16 G. Wang, R. Wu Insurance: Mathematics and Economics 26 2000 15–24
1997 and so on. Many results on ruin probability and other ruin problems have been obtained by above-mentioned works. The purpose of this paper will be to discuss the supremum distribution of the risk process before ruin
when ruin occurs and the surplus distribution at the same time of ruin. We first prove some properties of these two distributions and then give some special examples for those distributions we consider. The ideas and methods we
used in this paper are motivated by Grandell 1991, Paulsen 1993 and Paulsen and Gjessing 1997, can be used to the risk model with return on investment, see Wang and Wu 1998.
In Section 2, we introduce the supremum distribution of the risk process before ruin when ruin occurs. We first derive the integral equation satisfied by it. Then by using the integral equation we prove its differentiability. Finally
we give the integro-differential equation satisfied by it. In the last part of this section we define the supremum distribution of the risk process before ruin.
In Section 3, we discuss the surplus distribution at the time of ruin. We also derive the integral equation and the integro-differential equation satisfied by it. The contents in this section are parallel those in Section 2.
In Section 4, we get the simple and explicit expression for the ruin probability, the supremum distribution of the risk process before ruin when ruin occurs and the surplus distribution at the time of ruin by solving boundary value
problems with same differential equation of third order but with different boundary values with each other when the claims are exponentially distributed. For general expressions of these distributions, we can refer to Dufresne
and Gerber 1991 for the series expression of ruin probability and to Zhang 1997 for the series expression of the surplus distribution at the time of ruin.
2. Supremum distribution before ruin
Let T
u
= inf{t ≥ 0 : R
t
0} and T
u
= +∞ if R
t
≥ 0 for all t ≥ 0. Then T
u
is the time of ruin. The ruin probability 9u is defined by 9u = P inf
t ≥0
R
t
0 and the non-ruin probability 8u = 1 − 9u = P inf
t ≥0
R
t
≥ 0. We have 9u = P T
u
+∞. For u 0, we assume throughout this paper that E[R
t
] = u + c − λµt 0, i.e., we assume c − λµ 0, so that we have 9u 1.
Let x u 0, set Gu, x = P sup
0≤t≤T
u
R
t
≥ x, T
u
+∞ 2.1
Gu, x stands for the probability that the supremum value of the risk process before ruin reaches or surpasses the level x when ruin occurs. We call it the supremum distribution of the risk process before ruin when ruin occurs. It
is obvious that
Gu, x = 0,
u ≤ 0, x 0,
9u, u ≥ x 0.
2.2 Let τ
a
= inf{s : |W
s
| = a}. Put H a, t, x = 2πt
−12 +∞
X
k=−∞
exp −
1 2t
x + 4ka
2
− exp −
1 2t
x − 2a + 4ka
2
, 2.3
ha, t = 1
2 √
2π at
−32 +∞
X
k=−∞
4k + 1 exp −
a
2
2t 4k + 1
2
+ 4k − 3 exp −
a
2
2t 4k − 3
2
− 24k − 1 exp −
a
2
2t 4k − 1
2
2.4 It follows from Revuz and Yor 1991 pp. 105–106 that P W
s
∈ dx, τ
a
s = H a, s, x dxand P τ
a
∈ ds = ha, s ds.
G. Wang, R. Wu Insurance: Mathematics and Economics 26 2000 15–24 17
For t ≥ 0 let θ
t
be the shift operators from to itself defined by R
s
θ
t
ω = R
s+t
ω. For finite stopping time T, we define the map θ
T
from to itself by θ
T
ω = θ
t
ω if T ω = t see Revuz and Yor 1991, p. 34 and p. 97. Clearly, we have R
t
◦ θ
T
= R
t +T
.
Theorem 2.1. Let x u 0, then Gu, x satisfies the following integral equation:
Gu, x = exp{−λt }
Z
a −a
Gu + ct + σy, xH a, t
, y dy +
Z
t
λ exp{−λs} ds Z
a −a
H a, s, y dy Z
u+cs+σy
Gu + cs + σy − z, x dF z +
1 2
Z
t
Gu + ct + σ a, x + Gu + ct − σ a, x exp{−λt}ha, t dt, 2.5
where t ≤ 12cx − u, 0 a ≤ 12σ x − u ∧ u.
Proof. Let τ
a
= inf{t : |W
t
| = a}, set T = t ∧τ
a
∧T
1
. For t ∈ 0, T , we have 0 R
t
x. Thus P T ≤ T
u
= 1. Since T ≤ t
and on {T
u
+∞} we have T
u
= T + T
u
◦ θ
T
, therefore, by homogeneous strong Markovian property of R
t
, we get Gu, x = E
I sup
0≤t≤T
u
R
t
≥ x, T
u
+∞ = E
I sup
T ≤t≤T
u
R
t
≥ x, T ≤ T
u
+∞ = E
I sup
0≤t≤T
u
◦θ
r
R
t
◦ θ
T
≥ x, 0 ≤ T
u
◦ θ
T
+∞ = E
I sup
0≤t≤T
u
R
t
≥ x, T
u
+∞ ◦ θ
T
= E [GR
T
, x] 2.6
We now compute the expectation on the right-hand side of equality 2.6. E[GR
T
, x] = E G R
t
, x , t
τ
a
∧ T
1
+ E G R
τ
a
, x , τ
a
≤ T
1
, τ
a
≤ t +E
G R
T
1
, x , T
1
≤ τ
a
, T
1
≤ t = I
1
+ I
2
+ I
3
. 2.7
By assumption of independence we have I
1
= E G u + ct
+ σ W
t
, x I t
τ
a
I t T
1
= E [I T
1
t ] E
G u + ct + σ W
t
I τ
a
t = exp{−λt
} Z
a −a
Gu + ct + σy, xH a, t
, y dy. 2.8
By Proposition 2.8.3 in Port and Stone 1978 we get P W
τ
a
= a, τ
a
∈ dt = P W
τ
a
= −a, τ
a
∈ dt =
1 2
ha, t dt 2.9
18 G. Wang, R. Wu Insurance: Mathematics and Economics 26 2000 15–24
By equality 2.9 we get I
2
= E G u + cτ
a
+ σ W
τ
a
, x I τ
a
≤ T
1
I τ
a
≤ t = E
G u + cτ
a
+ σ a, x I τ
a
≤ T
1
I τ
a
≤ t I W
τ
a
= a +E
Gu + cτ
a
− σ a, xI τ
a
≤ T
1
I τ
a
≤ t I W
τ
a
= −a =
Z
t
G u + ct + σ a, x exp{−λt}P W
τ
a
= a, τ
a
∈ dt +
Z
t
G u + ct − σ a, x exp{−λt}P W
τ
a
= −a, τ
a
∈ dt =
1 2
Z
t
G u + ct + σ a, x + G u + ct − σ a, x exp{−λt}h a, t dt.
