On some characteristics of the claim surplus process
The distribution of characteristics, which describes the behaviour of the claim surplus process after the ruin, is investigated and compared with some previous results on distributions of risk functionals. Main attention is given to total durations of a sojourn time for risk processes in a risk (red...
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irk-123456789-45352009-11-26T12:00:34Z On some characteristics of the claim surplus process Gusak, D. The distribution of characteristics, which describes the behaviour of the claim surplus process after the ruin, is investigated and compared with some previous results on distributions of risk functionals. Main attention is given to total durations of a sojourn time for risk processes in a risk (red) zone and in a survival (green) zone. 2008 Article On some characteristics of the claim surplus process / D. Gusak // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 49–59. — Бібліогр.: 14 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4535 519.21 en Інститут математики НАН України |
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The distribution of characteristics, which describes the behaviour of the claim surplus process after the ruin, is investigated and compared with some previous results on distributions of risk functionals. Main attention is given to total durations of a sojourn time for risk processes in a risk (red) zone and in a survival (green) zone. |
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Gusak, D. On some characteristics of the claim surplus process |
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Gusak, D. |
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Gusak, D. |
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On some characteristics of the claim surplus process |
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On some characteristics of the claim surplus process |
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On some characteristics of the claim surplus process |
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On some characteristics of the claim surplus process |
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On some characteristics of the claim surplus process |
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on some characteristics of the claim surplus process |
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Інститут математики НАН України |
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2008 |
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On some characteristics of the claim surplus process / D. Gusak // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 49–59. — Бібліогр.: 14 назв.— англ. |
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Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 49–59
UDC 519.21
DMYTRO GUSAK
ON SOME CHARACTERISTICS
OF THE CLAIM SURPLUS PROCESS
The distribution of characteristics, which describes the behaviour of the claim surplus
process after the ruin, is investigated and compared with some previous results on
distributions of risk functionals. Main attention is given to total durations of a
sojourn time for risk processes in a risk (red) zone and in a survival (green) zone.
The characteristics of the behaviour of classic risk processes after a ruin are closely
related to the overjump functionals of semicontinuous Poisson processes. The distribu-
tions of the overjump functionals for processes with stationary independent increments
are studied by many authors (particularly in articles [1]–[3] and in monographs [4]–[6]).
But a lot of results of the mentioned distributions (obtained in the boundary-value prob-
lems for processes) sometimes is ignored in the studies of applied problems. Some risk
characteristics under consideration are studied in [7]–[11].
In [10], the following classic risk reserve process U(t) is considered:
U(t) = u+ ct− S(t), S(t) =
∑
k≤ν(t)
ξk, u ≥ 0,
F (x) = P{ξk < x}, x ≥ 0; m(z) = Eezξk , ∀k ≥ 1; F (x) = 1− F (x).
Let ν(t) be a simple Poisson process with intensity λ > 0. The ruin time, the sever-
ity (penalty) of a ruin, and the red time for u = 0 (according to [10]) were denoted
correspondingly
T = τ−(0), Y = U(T ), T̃1 = τ ′(0)− T, τ ′(0) = inf{t > T, ct− S(t) > 0}.
Instead of U(t), we consider a claim surplus process ζ(t) (which is lower semicontinuous)
with initial capital u > 0
ζ(t) = S(t)− ct, ζ+(t) = sup
0≤t′≤t
ζ(t′) −→
t→∞ ζ+,
τ+(u) = inf{t > 0; ζ(t) > u}, τ+(0)=̇T ;
γ+(u) = ζ(τ+(u))− u, u ≥ 0,
γ+(u) = u− ζ(τ+(u)− 0), γ+
u = γ+(u) + γ+(u);
τ ′(u) = inf{t > τ+(u), ζ(t) < u};
and, instead of T̃1, we introduce the red time T ′(u) (u ≥ 0, T ′(0)=̇T̃1)
T ′(u) =
{
τ ′(u)− τ+(u), τ+(u) <∞;
∞, τ+(u) =∞.
2000 AMS Mathematics Subject Classification. Primary 60G50; Secondary 60K10.
Key words and phrases. Claim surplus process, risk reserve process; sojourn time in red zone and in
survival zone; multivariate and marginal ruin functions.
This article was partly supported by Deutshe Forschungsgemeinschaft.
