On the exit from a finite interval for the risk processes with stochastic premiums
We consider the almost semicontinuous step-process ξ(t). The conditional characteristic functions of the jumps of ξ(t) have the form E [eiαξk /ξk > 0] = c(c − iα)−1. For such processes, the boundary functionals related to the exit from a finite interval are investigated.
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Інститут математики НАН України
2005
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| Cite this: | On the exit from a finite interval for the risk processes with stochastic premiums / D.V. Gusak, E.V. Karnaukh // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 71–81. — Бібліогр.: 11 назв.— англ. |
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| author | Gusak, D.V. Karnaukh, E.V. |
| author_facet | Gusak, D.V. Karnaukh, E.V. |
| citation_txt | On the exit from a finite interval for the risk processes with stochastic premiums / D.V. Gusak, E.V. Karnaukh // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 71–81. — Бібліогр.: 11 назв.— англ. |
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| description | We consider the almost semicontinuous step-process ξ(t). The conditional characteristic
functions of the jumps of ξ(t) have the form E [eiαξk /ξk > 0] = c(c − iα)−1.
For such processes, the boundary functionals related to the exit from a finite interval
are investigated.
|
| first_indexed | 2025-12-02T09:16:04Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 71–81
UDC 519.21
D. V. GUSAK AND E. V. KARNAUKH
ON THE EXIT FROM A FINITE INTERVAL FOR THE
RISK PROCESSES WITH STOCHASTIC PREMIUMS
We consider the almost semicontinuous step-process ξ(t). The conditional character-
istic functions of the jumps of ξ(t) have the form E eiαξk /ξk > 0 = c(c − iα)−1.
For such processes, the boundary functionals related to the exit from a finite interval
are investigated.
The problems on the exit from a finite interval for the process ξ(t) (t ≥ 0, ξ(0) =
0) with stationary independent increments were considered by many authors (see, for
example [1, ch. IV, § 2]). In [1], the joint distributions of extrema and the distributions
of values of the process up to the exit from the interval were expressed in terms of rather
complicate series of the ”convolutions” of
Γ±(s, x, y) = E
[
e−sτ±(±x), γ±(±x) ≤ y
]
,
where
τ±(±x) = inf {t > 0 : ±ξ(t) > x} , γ±(±x) = ±ξ(τ±(±x)) ∓ x, x > 0.
Simpler relations for the Wiener processes are established in [1, p. 463] and in [2,
§ 27]. In [3] - [6], the mentioned problems were investigated for semicontinuous processes
ξ(t) (ξ(t) have jumps of one sign). For these processes, the distribution density of ξ(t) up
to the exit from the interval was represented [7], [8] in terms of the resolvent functions
Rs(x) (introduced by V.S. Korolyuk in [3]).
We consider the compound Poisson process
ξ(t) =
∑
k≤ν(t)
ξk,
where ν(t) is the Poisson process with rate λ > 0. The distributions of ξk satisfy the
next condition (F (x) is a cumulative distribution function)
(1) P {ξk < x} = qF (x)I {x ≤ 0} + (1 − pe−cx)I {x > 0} , c > 0, p + q = 1.
The process ξ(t) is the almost upper semicontinuous piecewise constant process. We
can represent ξ(t) as the claim surplus process ξ(t) = C(t) − S(t) with the stochastic
premium function
C(t) =
∑
k≤ν1(t)
ηk, ηk > 0, E eiαηk =
c
c − iα
, c > 0,
and with the process of claims S(t) =
∑
k≤ν2(t) ξ′k, ξ′k > 0. Here, ν1(t), ν2(t) are the
independent Poisson processes with rates λ1, λ2 > 0, λ1 + λ2 = λ (for details, see [8] ).
2000 AMS Mathematics Subject Classification. Primary 60G50; Secondary 60K10.
Key words and phrases. Almost semicontinuous processes, risk process with stochastic premiums,
functionals connected with the exit from an interval.
71
72 D. V. GUSAK AND E. V. KARNAUKH
Note that C(t) → 0 and ξ(t) → −S(t) as c → ∞, where −S(t) is a non-increasing
process.
