Exact non-ruin probabilities in arithmetic case
Using the Wiener-Hopf method, for the model with arithmetic distributions of waiting times Ti and claims Zi in ordinary renewal process, an exact non-ruin probabilities for an insurance company in terms of the factorization of the symbol of the discrete Feller-Lundberg equation, are obtained. The de...
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irk-123456789-45672009-12-08T12:00:39Z Exact non-ruin probabilities in arithmetic case Chernecky, V. Using the Wiener-Hopf method, for the model with arithmetic distributions of waiting times Ti and claims Zi in ordinary renewal process, an exact non-ruin probabilities for an insurance company in terms of the factorization of the symbol of the discrete Feller-Lundberg equation, are obtained. The delayed stationary process is introduced and generating function for delay is given. It is proved that the stationary renewal process in arithmetic case is ordinary if and only if, when the inter-arrival times have the shifted geometrical distribution. A formula for exact non-ruin probabilities in delayed stationary process is obtained. Illustrative examples when the distributions of Ti and Zi are shifted geometrical or negative binomial with positive integer power are considered. In these cases the symbol of the equation is rational functions what allows us to obtain the factorization in explicit form. 2008 Article Exact non-ruin probabilities in arithmetic case / V. Chernecky // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 39-52. — Бібліогр.: 6 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4567 en Інститут математики НАН України |
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Using the Wiener-Hopf method, for the model with arithmetic distributions of waiting times Ti and claims Zi in ordinary renewal process, an exact non-ruin probabilities for an insurance company in terms of the factorization of the symbol of the discrete Feller-Lundberg equation, are obtained. The delayed stationary process is introduced and generating function for delay is given. It is proved that the stationary renewal process in arithmetic case is ordinary if and only if, when the inter-arrival times have the shifted geometrical distribution. A formula for exact non-ruin probabilities in delayed stationary process is obtained. Illustrative examples when the distributions of Ti and Zi are shifted geometrical or negative binomial with positive integer power are considered. In these cases the symbol of the equation is rational functions what allows us to obtain the factorization in explicit form. |
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Chernecky, V. |
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Chernecky, V. Exact non-ruin probabilities in arithmetic case |
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Chernecky, V. |
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Chernecky, V. |
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Exact non-ruin probabilities in arithmetic case |
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Exact non-ruin probabilities in arithmetic case |
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Exact non-ruin probabilities in arithmetic case |
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Exact non-ruin probabilities in arithmetic case |
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Exact non-ruin probabilities in arithmetic case |
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exact non-ruin probabilities in arithmetic case |
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Інститут математики НАН України |
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2008 |
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Exact non-ruin probabilities in arithmetic case / V. Chernecky // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 39-52. — Бібліогр.: 6 назв.— англ. |
| work_keys_str_mv |
AT cherneckyv exactnonruinprobabilitiesinarithmeticcase |
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2025-07-02T07:46:46Z |
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2025-07-02T07:46:46Z |
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1836520472668274688 |
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Theory of Stochastic Processes
Vol.14 (30), no.3-4, 2008, pp.39-52
VASILY CHERNECKY
EXACT NON-RUIN PROBABILITIES IN
ARITHMETIC CASE
Using the Wiener-Hopf method, for the model with arithmetic distri-
butions of waiting times Ti and claims Zi in ordinary renewal process,
an exact non-ruin probabilities for an insurance company in terms of
the factorization of the symbol of the discrete Feller-Lundberg equa-
tion, are obtained. The delayed stationary process is introduced and
generating function for delay is given. It is proved that the stationary
renewal process in arithmetic case is ordinary if and only if, when
the inter-arrival times have the shifted geometrical distribution. A
formula for exact non-ruin probabilities in delayed stationary process
is obtained. Illustrative examples when the distributions of Ti and
Zi are shifted geometrical or negative binomial with positive integer
power are considered. In these cases the symbol of the equation is ra-
tional functions what allows us to obtain the factorization in explicit
form.
