Projective method for equation of risk theory in the arithmetic case
We consider a discrete model of operation of an insurance company whose initial capital can take any integer value. In this statement, the problem of nonruin probability is naturally solved by the Wiener-Hopf method. Passing to generating functions and reducing the fundamental equation of risk the...
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Institute of Mathematics, NAS of Ukraine
2013
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| author | Chernetskii, V. A. Чернецький, В. А. |
| author_facet | Chernetskii, V. A. Чернецький, В. А. |
| author_sort | Chernetskii, V. A. |
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| datestamp_date | 2020-03-18T19:15:36Z |
| description | We consider a discrete model of operation of an insurance company whose initial capital can take any integer value.
In this statement, the problem of nonruin probability is naturally solved by the Wiener-Hopf method.
Passing to generating functions and reducing the fundamental equation of risk theory to a Riemann boundary-value problem on the unit circle,
we establish that this equation is a special one-sided discrete Wiener-Hopf equation whose symbol has a unique zero, and, furthermore, this zero is simple.
On the basis of the constructed solvability theory for this equation, we justify the applicability of the projective method to the approximation of ruin probabilities in the spaces $l^{+}_1$ and $\textbf{c}^{+}_0$.
Conditions for the distributions of waiting times and claims under which the method converges are established.
The delayed renewal process and stationary renewal process are considered, and approximations for the ruin probabilities in these processes are obtained. |
| first_indexed | 2026-03-24T02:23:29Z |
| format | Article |
| fulltext |
UDC 368.01; 517.44; 519.6
V. A. Chernecky (Odessa State Acad. Refrigeration, Ukraine)
PROJECTIVE METHOD FOR EQUATION OF RISK THEORY
IN THE ARITHMETIC CASE
ПРОЕКТИВНИЙ МЕТОД ДЛЯ РIВНЯННЯ ТЕОРIЇ РИЗИКУ
В АРИФМЕТИЧНОМУ ВИПАДКУ
We consider a discrete model of operation of an insurance company whose initial capital can take any integer value. In
this statement, the problem of nonruin probability is naturally solved by the Wiener – Hopf method. Passing to generating
functions and reducing the fundamental equation of risk theory to a Riemann boundary-value problem on the unit circle,
we establish that this equation is a special one-sided discrete Wiener – Hopf equation whose symbol has a unique zero,
and, furthermore, this zero is simple. On the basis of the constructed solvability theory for this equation, we justify the
applicability of the projective method to the approximation of ruin probabilities in the spaces l+1 and c+0 . Conditions for the
distributions of waiting times and claims under which the method converges are established. The delayed renewal process
and stationary renewal process are considered, and approximations for the ruin probabilities in these processes are obtained.
Розглядається дискретна модель функцiонування страхової компанiї, початковий капiтал якої може набувати довiль-
ного цiлого значення. У такiй постановцi проблема обчислення ймовiрностi стiйкостi компанiї природно розв’язу-
ється методом Вiнера – Хопфа. При переходi до твiрних функцiй i зведеннi фундаментального рiвняння теорiї ризику
до граничної задачi Рiмана на одиничному колi з’ясовано, що розглядуване рiвняння є особливим одностороннiм
дискретним рiвнянням Вiнера – Хопфа, символ якого має єдиний нуль i цей нуль є простим. На базi побудованої
теорiї розв’язностi цього рiвняння обґрунтовано застосування проективного методу до апроксимацiї ймовiрно-
стей банкрутства у просторах l+1 i c+0 . Отримано умови на розподiли часiв очiкування вимог i розмiрiв виплат
для збiжностi методу. Розглянуто процес вiдновлення iз запiзненням i стацiонарний процес вiдновлення, а також
наближення для ймовiрностей банкрутства у цих процесах.