Let Cc(t) be the process with the cumulant
ψc(α) = λc
(
c
c − iα
− 1
)
, λc = ac, a > 0,
then ψc(α) −→
c→∞ iαa, consequently Cc(t) −→
c→∞ at, and ξc(t) = Cc(t) − S(t) → ξ0(t) =
at−S(t), where the limit process ξ0(t) is the classical upper semicontinuous risk process
with the non-stochastic premium function C(t) = at.
Let θs be the exponentially distributed random variable (P{θs > t} = e−st; s, t > 0).
Then the randomly stopped process ξ(θs) has the characteristic function (ch.f.)
ϕ(s, α) = Eeiαξ(θs) =
s
s − ψ(α)
,
where
(2) ψ(α) = λp(c(c − iα)−1 − 1) + λq(ϕ(α) − 1), ϕ(α) =
∫ 0
−∞
eiαxdF (x).
Let us denote the first exit time from the interval (x − T, x), 0 < x < T , T > 0:
τ(x, T ) = inf {t > 0 : ξ(t) /∈ (x − T, x)} ,
and the events
A+(x) = {ω : ξ(τ(x, T )) ≥ x} , A−(x) = {ω : ξ(τ(x, T )) ≤ x − T } .
Then
τ(x, T )=̇
{
τ+(x, T ) = τ+(x), ω ∈ A+(x);
τ−(x, T ) = τ−(x − T ), ω ∈ A−(x).
Overshoots at the moments of the exit from the interval are denoted by the following
relations:
γ−
T (x) = x − T − ξ(τ−(x, T )), γ+
T (x) = ξ(τ+(x, T )) − x.
The main task of our paper is the finding of the following moment generating functions
(m.g.f.) of the functionals connected with the exit from the interval:
Q(T, s, x) = E e−sτ(x,T ),
QT (s, x) = E
[
e−sτ+(x,T ), A+(x)
]
,
QT (s, x) = E
[
e−sτ−(x,T ), A−(x)
]
,
V ±(s, α, x, T ) = E
[
eiαγ±
T (x)−sτ±(x,T ), A±(x)
]
,
V±(s, α, x, T ) = E
[
eiαξ(τ±(x,T ))−sτ±(x,T ), A±(x)
]
,
V (s, α, x, T ) = E
[
eiαξ(θs), τ(x, T ) > θs
]
,
Let us denote the extrema ξ±(t) = sup
0≤s≤t
(inf)ξ(s), ξ± = sup
0≤s<∞
(inf)ξ(s), the joint
distribution of {ξ(θs), ξ+(θs), ξ−(θs)}:
Hs(T, x, y) = P
{
ξ(θs) < y, ξ+(θs) < x, ξ−(θs) > x − T
}
= P {ξ(θs) < y, τ(x, T ) > θs} ,
ON THE EXIT FROM THE INTERVAL FOR THE RISK PROCESS 73
and
P±(s, x) = P
{
ξ±(θs) < x
}
, x ≷ 0, p±(s) = P
{
ξ±(θs) = 0
}
, q±(s) = 1 − p±(s);
ϕ±(s, α) = ±
∫ ±∞
0
eiαxdP±(s, x),
T±(s, x) = E
[
e−sτ±(x), τ±(x) < ∞
]
, x ≷ 0.
Lemma 1. For the process ξ(t) with cumulant (2), the main factorization identity is
represented by the relations
(3) ϕ(s, α) = ϕ+(s, α)ϕ−(s, α), �α = 0;
(4) ϕ+(s, α) =
p+(s)(c − iα)
ρ+(s) − iα
,
where ρ+(s) = cp+(s) is the positive root of Lundberg’s equation ψ(−ir) = s, s > 0.
(5) P
{
ξ+(θs) > x
}
= T +(s, x) = q+(s)e−cρ+(s)x, x > 0.
If m > 0 :
(6) lim
s→0
ρ+(s)s−1 = ρ′+(0) = m−1, lim
s→0
P−(s, x) = P
{
ξ− < x
}
, x < 0.
If m < 0 :
(7) lim
s→0
ρ+(s) = ρ+ > 0; lim
s→0
s−1P
{
ξ−(θs) > x
}
= E τ−(x), x < 0.