1. Introduction
Let F (u) be the distribution function of claims Zj(= Z), a sequence of
independent identically distributed (i.i.d.) random variables with expecta-
tion EZj = μ, K(u) be the distribution of waiting time Tj(= T ), a sequence
of i.i.d. random variables, ETj = 1/α, and c > αμ be the gross premium
rate, j ∈ N. Random variables Zj and Tj are supposed to be mutually
independent.
Regardless of the continuous or discrete distributions, probability of sol-
vency of an insurance company, ϕ(u), with initial capital u, in ordinary
renewal process satisfies the Feller-Lundberg integral equation, [1,4]:
(Lϕ)(u) ≡ ϕ(u) −
∫ ∞
0
dK(v)
∫ u+cv
0
ϕ(u+ cv − z) dF (z) = 0. (1)
2000 Mathematics Subject Classifications. 60G35, 45E10, 62P05.
Key words and phrases. Fundamental equation of the risk theory, ordinary/stationary
renewal process, delayed renewal processes, stationarity, discrete analog of one-sided
Wiener-Hopf integral equation, Riemann boundary-value problem, Wiener-Hopf factori-
zation method.
39
40 VASILY CHERNECKY
Note, that the initial capital u of the company can accept both the
non-negative or negative integer values. The non-ruin probability ϕ(u) in
the latter case means the probability that the company begins the activity
having a debt (−u), however to the moment of the first claim the capital of
the company becomes positive and is hereinafter not below of zero.
Therefore we are interested by the solution ϕ(u) which is a monotone
nondecreasing function of u, satisfying the conditions
ϕ(u) ↗ 1 when u→ +∞, and ϕ(u) ↘ 0 when u→ −∞. (2)
In this work we investigate the case when K(x) and F (x) are arithmetic
distributions with step d = 1, and c ∈ N.
Let m be the space of all real bounded number sequences ϕ = {ϕn}∞−∞
which after introduction of the norm
‖ϕ‖ = sup
n
|ϕn| (3)
turns into the Banach space. Let c be the subspace of all convergent se-
quences ϕ ∈ m, and c0 be the subspace of all sequences convergent to zero.
Introduce the norm in c and c0 by the formula (3) as for m. Then c and
c0 become the Banach spaces.
Let gT (z) =
∑∞
n=1 pnz
n and gZ(z) =
∑∞
n=1 qnz
n be the generating func-
tions of the random variables Z and T , respectively. In the arithmetic case,
we can rewrite (1)-(2) in the discrete form
Aϕ ≡ ϕu −
∞∑
v=1
qv
u+cv∑
k=1
ϕu+cv−kpk = 0, u ∈ Z, c ∈ N, (4)
ϕu ↗ 1 when u→ +∞, and ϕu ↘ 0 when u → −∞. (5)
Similar so called compound binomial model was earlier considered by
A. Melnikov in the monograph [5], which can be interpreted as model with
random variable T having the shifted geometrical distribution with genera-
ting function
gT (z) =
pz
1 − qz
, p+ q = 1, (6)
c = 1, Z has an arbitrary arithmetic distribution with P{Z = 0} = 0 and
α = q. A.Melnikov reduces the solution of the problem to the solution of
a system of infinite number of linear algebraic equations with the Toeplitz
matrix of coefficients using some recurrence relations. The solution of the
problem was received in the terms of a generating function.
In this paper we use the Wiener-Hopf method for the solution of the
problem (4)-(5).
EXACT NON-RUIN PROBABILITIES 41
2. The Wiener-Hopf method in arithmetic case.
We shall seek the solution of the problem (4)-(5) in the form of the
infinite-dimensional vector ϕ = {ϕn}∞−∞.
