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Transcript
The joint distribution of the time to ruin and
the number of claims until ruin in the
classical risk model
David C M Dickson
Abstract
We use probabilistic arguments to derive an expression for the joint
density of the time of ruin and the number of claims until ruin in the
classical risk model. From this we obtain a general expression for
the probability function of the number of claims until ruin. We also
consider the moments of the number of claims until ruin and illustrate
our results in the case of exponentially distributed individual claims.
We find a very strong correlation between the number of claims until
ruin and the time of ruin in this case. Finally, we briefly discuss joint
distributions involving the surplus prior to ruin and deficit at ruin.
1
Introduction
The distribution of the number of claims until ruin has been studied by a
number of authors over the years. One of the earliest references is Beard
(1971). A more fruitful approach was that of Stanford and Stroiński (1994)
who derive some recursive procedures to calculate the probability of ruin at
the nth claim in the classical risk model. Egídio dos Reis (2002) derives
the Laplace transform of the probability function of the number of claims
until ruin in the classical risk model. He inverts this for certain claim size
distributions, and, using a duality argument, finds moments of the number
of claims until ruin when the initial surplus is 0.
In a recent paper, Landriault et al (2011) consider a Sparre Andersen risk
model with exponential claims. Using Gerber-Shiu type analysis (see Gerber
and Shiu (1998)) they derive a number of results including an expression
for the probability function of the number of claims until ruin. They also
consider the classical risk model and apply the approach adopted by Dickson
and Willmot (2005) to find the density of the time to ruin in order to derive an
1
expression for the joint density of the time to ruin and the number of claims
until ruin. (For convenience, we use the term joint density throughout when
referring to two or more variables, even if one of the variables is discrete.) An
important idea in Landriault et al (2011) is that some known results relating
to the time to ruin can be interpreted in terms of the number of claims until
ruin, and we show that this idea has applications to other results in ruin
theory.
Dickson (2007) uses probabilistic arguments to find an expression for the
density of the time to ruin in the classical risk model, and we now adapt this
approach to obtain an expression for the joint density of the time to ruin
and the number of claims until ruin, from which we can easily extract the
marginal distributions. We extend the results of Egídio dos Reis (2002) by
considering the moments of the number of claims until ruin when the initial
surplus is greater than 0. Central to our analysis is the case when the initial
surplus is 0. We can obtain all results of interest for this case using GerberShiu type analysis, with the function introduced by Landriault et al (2011),
and Lagrange’s implicit function theorem, as in Dickson and Willmot (2005).
2
Notation
Throughout this paper we consider the classical risk model. Let {U(t)}t≥0
denote the surplus process of an insurer with
U(t) = u + ct − S(t),
where u ≥ 0 is the initial surplus, c is the rate of premium income per
unit time, and {S(t)}t≥0 is the aggregate claims process. We have S(t) =
PM(t)
i=1 Xi , where {M(t)}t≥0 is a Poisson process with Poisson parameter λ, Xi
denotes the amount of the ith claim, and {Xi }∞
i=1 is a sequence of independent
and identically distributed random variables, each with distribution function
F , such that F (0) = 0, and probability density function f. We denote E[X1k ]
by mk and we assume that m1 and m2 exist and that c > λm1 .
For this process, let Tu denote the time to ruin from initial surplus u, so
that
Tu = inf{t: U(t) < 0}
and define ψ(u) = Pr(Tu < ∞) = 1 − χ(u) and ψ(u, t) = Pr(Tu ≤ t), and let
d
ψ(u, t)
dt
denote the defective density of the time to ruin. Let NTu denote the number of
claims until ruin (including the claim causing ruin), and let wn (u, t) denote
w(u, t) =
2
the joint density of Tu and NTu , given initial surplus u, defined for n =
1, 2, 3, . . . and t > 0. Further let pn (u) denote the probability of ruin at the
nth claim given initial surplus u, so that
Z ∞
pn (u) =
wn (u, t)dt.
0
As in Landriault et al (2011) we define
£
¤
φr,δ (u) = E rNTu e−δTu I(Tu < ∞)
for 0 < r ≤ 1 and δ ≥ 0, where I is the indicator function. Thus
Z ∞
∞
X
n
r
e−δt wn (u, t)dt.