If σ2
1 = Dξ(1) < ∞ and m = λ
(
pc−1 − qF̃ (0)
)
= 0
(
F̃ (0) =
∫ 0
−∞ F (x)dx
)
, then
(8)
lim
s→0
ρ+(s)s−1/2 =
√
2
σ1
; lim
s→0
s−1/2P ′
−(s, x) = f0(x), x < 0,
f0(x) = k0
∂
∂x
(∫ ∞
0
P
{
ξ̃0(t) < x
}
dt
)
= −k0
∂
∂x
E τ0(x), x < 0;
where k0 = cσ1
(√
2
)−1
, τ0(x) = inf
{
t > 0 : ξ̃0(t) < x
}
, x < 0; ξ̃0(t) is the decreasing
process with the spectral measure
Π0(dx) = λq (cF (x)dx + dF (x)) , x < 0.
Proof. Relations (3) - (7) were proved in [7] - [8]. If m = 0
(
p = cqF̃ (0)
)
, then
ϕ(s, α) =
s(c − iα)
s(c − iα) − iαλ(p − qF̃ (α)(c − iα))
, F̃ (α) =
∫ 0
−∞
eiαxF (x)dx.
On the basis of the factorization identity (3) as s → 0, we get
1√
s
ϕ−(s, α) =
√
s
p+(s)
ρ+(s) − iα
s(c − iα) − iαλ
(
p − qF̃ (α)(c − iα)
) → f̃0(α),
f̃0(α) =
cσ1√
2
1
−λq
[(
F̃ (α) − F̃ (0)
)
c + ϕ(α) − 1
] =
cσ1√
2
1
−ψ̃0(α)
,
ψ̃0(α) =
∫ 0
−∞
(
eiαx − 1
)
Π0(dx), Π0(dx) = λq (cF (x)dx + dF (x)) , x < 0.
74 D. V. GUSAK AND E. V. KARNAUKH
Let’s denote
ϕ0(s, α) = E eiαξ0(θs) =
s
s − ψ̃0(α)
,
where ξ̃0(t) is the decreasing process with the cumulant ψ̃0(α). Since
cσ1√
2
ϕ0(s, α)s−1 → f̃0(α) =
∫ 0
−∞
eiαxf0(x)dx, s → 0,
we get that
f0(x) = k0
∂
∂x
(∫ ∞
0
P
{
ξ̃0(t) < x
}
dt
)
,
or
−f0(x) = k0
∂
∂x
∫ ∞
0
P {τ0(x) > t} dt = k0
∂
∂x
E τ0(x), x < 0.
Let’s introduce the set of boundary functions on the interval I ⊂ (−∞,∞)
L(I) =
{
G(x) :
∫
I
|G(x)|dx < ∞
}
and the set of integral transforms
R0(I) =
{
g0(α) : g0(α) = C +
∫
I
eiαxG(x)dx
}
.
Let’s denote the projection operations on R0((−∞,∞)) by the following relations:[
g0(α)
]
I
=
∫
I
eiαxG(x)dx,
[
g0(α)
]0
I
= C +
∫
I
eiαxG(x)dx,[
g0(α)
]
− =
[
g0(α)
]
(−∞,0)
,
[
g0(α)
]
+
=
[
g0(α)
]
(0,∞)
.
The main results of our paper are included in the following two assertions.
Theorem 1. For the process ξ(t) with cumulant (2), QT (s, x) has the form (0 < x < T )
(9) QT (s, x) = q+(s)e−ρ+(s)x
∫ 0
x−T
eρ+(s)ydP−(s, y)×
×
[
e−ρ+(s)T
∫ −T
−∞
ec(T+y)dP−(s, y) +
∫ 0
−T
eρ+(s)ydP−(s, y)
]−1
.
Theorem 2. For the process ξ(t) with cumulant (2), the joint distributions of{
τ+(x, T ), γ+
T (x)
}
and {τ+(x, T ), ξ(τ+(x, T ))} are determined by the relations
(10)
⎧⎪⎨⎪⎩
V +(s, α, x, T ) =
c
c − iα
QT (s, x), 0 < x < T,
V+(s, α, x, T ) = eiαxV +(s, α, x, T ) =
c eiαx
c − iα
QT (s, x).