Let S be the unit circle in the complex plane, S = {t : |t| = 1}, B+ =
{z : |z| < 1}, B− = {z : |z| > 1}. Associate with the vector ϕ the formal
Laurent trasform,
ϕ(t) ∼
+∞∑
k=−∞
ϕkt
k, t ∈ S,
where the series
ϕ+(z) =
+∞∑
k=0
ϕkz
k, −ϕ−(z) =
−∞∑
k=−1
ϕkz
k, (7)
converge at z ∈ B+ and z ∈ B−, respectively, and there exist everywhere
on S, except z = 1, the boundary values ϕ+(t) unbounded at t = 1, and
ϕ−(t) bounded on S (without singularities). It is obvious that
ϕ(t) = ϕ+(t) − ϕ−(t), t ∈ S. (8)
Theorem 1. The symbol a(t) of the operator A is equal
a(t) = 1 − gT (t−c) gZ(t), t ∈ S. (9)
Proof. Using the formulas (8) and going over to the generating functions in
(4), we obtain the homogeneous Riemann boundary-value problem on S
ϕ+(t) − ϕ−(t) − gT (t−c) gZ(t)ϕ+(t) = 0, t ∈ S, (10)
or
ϕ+(t) =
1
a(t)
ϕ−(t), t ∈ S, (11)
where a(t) is given by the formula (9).
Remark the following properties of the symbol.
Property 1. The symbol a(t) is differentiable function on S.
This follows from the existence of the expectations of the random vari-
ables T and Z.
Property 2. The point t = 1 is the unique root of the symbol a(t) on S,
and this root is of the first order.
The notation ‘−ϕ−(t)’ is the tribute to the tradition of the theory of boundary
problem for analytical function to present a function on a contour in the form (8) (the
Sokhotsky-Plemelj formula).
42 VASILY CHERNECKY
Uniqueness follows from the fact that the step of the random variables
T and Z is assumed to be equal 1, [2].
The first order of the root t = 1 follows from the L’Hospital rule and
the condition c > αμ,
lim
t→1
a(t)
t− 1
= lim
t→1
(
g′T (t−c)ct−c−1 gZ(t) − gT (t−c) g′Z(t)
)
=
c
α
− μ > 0.
So, the Riemann problem (11) is singular (non-Noether).
Property 3.
|1 − a(t)| ≤ 1 for t ∈ S. (12)
The inequality (12) follows from the property of the generating functions.
Geometrical sense of this inequality is that the plot of the symbol a(t), t ∈ S,
is situated in the circle of the radius 1 with the center in the point z = 1
and is tangent to the axis of ordinates at the point z = 0.
Let [arg a(t)]S be the increment of the argument of a(t)] when t passes
S in positive direction (counter-clockwise).
Property 4. The function b(t) = a(t)/(1 − t), t ∈ S, has not the roots on
S and
indS b(t) =
1
2π
[arg b(t)]S = −1, (13)
Proof. On the one hand we have
indSa(t) = −1
2
. (14)
Really, let Γ be the plot of the function a(t). Then(
a(eiτ )
)′ |τ=0 = i(c/α− μ),
what means that the point a(t) = a(eiτ ) on the plot Γ bypasses through the
origin of coordinates in negative direction when τ ∈ [0, 2π] bypasses from
0 to 2π. From this follows (14), since the a(t) at t = 1 has the root of the
first order and the plot of function a(t) is smooth.
On the other hand we have
indSa(t) = indS(1 − t) + indSb(t) =
1
2
+ indSb(t),
whence, in view of (14), we have (13).
From the above Properties and results of [3,6] follows
Property 5. For the symbol a(t) the factorization exists
a(t) = (1 − t) a+(t) t−1 a−(t), (15)
EXACT NON-RUIN PROBABILITIES 43
where
a+(z)
= 0, |z| ≤ 1, a+(1) = 1,
a−(z)
= 0, |z| ≥ 1, a−(1) =
c
α
− μ.
The formulas for a±(t) are given in [3] in terms of the Cauchy type in-
tegrals. The value a−(1) = c
α
− μ is obtained from the condition a+(1) = 1
by the L’Hospital rule applied to the symbol a(t) at t = 1.
Theorem 2. The equation (4) has two linear independent solution generated
by the factorization (12), one in the space c+ and another (irrelevant) in c0
+.
Proof. Proof follows from the solvability of discrete Wiener-Hopf equa-
tion with the symbol of entire order, [6], Theorem 1.5.
The desired solution of the problem (4)-(5) is generated by the functions
ϕ+(z) =
1
(1 − z) a+(z)
, z ∈ B+, −ϕ−(z) =
a−(z)
z
, z ∈ B−,
from which the probabilities ϕu are obtained by the expansions of the func-
tions ϕ+(z) and −ϕ−(z) into a Taylor series about z and 1/z, respectively.