49
50 DMYTRO GUSAK
The risk level y = u > 0 divides the right half-plane on a risk ”red” zone {y > u} and
on a survival ”green” zone {y ≤ u}. The ruin time τ+(u) defines the duration of the first
survival ”green” period, whereas T ′(u) does the duration of the first ”red” period.
In what follows, we use the notation of a random variable θs > 0 :
P{θs > t} = e−st, s > 0, t > 0.
All risk characteristics for U(t) and ζ(t) are considered under the condition of the posi-
tiveness of a security load δ > 0,
δ =
c− λμ
λμ
=
c
λμ
− 1 (μ = Eξ1, μ2 = Eξ21 , m = Eζ(1)).
We need some relations from [3] and [13] (see Theorem 3 in [13] and Corollary 1 in [3]
or in [13]) which are gathered in
Lemma. If m = λμ− c < 0 (δ > 0), then, for u ≥ 0,
gu(s) =: E[e−sT ′(u), T ′(u) <∞] =
λ
c
∫ ∞
0
e−zρ−(s)F (u+ z)dz+
+
λ
|m|
∫ ∞
0
e−zρ−(s)
∫ u
0
F (u+ z − y)dP+(y)dz, (1)
where we denote P+(u) = P{ζ+ < u} = δ(u) = φ(u), Ψ(u) = φ(u),
F (x) = 1− F (x), F (x) =
∫ ∞
x
F (y)dy, x > 0.
gu(0) = P{T ′(u) <∞} =
λ
c
F (u) +
λ
|m|
∫ u
0+
F (u− y)dP+(y), (2)
where f(s) = −ρ−(s) is the root of Lundberg’s equation
s = λ[m(f(s)) − 1]− cf(s), m(f) = λ
∫ ∞
0
efxdF (x).
For u = 0 and δ > 0, P+(u) −→
u→0
= p+ = P{ζ+ = 0}
gu(s) −→
u→0
g(s) =: E[e−sT ′(0), T ′(0) <∞] =
λ
c
∫ ∞
0
e−xρ−(s)F (x)dx. (3)
The m.g.f. of {τ+(u), γ+(u)}
q+(s, u, z) =: E[e−sτ+(u)−zγ+(u), τ+(u) <∞]
is defined for u ≥ 0 by the relations
q+(s, u, z) =
∫ u
0−
G0(s, u− y, z)dP+(s, y),
q+(s, 0, z) = E[e−sτ+(0)−zγ+(0), τ+(0) <∞] = p+(s)G0(s, 0, z), (4)
where p+(s) = P{ζ+(θs) = 0}, P+(s, y) = P{ζ+(θs) < y}, y > 0; moreover,
G0(s, u, z) =
s−1λρ−(s)
ρ−(s)− z
∫ ∞
0
[e−zy − e−ρ−(s)y ]dF (u+ y),
G0(0, u, z) =
λ
c
∫ ∞
0
e−zyF (u + y)dy (cρ−(s)p+(s) = s). (5)
From (4)− (5) (as s→ 0), the m.g.f. of γ+(u) is defined as
g+(u, z) = q+(0, u, z) =
λ
c
∫ ∞
0
e−zxF (u+ x)dx+
+
λ
|m|
∫ ∞
0
e−zxF (u− y + x)dxdP+(y). (6)
ON SOME CHARACTERISTICS OF THE CLAIM SURPLUS PROCESS 51
By comparing (1) and (6), the corrected Dos Reis relation is established:
gu(s) = g+(u, ρ−(s)) = E[e−γ+(u)ρ−(s), τ+(u) <∞]. (7)
The m.g.f. of τ+(u),
q+(s, u) = q+(s, u, 0) = P+(s, u) = P{ζ+(θs) > u},
is represented by the convolution (see (4) for z = 0)
q+(s, u) = p+(s)G0(s, u) +
∫ u
0+
G0(s, u− y)dP+(s, y), (8)
G0(s, u) = G0(s, u, 0) = s−1λρ−(s)
∫ ∞
0
e−ρ−(s)yF (u+ y)dy.