1. Introduction. Consider the discrete ordinary renewal model for functioning of an insurance
company in the arithmetic1 case:
Assumptions. The renewal model is given by following conditions:
(a) the claim sizes {Zn}n∈N are positive integer-valued independent identically distributed (iid)
random variables (rvs), having common generating function gZ(z) =
∑∞
n=1
qnz
n and finite mean2
µ = EZ1;
(b) the inter-arrival times {Tn}n∈N are positive integer-valued iid rvs having common generating
function gT (z) =
∑∞
n=1
pnz
n and finite mean ET1 = 1/α;
(c) the gross premium rate c > αµ, c ∈ N;
(d) the sequences {Zn}n∈N and {Tn}n∈N are independent of each other.
Let FZ(v) and FT (v) be the distributions of the random variables Z1 and T1, respectively. It is
known that probability of solvency of the insurance company, ϕ(u), with initial capital u ≥ 0 in
ordinary renewal process, in general case, satisfies the fundamental equation of risk theory [2, 6, 14]:
ϕ(u)−
∞∫
0
dFT (v)
u+cv∫
0
ϕ(u+ cv − w) dFZ(w) = 0, u ∈ R+. (1)
1In the literature it is also used the term “periodic” [1].
2We denote by N the set of positive integers.
c© V. A. CHERNECKY, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4 565
566 V. A. CHERNECKY
At the derivation of this equation, nothing interferes us to consider u to be also negative. Really,
the insurance company can start its activity having a debt, u < 0, i.e., being in the state of ruin.
At the favorable concurrence of circumstances, the company can leave this state for the moment of
arrival of first claim and more in it can return not. It happens, for example, when the random variable
u+ cT1−Z1 accepts a nonnegative value and hereinafter the company will not be ruined. Therefore,
it is reasonable to state the more general problem on the calculation of the nonruin probability for the
company with initial capital u ∈ R, being in the state of nonruin later on, that is, we are interested
by the probability ϕ(u) of the following event:
u+
n∑
k=1
(cTk − Zk) ≥ 0 ∀n ∈ N, u ∈ R.
In such setting it is naturally to apply the Wiener – Hopf method [8, 16] for the solution of this
problem.
For the nonruin probability ϕ(u), u ∈ R, introduce in consideration the probabilities ϕ±(u) by
the formulas
ϕ±(u) = H(±u)ϕ(u), ϕ(u) = ϕ+(u) + ϕ−(u),
where H(u) is the Heaviside function.
Following Feller, [2, 6, 13, 14], derive the equation for ϕ−(u),
ϕ−(u)−
∞∫
0
dFT (v)
u+cv∫
0
ϕ+(u+ cv − w) dFZ(w) = 0, u ∈ R−. (2)
Here we take into account that ϕ+(u) ≡ 0 for u < 0, and therefore integration in internal integral,
as a matter of fact, is over a set on which u+ cv − w > 0.
Joining the equations (1) and (2), the equation for ϕ±(u), u ∈ R, can be written in the form of
one equation,
ϕ+(u) + ϕ−(u)−
∞∫
0
dFT (v)
u+cv∫
0
ϕ+(u+ cv − w) dFZ(w) = 0, u ∈ R. (3)
We are interested by the solution ϕ(u) satisfying the conditions
ϕ+(u)↗ 1 when u→ +∞, and ϕ−(u)↘ 0 when u→ −∞. (4)
Denote by Z the set of integers, Z+ = {0, 1, 2, . . . }, Z− = {−1,−2, . . . }.
Consider the linear space of all two-sided sequences of complex numbers ξ = {ξn}u∈Z. Denote
by l1 the Banach space of all sequences of complex numbers ξ = {ξn}n∈Z with the finite norm
‖ξ‖l1 =
∞∑
n=−∞
|ξn| <∞
and by c the space of all convergent sequences ξ which after introduction of the norm
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
PROJECTIVE METHOD FOR EQUATION OF RISK THEORY IN THE ARITHMETIC CASE 567
‖ξ‖c = sup
n
|ξn| <∞
also becomes the Banach space. Let c0 ⊂ c be the subspace of all sequences convergent to zero.