φr,δ (u) =
n=1
(1)
0
This function allows us to find the joint density of Tu and NTu when u = 0,
as shown in the next section.
Finally, we use the notation ã for the Laplace transform of a function a.
3
The case u = 0
In this section we consider the case u = 0. Although formulae (4) to (7) below
are obtained by Landriault et al (2011) we include them for completeness as
results for the case u = 0 are required for the more general case when u > 0.
Also, the approach below is more direct than theirs if we are interested in
using transform inversion techniques only for the case u = 0, and not for the
case u > 0 as in Landriault et al (2011). We apply a different approach to
the case u > 0 in the next section.
By the standard argument of conditioning on the time and the amount
of the first claim we obtain
Z
d
λ+δ
λr
λr u
φ (u − x)f(x)dx, (2)
φ (u) =
φr,δ (u) − F̄ (u) −
du r,δ
c
c
c 0 r,δ
and taking the Laplace transform of this equation with respect to u we obtain
R∞
c φr,δ (0) − λr 0 e−su F̄ (u)du
φ̃r,δ (s) =
.
(3)
cs − (λ + δ) + λrf˜(s)
Now let ρ be the unique positive solution of
cs − (λ + δ) + λrf˜(s) = 0.
3
(See Landriault et al (2011).) Then, as ρ is a zero of the denominator of
equation (3), it must also be a zero of the numerator, giving
Z
λr ∞ −ρu
φr,δ (0) =
e F̄ (u)du.
c 0
Landriault et al (2011) adapt the technique in Dickson and Willmot (2005) of
employing Lagrange’s implicit function theorem to express a Laplace transform with transform parameter ρ as a Laplace transform with transform
parameter δ. Formula (44) of Landriault et al (2011) allows us to invert
φr,δ (0) with respect to δ as
−λt
λre
F̄ (ct) +
∞
X
n+1 n−1
t
−λt (λr)
e
n!c
n=1
Z
ct
y f n∗ (ct − y)F̄ (y)dy,
0
and hence by equation (1), we have
w1 (0, t) = λe−λt F̄ (ct)
(4)
and for m = 2, 3, 4, . . . ,
−λt
wm (0, t) = e
λm tm−1
(m − 1)!
Z
ct
0
y (m−1)∗
f
(ct − y)F̄ (y)dy.
ct
(5)
Summing wm (0, t) over m yields the known formula (see Dickson and Willmot
(2005))
−λt
w(0, t) = λe
F̄ (ct) +
∞
X
m=2
−λt
e
λm tm−1
(m − 1)!
Z
0
ct
y (m−1)∗
(ct − y)F̄ (y)dy.
f
ct
Similarly, integrating wm (0, t) over t yields
Z ∞
p1 (0) =
λe−λt F̄ (ct)dt
(6)
0
and for m = 2, 3, 4, . . . ,
Z ∞
m m−1 Z ct
y (m−1)∗
−λt λ t
pm (0) =
e
(ct − y)F̄ (y)dydt.
f
(m − 1)! 0 ct
0
(7)
Formula (5) is required to find wm (u, t), and formula (7) is required to find
pm (u), both for m = 2, 3, 4, . . ., as shown in the next section.
4
4
The case u > 0
In this section we consider the case when u > 0. We adapt ideas in Dickson
(2007) where a formula for w(u, t) is obtained by applying probabilistic arguments from Prabhu (1961). Consider first w1 (u, t). For small dt, we think of
w1 (u, t) dt as representing the probability that ruin occurs on the first claim
and in the interval (t, t + dt). For this to occur, we require no claims up to
time t, and a claim exceeding u + ct in the interval (t, t + dt). Hence
w1 (u, t) = λe−λt F̄ (u + ct),
with formula (4) as a special case. Similarly, for n = 1, 2, 3, . . . we can
construct the formula
n Z u+ct
−λt (λt)
wn+1 (u, t) = e
f n∗ (u + ct − x)λF̄ (x)dx
n! 0
n Z t
X
(λs)j j∗
f (u + cs)wn+1−j (0, t − s)ds. (8)
e−λs
−c
j!