By substitution G0 in (8), q+(s, u) is defined by the relation
q+(s, u) =
λ
c
∫ ∞
0
e−ρ−(s)yF (u+ y)dy+
+
λ
s
ρ−(s)
∫ u
0+
∫ ∞
0
e−ρ−(s)yF (u+ y − z)dP+(s, z)dy. (9)
After the Laplace transformation with respect to u, (8) is reduced to the relation
q̃+(s, ν) =:
∫ ∞
0
e−νuq+(s, u)du =
G̃0(s, ν)
1 + νG̃0(s, ν)
, (10)
G̃0(s, ν) =
λρ−(s)s−1
ρ−(s)− ν
∫ ∞
0
[e−νz − e−ρ−(s)z]F (z)dz.
It should be remarked that, for s→ 0,
q̃+(ν) = lim
s→0
q̃+(s, ν) =
1
ν
(1− Ee−νζ+
) =
C̃0(0, ν)
1 + νC̃0(0, ν)
, (11)
Ee−νζ+
=
1
1 + νC̃0(0, ν)
, G̃0(0, ν) =
λ
|m|ν
∫ ∞
0
[1− e−νz]F (z)dz.
The last relation is a rephrasing of the Pollaczec–Khinchin formula for Ee−νζ+
. Com-
paring (3) and (9) for u = 0, we obtain the relation
q+(s, 0) = q+(s) = g(s) =
λ
c
∫ ∞
0
e−zρ−(s)F (z)dz, (12)
which yields τ+(0)=̇T ′(0).
Comparing (1) and (9) for u > 0, we can say that τ+(u) and T ′(u) have a com-
mon “defect” of the distribution. But their m.g.f. are different. Hence, the following
important assertion is true.
Corollary 1. For u = 0, the first regeneration period of ζ(t)
τ ′1(0) = inf{t > τ+
1 (0) : ζ(t) < 0}=̇τ+
1 (0) + T ′
1(0) (13)
consists of two random variables with a common m.g.f. (12) and
ϕ(s) = E[e−sτ ′
1(0), τ ′1(0) <∞] = E[e−sT ′
1(0), ζ+(θs) > 0]. (14)
For u > 0, the first regeneration period
τ ′1(u) = inf{t > τ+
1 (u) : ζ(t) < u}=̇τ+
1 (u) + T ′
1(u) (15)
consists of two random variables τ+
1 (u), T ′
1(u) which have a common defect of the dis-
tribution (see (2)), but their m.g.f. q+(s, u) and gu(s) are different (see (1) and (9))
and
ϕu(s) = E[e−sτ ′
1(u), τ ′1(u) <∞] = E[e−sT ′
1(u), ζ+(θs) > u] < q+(s, u). (16)
52 DMYTRO GUSAK
The Laplace transform of q+(s, u) is defined by (10). The regeneration periods τ ′k(u) for
u ≥ 0
τ ′k(u)=̇τ+
k (0) + T ′
k(0), k ≥ 2, (17)
consist of two random variables and
E[e−sτ ′
k(u), τ ′k(u) <∞] = E[e−sT ′
k(0), ζ+(θs) > 0] < q+(s). (18)
Now let us compare (1)–(3) with some results in [10], where, for u = 0, the m.g.f. of
Y =̇γ+(0) (according to the notation in [10]) is defined by the relation
MY (z, 0) = E[ezY |u = 0] =
1
μz
[m(z)− 1]. (19)
It should be remarked that Y = U(τ−(0)) depends on τ−(0) (γ−(x) depends on τ−(x)
for x < 0). In the case where claims ξk > 0 are exponentially distributed,
E[ezγ−(x)/τ−(u)<∞] =
b
b− z = m(z), ∀x ≤ 0,m(z)− 1 =
z
b− z .
In the general case, relation (19) and the following two relations for the red time T̃1=̇T ′
1(0)
E[e−s�T1 |u = 0] = M �T1
(s, 0) =
1
μf(s)
[m(f(s))− 1], (20)
M �T (s, 0) = 1 + θ − s
sμf(s)
, θ = δ =
c
λμ
− 1 > 0,
(relations (20) in [10], p. 27, rows 14,16) do not agree with (6), (9), and (12). Really,
relation (6) yields
g+(0, z) = E[e−zγ+(0), ζ+ > 0] =
λ
c
∫ ∞
0
e−zxF (x)dx,m = Eζ(1) < 0. (21)
The last relation does not accord with (19), because, for m < 0,
MY (z, 0) −→
z→0
1, but E[e−zγ+(0), ζ+ > 0] −→
z→0
λμ
c
= q+ = Ψ(0) < 1,
E[e−sτ+(0), τ+(0) <∞] −→
s→0
P{τ+(0) <∞} = q+ < 1.