Note that −ϕ−(1) = c
α
− μ.
The second irrelevant solution of the equation (4) is generated by the
functions
φ+(z) =
1
a+(z)
, |z| ≤ 1, −φ−(z) =
(1 − z)a−(z)
z
, |z| ≥ 1.
Note that we can attach to the solution φ+(z) − φ−(z) certain probabi-
listic sense. Really, the following obvious statement takes place.
Let g(z) be the generating function of a random variable X. Then the
generating function of the distribution function of X is given by the formula
G(z) =
g(z)
1 − z
.
Comparing two solutions of the Riemann boundary problem we see that
they are connected by the relations
ϕ±(z) =
φ±(z)
1 − z
, z ∈ B±.
The solution ϕ(t) = ϕ+(t) − ϕ−(t), t ∈ S, can be considered as a
generating function for the distribution function of some imaginary ran-
dom variable X. The component ϕ+(t) can be interpreted as a genera-
ting function for the distribution function of the random variable X+ with
44 VASILY CHERNECKY
P{X+ = n} = ϕn, n = 0, 1, . . .. Then φ+(t) is a generating function of the
random variable X+ with
P{X+ = n} = φ+
n = ϕn−ϕn−1, n ∈ N, P{X+ = 0} = φ+
0 = ϕ0−0 = ϕ0.
Note that
∑∞
n=0 φ
+
n = a+(1) = 1. Namely like this we selected the factor
a+(t) in (16). To the component −ϕ−(t) we can not attach any probabilistic
sense. The coefficient of the decomposition of this function into power series
about 1/z are
φ−
−n = ϕ−n − ϕ−n−1, n ∈ N, φ−
0 = −ϕ−1(< 0!).
But if we consider the function φ(t) = φ+(t) − φ−(t) having the formal
decomposition into the Laurant series
φ(t) =
+∞∑
n=−∞
φnt
n =
+∞∑
n=−∞
(ϕn − ϕn−1)t
n, t ∈ S,
we obtain the generating function of the random variable X. Here
φ±n = φ±
n , n ∈ N, φ0 = φ+
0 + φ−
0 = ϕ0 − ϕ−1.
Note that analogous fact takes place also in the non-arithmetic case.
In that case the equation (1) also has two linear independent solutions
connected by the relation
φ(u) = φ+(u) − φ−(u) = ϕ′(u) = ϕ+′
(u) − ϕ−′
(u), u ∈ R,
[6], Theorem 2.3. So ϕ(u) and φ(u) can be interpreted as the distribution
function and density of some imaginary random variable X, respectively.
3. Delayed renewal processes and stationarity
in arithmetic case.
In addition to the sequence Tn there is defined a non-negative lat-
tice variable S0 with a generating function g0(z) =
∑∞
n=1 rnz
n. The vari-
ables Sn = S0 + T1 + . . . + Tn are called renewal epochs, [2]. The re-
newal process {Sn} is called delayed if S0
= 0. The expected number
V (m) =
∑∞
k=0 P{Sk ≤ m} of renewal epochs in [0, m] , m ∈ N, has the
generating function
gV (z) =
g0(z)
(1 − z)(1 − gT (z)
. (16)
a+(t) in (16) is selected up to constant factor C
= 0. If a+(t) is multiplied by C,
then a−(t) is divided into C.
The derivatives are understood in the generalized sense.
EXACT NON-RUIN PROBABILITIES 45
The expected number of renewal epochs within [m,m+1], m ∈ N, tends
to α,
V (m+ 1) − V (m) → α.
It follows that V (m) ∼ αm asm→ ∞. It is natural to ask whether g0(z) can
be chosen as to get the identity V (m) = αm, m ∈ N, meaning a constant
renewal rate. Noticing that the generating function for the sequence {αm}
equals
gV (z) =
αz
(1 − z)2
,
from (16) we obtain that g0(z) satisfies now the equation
gV (z) =
1
1 − z
g0(z) + gT (z) gV (z)
and thus equals
g0(z) = (1 − gT (z)) gV (z) (1 − z) =
αz(1 − gT (z))
1 − z
. (17)
This g0(z) is a generating function of a probability distribution and so
the answer is affirmative:
With the initial random variable S0 having the generating function (17)
the renewal rate is constant, V (m) = αm, m ∈ N.