0
j=1
The arguments behind this formula are as follows. We consider wn+1 (u, t) dt
as representing the probability that ruin occurs on the (n + 1)th claim and
in the interval (t, t + dt). The surplus falls below 0 on the (n + 1)th claim
and in the interval (t, t+dt) it there are n claims up to time t of total amount
u + ct − x, so that the surplus is x at time t, and if a claim exceeding x occurs
in (t, t + dt). The first term in formula (8) covers this situation, but makes
no allowance for the possibility that the surplus was below 0 prior to time t.
The second term in formula (8) allows for this possibility. It is constructed
by adapting Prabhu’s (1961) argument. Suppose that at time s, 0 < s < t,
there have been j claims, 1 ≤ j ≤ n, whose amount is u + cs, so that the
surplus at time s is 0. Then the joint density associated with the surplus
next falling below 0 at time t and on the (n + 1 − j)th claim from time s
is wn+1−j (0, t − s), meaning that the surplus falls below 0 on the (n + 1)th
claim. Summing over j and integrating over t gives the adjustment required
to the first term in formula (8). Formula (8) is considerably simpler than the
formula for wn (u, t) given by Landriault et al (2011).
It is straightforward to show that
∞
X
−λt
wn+1 (u, t) = λe
n=0
−c
Z
0
F̄ (u + ct) + λ
Z
0
t
u+ct
g(u + ct − x, t)F̄ (x)dx
g(u + cs, s)w(0, t − s)ds
5
which is the formula for w(u, t) in Dickson (2007).
Now consider the probability function pn (u). We have
Z ∞
p1 (u) =
λe−λt F̄ (u + ct)dt
(9)
0
and for n = 1, 2, 3, . . .
Z ∞
n Z u+ct
−λt (λt)
e
f n∗ (u + ct − x)λF̄ (x)dxdt
pn+1 (u) =
n!
0
0
n Z ∞Z t
X
(λs)j j∗
f (u + cs)wn+1−j (0, t − s)dsdt
e−λs
−c
j!
0
j=1 0
Z ∞
n Z u+ct
−λt (λt)
=
e
f n∗ (u + ct − x)λF̄ (x)dxdt
n!
0
0
n Z ∞Z ∞
X
(λs)j j∗
−c
f (u + cs)wn+1−j (0, t − s)dtds
e−λs
j!
s
j=1 0
Z ∞
n Z u+ct
−λt (λt)
=
e
f n∗ (u + ct − x)λF̄ (x)dxdt
n! 0
0
Z ∞
n Z ∞
j
X
−λs (λs)
j∗
−c
e
wn+1−j (0, t − s)dtds
f (u + cs)
j!
0
s
j=1
Z ∞
n Z u+ct
−λt (λt)
=
e
f n∗ (u + ct − x)λF̄ (x)dxdt
n!
0
0
Z ∞
n
X
(λs)j j∗
f (u + cs)ds.
pn+1−j (0)
e−λs
(10)
−c
j!
0
j=1
This expression is quite different to what we obtain when we invert the
formula given by Egídio dos Reis (2002) for the Laplace transform of pn (u).
Interestingly, both his formula and ours have the property that we require
values of pn (0) to calculate values of pn (u).
5
Moments of NTu
Let δ = 0 in this section and consider
£
¤
φr (u) = E rNTu I(Tu < ∞) .
Then
¯
¯
dk
¯
φ
(u)
= E [NTu (NTu − 1) . . . (NTu − k + 1)I(Tu < ∞)] .
r
¯
drk
r=1
6
In particular, setting u = 0 we can find the moments of NT0 by differentiating
Z
λr ∞ −ρu
φr (0) =
e F̄ (u)du
c 0
where
cρ − λ + λrf˜(ρ) = 0.