Now we consider a renewal scheme generated by the sums
Sn =
∑
k≤n
τ ′k(0), u = 0; S(u)
n = τ ′1(u) +
n∑
k=2
τ ′k(0), u > 0. (22)
For u = 0, we denote
N(t) = max{n : Sn ≤ t}, H(t) = EN(t),
N(θs) – the randomly stopped renewal process.
Proposition 1. The Laplace–Stieltjes transform of the renewal function H(t) is de-
fined by the relation (in terms ϕ(s) from (14))
h̃(s) =:
∫ ∞
0
e−stdH(t) =
ϕ(s)
1− ϕ(s)
. (23)
The distribution and the m.g.f. of N(θs) are defined by the relations
p(k, s) =: P{N(θs) = k} = (1− ϕ(s))ϕ(s)k , k ≥ 0,
π(z, s) =: EzN(θs) =
1− ϕ(s)
1− zϕ(s)
, |z| ≤ 1. (24)
ON SOME CHARACTERISTICS OF THE CLAIM SURPLUS PROCESS 53
For m < 0, the number of returns of ζ(t) on [0,∞) into the survival zone {y ≤ u}
(N = N(∞)) has a geometric distribution with the m.g.f.
π(z) =: EzN = lim
s→0
π(z, s) =
1− q+
1− zq+ , q+ = Ψ(0), (25)
p(k) = lim
s→0
p(k, s) = P{N = k} = (1 − q+)qk
+, k ≥ 0.
To prove an analogous assertion for u > 0, we consider the second random walk in
(22) (with the m.g.f. ϕu(s) for τ ′1(u) see (16))
S(u)
n = τ ′1(u) +
n∑
k=2
τ ′k(0).
We denote, for u > 0,
Nu(t) = max{n : S(u)
n ≤ t}
Hu(t) = ENu(t), h̃u(s) =
∫ ∞
0
e−stdHu(t).
Theorem 1. The Laplace–Stieltjes transform h̃u(s) is defined by the relation
h̃u(s) =
ϕu(s)
1− ϕ(s)
, ϕu(s) = E[e−sT ′(u), T ′(u) <∞], u ≥ 0. (26)
The distribution and the m.g.f. of Nu(θs) are defined by the relations
pu(0, s) =: P{Nu(θs) = 0} = 1− ϕu(s), (k = 0),
pu(k, s) =: P{Nu(θs) = k} = (1 − ϕ(s))ϕk−1(s)ϕu(s), k ≥ 1; (27)
πu(z, s) =: EzNu(θs) = 1− ϕu(s)
1− z
1− zϕ(s)
. (28)
If m < 0, then the m.g.f. and the distribution of Nu = Nu(∞), being the total number
of negative surpluses, are defined by the relations
πu(z) =: EzNu = 1− ϕu(0)
1− z
1− zq+ ,
ϕu(0) = P+(u) = P{ζ+ > u}, (29)
pu(0) = P{Nu = 0} = 1− ϕu(0), k = 0,
pu(k) = P{Nu = k} = (1− q+)ϕu(0)q(k−1)
+ , k ≥ 1. (30)
The first two moments of Nu(θs) are defined as
ENu(θs) =
ϕu(s)
1− ϕ(s)
= h̃u(s), EN2
u(θs) = h̃u(s)
1 + ϕ(s)
1 − ϕ(s)
, (31)
DNu(θs) = V Nu(θs) = h̃u(s)
1 + ϕ(s)− ϕu(s)
1− ϕ(s)
,
and those of Nu look as
ENu =
1
1− q+P+(u),
DNu =
1
1− q+ ENu[1 + q+ − P+(u)].
(32)
For u = 0, relations (31) and (32) yield
EN(θs) = h̃(s), DN(θs) = V N(θs) =
h̃(s)
1− ϕ(s)
,
EN = h̃(0), DN =
h̃(0)
1− q+ , h̃(0) =
q+
1− q+ .
(33)
54 DMYTRO GUSAK
Proof. Relation (26) is established after the Laplace–Stieltjes transformation of the
renewal function
H(t) =
∑
n≥1
P{Sn ≤ t}.
