
?
ki
?
,i ? 1, . . . , n,(9.5)
i
C ? ? ? ? ? C
1
? w .(9.6)
n?10
The latter condition follows from ?(0) ? w0. The solutions of (9.4) are analyzed in Zhang et al. (2010).
In analogy to (4.1), we define the rational function Q(r) as
?r ? r
k?1
n?1
Ck
Q(r) ?
.(9.7)
k
Knowing this function, we can obtain
C ? lim (r ? r )Q(r),
h
r→rh
h ? 1, . . . , n ? 1(9.8)
h
and
ˆ ?(?) ? ?Q(??),
? ? 0,(9.9)
the Laplace transform of ?(u). The function Q(r) is characterized by the following four properties:
P1. It is a rational function of the type with a polynomial of degree at most n divided by one of
degree n ? 1.
P2. Its poles are r1, . . . , r.
n?1
P3. Q(?i) ? ?i/?i, i ? 1, . . . , n, according to (9.5).
P4.rQ(r) ? w0, according to (9.6).limr→??
These properties are satisfied by the rational function
? ??
?
j?1k?1i?1,i?j
j
Q (r) ?
n?1
1
?
k?1
nn?1nn
?
r ? ?
? ? ?
j
j
i
(? ? r )
j
? w(r ? ?)
?
i?1
k0i
i
,(9.10)
(r ? r )
k
and hence Q1(r) ? Q(r). Application of (9.8) yields
? ?
?
j?1k?1
j
C ?
h
nn?1nn
?
r ? ?
h
? ? ?
j
j
i
(? ? r )
j
? w(r ? ?)
h
?
?
k?1,k?h
?
i?1
k0i
i?1,i?j
n?1
i
,h ? 1, . . . , n ? 1.(9.11)
(r ? r )
hk
As in Section 5, we make the additional assumption that q(x) ? ?e??x. Then (9.4) reduces to the
equation

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430NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 4
Figure 2
The Zeros of L(r)
n
??
i
2
?
A
? ?
? (? ? ? ? ?) ? Dr ? 0.(9.12)
?
i?1
i? ? r
i
? ? r
This is equivalent to a polynomial equation of degree n ? 3. Its n ? 3 solutions are r1, . . . ,
a positive real part and the two negative solutions ?? and ??dwith ?? ? ??d? ?? ? ?? ? 0. Then
the function
?
1
? ? r
i?1
i
Q (r) ?
n
2
? ? r
?
i
?
A
? ?
?
i?1
i
withrn?1
nn
??
ii
?(? ? r)A
? ?(? ? ?)A
? Dw (r ? ?)(r ? ?)
0
?
i?1
ii
? ? ?
i
(9.13)
?
2
? (? ? ? ? ?) ? Dr
i? ? r
? ? r
satisfies properties P1–P4, provided that ? is chosen such that the numerator vanishes for r ? ??d.
For an arbitrary p(x), the function Q2(r) can be written as
1(? ? r)N(r) ? (? ? ?)N(??) ? Dw (r ? ?)(r ? ?)
L(r)
0
Q (r) ?
2
,(9.14)
? ? r
where N(r) is defined as in (6.5),
(? ? ?)N(??) ? (? ? ? )N(?? )
Dw (? ? ? )
0
dd
? ? ? ?
,(9.15)
d
d
and
?
2
L(r) ? ?M(r) ? ?
? (? ? ? ? ?) ? Dr .(9.16)
? ? r
The zeros of L(r) are illustrated by Figure 2.
From (9.13) or (9.14) combined with (9.8) and (9.9), expressions for the Laplace transform and the
asymptotic formula of ?(u) can be calculated in a straightforward manner.
As in Section 7, we consider the model with deterministic premium ct, which is now
U(t) ? u ? ct ? S (t) ? ?W(t).(9.17)
1

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A DIRECT APPROACH TO THE DISCOUNTED PENALTY FUNCTION
431
This is the model examined by Gerber and Landry (1998). Now,
nn
??
ii
?
A
? ?
A
? Dw (r ? ?)
0
?
i?1
?
i?1
i
?
i?1
i
? ? r
i
n
A
? ? ?
i
Q (r) ?
2
(9.18)
?i
2
?
? cr ? (? ? ?) ? Dr
i? ? r
i
does the job, that is, Q2(r) ? Q(r). It is possible to show that the function (9.13) converges to the
function (9.18) if ? → ?, ? → ? with ?/? ? c. For this, one has to show that ?d/? → 1, if ? → ?. This
is proved in the Appendix.