(11)
For example,
¯
µ Z ∞
µ ¶
¶¯
Z
¯
¯
d
λ
dρ −ρu
λr ∞
−ρu
¯
=
e F̄ (u)du −
u
φr (0)¯
e F̄ (u)du ¯¯ ,
dr
c 0
c 0
dr
r=1
r=1
and from formula (11), ρ = 0 when r = 1, and differentiating formula (11)
with respect to r gives
−λ
ρ0 |r=1 =
c − λm1
and hence
λm2
λm1 λ
E[NT0 I(T0 < ∞)] =
+
.
c
c 2 (c − λm1 )
We remark that this formula can be easily obtained from Egídio dos Reis
(2002) who considers the moments of the number of claims from the ruin
time to the time of recovery when u = 0.
We can now apply ideas in Albrecher and Boxma (2005). From equation
(3) we have
R∞
cφr (0) − λr 0 e−su F̄ (u)du
φ̃r (s) =
cs − λ + λrf˜(s)
and so
R∞
d
φr (0) − λ 0 e−su F̄ (u)du
c dr
d
φ̃ (s) =
dr r
cs − λ + λrf˜(s)
¡
¢
R∞
˜
cφr (0) − λr 0 e−su F̄ (u)du λf(s)
−
³
´2
˜
cs − λ + λrf(s)
R∞
d
φr (0) − λ 0 e−su F̄ (u)du − λf˜(s)φ̃r (s)
c dr
=
.
˜
cs − λ + λrf(s)
Then
R ∞ −su
¯
¯
I(T
<
∞)]
−
λ
e F̄ (u)du
cE[N
d
T
0
0
0
=
φ̃r (s)¯¯
˜
dr
cs − λ + λf(s)
r=1
7
(12)
˜ ψ̃(s)
λf(s)
cs − λ + λf˜(s)
µ
¶
Z
λ ∞ −su
χ̃(s)
E[NT0 I(T0 < ∞)] −
e F̄ (u)du
=
χ(0)
c 0
−
−
χ̃(s)λf˜(s)ψ̃(s)
,
cχ(0)
(13)
where we have used the well-known formula for the Laplace transform of χ
— see, for example, Dickson (2005). Now define
Z x
ψ(y)χ(x − y)dy.
b(x) =
0
Then by inverting equation (13) we obtain
E[NTu I(Tu < ∞)]
λ/c
χ(u)
E[NT0 I(T0 < ∞)] −
=
χ(0)
χ(0)
Z u
λ
b(u − x)f (x)dx
−
cχ(0) 0
=
Z
0
u
χ(u − x)F̄ (x)dx
χ(u) − χ(0)
χ(u)
E[NT0 I(T0 < ∞)] −
χ(0)
χ(0)
Z u
λ
b(u − x)f (x)dx
−
cχ(0) 0
χ(u)
(E[NT0 I(T0 < ∞)] − 1)
χ(0)
Z u
λ
b(u − x)f (x)dx.
−
cχ(0) 0
= 1+
(14)
We can find higher moments of NTu in a similar fashion. Differentiating
formula (12) with respect to r we obtain
d2
˜ d
c dr
d2
2 φr (0) − 2λf (s) dr φ̃r (s)
.
(s)
=
φ̃
˜
dr2 r
cs − λ + λrf(s)
Then
¯
¯
d2
cE[NT0 (NT0 − 1) I(T0 < ∞)]
¯
φ̃
(s)
=
r
¯
dr2
cs − λ + λf˜(s)
r=1
8
(15)
¯
˜
¯
d
2λf(s)
−
φ̃r (s)¯¯
˜
dr
cs − λ + λf(s)
r=1
χ̃(s)
E[NT0 (NT0 − 1) I(T0 < ∞)]
χ(0)
¯
˜
¯
d
2λf(s)χ̃(s)
φ̃r (s)¯¯
−
cχ(0)
dr
r=1
=
giving
χ(u)
E[NT0 (NT0 − 1) I(T0 < ∞)]
χ(0)
Z u
2λ
m(u − x)f (x)dx,
(16)
−
cχ(0) 0
E[NTu (NTu − 1) I(Tu < ∞)] =
where
m(x) =
Z
x
0
χ(x − y)E[NTy I(Ty < ∞)] dy.
We remark that this argument easily extends to finding higher moments.