The distribution of Nu(θs) (27) follows from the relation
pk,u(t) = P{Nu(t) = k} = P{S(u)
k ≤ t} − P{S(u)
k+1 ≤ t}
after the Laplace–Carson transformation (with the use of the integration by parts)
pu(k, s) = s
∫ ∞
0
e−stpk,u(t)dt = −
∫ ∞
0
pk,u(t)de−st.
On the basis of (27), the m.g.f. of Nu(θs) is defined in (28). From (27)–(28) after the
limit passage (s→ 0), relations (29)–(30) are established.
To calculate the moments in (31)–(33), the derivatives of the corresponding m.g.f. are
used (particularly, ENu(θs) = ∂
∂zπu(z, s)|z=1)
EN2
u(θs) =
∂2
∂z2
πu(z, s)|z=1 + ENu(θs);
∂
∂z
πu(z, s) =
(1 − ϕ(s))ϕu(s)
(1− zϕ(s))2
−→
z→1
ϕu(s)
1− ϕ(s)
, (34)
∂2
∂z2
πu(z, s) =
2(1− ϕ(s))ϕu(s)ϕ(s)
(1− zϕ(s))3
−→
z→1
2ϕu(s)ϕ(s)
(1− ϕ(s))2
.
Let us separate the random walk (22) into two partial walks
θ(u)
n =̇τ+
1 (u) +
n∑
k=2
τ+
k (0) (θn =
n∑
k=1
τ+
k (0), u = 0),
σ(u)
n =̇T ′
1(u) +
n∑
k=2
T ′
k(0) (σn =
n∑
k=1
T ′
k(0), u = 0).
(35)
For a claim surplus process ζ(t) with risk level u > 0, we denote
θNu = θ
(u)
Nu
– the total duration of survival periods,
σNu = σ
(u)
Nu
– the total duration of red periods,
θN = θNu
∣∣
u=0
, σN = σNu
∣∣
u=0
.
The following assertion is true.
Theorem 2. For u = 0 and m < 0, the m.g.f. θN and σN are identical:
E[e−sθN , θN <∞] = E[e−sσN , σN <∞] =
=
1− q+
1− q+g(s) (q+(s) = g(s) see in (12)). (36)
For u > 0 and m < 0, the m.g.f. of θNu is defined in terms of q+(s, u) and ϕu(0) = Ψ(u)
as (see (9) and (16))
E[e−sθNu , θNu <∞] =
= 1−Ψ(u) +
(1− q+)Ψ(u)
1− q+q+(s)
q+(s, u).
(37)
ON SOME CHARACTERISTICS OF THE CLAIM SURPLUS PROCESS 55
For u > 0 and m < 0, the m.g.f. of σNu and SNu is defined in terms of gu(s) and ϕu(0)
as (see (1) and (16))
E[e−sσNu , σNu <∞] =
= 1− Ψ(u) +
p+Ψ(u)
1− q+q+(s)
gu(s), (gu(s)see (7))
E[e−sSNu , SNu <∞] =
= 1−Ψ(u) +
p+Ψ(u)
1− q+ϕ(s)
ϕu(s), ϕu(s)see in (16).
(38)
For s→ 0,
P{θN <∞} = P{σN <∞} = P{SN <∞} =
1
1 + Ψ(0)
,
P{θNu <∞} = P{σNu <∞} = P{SNu <∞} =
= 1−Ψ(u) +
Ψ2(u)
1 + Ψ(0)
−→
u→0
1
1 + Ψ(0)
, (39)
1
2
< P{θN <∞} = P{σN <∞} < 1.
Proof. Relations (36) are established by averaging the m.g.f. of θn and σn in (35) over
the geometric distribution (25). The proof of (37) and (38) follows from averaging the
m.g.f. of θ(u)
n , σ(u)
n , and S(u)
n in (35) over distribution (30) for Nu.
To analyze the last relation in extreme cases, we denote
A∗ = {ω : ζ(t)→ −∞}
and remark that δ = c−λμ
λμ = 1
q+
− 1 = p+
q+
.
1) Let q+ = 1− ε (ε is small), δ = ε
1−ε (|m| = O(ε)). Then we have
P{σN <∞} = P{θN <∞} =
1
2− ε ≈
1
2
for ω ∈ A∗,
P{σN =∞} = P{θN =∞} =
1− ε
2− ε ≈
1
2
for ω ∈ A∗.