The following statement takes place:
The stationary renewal process in arithmetic case is ordinary if and only
if, when the inter-arrival times Tn have the shifted geometrical distribution
(6).
This shifted geometrical distributions is analog of exponential distribu-
tion for Tn in non-arithmetic case.
For the problem (4)-(5) we introduce the accompanying delayed stationa-
ry renewal process {Sn} with generating function for S0, given by (17). Then
the generating function ϕs(z) for non-ruin probabilities of the company in
such process is built as follows
ϕs(t) = g0(t
−c) gZ(t)ϕ+(t) =
αt−c(1 − gT (t−c)) gZ(t)ϕ+(t)
1 − t−c
=
=
α(1 − gT (t−c)) gZ(t)ϕ+(t)
tc − 1
, t ∈ S, (18)
similarly as it is done in the non-arithmetic case.
Solve the jump problem for ϕs(t), [3],
ϕs(t) = ϕ+
s (t) − ϕ−
s (t),
where ϕ+
s (t) is the boundary value on S of an analytic function in B+,
having the pole of the first order at t = 1, and ϕ−
s (t) is the boundary
46 VASILY CHERNECKY
value on S of an analytic function in B−, lacking the singularities on S
and ϕ−
s (∞) = 0. Whence the probabilities ϕsn, n ∈ Z, are obtained by the
decompositions of the functions ϕ+
s (z) and −ϕ−
s (z) into power series about
z and 1/z, respectively.
Using (10) we can simplify the formula (18), excluding gT (t−c),
ϕs(t) =
α(1 − gZ(t))ϕ+(t) − αϕ−(t)
1 − tc
, t ∈ S. (19)
The case c = 1. Multiplying both sides of the equation (19) on (1 − t),
(1 − t)ϕs(t) = α(1 − gZ(t))ϕ+(t) − αϕ−(t), t ∈ S, (20)
and project both parts of this equation on the space of the functions Φ−(z)
holomorphic in B−, Φ−(∞) = 0, having a boundary values on S without
singularities. Equating the coefficients in the decompositions of these pro-
jections into the power series about 1/z, we obtain the system of equations
for ϕs−n, n ∈ N, ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
ϕs−1 − ϕs−2 = αϕ−1
ϕs−2 − ϕs−3 = αϕ−2
· · · · · · · · · · · · · · · · · · · · · ·
ϕs−n − ϕs−(n+1) = αϕ−n
· · · · · · · · · · · · · · · · · · · · · ·
Summing these equations, we obtain
ϕs−1 = α
−∞∑
n=−1
ϕn = −αϕ−(1) = α
(
c
α
− μ
)
= 1 − αμ, (c = 1),
and the values ϕs−n form the geometric progression with the denominator
1 − α, ϕs−(n+1) = (1 − α)ϕs−n, n ∈ N.
Note that in non-arithmetic case 1 − αμ is the value of ϕs(0).
Using the value ϕs−1, we can construct the truncated form ϕ+
s (z) of ϕs(z).
Observing that the right part of (20) can be considered as the generating
function of some imaginary random variableX with P{X = n} = ϕsn−ϕsn−1,
n ∈ Z, ϕ+
s (z) as the truncation of ϕs(z) that is the generating function of
distribution function of X, and projecting both parts of the equation (20)
on the space of the functions Φ+
0 (z) holomorphic in B+, we obtain
ϕ+
s (z) =
ϕs−1 + α(1 − gZ(z))ϕ+(z)
1 − z
, z ∈ B+. (21)
Here the summand ϕs−1 appears to annihilate the corresponding term in the
expression ϕs0 − ϕs−1 = P{X = 0}, if we wish to remain in the space of
analytical functions in B+.