The ch.f. of ξ(θs) before the exit time from the interval has the form
V (s, α, x, T ) = ϕ+(s, α) [ϕ−(s, α) (1 − V+(s, α, x, T ))][x−T,∞)
= ϕ+(s, α)
[
ϕ−(s, α)
(
1 − c eiαx(c − iα)−1QT (s, x)
)]
[x−T,∞)
,
(11)
ON THE EXIT FROM THE INTERVAL FOR THE RISK PROCESS 75
the corresponding distribution has the density (x − T < z < x, z �= 0)
(12) hs(T, x, z) =
∂
∂z
Hs(T, x, z) =
=
(
p+(s)P ′
−(s, z) − q+(s)ρ+(s)
∫ 0
z
eρ+(s)(y−z)dP−(s, y)
)
I {z < 0}+
+ ρ+(s)QT (s, x)
∫ 0
z−x
eρ+(s)(y−(z−x))dP−(s, y),
and the following atomic probability
P {ξ(θs) = 0, τ(x, T ) > θs} = P {ξ(θs) = 0} = p−(s)p+(s) =
s
s + λ
.
Proof of Theorem 1. From the stochastic relations for τ+(x, T ), γ+
T (x) (ξ = ξ1 has the
cumulative distribution function F1(x), ζ is the moment of the first jump of ξ(t)),
τ+(x, T )=̇
{
ζ, ξ > x,
ζ + τ+(x − ξ, T ), x − T < ξ < x,
γ+
T (x)=̇
{
ξ − x, ξ > x,
γ+
T (x − ξ), x − T < ξ < x,
we have the following equation for V +(s, α, x) = V +(s, α, x, T ):
(13) (s + λ)V +(s, α, x) =
λpc
c − iα
e−cx + λ
∫ x
x−T
V +(s, α, x − z)dF1(z), 0 < x < T.
If α = 0, then, from (13), we obtain the equation for QT (s, x)
(14) (s + λ)QT (s, x) = λpe−cx + λ
∫ x
x−T
QT (s, x − z)dF1(z), 0 < x < T.
Since P (A+(x)) = 1 for x < 0, we have the boundary conditions
QT (s, x) =
{
0, x > T,
1, x < 0.
After the replacement
Q T (s, x) = 1 − QT (s, x)
, relation (14) yields the equation for Q T (s, x) (0 < x < T )
(s + λ)Q T (s, x) = s + λF (x − T ) + λ
∫ T
0
Q T (s, z)F ′
1(x − z)dz,
which, after prolonging for x > 0, has the form
(15) (s + λ)Q T (s, x) = sC(x) + λ
∫ ∞
−∞
Q T (s, z)F ′
1(x − z)dz + C>
T (s, x),
C(x) = I {x > 0} , C>
T (s, x) = CT (s)e−cx, x > 0,
(16) CT (s) = λp
[
ecT − cQ ∗
s(T )
]
, Q ∗
s(T ) =
∫ T
0
ecxQ T (s, x)dx.
Let’s introduce the function Cε(x) = e−εxC(x), x > 0, and consider, instead of (15), the
equation for Yε(T, s, x) (ε > 0):
(17) (s + λ)Yε(T, s, x) = sCε(x) + λ
∫ ∞
−∞
Yε(T, s, x − z)dF1(z) + C>
T (s, x), x > 0.
76 D. V. GUSAK AND E. V. KARNAUKH
Denote
yε(T, s, α) =
∫ ∞
0
eiαxYε(T, s, x)dx, C̃ε(α) =
∫ ∞
0
eiαxCε(x)dx,
C̃T (s, α) =
∫ ∞
0
eiαxC>
T (s, x)dx.
By performing the integral transformation of (17), we obtain the equation
(s − ψ(α))yε(T, s, α) = sC̃ε(α) + C̃T (s, α) − [yε(α)ϕ(α)]−
or
(18) syε(T, s, α)ϕ−1(s, α) = sC̃ε(α) + C̃T (s, α) − [yε(α)ϕ(α)]− .
After using the factorization decomposition (3) and the projection operation [ ]+, relation
(18) yields
syε(T, s, α)ϕ−1
+ (s, α) =
[
ϕ−(s, α)
(
sC̃ε(α) + C̃T (s, α)
)]
+
or
(19) syε(T, s, α) = ϕ+(s, α)
[
ϕ−(s, α)
(
sC̃ε(α) + C̃T (s, α)
)]
+
.