2) Let q+ = Ψ(0) = ε, δ = 1−ε
ε (|m| = O(1
ε )). Then we have
P{σN <∞} = P{θN <∞} =
1
1 + ε
≈ 1 for ω ∈ A∗,
P{σN =∞} = P{θN =∞} =
ε
1 + ε
≈ 0 for ω ∈ A∗.
3) Let p+ = q+ = 1
2 . Then we have
P{σN <∞} = P{θN <∞} =
2
3
for ω ∈ A∗,
P{σN =∞} = P{θN =∞} =
1
3
for ω ∈ A∗.
Namely the last case δ = 1 is regarded as the extreme one in the risk theory, because,
as in the case of a ”fair play” in the classic ruin problem, p+ = q+ = 1
2 .
Let us consider the multivariate ruin function which is determined by the joint m.g.f.
for
{τ+(u), γ1(u), γ2(u), γ3(u)}; γ1(u) = γ+(u), γ2(u) = γ+(u), γ3(u) = γ+
u :
V (s, u, u1, u2, u3) = E
[
e−sτ+(u)−�3
k=1 ukγk(u), τ+(u) <∞
]
=
= E[e−
�3
k=1 ukγk(u), ζ+(θs) > u].
56 DMYTRO GUSAK
The results obtained in [3], [12] yield the following theorem for a semicontinuous process
ζ(t).
Theorem 3. The joint m.g.f. of {τ+(u), γ1(u), γ2(u), γ3(u)} and {τ+(u), γk(u)}k=1,3
is determined in terms of the convolutions
G(s, x, u1, u2, u3) = s−1
∫ 0
−∞
Ax−y(u1, u2, u3)dP−(s, y),
P±(s, y) = P
{
ζ±(θs) < y
}
,±y > 0, P−(s, y) = q−(s)eyρ−(s),
Ax(u1, u2, u3) = λ
∫ ∞
x
e(u1−u2)x−(u1+u3)zdF (z), x > 0,
by the relations
V (s, u, u1, u2, u3) =
∫ u
0
G(s, u − y, u1, u2, u3)dP+(s, y);
Vk(s, u, uk) =
∫ u
0
Gk(s, u− y, uk)dP+(s, z),
(40)
Gk(s, u, uk) = G(s, u, u1, u2, u3)|ur=0,r �=k (1 ≤ k ≤ 3).
For the triple {τ+(u), γ1(u), γ2(u)}, the m.g.f. is determined by the relation
V (s, u, u1, u2) =: V (s, u, u1, u2, 0) =
∫ u
0
G(s, u− z, u1, u2)dP+(s, z), (41)
G(s, u, u1, u2) =: G(s, u, u1, u2, 0) =
s−1λρ−(s)e−u2u
ρ−(s)− u1 + u2
∫ ∞
0
[
e−u1y − e−(ρ−(s)+u2)y
]
dF (u + y).
It should be remarked that G(s, u, u1, u2) and Gk(s, u, uk) are invertible with respect
to uk, particularly, Gk(s, u, uk) have inversions
g1(s, u, x) = s−1λρ−(s)
∫ ∞
x
eρ−(s)(x−y)dF (u+ y),
g2(s, u, y) = s−1λρ−(s)eρ−(s)(u−y)F (y)I{y > u},
g3(s, u, z) = s−1λF ′(z)[1− eρ−(s)(u−z)]I{z > u}.
On the basis of Theorem 3, the following assertion (see Corollary 1 in [3]) is proved.
Corollary 2. For the claim surplus process ζ(t), the densities of multivariate ruin
functions (a prelimit for s > 0 and the limit for s = 0) are determined for y > 0 (y �= u)
by the relations⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
s
∂2
∂x∂y
P
{
ζ+(θs) > u, γ+(u) < x, γ+(u) < y
}
=
=
{
λρ−(s)F ′(x + y)
∫ u
0 e
ρ−(s)(u−y−z)dP+(s, z), y > u,
λρ−(s)F ′(x + y)
∫ u
u−y
eρ−(s)(u−z−y)dP+(s, z), 0 < y < u.
∂2
∂x∂y
P
{
ζ+ > u, γ+(u) < x, γ+(u) < y
}
=
=
{
λ|m|−1F ′(x+ y)P{ζ+ < u}, y > u,m = λμ− c < 0,
λ|m|−1F ′(x+ y)P {u− y < ζ+ < u} , 0 < y < u.