EXACT NON-RUIN PROBABILITIES 47
Assume now that the inter-arrival times Tn have the shifted geometrical
distribution (6), i.e., the initial renewal process is stationary and ϕs(t) =
ϕ(t). Using (21) and ϕ+(z) = ϕ+
s (s) we obtain the equation for ϕ+
s (z)
ϕ+
s (z) =
ϕs−1 + α(1 − gZ(z))ϕ+
s (z)
1 − z
, z ∈ B+.
whence
ϕ+
s (z) =
ϕs−1
1 − z − α(1 − gZ(z))
, z ∈ B+. (22)
In [5] the formula for ϕ+
s (z) is written in some cumbersome form.
Project both parts of the equation (20) on the space of the functions
Φ−(z) holomorphic outside of S, Φ−(∞)
= 0, having a boundary values on
S without singularities. Equating the coefficients in the decompositions of
these projections into the power series about 1/z, we obtain the series of
equations for ϕs−n, n ∈ N ∪ {0},⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
ϕs0 − ϕs−1 = αϕs0
ϕs−1 − ϕs−2 = αϕs−1
· · · · · · · · · · · · · · · · · · · · · ·
ϕs−n − ϕs−(n+1) = αϕs−n
· · · · · · · · · · · · · · · · · · · · · ·
Summing these equations, we obtain
ϕs0 = αϕs0 − αϕ−(1),
whence we receive the formula derived by A.Melnikov [5]
ϕs0 =
1 − αμ
1 − α
=
1 − qμ
1 − q
.
The case c > 1. Multiplying both sides of the equation (19) on (1 − tc),
(1 − tc)ϕs(t) = α(1 − gZ(t))ϕ+(t) − αϕ−(t), t ∈ S. (23)
Observing that the right part of (23) can be considered as the distinctive
generating function of some imaginary random variable X with P{X =
n} = ϕsn − ϕsn−c, n ∈ Z, and ϕ+
s (z) as the truncation of ϕs(z) that is the
generating function of distribution function of X. Projecting both parts of
the equation (20) on the space of the functions Φ+
0 (z) holomorphic in B+,
we obtain
ϕ+
s (z) =
ϕs−1z
c−1 + ϕs−2z
c−2 + · · ·+ ϕs−c + α(1 − gZ(z))ϕ+(z)
1 − zc
, z ∈ B+.
(24)
48 VASILY CHERNECKY
This formula is the generalization of (22). Here the summand ϕs−1z
c−1 +
ϕs−2z
c−2 + · · ·+ϕs−c appears to annihilate corresponding subtrahends in the
generating function for the distribution function of the imaginary random
variable X, if we wish to remain in the space of analytical functions in B+.
Unfortunately we have not succeeded to obtain the explicit formulas
for ϕs−1, ϕ
s
−2, · · · , ϕs−c. Their numerical values it is possible obtain from the
decomposition of ϕ−
s (z) into power series about 1/z. Note only that using
the reasoning as at the derivation of the formula for ϕs−1 in the case of c = 1,
we can receive the expression for the sum
ϕs−1 + ϕs−2 + · · ·+ ϕs−c = c− αμ.
Then the sums of the form
ϕsnc−1 + ϕsnc−2 + · · ·+ ϕsnc−c, n ∈ N,
makes up the geometrical progression with the denominator 1 − α.
Assume now that the inter-arrival times Tn have the shifted geometrical
distribution (6). ϕs(t) = ϕ(t).
Using (24) and ϕ+(z) = ϕ+
s (s) we obtain the equation for ϕ+
s (z)
ϕ+
s (z) =
ϕs−1z
c−1 + ϕs−2z
c−2 + · · · + ϕs−c + α(1 − gZ(z))ϕ+
s (z)
1 − zc
, z ∈ B+.
whence
ϕ+
s (z) =
ϕs−1z
c−1 + ϕs−2z
c−2 + · · ·+ ϕs−c
1 − zc − α(1 − gZ(z))
, z ∈ B+. (25)
This formula is the generalization of (22).
4. Examples.
Especially simply the Wiener-Hopf method works when the random vari-
ables Tk and Zk are shifted uniform discrete, binomial, geometrical, negative
binomial (with entire exponent). In these cases the symbol a(t) is a ratio-
nal function that allows us obtain its factorization in explicit form. For
example, in the case of the shifted Poisson distribution, the problem of fac-
torization is reduced to solution of some transcendental equation.