By inverting relation (19), we obtain
(20) sYε(T, s, x) = s
∫ x
0
Bε(x − y)dP+(s, y) +
∫ x
0
B(s, x − y, T )dP+(s, y),
Bε(x) =
∫ x
−∞
e−ε(x−y)dP−(s, y) =
∫ 0
−∞
e−ε(x−y)dP−(s, y) =
= e−εxE eεξ−(θs),
B(s, x, T ) = CT (s)
∫ x−T
−∞
e−c(x−y)dP−(s, y), x > 0.
Taking into account that Cε(x) → I {x > 0} as ε → 0, Yε(T, s, x) → Q T (s, x) as ε → 0,
0 < x < T . So Eq. (20) yields
sQ T (s, x) = sP+(s, x) + p+(s)B(s, x, T ) +
∫ x
+0
B(s, x − z, T )P ′
+(s, z)dz.
Taking into account that
q+(s)ρ+(s)
∫ x
0
∫ z−T
−∞
e−c(z−y)dP−(s, y)e−ρ+(s)(x−z)dz =
=q+(s)ρ+(s)
∫ x−T
−∞
e−ρ+(s)x+cydP−(s, y)
∫ x
max(0,y+T )
e−cq+(s)zdz
=p+(s)
[∫ −T
−∞
ecy−ρ+(s)xdP−(s, y)+
+
∫ x−T
−T
eρ+(s)(y+T−x)−cTdP−(s, y) −
∫ x−T
−∞
e−c(x−y)dP−(s, y)
]
,
we have
sQ T (s, x) = sP+(s, x) + p+(s)CT (s)e−ρ+(s)x×
×
[∫ −T
−∞
ecydP−(s, y) +
∫ x−T
−T
e−cT+ρ+(s)(y+T )dP−(s, y)
]
.
ON THE EXIT FROM THE INTERVAL FOR THE RISK PROCESS 77
From the last equation, we can find CT (s) and Q ∗
s(T ) and then get (9).
Let’s note that QT (s, x) → P+(s, x) as T → ∞ and QT (s, x) → 0 as c → ∞. If we
consider, instead of ξ(t), the process ξc(t) = Cc(t) − S(t), then relation (9) yields
QT
c (s, x) = qc
+(s)E
[
eρc
+(s)(ξ−
c (θs)+T−x), ξ−c (θs) + T − x > 0
]
×
×
(
E
[
ec(ξ−
c (θs)+T ), ξ−c (θs) + T < 0
]
+ E
[
eρc
+(s)(ξ−
c (θs)+T ), ξ−c (θs) + T > 0
])−1
.
Taking into account that, for x > 0, P {ξ+
c (θs) > x} = qc
+(s)e−ρc
+(s)x −→
c→∞ e−ρ+
0 (s)x,
where ρ+
0 (s) is the positive solution of the equation
ψ0(−ir) := ar − λ2
(∫ 0
−∞
erxdF (x) − 1
)
= 0,
we get QT
c (s, x) → QT
∞(s, x) as c → ∞. If we denote
ξ0
±(t) = sup
0≤u≤t
(inf)ξ0(u),
then
QT
∞(s, x) = E
[
eρ+
0 (s)(ξ0
−(θs)+T−x), ξ0
−(θs) + T − x > 0
]
×
×
(
E
[
eρ+
0 (s)(ξ0
−(θs)+T ), ξ0
−(θs) + T > 0
])−1
=
∫ T−x
0
eρ+
0 (s)(T−x−y)dP
{−ξ0
−(θs) < y
}×(∫ T
0
eρ+
0 (s)(T−y)dP
{−ξ0
−(θs) < y
})−1
= Rs(T − x)R−1
s (T ),
where the last relation is the well-known formula (see [3]) for the upper semicontinuous
processes.
Proof of Theorem 2. The first relation in (10) follows from Eqs. (13) and (14). The
second relation follows from the first one. The first equality in (11) was proved in [9].
After inverting (11), we get
hs(T, x, z) =p+(s)
∂
∂z
P−(s, z)I {z < 0} + q+(s)ρ+(s)
∫ min{z,0}
x−T
e−ρ+(s)(z−y)dP−(s, y)−
− QT (s, x)
[
p+(s)
∂
∂z
P
{
ξ−(θs) + θ′c + x ≤ z
}
+
+ q+(s)ρ+(s)
∫ z
x−T
e−ρ+(s)(z−y)dP
{
ξ−(θs) + θ′c + x < z
}]
.
(21)
Using the integral transformation of (21) with respect to the distribution of θ′c, we get
formula (12).