Let E be either l1 or c0 and let m be some real number. Denote by Em the space of all numerical
sequences of the form f = {(1 + |n|−mξn)}n∈Z , where ξ ∈ E. Introduction of the norm |f | = ‖ξ‖E
converts Em in the Banach space isometric and isomorphic to E [16]. In the case when m is a
positive integer, the sequence f = {fn}n∈Z belongs to Em if and only if h(k) = {nkfn}n∈Z ∈ E,
k = 0, 1, . . . ,m.
Each of the spaces Em has two distinguished subspaces: E+
m is the subspace of sequences ξ+ =
= {ξ+n }n∈Z+ (or {ξn}n∈Z+) characterized by the condition ξ+n = 0 for n ∈ Z−, and E−m is the
subspace of sequences ξ− = {ξ−n }n∈Z (or {ξn}n∈Z−) for which ξ−n = 0, n ∈ Z+.
Let T = {t ∈ C : |t| = 1} denote the complex unit circle, B+ = {z : |z| < 1} the complex unit
disk and B− = {z : |z| > 1} the complementary disk to B+ ∪ T.
For the space Em of sequences ξ = {ξn}n∈Z denote by Êm the space of generating functions
(called also Laurent or Fourier transforms) of the form
Ξ(t) =
∞∑
n=−∞
ξnt
n, t ∈ T, (5)
being generalized functions on T (Schwartz distributions) [9].
Denote by W the Wiener algebra of all functions of the form (5) on T, expanding in absolutely
convergent Fourier series. Let Wm, 0 ≤ m < ∞, be the algebra of all functions of the form (5) for
which {ξn}n∈Z ∈ l1,m, W0 = W. As it is known, for m ∈ N, the algebra Wm contains any m times
differentiable function Ξ(t) the derivatives of which belong to W [16]. In what follows, it will be
useful for us the fact that if the point t0 ∈ T is zero of the order m for Ξ(t) ∈ Wm, then Ξ(t) is
representable in the form Ξ(t) = (t− t0)mb(t) with b(t) ∈W [16].
Denote by Ê+
m the subspace of Êm, consisting, generally speaking, of the Schwartz distributions
of the form Ξ+(t) =
∑∞
n=0
ξnt
n, t ∈ T, which are the boundary values of analytic functions in B+,
expandable into a Taylor series about z,
Ξ+(z) =
∞∑
n=0
ξnz
n, z ∈ B+,
and let Ê−m be the subspace of Êm, consisting, generally speaking, of the Schwartz distributions
Ξ−(t) =
∑−1
n=−∞
ξnt
n, t ∈ T, being the boundary values of analytic functions in B−, expandable
into a Taylor series about 1/z,
Ξ−(z) =
−1∑
n=−∞
ξnz
n, z ∈ B−.
We say that Ξ(t) = Ξ+(t) + Ξ−(t) ∈ {Ê+
m, Ê
−
m} if Ξ+(t) ∈ Ê+
m and Ξ−(t) ∈ Ê−m, t ∈ T.
Consider the sequence of nonruin probabilities ϕ = {ϕn}n∈Z ∈ c. Following Feller, [2, 6, 13, 14],
going over to the sums in the repeated Stieltjes integral in (3), the problem (3), (4) in the arithmetic
case can be rewritten in the discrete form
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
568 V. A. CHERNECKY
Aϕ := ϕ+
u + ϕ−u −
∞∑
v=1
qv
u+cv∑
k=1
pkϕ
+
u+cv−k = 0, u ∈ Z, (6)
ϕ+
u ↗ 1 when u→ +∞, and ϕ−u ↘ 0 when u→ −∞. (7)
Here we take into account that ϕ+
u = 0 for u ∈ Z−, and therefore summation in internal sum, as a
matter of fact, is proceeded on a set on which u+cv−w ∈ Z+. As we will see, the second condition
in (7) is a direct consequence of the first one. We will say that the vector ϕ = ϕ+ + ϕ− belongs to
the class {c+, c−0 }, if ϕ+ ∈ c+ and ϕ− ∈ c−0 . Thus, we shall seek the solution of the problem (6),
(7) in the space ϕ ∈ {c+, c−0 }.
