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
We consider the surplus distribution at the time of ruin Du, y defined by Du, y = P T
u
+∞, R
T
u
≥ −y ,
y 0. 3.1
Du, y indicates the probability that the surplus of an insurance company at the time of ruin is in the interval [−y, 0. Evidently, we have
20 G. Wang, R. Wu Insurance: Mathematics and Economics 26 2000 15–24
Du, y = I
[0,y]
−u, u ≤ 0, D+∞, y = 0.
3.2
Theorem 3.1. Let u 0, then Du, y satisfies the following integral equation
Du, y = 1
2 Z
+∞
Dct, y + D2u + ct, y exp{−λt}h u
σ , t
dt +
Z
+∞
λ exp{−λs}ds Z
+uσ −uσ
H u
σ , s, x
dx Z
u+cs+σ x
Du + cs + σ x − z, y dF z + λ
Z
u+cs+σ x+y u+cs+σ x
dF z .
3.3
Proof. Let T = τ
uσ
∧ T
1
. For t ∈ 0, T , we have R
t
0, thus P T ≤ T
u
= 1. Therefore, we have T
u
= T + T
u
◦ θ
T
on {T
u
+∞}. By homogeneous strong Markov property for R
t
we get Du, y = P T
u
+∞, R
T
u
≥ −y = E I T
u
+∞, R
T
u
≥ −y = E
I T ≤ T
u
+∞, R
T
u
≥ −y = E I T
u
◦ θ
T
+∞, R
T +T
u
◦θ
T
≥ −y = E[I T
u
+∞, R
T
u
≥ −y ◦ θ
T
] = E[DR
T
, y] 3.4
By equality 3.4, we get Du, y = E
h D
R
τ
uσ ∧T1
, y i
= E D u + cτ
uσ
+ σ W
τ
uσ
, y , τ
uσ
T
1
+E D u + cT
1
+ σ W
T
1
− Z
1
, y , τ
uσ
≥ T
1
= I
1
+ I
2
. 3.5
By equality 2.9 we get I
1
= E[Du + cτ
uσ
+ σ W
τ
uσ
, yI τ
uσ
T
1
] = 1
2 E[D2u + cτ
uσ
, y + Dcτ
uσ
, yI τ
uσ
T
1
] =
1 2
Z
+∞
D2u + ct, y + Dct, y exp{−λt}h u
σ , t
dt. 3.6
Similar to 2.11 we have I
2
= E D u + cT
1
+ σ W
T
1
− Z
1
, y I τ
uσ
≥ T =
Z
+∞
λ exp{−λs} ds Z
+∞
dF z Z
+uσ −uσ
Du + cs + σ x − z, yH u
σ , s, x
dx =
Z
+∞
λ exp{−λs} ds Z
+uσ −uσ
H u
σ , s, x
dx Z
u+cs+σ x
Du + cs + σ x − z, y dF z + λ
Z
u+cs+σ x+y u+cs+σ x
dF z 3.7
Formula 3.3 now follows from equalities 3.4, 3.5, 3.6 and 3.7. Corresponding to Theorem 2.2, we have the following Theorem 3.2.
Theorem 3.2. Suppose F z has continuous density function on [0, +∞. Then Du, x is twice continuously differentiable in u in interval 0, +∞.
G. Wang, R. Wu Insurance: Mathematics and Economics 26 2000 15–24 21
Theorem 3.3. Let u 0, suppose F z has continuous density function on [0, +∞. Then Du, x satisfies the following integro-differential equation
1 2
σ
2
D
′′ u
u, y + cD
′ u
u, y = λDu, y − λ Z
u
Du − z, y dF z − λ Z
u+y u
dF z. 3.8
Proof. Let ε, t, M 0 such that ε u M and T
ε,M t
= inf{s 0 : u + cs + σ W
s
∈ ε, M} ∧ t. Note that D
′ u
u, y and D
′′ u
u, y are bounded on [ε, M] and therefore, R
s∧T
ε,M t
σ D
′ u
u + cv + σ W
v
, y dW
v
is a martingale. Put T = T
ε,M t
∧ T
1
. Similar to 2.13 we have Du, x = E
h D
R
T
ε,M t
∧T
1
, y i
. 3.9
Similar to 2.15 we have 1 − exp{−λt}
t Du, y = exp{−λt}
E 1
t Z
T
ε,M t
cD
′ u
u + cs + σ W
s
, y+ σ
2
2 D
′′ u
u + cs + σ W
s
, y ds +
1 t
Z
t
λ exp{−λs} n
E h
D u + cT
ε,M s
+ σ W
T
ε,M s
, y ; T
ε,M s
s i
+ Z
+∞
E h
Du + cs + σ W
s
− z, x; T
ε,M s
= s i
dF z ds
3.10 Letting t → 0 gives formula 3.8.
4. Examples