Example 1. Consider from the point of view of the Wiener-Hopf method
the example examined by A. Melnikov [5].
Let T be shifted geometric random variable (6), Z constant variable
with the generating functions
gT =
z
5 − 4z
, gZ = z2,
EXACT NON-RUIN PROBABILITIES 49
respectively, and c = 1. Then
α = 0.2, μ = 2, ϕ−1 = 0.6, ϕ0 = 0.75,
a(t) =
(1 − t)(4 − t)
5t− 4
= (1 − t)
4 − t
3
t−1 3
5 − 4
t
, t ∈ S.
Here
a+(t) =
4 − t
3
, a−(t) =
3
5 − 4
t
, t ∈ S.
ϕ+(z) = ϕ+
s (z) =
3
(1 − z)(4 − z)
, z ∈ B+,
−ϕ−(z) = −ϕ−
s (z) =
3
5z − 4
, z ∈ B−.
The formula (22) for ϕ+
s (z) gives the same result.
Expanding the functions ϕ+(z) and −ϕ−(z) in the series about z and
1/z, respectively, we obtain the solution of the problem (4)-(5).
The following table gives the probabilities of solvency and the results for
irrelevant solution with accuracy 0.000001.
u ϕu φu u ϕu φu
-9 .100663 .201324e-1 1 .937500 .187500
-8 .125829 .251658e-1 2 .984375 .468750e-1
-7 .157286 .314574e-1 3 .996094 .117188e-1
-6 .196608 .393216e-1 4 .999023 .292969e-2
-5 .245760 .491520e-1 5 .999756 .732422e-3
-4 .307200 .614400e-1 6 .999939 .183105e-3
-3 .384000 .768000e-1 7 .999985 .457764e-4
-2 .480000 .960000e-1 8 .999996 .114441e-4
-1 .600000 .120000 9 .999999 .286102e-5
0 .750000 .150000 10 .999999 .715256e-6
Here φ0 = φ+
0 + φ−
0 = ϕ0 − ϕ−1 = .750000 − .600000 = .150000.
Our results coincides with the ones of A. Melnikov if take into account
that in [4] the probabilities 1 − ϕu are computed only for u = 0, 1, . . . , 10.
Example 2. Let T be shifted negative binomial variable, Z be shifted
geometric random variable with the generating functions
gT =
16z
(5 − z)2
, gZ =
9z
10 − z
,
respectively, and c = 1. Then
α =
2
3
, μ =
10
9
, a(t) =
(25t2 − 91t+ 10)(t− 1)
(5t− 1)2(t− 10)
,
50 VASILY CHERNECKY
a+(t) =
.280730(t− 10)
(t− 3.52658)
, a−(t) =
7.01825t(t− .11342)
(5t− 1)2
, t ∈ S.
ϕ+(z) =
(z − 3.52657)
(1 − z).280730(z − 10)
, z ∈ B+,
−ϕ−(z) =
7.01825(z − .11342)
(5z − 1)2
, z ∈ B−.
The generating function ϕs(z) for non-ruin probabilities in the delayed sta-
tionary renewal process is built as follows
ϕs(t) =
1.68438(1 − 25t)t
(5t− 1)2(1 − t)(t− 3.52658)
, t ∈ S.
Expanding this function in the sum of partial fractions and using the So-
khotski-Plemelj decomposition we obtain
ϕ+
s (z) =
1
1 − z
+
.740738
z − 3.52658
, z ∈ B+,
−ϕ−
s (z) =
.4(.648149z − .0789956)
z2 − .4z + .04
, z ∈ B−.