That problem, called compound binomial model, was earlier considered in the monograph of
A. Mel’nikov [15] only for stationary process and only in the case when c = 1. This model can be
interpreted as a model with inter-arrival times Tn having the shifted geometrical distribution with
generating function
gT (z) =
qz
1− (1− q)z
, 0 < q = α < 1. (8)
A. Mel’nikov reduces the solution of such problem to the solution of infinite system of linear algebraic
equations with a Toeplitz matrix of coefficients using some recurrence relations. The solution of the
problem is received in the terms of a generating function only for u ∈ Z+.
In the present paper, going over to generating functions and reducing the equation (6) to a Rie-
mann boundary-value problem on the unit circle T, we will see that this equation turns out to be
nonnormal3 one-sided discrete Wiener – Hopf equation. Nonnormality of the equation (6) imposes
some difficulties on the construction of its solvability theory and additional restrictions on the distri-
butions of Tn and Zn for the convergence of the projective method for the approximate solution of
the problem. Using the Wiener – Hopf method, the solvability theory for this equation is constructed,
on the base of which the applicability of the projective method is justified, and the conditions on
distributions of the waiting times and claims are obtained for the convergence of the method in the
spaces l+1 and c+0 . Illustrative example is given.
The paper is organized as follows. In Section 2, we investigate solvability of the problem (6), (7)
reducing the equation (6) to a Riemann boundary-value problem which is solved by the factorization
method, and obtain exact solution of the problem (6), (7) in terms of the generating functions.
In Section 3, the formulas for nonruin probabilities in accompanying delayed renewal (stationary)
process are also given in terms of generating functions. Solvability theory of the problem for delayed
ordinary renewal process in arithmetic case slightly differs of that in the nonarithmetic case. It
concerns the value ϕs(0) = 1 − αµ
c
in the nonarithmetic case, which, in arithmetic case for c = 1,
is accepted for the value u = −1, and, in the general case, when c ∈ N the values ϕsu, u =
= −1,−2, . . . ,−c, are expressed in the terms of c th roots of unity. The formulas for the solution
of the problem (6), (7) in the stationary case are also given. Earlier some results on given problem
was announced by the author in [3]. Section 4 presents the results from [16] on the convergence of
projective method for degenerated discrete Wiener – Hopf equation, and in Section 5, relying on these
3In functional analysis, the terms ‘non-Noetherian’, ‘singular’, ‘nonelliptic’, or ‘degenerated’ are also used.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
PROJECTIVE METHOD FOR EQUATION OF RISK THEORY IN THE ARITHMETIC CASE 569
results, we obtain sufficient conditions on the distributions of random variables Tn and Zn for the
convergence of the projective method in the spaces l+1 and c+0 . Approximation for ruin probabilities
in delayed process is also given. Illustrative example is considered.
2. Solvability of the fundamental equation. Consider the sequence of nonruin probabilities
ϕ = {ϕn}n∈Z ∈ {c+, c−0 }, and define the generating functions Φ(t) ∈ {ĉ+, ĉ−0 }, t ∈ T, for the
sequence ϕ by the formula
Φ(t) =
+∞∑
u=−∞
ϕut
u, t ∈ T,
considered as a generalized function on T. In reality, we are interested by the convergent series
Φ+(z) =
+∞∑
u=0
ϕuz
u, z ∈ B+, Φ−(z) =
−1∑
u=−∞
ϕuz
u, z ∈ B−.
Theorem 1. If the conditions of Assumptions is fulfilled, then the symbol A(t) of the operator
A in (6) is given by the formula
A(t) = 1− gT (t−c) gZ(t) ∈W1, t ∈ T. (9)
Proof. Going over to generating functions, reduce the equation (6) to a Riemann boundary-value
problem. We have
Φ+(t) + Φ−(t)−
∞∑
u=−∞
( ∞∑
v=1
qv
u+cv∑
k=1
pkϕ
+
u+cv−k
)
tu = 0, t ∈ T.