Results when u = 0 can also be found using arguments given by Albrecher
and Boxma (2005). From formula (12) we obtain
µ
¶³
Z ∞
´
d
d
˜
φ̃r (s) cs − λ + λrf(s) = c φr (0) − λ
e−su F̄ (u)du − λf˜(s)φ̃r (s).
dr
dr
0
Setting r = 1 and s = 0 we obtain
¯
Z
¯
λ
d
=
φ (0)¯
dr r ¯
c
r=1
which gives
∞
0
λ
F̄ (u)du + φ̃1 (0),
c
Z
λm1 λ ∞
ψ(u)du
E[NT0 I(T0 < ∞)] =
+
c
c 0
λm2
λm1 λ
+
.
=
c
c 2(c − λm1 )
Similarly, formula (15) yields
¯
¯
¯
¯
d2
d
2λ
¯
¯
φ̃
φ
(0)
=
(0)
,
r
r
¯
¯
dr2
c dr
r=1
r=1
so that
E[NT0 (NT0
2λ
− 1) I(T0 < ∞)] =
c
9
Z
0
∞
E[NTu I(Tu < ∞)]du.
6
Exponential claims
Let p(x) = βe−βx for x > 0. We start with the case u = 0 and derive a
known formula for pn (0) — see Landriault et al (2011). First, by formula (6)
we have
Z ∞
λ
p1 (0) =
λe−λt e−βct du =
.
λ + βc
0
Next, consider the inner integral in formula (7). It is straightforward to show
that
Z ct
β n−1 (ct)n e−βct
,
u f (n−1)∗ (ct − u)F̄ (u)du =
n!
0
and so for n = 2, 3, 4, . . .
Z ∞
n−1
(ct)n e−βct
λn
−λt n−2 β
dt
e t
pn (0) =
c (n − 1)! 0
n!
=
λn (βc)n−1 (2n − 2)!
.
n! (n − 1)! (λ + βc)2n−1
Thus, for n = 1, 2, 3, . . . ,
(2n − 2)!
pn (0) =
n! (n − 1)!
µ
λ
λ + βc
¶n µ
βc
λ + βc
¶n−1
.
Summing pn (0) over n yields ψ(0) = λ/(βc), so we have
µ
¶n µ
¶n−1
∞
X
(2n − 2)!
βc
λ
λ
= ,
n! (n − 1)! λ + βc
λ + βc
βc
n=1
an identity which can also be obtained from the general result (e.g. Graham
et al (1994))
√
¶ k
∞ µ
X
2k + 1
z
1 − 1 − 4z
=
.
k
2k + 1
2z
k=0
In the case u > 0 we obtain the following from formulae (9) and (10):
p1 (u) =
λe−βu
λ + cβ
and for n = 2, 3, 4, . . . ,
¶
n−1 µ
λn β n−1 e−βu X n − 1 cj un−1−j (n + j − 1)!
pn (u) =
(n − 1)!2 j=0
(λ + cβ)n+j
j
10
¶
j−1 µ
(λβ)j e−βu X j − 1 cr uj−1−r (r + j)!
−c
pn−j (0)
.
r+j+1
r
j!
(j
−
1)!
(λ
+
cβ)
r=0
j=1
n−1
X
Moments can be obtained from formulae (14) and (16) as
E[NTu I(Tu < ∞)] =
λ(c + λu) −(β−λ/c)u
e
c(cβ − λ)
giving
E[NTu |Tu < ∞] =
and
E[NTu (NTu − 1) I(Tu < ∞)] =
µ
β(c + λu)
,
cβ − λ
2βcλ2 (1 + βu)
βu2 λ3
+
(cβ − λ)3
c(cβ − λ)2
¶
e−(β−λ/c)u
giving
E[NTu (NTu
2 (βcλ)2 (1 + βu)
(βuλ)2
− 1)|Tu < ∞] =
+
λ(cβ − λ)3
(cβ − λ)2
and
V [NTu |Tu < ∞] =
We remark that
βλ(c(βc + λ) + u(β 2 c2 + λ2 ))
.
(cβ − λ)3
E[NTu |Tu < ∞] = βcE[Tu |Tu < ∞].
See, for example, Lin and Willmot (2000) or Dickson (2005).
Now consider the covariance between NTu and Tu given that ruin occurs.