2.10 Now,
I
3
= E G u + cT
1
+ σ W
T
1
− Z
1
, x I T
1
≤ τ
a
I T
1
≤ t =
Z
t
λ exp{−λs} ds Z
+∞
dF zE [G u + cs + σ W
s
− z, x I τ
a
≥ s] =
Z
t
λ exp{−λs}ds Z
+∞
dF z Z
a −a
G u + cs + σy − z, x H a, s, y dy =
Z
t
λ exp{−λs} ds Z
a −a
H a, s, y dy Z
u+cs+σy
Gu + cs + σy − z, x dF z. 2.11
Formula 2.5 now follows from equalities 2.6,2.7 and 2.8 and 2.10 and 2.11.
Theorem 2.2. Suppose F z has continuous density function on [0, +∞. Then Gu, x is twice continuously differentiable in u in interval 0, x.
Proof. For arbitrary ε 0, it is sufficient to show that Gu, x is twice continuously differentiable in u in in-
terval ε , x − ε
. If we set t 12cε
and a 12σ ε note that t
and a do not depend on u, we still have formula 2.5. By changing the variable of integration, we can move u in G’s in the integrand on the
right-hand side of equality 2.5 into H or h. Then by the fine properties H and h we can verify that the theorem holds.
Theorem 2.3. Suppose F z has continuous density function on [0, +∞, let x u 0. Then Gu, x satisfies the following integro-differential equation:
1 2
σ
2
G
′′ u
u, x + cG
′ u
u, x = λGu, x − λ Z
u
Gu − z, x dF z. 2.12
Proof. Let ε, t 0 such that ε u x − ε and T
ε t
= inf{s 0 : u + cs + σ W
s
∈ ε, x − ε} ∧ t. Note that G
′ u
u, x and G
′′ u
u, x are bounded on [ε, x − ε] and therefore R
s∧T
ε t
σ G
′ u
u + cv + σ W
v
, x dW
v
is a martingale. Put T = T
ε t
∧ T
1
. Similar to 2.6 we have Gu, x = E
G R
T
ε t
∧T
1
, x .
2.13
G. Wang, R. Wu Insurance: Mathematics and Economics 26 2000 15–24 19
Therefore, by Itô formula, we get Gu, x = exp{−λt}E
G u + cT
ε t
+ σ W
T
ε t
, x +
Z
t
λ exp{−λs} E
G u + cT
ε s
+ σ W
T
ε s
, x ; T
ε s
s +
Z
+∞
E Gu + cs + σ W
s
− z, x; T
ε s
= s dF z
ds = exp{−λt}
Gu, x + E Z
T
ε t
cG
′ u
u + cs + σ W
s
, x + σ
2
2 G
′′ u
u + cs + σ W
s
, x ds +
Z
t
λ exp{−λs} E
G u + cT
ε s
+ σ W
T
ε s
, x ; T
ε s
s +
Z
+∞
E Gu + cs + σ W
s
− z, x; T
ε s
= s dF z
ds. 2.14
Dividing by t gives 1 − exp{−λt}
t Gu, x = exp{−λt}
E 1
t Z
T
ε t
cG
′ u
u + cs + σ W
s
, x + σ
2
2 G
′′ u
u + cs + σ W
s
, x ds +
1 t
Z
t
λ exp{−λs} E
Gu + cT
ε s
+ σ W
T
ε s
, x; T
ε s
s +
Z
+∞
E Gu + cs + σ W
s
− z, x; T
ε s
= s dF z
ds. 2.15
Letting t → 0 gives formula 2.12. We now consider the distribution Ŵu, x defined by Ŵu, x = P sup
0≤tT
u
R
t
≥ x.Ŵu, x can be decomposed as follows:
Ŵu, x = P sup
0≤tT
u
R
t
≥ x, T
u
+∞ + P
sup
0≤tT
u
R
t
≥ x, T
u
= +∞ = Gu, x + 8u.
2.16 Indeed, since R
t
is a process with stationary and independent increments and E[R
t
] = c − λut 0, thus lim
t ↑+∞
R
t
= +∞ P -a.e. Therefore, P
sup
0tT
u
R
t
≥ x, T
u
= +∞ = P T
u
= +∞ = 8u. 2.17
Ŵu, x is the probability that the supremum value of the risk process before ruin reaches or surpasses the level x. We call it the supremum distribution of the risk process before ruin.
3. Surplus distribution at the time of ruin