Interchanging order of summation in the sum term, we obtain
∞∑
u=−∞
( ∞∑
v=1
qv
u+cv∑
k=1
pkϕ
+
u+cv−k
)
tu =
∞∑
v=1
qv
∞∑
k=1
pk
∞∑
u=k−cv
ϕ+
u−(k−cv)t
u =
=
∞∑
v=1
qvt
−cv
∞∑
k=1
pkt
k
∞∑
u=k−cv
ϕ+
u−(k−cv)t
u−(k−cv) =
=
∞∑
v=1
qvt
−cv
∞∑
k=1
pkt
k
∞∑
n=0
ϕ+
n t
n = gT (t−ct) gZ(t) Φ+(t), t ∈ T.
Thus, the equation (6) is reduced to the Riemann boundary-value problem [7, 8, 16],(
1− gT (t−c)gZ(t)
)
Φ+(t) = −Φ−(t), t ∈ T. (10)
According to [7, 8, 16], the symbol A(t) is the coefficient at Φ+(t).
Note that we seek a solution to (10) at additional conditions: Φ−(∞) = 0 and Φ+(t) has simple
pole with the residue +1 at t0 = 1, since ϕn → 1 when n→∞.
The belonging A(t) ∈ W1 follows from the existence of finite means for the random variables
Tn and Zn. Thus, A(t) is differentiable function on T.
Theorem 1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
570 V. A. CHERNECKY
It is interesting to observe that gT (t−ct) gZ(t) is the generating function for the random variable
U = Z1 − cT1.
In this way, the equation (6) can be considered as one-sided Wiener – Hopf type discrete equation
∞∑
j=0
ak−jϕ
+
j = 0, k ∈ Z+, (11)
the coefficient matrix of which, {ak−j}k,j∈Z+ , is determined by the decomposition of the symbol
A(t) into Fourier series
A(t) =
∞∑
j=−∞
ajt
j , t ∈ T.
Observe that at solution of the Wiener – Hopf equation, the introduction in consideration of the
function Φ−(z), z ∈ B−∪T, is a successful artificial method [8, 10, 17]. As in our case, the function
Φ−(z), z ∈ B−, bears, in addition, the completely definite probabilistic sense load as the component
of generating function Φ(t), t ∈ T.
Let [argA(t)]T be the increment of the argument of A(t) when t passes T in positive direction
(counter-clockwise) and
indTA(t) :=
1
2π
[argA(t)]T.
It should be noted the following properties of the symbol A(t).
Theorem 2. If conditions of Assumptions is fulfilled, then:
1) the point t0 = 1 is the unique zero of the symbol A(t) on T, and this zero is simple;
2) |1−A(t)| ≤ 1, t ∈ T;
3) for the function B(t) = A(t)/(1− t) ∈W, we have B(t) 6= 0, t ∈ T, and
indT B(t) = −1; (12)
4) for the symbol A(t), the following factorization exists:
A(t) = A+(t)
1− t
t
A−(t), t ∈ T, (13)
with
A±(z) 6= 0, z ∈ B± ∪ T, A±(t) ∈W±, t ∈ T, A+(1) = 1, A−(1) = − c
α
+ µ;
5) the unique solution of the problem (6), (7) is generated by the functions
Φ+(z) =
1
(1− z)A+(z)
, z ∈ B+ ∪ T, Φ−(z) = −A
−(z)
z
, z ∈ B− ∪ T. (14)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
PROJECTIVE METHOD FOR EQUATION OF RISK THEORY IN THE ARITHMETIC CASE 571
Proof. 1. Uniqueness of zero of A(t), t ∈ T, follows from the fact that the steps of the random
variables Tn and Zn are assumed to be equal 1 [6]. The first order of the root t0 = 1 follows from
the L’Hospital rule and the condition c > αµ:
lim
t→1
A(t)
1− t
= lim
t→1
(
(gT (t−c))′ gZ(t) + gT (t−c) g′Z(t)
)
= − c
α
+ µ 6= 0.