We can find E[Tu NTu I(T < ∞)] as
¯
¯
d d
.
φr,δ (u)¯¯
−
dr dδ
r=1,δ=0
For this particular claim size distribution a straightforward approach to obtaining this is to note that equation (2) for φr,δ (u) can be solved as
¶
µ
Rδ,r
exp {−Rδ,r u}
(17)
φr,δ (u) = 1 −
β
where
−cRδ,r − (λ + δ) +
11
λrβ
= 0.
β − Rδ,r
(18)
If we differentiate equation (17) with respect to both δ and r we obtain
´
³
(1,0) (0,1)
(1,1)
))R
R
−
((β
−
R
)u
−
1)R
exp {−Rδ,r u}
u(2
+
u
(β
−
R
δ,r
δ,r
δ,r
δ,r
δ,r
d d
φr,δ (u) =
dr dδ
β
where
d
d
d d
(0,1)
(1,1)
Rδ,r , Rδ,r = Rδ,r , Rδ,r =
Rδ,r .
dr
dδ
dr dδ
If we differentiate equation (18) with respect to δ, and then set δ = 0 and
r = 1 we obtain
¯
λ
(0,1) ¯
,
=
Rδ,r ¯
c(βc − λ)
r=1,δ=0
(1,0)
Rδ,r =
if we differentiate equation (18) with respect to r, and then set δ = 0 and
r = 1 we obtain
¯
−βλ
(1,0) ¯
Rδ,r ¯
,
=
βc − λ
r=1,δ=0
and if we differentiate equation (18) with respect to δ, and then r, then set
δ = 0 and r = 1 we obtain
¯
λβ(λ + βc)
(1,1) ¯
Rδ,r ¯
=
.
(βc − λ)3
r=1,δ=0
As Rδ,r = β − λ/c when δ = 0 and r = 1, we obtain
¢
¡
β c3 β + cu(βu − 1)λ2 − λ3 u2 + c2 λ(1 + 3uβ)
E[Tu NTu |Tu < ∞] =
c(βc − λ)3
leading to
Cov [Tu , NTu |Tu < ∞] =
βλ(2c + u(βc + λ))
.
(βc − λ)3
A formula for V [NTu |Tu < ∞] can be found in Lin and Willmot (2000) or
Dickson (2005), and so we can calculate the correlation coefficient between
Tu and NTu given that Tu < ∞. Table 1 shows some values of this correlation
coefficient when λ = β = 1 and c = 1.1, 1.2 and 1.3. We observe that the
value of the correlation coefficient is very high for each value of c. Whilst we
would expect the time to ruin and the number of claims until ruin to show
a strong positive correlation, the values in Table 1 are higher than we might
expect.
12
u
0
5
10
15
20
25
c = 1.1
0.998866
0.998867
0.998868
0.998868
0.998868
0.998868
c = 1.2
0.995859
0.995882
0.995887
0.995889
0.995890
0.995890
c = 1.3
0.991457
0.991552
0.991573
0.991581
0.991585
0.991588
Table 1: Values of the correlation coefficient
7
Other joint distributions
The arguments in Section 4 together with arguments in Dickson (2007) lead
to expressions for joint densities involving the number of claims until ruin,
the deficit at ruin and the surplus prior to ruin. Many of the arguments
in this section come from Dickson (2007), and consequently we state rather
than derive results if no new ideas are involved.
The starting point is a joint density dn (u, t, x) which is the density associated with n claims in (0, t), non-ruin over (0, t), and a surplus of x at time
t. From Gerber (1988) we have that
dn (0, t, x) =
x −λt (λt)n n∗
e
f (ct − x)
ct
n!
for n = 1, 2, 3 . . . and 0 < x < ct, with d0 (0, t, x) = e−λt for x = ct. Adapting
arguments in Dickson and Waters (2006) we have
d0 (u, t, x) = e−λt
for x = ct, and for n = 1, 2, 3, . . . and 0 < x < u + ct,
(λt)n n∗
f (u + ct − x)
n!
n−1 Z t−x/c
X
(λt)r r∗
−c I(t > x/c)
f (u + cs) dn−r (0, t − s, x) ds
e−λt
r!
r=1 0
dn (u, t, x) = e−λt
n
−λt λ (t
−I(t > x/c) e
− x/c)n n∗
f (u + ct − x).
n!