Corollary 1. For the joint distribution {τ−(x, T ), ξ(τ−(x, T ))}, we have
(22) sE
[
e−sτ−(x,T ), ξ(τ−(x, T )) < z, A−(x)
]
=
∫ x
x−T
Π−(z−y)dHs(T, x, y), z ≤ x−T,
where Hs(T, x, y) is determined by its density (12) and Π−(x) =
∫ x
−∞ Π(dy), x < 0.
78 D. V. GUSAK AND E. V. KARNAUKH
The probability of the lack of exit (non-exit) from the interval (x − T, x) has the form
(23) P {τ(x, T ) > θs} = P
{
ξ−(θs) > x − T
}−
− QT (s, x)
[∫ −T
−∞
ec(z+T )dP−(s, z) + P
{
ξ−(θs) > −T
}]
.
The m.g.f. for τ(x, T ) and τ−(x, T ) are determined in the following way:
(24)
{
Q(T, s, x) = 1 − P {τ(x, T ) > θs} , 0 < x < T,
QT (s, x) = Q(T, s, x) − QT (s, x), 0 < x < T.
Proof. Formula (22) follows from [6, Theorem 7.3]. By substituting (12) in (22), we
obtain the relation in terms of QT (s, x) and the truncated distribution of ξ−(θs) + θ′c.
Taking into account that
P {τ(x, T ) > θs} =
∫ x
x−T
dHs(T, x, z) =
= P
{
ξ−(θs) > x − T
}− q+(s)
∫ 0
x−T
eρ+(s)(y−(x−T ))dP−(s, z)+
+ QT (s, x)
[∫ 0
−T
eρ+(s)(z+T )dP−(s, z) − P
{
ξ−(θs) > −T
}]
,
and using formula (9), we obtain (23) after some simple transformations. Substitut-
ing (23) into the first relation of (24), we find the m.g.f. of τ(x, T ), and then we can get
the m.g.f. of τ−(x, T ) (see the second relation in (24)).
On the basis of formulas (6) - (8), we can get the following statement about the limit
behavior of QT (s, x) and hs(T, x, z) as s → 0.
Corollary 2. The function h′
0(T, x, z) = lims→0 s−1hs(T, x, z) (x − T < z < x, z �=
0, 0 < x < T ) according to the sign of m has the following forms:
if m > 0
(25) h′
0(T, x, z) =
1
m
(
c−1 ∂
∂z
P
{
ξ− < z
}− P
{
ξ− > z
})
I {z < 0}+
+
1
m
QT (x)P
{
ξ− > z − x
}
;
if m < 0
(26) h′
0(T, x, z) =
(
−p+
∂
∂z
E τ−(z) + q+ρ+
∫ 0
z
eρ+(y−z)dE τ−(y)
)
I {z < 0}−
− QT (x)ρ+
∫ 0
z−x
eρ+(y−(z−x))dE τ−(y);
if m = 0
(27) h′
0(T, x, z) =
(
− ∂
∂z
E τ0(z) − cλ−1 + c
∫ 0
z
∂
∂y
E τ0(y)dy
)
I {z < 0}+
+ cQT (x)
(
λ−1 −
∫ 0
z−x
∂
∂y
E τ0(y)dy
)
.
The ruin probability
QT (x) = lim
s→0
QT (s, x)
ON THE EXIT FROM THE INTERVAL FOR THE RISK PROCESS 79
(according to the sign of m) is determined from (9) in the following way:
(28)
QT (x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
∫ 0
x−T
dP
{
ξ− < y
}×
×
[∫ −T
−∞
ec(T+y)dP
{
ξ− < y
}
+
∫ 0
−T
dP
{
ξ− < y
}]−1
, m > 0,
q+e−ρ+x
(
1
λp+
−
∫ 0
x−T
eρ+y ∂
∂y
E τ−(y)dy
)
×
×
[
1
λp+
− e−ρ+T
∫ −T
−∞
ec(T+y) ∂
∂y
E τ−(y)dy−
−
∫ 0
−T
eρ+y ∂
∂y
E τ−(y)dy
]−1
, m < 0,
(
λ−1 −
∫ 0
x−T
∂
∂y
E τ0(y)dy
)
×
×
[
λ−1 −
∫ −T
−∞
ec(T+y) ∂
∂y
E τ0(y)dy −
∫ 0
−T
∂
∂y
E τ0(y)dy
]−1
, m = 0.