For an arbitrary p(x), (9.18) becomes
N(r) ? N(??) ? Dw (r ? ?)
L(r)
0
Q (r) ?
2
,(9.19)
with L(r) given by
2
L(r) ? ?M(r) ? cr ? (? ? ?) ? Dr ,(9.20)
N(r) defined as in (6.5), and ?? the negative zero of L(r). From (9.9) we obtain the Laplace transform
of ?(u):
N(??) ? N(??) ? Dw (? ? ?)
L(??)
0
ˆ ?(?) ?
,
? ? 0.(9.21)
From (9.8) with h ? 1 (r1? R), we obtain
N(R) ? N(??) ? Dw (R ? ?)
L?(R)
0
C ?
,(9.22)
which is needed in the asymptotic formula for ?(u). Let us consider the special case w(x) ? w0? 1.
Then (9.22) simplifies to
?
1
R
1
?
C ??
.(9.23)
??
L?(R)
To see this, use (6.10) for r ? R and r ? ?? and remember that L(R) ? L(??) ? 0. Note that (9.23)
is formally the same as (7.7). Of course, D is contained implicitly in R, ?, and L?(R).
To obtain results for the probability of ruin, we assume c ? ?p1and take the limit ? → 0. Then
? → 0 and ?/? → c ? ?p1. We take the limit in (9.22) and see that in the asymptotic formula,
?Ru
?(u) ? Cefor u → ?,
the coeffcient C has formally the same value as in (8.9). The Laplace transform of ?(u) is
N(0) ? N(??) ? D?
L(??)
ˆ ?(?) ?
,
? ? 0(9.24)
with
2
L(??) ? ?M(??) ? c? ? ? ? D? ,
N(0) ? ?p ,
1
?
?
N(??) ? ?
[M(??) ? 1].

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432NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 4
As a check, set ? ? 0. Then (9.24) reduces to
D1
ˆ ?(?) ??
,
? ? 0,(9.25)
c ? D?
R ? ?
with R ? c/D. This is indeed the Laplace transform of the well-known expression ?(u) ? e?Rufor a
diffusion process. For another check, consider the limit D → 0 in (9.24). We obtain the well-known
expression for the Laplace transform of the probability of ruin in the classical model; see, for example,
formula (2.60) in Gerber and Shiu (1998) or formula (2.9) in Feller (1971, Chapter XIV.3).
As in Dufresne and Gerber (1991), the probability of ruin is decomposed as
?(u) ? ? (u) ? ? (u),
s
(9.26)
d
the probability of ruin by a claim plus the probability of ruin by oscillation. (The index s is from the
French word for claim, sinistre, and d stands for diffusion). To obtain ?d(u), we set w(x) ? 0, w0? 1.
This leads to
D?
ˆ ? (?) ?
d
,
? ? 0.(9.27)
L(??)
For the asymptotic formula ?d(u) ? Cde?Ru, u → ?, we find
DR
L?(R)
C ?
d
.(9.28)
From (9.24) and (9.26) it follows that
N(0) ? N(??)
L(??)
ˆ ? (?) ?
s
,
? ? 0.(9.29)
Furthermore Cs? C ? Cdfor the asymptotic formula. The expressions for C, Cd, and Cscan be found
as formulas (7.30)–(7.32) in Dufresne (1989).
One should note that the diffusion perturbation W(t) in (9.1) could itself be obtained as a limit of
a family of independent compound Poisson processes; see, for instance, Sarkar and Sen (2005).
10. CONCLUDING REMARKS
This paper presents a new and in some sense elementary and pedagogical approach to obtain a series
of results for the discounted penalty function. The method could be applied directly in the classical
model of collective risk theory. However, we prefer to explain the method in a model where the deter-
ministic premium income is replaced by an independent compound Poisson process with exponentially
distributed jumps, because this more advanced model can be treated with the same direct approach,
and it contains the classical model as a limit. Also, because there are upward and downward jumps in
this model, the results may have applications in finance, in particular the pricing of barrier options
and American options. Finally, we note that this more general model facilitates a simple interpretation
of the dividends-penalty identity, again with the classical identity as a natural limit; see Gerber and
Yang (2010).