(42)
After inversion (40) with respect to uk (k = 1, 3), the marginal densities of ruin
functions
pk(s, u, xk) =
∂
∂xk
P{ζ+(θs) > u, γk(u) < xk}, k = 1, 3,
are determined by the relations
ON SOME CHARACTERISTICS OF THE CLAIM SURPLUS PROCESS 57
(k = 1) for x > 0⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
p1(s, u, x) = λc−1
∫ ∞
x
eρ−(s)(x−y)dF (u+ y)+
+ s−1λρ−(s)
∫ u
0
∫ ∞
x
eρ−(s)(x−y)dF (u+ y − z)dP+(s, z);
p1(0, u, x) = ∂
∂xP {ζ+ > u, γ+(u) < x} =
λ
cF (u+ x) + λ
|m|
∫ u
0 F (u+ x− z)dP {ζ+ < z} ,m < 0;
(43)
(k = 2) for y > 0 (y �= u)⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
p2(s, u, y) =
{
s−1λρ−(s)F (y)
∫ u
−0 e
ρ−(s)(u−y−z)dP+(s, z), y > u,
s−1λρ−(s)F (y)
∫ u
u−y
eρ−(s)(u−y−z)dP+(s, z), 0 < y < u.
p2(0, u, y) =
{
λ|m|−1F (y)P{ζ+ < u}, y > u,m < 0,
λ
|m|F (y)P {u− y < ζ+ < u} , 0 < y < u;
(44)
(k = 3) for z > 0 (z �= 0)⎧⎪⎪⎪⎨⎪⎪⎪⎩
p3(s, u, z) = p+(s)g3(s, u, z) +
∫ u
0+ g3(s, u − y, z)dP+(s, y),
p3(0, u, z) = λ
|m|F
′(z)
∫ u
0
(z − u+ y)dP{ζ+ < y}I{z > u}+
λF ′(z)
|m|
z∫
0
(z + v)dP{ζ+ < v − u}I{0 < z < u}.
(45)
For u = 0, the marginal ruin functions are determined by the relations
P{ζ+(θs) > 0, γ+(0) > x} =
λ
c
∫ ∞
x
eρ−(s)(x−y)F (y)dy −→
s→0
λ
c
F (x),
P{ζ+(θs) > 0, γ+(0) > x} =
λ
c
∫ ∞
x
e−ρ−(s)(y)F (y)dy −→
s→0
λ
c
F (x), (46)
P{ζ+(θs) > 0, γ+
0 > x} =
λ
cρ−(s)
∫ ∞
x
(1− e−ρ−(s)ydF (y) −→
s→0
λ
c
∫ ∞
x
ydF (y).
This implies that, only for s→ 0, u = 0,
p1(0, 0, x) = p2(0, 0, x) =
λ
c
F (x), x > 0. (47)
But, for s > 0 and u = 0, the densities of γ1(0) = γ+(0) and γ2(0) = γ+(0) on {ζ+(θs) >
0} are different:
p1(s, 0, x) =
λ
c
∫ ∞
0
e−ρ−(s)ydF (x+ y), x > 0;
p2(s, 0, x) =
λ
c
e−ρ−(s)xF (x) �= p1(s, 0, x); (48)
p3(s, 0, x) =
λ
cρ−(s)
(1− e−xρ−(s))F ′(x) −→
s→0
λ
c
xF ′(x), x > 0.
Some results on the distributions of overjump functionals for semicontinuous processes
were stated in our doctoral-degree thesis ([12], particularly, relations (5), (6), (40), (41),
and (46).
The more complete results on the overjump functionals for the processes with sta-
tionary independent increments are stated in [3] and [12], on the basis of which the
multivariate ruin function and other risk characteristics are studied in [13] and in our
monograph [14].
58 DMYTRO GUSAK
Example (see [13], [14]). Let ζ(t) =
∑
k≤ν(t) ξk−ct be the claim surplus process with
exponentially distributed claims ξk > 0, c > 0. We have
F (x) = P{ξk > x} = e−bx, x > 0, μ = Eξk = b−1.