Now let wn (u, y, t) denote the joint density of the time to ruin (t), deficit
at ruin (y), and number of claims until ruin (n). Then following Dickson
(2007), we have
w1 (0, y, t) = λe−λt f(ct + y)
13
and for n = 2, 3, 4, . . .,
wn (0, y, t) = λ
Z
ct
0
x −λt (λt)n−1 (n−1)∗
e
f
(ct − x)f(x + y)dx.
ct
(n − 1)!
Similarly,
w1 (u, y, t) = λe−λt f(u + ct + y)
and for n = 2, 3, 4, . . .,
Z
wn (u, y, t) = λ
−c
u+ct
e−λt
0
n−1 Z t
X
r=1
0
(λt)n−1 (n−1)∗
f
(u + ct − x)f(x + y)dx
(n − 1)!
e−λs
(λs)r r∗
f (u + cs) wn−r (0, y, t − s) ds.
r!
If we define the joint density wn (u, x, y, t) where we have now introduced the
surplus prior to ruin (x), following Dickson (2007) we have
w1 (u, x, y, t) = λe−λt f(u + ct + y)
for x = u + ct, and for n = 2, 3, 4, . . . and 0 < x < u + ct,
wn (u, x, y, t) = λ dn−1 (u, t, x) f(x + y).
8
Concluding remarks
It was convenient to solve for φr,δ (u) in order to find E[Tu NTu I(Tu < ∞)] in
Section 6. This approach equally applies to other individual claim amount
distributions. Similarly, the arguments from Albrecher and Boxma (2005)
can be used to obtain a general expression for E[Tu NTu I(Tu < ∞)]. As no
new techniques are involved, we have omitted the details.
References
[1] Albrecher, H. and Boxma, O.J. (2005) On the discounted penalty function in a Markov-dependent risk model. Insurance: Mathematics & Economics 37, 650—672.
[2] Beard, R. E. (1971) On the calculation of the ruin probability for a finite
time interval. ASTIN Bulletin 6, 129—133.
[3] Dickson, D.C.M. (2005) Insurance Risk and Ruin. Cambridge University
Press, Cambridge.
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[4] Dickson, D.C.M. (2007) Some finite time ruin problems. Annals of Actuarial Science 2, 217—232.
[5] Dickson, D.C.M. and Waters, H.R. (2006) Optimal dynamic reinsurance.
ASTIN Bulletin 36, 415—432.
[6] Dickson, D.C.M. and Willmot, G.E. (2005) The density of the time to
ruin in the classical Poisson risk model. ASTIN Bulletin 35, 45—60.
[7] Egídio dos Reis, A.D. (2002) How many claims does it take to get ruined
and recovered? Insurance: Mathematics & Economics 31, 235—248.
[8] Gerber, H.U. (1988) Mathematical fun with ruin theory. Insurance:
Mathematics & Economics 7, 15—23.
[9] Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin. North
American Actuarial Journal 2, 1, 48-78.
[10] Graham, R.L., Knuth, D.E. and Patashnik, O. (1994) Concrete Mathematics, 2nd edition. Addison-Wesley, Upper Saddle River, NJ.
[11] Landriault, D., Shi, T. and Willmot, G.E. (2011) Joint density involving
the time to ruin in the Sparre Andersen risk model under exponential
assumptions. Insurance: Mathematics & Economics, to appear.
[12] Lin, X. and Willmot. G.E. (2000) The moments of the time of ruin, the
surplus before ruin, and the deficit at ruin. Insurance: Mathematics &
Economics 27, 19—44.
[13] Prabhu, N.U. (1961) On the ruin problem of collective risk theory. Annals of Mathematical Statistics 32, 757—764.
[14] Stanford, D.A. and Stroiński, K.J. (1994) Recursive methods for computing finite-time ruin probabilities for phase-distributed claims. ASTIN
Bulletin 24, 235—254.
David C M Dickson
Centre for Actuarial Studies
Department of Economics
University of Melbourne
Victoria 3010
Australia
[email protected]
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