The distribution of ξ(τ−(x, T )) has the form
(29)
P
{
ξ(τ−(x, T )) < z, A−(x)
}
=
1
λ
Π−(z) +
∫ 0−
x−T
Π−(z − y)h′
0(T, x, y)dy+
+
∫ x
0+
Π−(z − y)h′
0(T, x, y)dy, z < x − T.
Corollary 3. For the process ξ(t) with the cumulant function
(30) ψ(α) = λp(c(c − iα)−1 − 1) + λq(b(b + iα)−1 − 1),
QT (x) is represented in the following way (0 < x < T ):
(31)
QT (x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
(
1 − q−eρ−(x−T )
)(
1 − q−c (c + ρ−)−1
e−ρ−T
)−1
, m > 0,
q+e−ρ+x
(
1 − b(ρ+ + b)−1eρ+(x−T )
) (
1 − b(ρ+ + b)−1q+e−ρ+T
)−1
, m < 0,
c(1 + b(T − x))
b + c + bcT
, m = 0.
If ξ(t) is a symmetric process (p = q = 1/2, b = c), then
QT (x) =
1 + c(T − x)
2 + cT
, QT (x) =
1 + cx
2 + cT
, (0 < x < T ).
Proof. Let’s note that the process with cumulant (30) is the almost upper and lower
semicontinuous process. Then, in addition to relations (4) - (5), we have
(32) ϕ−(s, α) =
p−(s)(b + iα)
ρ−(s) + iα
,
where −ρ−(s) = −bp−(s) is the negative root of the equation ψ(−ir) = s, s > 0,
(33) P
{
ξ−(θs) < x
}
= T−(s, x) = q−(s)eρ−(s)x, x < 0.
80 D. V. GUSAK AND E. V. KARNAUKH
If m > 0, then
(34) P
{
ξ−(θs) < x
} −→
s→0
P
{
ξ− < x
}
= q−ebp−x, x < 0, p−(s) −→
s→0
p− > 0.
Taking into account that p+(s)p−(s) = s (s + λ)−1, we have, for m < 0, q′−(s) =
−p′−(s) → −(λp+)−1 as s → 0. Hence,
(35) E τ−(x) = − ∂
∂s
T−(s, x)|s=0 =
1 − bx
λp+
, x < 0.
If m = 0, then we have, for ξ̃0(t), Π0(dx) = λ0be
bxdx, x < 0, λ0 = λq(c + b)b−1.
Moreover,
ξ̃−0 (t) = ξ̃0(t), p0
−(s) = P
{
ξ̃0(θs) = 0
}
=
s
s + λ0
.
Hence, the m.g.f. of τ0(x) has the form
T−
0 (s, x) = E e−sτ0(x) = q0
−(s)ebp0
−(s)x, x < 0.
Since (p0
−)′(s) = −(q0
−)′(s) → λ−1
0 as s → 0, we get
(36) E τ0(x) = − ∂
∂s
T−
0 (s, x)
∣∣
s=0
=
1 − bx
λ0
, x < 0.
Substituting formulas (34) - (36) into the corresponding relations of (28), we get (31).
Remark. We should note that it is easy to get the representation of the m.g.f. of the
functionals related to the exit from the interval for the almost lower semicontinuous
process η(t) (with the parameter b > 0, by considering that ξ(t) = −η(t)). Particularly,
QT (s, x) = q−(s)
∫ x
0
eρ−(s)(x−y)dP+(s, y)×
×
[∫ ∞
T
eb(T−y)dP+(s, y) +
∫ T
0
eρ−(s)(T−y)dP+(s, y)
]−1
.
Let ξ(t) be the almost upper semicontinuous piecewise constant process. Then ξ1(t) =
at + ξ(t), a < 0, is the almost upper semicontinuous piecewise linear process. For the
process ξ1(t) on the basis of the stochastic relations for τ+(x, T ),
τ+(x, T )=̇
{
ζ, ξ + aζ > x,
ζ + τ+(x − ξ − aζ), x − T < ξ + aζ < x,
we have the integro-differential equation for QT (s, x)
a
∂
∂x
QT (s, x) = λ
∫ x
x−T
QT (s, x − z)dF1(z) − (s + λ)QT (s, x) + λpe−cx, 0 < x < T.