The m.g.f. of ζ(t), ζ(θs) are defined by the cumulant k(r)
Eerζ(θs) =
s
s− k(r) , Re r = 0,
k(r) =
λr − cr(b − r)
b− r ,m = Eζ(1) =
λ− cb
b
< 0. (49)
Lundberg’s equation
s− k(r) = 0 ∼ cr2 + (s− |m|b)r + sb = 0 (50)
has two roots r1(s) = −ρ−(s), r2(s) = ρ+(s) > 0, which define the m.g.f. of ζ±(θs):
Ee−zζ+(θs) =
p+(s)(b + z)
ρ+(s) + z
, ρ+(s) = bp+(s),
Ee−zζ−(θs) =
p−(s)
ρ−(s)− z , cρ−(s)p+(s) = s. (51)
Hence,
P−(s, x) = P{ζ−(θs) < x} = eρ−(s)x, x ≤ 0;
P+(s, x) = q+(s)e−ρ+(s)x, x > 0, p+(s) = P{ζ+(θs) = 0} > 0. (52)
If m < 0 and s→ 0, then ρ−(s)→ 0, ρ+(s)→ b|m|
c ,
Ψ(u) = P{ζ+ > u} = q+e
−ρ+u, p+(s)→ p+ =
|m|
c
, ρ′−(0) =
1
|m| .
The m.g.f. for τ+(u), γ+(u), T ′(u), and τ ′(u) are simply calculated as⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
q+(s, u) = E[e−sτ+(u), τ+(u) <∞] = q+(s)e−ρ+(s)u.
g+(z, u) = E[e−zγ+(u), τ+(u) <∞] = Ψ(u) b
b+z ,
gu(s) = E[e−sT ′(u), τ+(u) <∞] = Ψ(u) b
b+ρ−(s) ,
ϕu(s) = E[e−sτ ′(u), τ+(u) <∞] = q+(s, u) b
b+ρ−(s) ,
(53)
q+(s, u) −→
s→0
Ψ(u) = q+e
−ρ+u.
So the conditional m.g.f. of γ+
1 (u) and T ′
1(u),
E[e−zγ+
1 (u)/τ+(u) <∞] = Ee−zγ̃+
1 (u) =
b
b+ z
,
E[e−sT ′
1(u)τ+(u) <∞] = Ee−sT̃ ′
1(u) =
b
b+ ρ−(s)
,
(54)
are independent of u. For u = 0,
g+(z, 0) = Ψ(0)
b
b+ z
, g(s) = Ψ(0)
b
b+ ρ−(s)
ϕ(s) = q+(s)
b
b+ ρ−(s)
−→
s→0
Ψ(0) = q+. (55)
After the substitution of (53) in (30), we obtain
pu(0) = 1−Ψ(u),Ψ(u) = q+e
−ρ+u,
pu(k) = (1− q+)Ψ(u)qk−1
+ , k ≥ 1,
πu(z) = 1− q+e−ρ+u 1− z
1− zq+ . (56)
ON SOME CHARACTERISTICS OF THE CLAIM SURPLUS PROCESS 59
By substituting (53) and (54) in (37) and (38), it is easy to calculate the m.g.f. of
σ̃Nu =
∑
k≤Nu
T̃k(u), σNu , and SNu :
Ee−sσ̃Nu = Ee−s
�
k≤Nu
T̃k(u) = 1− ρ−(s)Ψ(u)
ρ−(s) + ρ+
, (57)
E[e−sσNu , σNu <∞] = 1−Ψ(u) +
p+Ψ2(u)
1− q+q+(s)
b
b+ ρ−(s)
, (58)
E[e−sSNu , SNu <∞] = 1−Ψ(u) +
p+Ψ(u)
1− q+q+(s)
bq+(s)
b+ ρ−(s)
e−ρ+(s)u,
P{σNn <∞} = P{SNu <∞} = 1−Ψ(u) +
Ψ(u)2
1 + q+
,Ψ(u) = e−ρ+u.
From the Vieta formulas, we get
ρ−(s)ρ+(s) =
bs
c
, ρ+(s)− ρ−(s) =
|m|b− s
c
, ρ+ =
|m|b
c
.
Then, according to §2.4 in [14], the total sojourn time of ζ(t) in {y > u} is defined by
the integral functional
Qu(∞) =
∫ ∞
0
I{ζ(t) > u}dt.
Its m.g.f. is defined by the relation (see (2.84) in [14])
D+
u (0, μ) = Ee−μQu(∞) = 1− ρ+(s)− ρ+
ρ+(s)
e−ρ+u
which is similar to (57) (σ̃Nu=̇Qu(∞)).
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