Expanding the functions ±ϕ±(z) and ±ϕ±
s (z) in the series about z and 1/z,
respectively, we obtain the solution of the problem (4)-(5). The following
table gives the probabilities of solvency in ordinary and stationary processes
with accuracy 0.000001.
u ϕu ϕsu u ϕu ϕsu
-9 .320745e-5 .27377e-5 1 .942163 .940440
-8 .144817e-4 .1239e-4 2 .983598 .983111
-7 .646313e-4 .5548e-4 3 .995348 .995211
-6 .284269e-3 .24499e-3 4 .998680 .998642
-5 .122691e-2 .10629e-2 5 .999624 .999615
-4 .516238e-2 .4504e-2 6 .999892 .999891
-3 .209510e-1 .1847e-1 7 .999968 .999969
-2 .804505e-1 .7210e-1 8 .999989 .999991
-1 .280730 .259260 9 .999995 .999997
0 .796041 .789956 10 .999997 .999999
Example 3. Let T be shifted negative binomial variable, Z be shifted
geometric random variable with the generating functions
gT =
81z
(10 − z)2
, gZ =
z
2 − z
,
respectively, and c = 2. Then α = 9
11
, μ = 2,
a(t) =
(20t3 − 16t2 − 11t− 2)(5t− 1)(t− 1)
(10t2 − 1)2(−2 + t)
, t ∈ S,
EXACT NON-RUIN PROBABILITIES 51
a+(t) =
.280730(t− 10)
(t− 3.52658)
, a−(t) =
7.01825t(t− .11342)
(5t− 1)2
, t ∈ S.
ϕ+(z) =
.2875080(z − 2)
(1 − z)(z − 1.287508)
,
−ϕ−(z) =
.2875080(5z − 1)(z + .2437537 − .1351057I)(z + .2437537 + .1351057I)
(10z2 − 1)2
.
The generating function ϕs(t) for non-ruin probabilities in the delayed sta-
tionary renewal process is built as follows
ϕs(t) =
.2352338(1− 100t2)t
(10t2 − 1)2(1 − t)(t− 1.287508)
, t ∈ S.
Expanding this function in the sum of partial fractions and using the So-
khotski-Plemelj decomposition we obtain
ϕ+
s (z) =
1
1 − z
+
.7153470
z − 1.287508
,
−ϕ−
s (z) =
62.01608
z + .3162917
− 61.97134
z + .3161638
− 94.09140
z − .3160941
+
94.33134
z − .3163616
.
Expanding the functions ±ϕ±(z) and ±ϕ±
s (z) in the series about z and 1/z,
respectively, we obtain the solution of the problem (4)-(5). The following
table gives the probabilities of solvency in ordinary and stationary processes
with accuracy 0.0000001.
u ϕu ϕsu u ϕu ϕsu
-2 .4133037e-2 .7898426e-1 10 .9557890 .9556118
-1 .1437540e-1 .2846521 11 .9656615 .9655239
0 .4466115 .4443941 12 .9733296 .9732226
1 .5701864 .5684641 13 .9792853 .9792022
2 .6661664 .6648286 14 .9839110 .9838465
3 .7407135 .7396743 15 .9875037 .9874536
4 .7986136 .7978065 16 .9902943 .9902553
5 .8435844 .8429575 17 .9924615 .9924314
6 .8785129 .8780260 18 .9941449 .9941215
7 .9056416 .9052635 19 .9954525 .9954342
8 .9267125 .9264187 20 .9964679 .9964538
9 .9430780 .9428498 21 .9972566 .9972457
Note that
ϕs−1 + ϕs−1 = c− αμ = 2 − 9
11
· 2 ≈ .3636364.
The computation carried out in Maple-V.
52 VASILY CHERNECKY
References
1. Bühlmann, H., Mathematical Methods in Risk Theory, Springer-
Verlag, 1970.
2. Feller, W., An Introduction to Probability Theory and its Applications,
John Wile & Sons, Inc., V.2, 1966.
3. Gakhov, F.D. and Chersky, Yu.I., Equations of convolution type,
(Russian) Nauka, Moskow, 1978.
4. Grandell, J., Aspects of Risk Theory, Springer-Verlag, 1991.
5. Melnikov, A., Risk Analysis in Finance and Insurance, Chapman &
Hall/CRC, 2003.
6. Prößdorf, S., Einige Klassen singulärer Gleichungen, Akademie-
Verlag-Berlin, 1974.
Department of Higher Mathematics,
Odessa State Academy of Refrigeration,
65026 Odessa, Ukraine
E-mail: chern.va@paco.net
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