The method consists of two steps. In the first step, results are obtained for the special case where
the claim amount distribution is a combination of exponential distributions. This family is dense in
the set of all claim amount distributions. Because the discounted penalty function can be expressed
solely through the negative root of the generalized Lundberg equation, one can then in a second step
express the results in general terms to obtain the results for an arbitrary claim amount distribution.
This recipe works for all sufficiently well-behaved penalty functions and in particular for all those of
practical interest. The method can in principle be extended to the case in which the distribution of
the upward jumps is a combination of m exponential distributions. Then, the resulting expression

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A DIRECT APPROACH TO THE DISCOUNTED PENALTY FUNCTION
433
depends on the m zeros with negative real part of the generalized Lundberg equation, and the results
are not quite as elegant as in the case m ? 1.
11. ACKNOWLEDGMENT
The authors thank two anonymous referees for their constructive remarks. Hansjo ¨rg Albrecher grate-
fully acknowledges financial support from the Swiss National Science Foundation Project 200021-
124635/1. Hailiang Yang acknowledges the Research Grants Council of the Hong Kong Special Ad-
ministrative Region, China (project HKU 7540/08H).
APPENDIX
PROOF OF ?d/? → 1
Let r1, . . . ,
?? be the solutions of the equation
?
i?1
r,
n?1
n
?i
2
?
A
? cr ? (? ? ?) ? Dr ? 0.(A.1)
i? ? r
i
By Vieta’s rule (for the product of the solutions of a polynomial equation) we have
n?1
?
k?1
n
?
r ? ?
k
?/D.
i
(A.2)
?
i?1
Now, let r1(?), . . . ,(?), ??(?), ??d(?) denote the solutions of the equationrn?1
?
? ? r
i?1
i
n
??
i
2
?
A
? ?
? (? ? ? ? ?) ? Dr ? 0,(A.3)
i? ? r
with ?/? ? c, such that rk(?) → rkand ?(?) → ? for ? → ?. Using again Vieta’s rule and (A.2) we see
that
?
k?1
n?1nn?1
?
k?1
?(?)? (?)r (?) ? ??
k
?/D ? ??
i
r .
k
(A.4)
?
i?1
d
Now divide this equation by ? and let ? → ? to see that ?d(?)/? → 1 for ? → ?. This shows that (9.18)
follows from (9.13) as a limit.
REFERENCES
ASMUSSEN, S., D. MADAN, AND M. R. PISTORIUS. 2008. Pricing Equity Default Swaps under an Approximation to the CGMY Le ´vy Model.
Journal of Computational Finance 11: 79–93.
BAO, T.-H. 2006. The Expected Discounted Penalty at Ruin in the Risk Process with Random Income. Applied Mathematics and
Computation 179: 559–566.
BOUCHERIE, R. J., AND O. J. BOXMA. 1996. The Workload in the M/G/1 Queue with Work Removal. Probability in Engineering and
Informational Systems 10: 261–277.
BOUCHERIE, R. J., O. J. BOXMA, AND K. SIGMAN. 1997. A Note on Negative Customers, GI/G/1 Workload, and Risk Processes. Probability
in Engineering and Informational Systems 11: 305–311.
BOWERS, N. L., H. U. GERBER, J. C. HICKMAN, D. A. JONES, AND C. J. NESBITT. 1997. Actuarial Mathematics. 2nd ed. Schaumburg, IL:
Society of Actuaries.
BURDEN, R. L., AND J. D. FAIRES. 1989. Numerical Analysis. 4th ed. Boston: PWS-Kent Publishing.
CAI, J., R. FENG, AND G. E. WILLMOT. 2009. On the Expectation of Total Discounted Operating Costs Up to Default and Its Applications.
Advances in Applied Probability 41: 495–522.
CRAME´R, H. 1955. Collective Risk Theory: A Survey from the Point of View of Stochastic Processes. Skandia Jubilee Volume. Stockholm.

Page 15

434NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 14, NUMBER 4
DUFRESNE, D. 2007. Fitting Combinations of Exponentials to Probability Distributions. Applied Stochastic Models in Business and
Industry 23(1): 23–48.
DUFRESNE, F. 1989. Probabilite ´ et se ´ve ´rite ´ de la ruine: Mode ´le classique de la the ´orie du risque et une de ses extensions. Doctoral
thesis, University of Lausanne.
DUFRESNE, F., AND H. U. GERBER. 1989. Three Methods to Calculate the Probability of Ruin. ASTIN Bulletin 19(1): 71–90.