Introducing the function Q T (s, x) = 1 − QT (s, x) and following the reasoning analo-
gous to that for the piecewise constant process ξ(t), we can get the representation of
the functionals related to the exit from the interval (x − T, x) for the piecewise linear
processes.
Two boundary-value problems for the integer-valued random walks are considered
in [10] and, for the process with stationary independent increments, are treated in [11].
ON THE EXIT FROM THE INTERVAL FOR THE RISK PROCESS 81
Bibliography
1. I.I. Gikhman, A.V. Skorokhod, Theory of Stochastic Processes, vol. 2, Nauka, Moscow, 1973,
pp. 620. (in Russian)
2. A.V. Skorokhod, Stochastic Processes with Independent Increments, Nauka, Moscow, 1964,
pp. 280. (in Russian)
3. V.S. Korolyuk, Boundary Problems for Compound Poisson Processes, Naukova Dumka, Kyiv,
1975, pp. 140. (in Russian)
4. V.M. Shurenkov, Limit distribution of the exit time from an expanding interval and of the
position at exit time of a process with independent increments and one-signed jumps, Prob.
Theory Appl. 23 (1978), no. 2, 402-407.
5. V.S. Korolyuk, N.S. Bratiichuk, B. Pirdjanov, Boundary-Value Problems for Random Walks,
Ilim, Ashkhabad, 1987, pp. 250. (in Russian)
6. N.S. Bratiichuk, D.V. Gusak, Boundary Problems for Processes with Independent Increments,
Naukova Dumka, Kyiv, 1990, pp. 264. (in Russian)
7. D.V. Gusak, Compound Poisson processes with two-sided reflection, Ukr. Math. J. 53 (2002),
no. 12, 1616-1625.
8. D.V. Gusak, Risk process with stochastic premiums and distributions of their functionals, The-
ory of Stoch. Process. 11(27) (2005), no. 1-2, 29-39.
9. E.A. Pecherskiy, Some identities related to the exit of a random walk out of a segment and a
semiinterval, Probab. Theory Appl. 19 (1974), no. 1, 104-119. (in Russian)
10. T.V. Kadankova, Two-sided boundary problems for the random walks with geometrically dis-
tributed negative jumps, Teor. Imov. and Mat. Stat. 68 (2003), 49-60. (in Ukrainian)
11. V.F. Kadankov, T.V. Kadankova, On the distribution of the moment of the first exit from an
interval and the value of overshoot through the boundary for the processes with independent
increments and random walks, Ukr. Math. J. 57 (2005), no. 10, 1359-1385.
E-mail : random@imath.kiev.ua
E-mail : kveugene@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4427 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-02T09:16:04Z |
| publishDate | 2005 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gusak, D.V. Karnaukh, E.V. 2009-11-09T15:33:32Z 2009-11-09T15:33:32Z 2005 On the exit from a finite interval for the risk processes with stochastic premiums / D.V. Gusak, E.V. Karnaukh // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 71–81. — Бібліогр.: 11 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4427 519.21 We consider the almost semicontinuous step-process ξ(t). The conditional characteristic functions of the jumps of ξ(t) have the form E [eiαξk /ξk > 0] = c(c − iα)−1. For such processes, the boundary functionals related to the exit from a finite interval are investigated. en Інститут математики НАН України On the exit from a finite interval for the risk processes with stochastic premiums Article published earlier |
| spellingShingle | On the exit from a finite interval for the risk processes with stochastic premiums Gusak, D.V. Karnaukh, E.V. |
| title | On the exit from a finite interval for the risk processes with stochastic premiums |
| title_full | On the exit from a finite interval for the risk processes with stochastic premiums |
| title_fullStr | On the exit from a finite interval for the risk processes with stochastic premiums |
| title_full_unstemmed | On the exit from a finite interval for the risk processes with stochastic premiums |
| title_short | On the exit from a finite interval for the risk processes with stochastic premiums |
| title_sort | on the exit from a finite interval for the risk processes with stochastic premiums |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4427 |
| work_keys_str_mv | AT gusakdv ontheexitfromafiniteintervalfortheriskprocesseswithstochasticpremiums AT karnaukhev ontheexitfromafiniteintervalfortheriskprocesseswithstochasticpremiums |