DUFRESNE, F., AND H. U. GERBER. 1991. Risk Theory for the Compound Poisson Process That Is Perturbed by Diffusion. Insurance:
Mathematics and Economics 10: 51–59.
FELLER, W. 1971. An Introduction to Probability Theory and Its Applications. New York: John Wiley & Sons.
GERBER, H. U., AND B. LANDRY. 1998. On the Discounted Penalty at Ruin in a Jump-Diffusion and the Perpetual Put Option. Insurance:
Mathematics and Economics 22(3): 263–276.
GERBER, H. U., S. X. LIN, AND H. YANG. 2006. A Note on the Dividends-Penalty Identity and the Optimal Dividend Barrier. ASTIN
Bulletin 36(2): 489–503.
GERBER, H. U., AND E. S. W. SHIU. 1997. The Joint Distribution of the Time of Ruin, the Surplus Immediately before Ruin, and the
Deficit at Ruin. Insurance: Mathematics and Economics 21: 129–137.
GERBER, H. U., AND E. S. W. SHIU. 1998. On the Time Value of Ruin. North American Actuarial Journal 2(1): 48–78.
GERBER, H. U., AND H. YANG. 2010. Obtaining the Dividends-Penalty Identities by Interpretation. Insurance: Mathematics and Eco-
nomics. 47(2): 206–207.
JACOBSEN, M. 2005. The Time to Ruin for a Class of Markov Additive Risk Process with Two-Sided Jumps. Advances in Applied
Probability 37: 963–992.
KOU, S., AND H. WANG. 2003. First Passage Times of a Jump Diffusion Process. Advances in Applied Probability 35: 504–531.
LABBE´, C., AND K. SENDOVA. 2009. The Expected Discounted Penalty Function under a Risk Model with Stochastic Income. Applied
Mathematics and Computation 215: 1852–1867.
LEVENDORSKII, S. Z. 2004. Pricing of the American Put under Le ´vy Processes. International Journal of Theoretical and Applied Finance
7(3): 303–335.
LEWIS, A. L., AND E. MORDECKI. 2008. Wiener-Hopf Factorization for Le ´vy Processes Having Positive Jumps with Rational Transforms.
Journal of Applied Probability 45(1): 118–134.
PERRY, D., W. STADJE, AND S. ZACKS. 2002. First-Exit Times for Compound Poisson Processes for Some Types of Positive and Negative
Jumps. Stochastic Models 18(1): 139–157.
SARKAR, J., AND A. SEN. 2005. Weak Convergence Approach to Compound Poisson Risk Processes Perturbed by Diffusion. Insurance:
Mathematics and Economics 36(3): 421–432.
SCHMIDLI, H. 2010. On the Gerber-Shiu Function and Change of Measure. Insurance: Mathematics and Economics 46(1): 3–11.
SEGERDAHL, C. O. 1939. On Homogeneous Random Processes and Collective Risk Theory. Thesis, Stockholm.
TEMNOV, G. 2004. Risk Process with Random Income. Journal of Mathematical Sciences 123(1): 3780–3794.
XING, X., W. ZHANG, AND Y. JIANG. 2008. On the Time to Ruin and the Deficit at Ruin in a Risk Model with Double-Sided Jumps.
Statistics and Probability Letters 78: 2692–2699.
ZHANG, Z., AND H. YANG. 2010. A Generalized Penalty Function in the Sparre Andersen Risk Model with Two-Sided Jumps. Statistics
and Probability Letters 80(7–8): 597–607.
ZHANG, Z., H. YANG, AND S. LI. 2010. The Perturbed Compound Poisson Risk Model with Two-Sided Jumps. Journal of Computational
and Applied Mathematics 233: 1773–1784.
DISCUSSIONS
VOLKMAR LAUTSCHAM*
In their article, Professors Albrecher, Gerber and Yang develop an elegant approach to establishing
results regarding the discounted penalty function, which is often called Gerber-Shiu function in the
literature. The purpose of this discussion is twofold. In the first part, the role of the parameter ? in
q(x) ? ?e??xis examined. In the second and more substantial part, it is shown how several results of
the paper can be generalized to the case in which q(x) is a combination of exponential probability
density functions.
* Volkmar Lautscham is a PhD candidate at the Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne,
CH-1015 Lausanne, Switzerland, volkmar.lautscham@unil.ch.