So, we have to do with the Riemann problem (10) in exceptional case and the corresponding
equation (6) is of nonnormal type [7, 8, 16].
2. The inequality |1 − A(t)| ≤ 1, t ∈ T, follows from the corresponding property of generating
functions. Geometrical sense of this inequality is that the plot Γ of the symbol ζ = A(t), t ∈ T, is
situated in the unite disk with the center in the point ζ1 = 1 and this plot is tangent to the axis of
ordinates at the point ζ0 = 0 in the plane of the complex variable ζ.
3. Since the point t0 = 1 ∈ T is the simple root of A(t) ∈W1, it follows that A(t) is representable
in the form A(t) = (t− 1)B(t) with B(t) ∈W [16].
Next, the following equality holds:
indTA(t) = −1
2
. (15)
Really, (
A(eiτ )
)′ |τ=0 = i(c/α− µ), τ ∈ (−π, π],
what means that the tangent vector to Γ at ζ0 = 0 is directed along the positive direction on the axis
of ordinates when τ ∈ (−π, π] bypasses from −π to π, i.e., in negative direction (clockwise) with
respect to the domain bounded by the curve Γ. This implies (15), since the A(t) at t0 = 1 has the
simple root, A(t) has not other roots on T, and the plot Γ of the symbol A(t) is smooth.
On the other hand we have
indTA(t) = indT (1− t) + indTB(t) =
1
2
+ indTB(t),
whence, in view of (15), we have (12).
4. Introduce into consideration the function C(t) = A(t)
t
1− t
which has the following properties:
C(t) 6= 0, t ∈ T, indTC(t) = 0, C(t) is smooth on T \ {1} and continuous4 on T.
Since the point t0 = 1 is a simple root of the function A(t) ∈W1 on T, the function C(t) belongs
to the space W, [16], Section 5.1.3, Corollary 1.4, and can be factored in the form
C(t) = C+(t) · C−(t), t ∈ T,
where C±(t) ∈ W± and, consequently, are continuous, C±(z) 6= 0, z ∈ B± ∪ T. As it is known,
C±(t) are defined up to constant factors.
On the other hand, since C(t) is smooth on T \ {1}, the factors C±(t) can be expressed in terms
of the Cauchy type integral on t ∈ T \ {1} [7]
4Smoothness is, generally speaking, violated at t0 = 1, and C(t) can be nothing but continuous at t0 = 1.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
572 V. A. CHERNECKY
C±(t) = exp
1
2
ln[C(t)]± 1
2πi
v.p.
∫
T
ln[C(τ)]
τ − t
dτ
, t ∈ T \ {1}.
Whether these formulas work for the point t0 = 1 remains the open problem, but since C±(t) are
continuous on T, the values C±(1) can be determined by continuity.
Setting
A+(t) =
C+(t)
C+(1)
, A−(t) = C−(t)C+(1), t ∈ T,
we obtain the existence of the unique factorization of the form (13) with A+(1) = 1. The value
A−(1) = − c
α
+ µ follows from the condition A+(1) = 1 by the L’Hospital rule applied to the
representation A(t) = A+(t)
1− t
t
A−(t) at the point t0 = 1.
5. Note that in [7], the Riemann problem (10) is written in classical form as
Φ+(t) = − 1
A(t)
Φ−(t), t ∈ T,
and we seek the solution of the problem in exceptional case with coefficient (−1/A(t)) having a
simple pole at t0 = 1, and on the solution of the problem additional conditions are imposed: Φ+(t)
has a simple pole at t0 = 1, Φ−(t) ∈ W− on T and Φ−(∞) = 0. Using the results of the Gakhov
monograph [7] on exceptional case of the Riemann problem, observing that in our case (in notation
of [7]) κ = 1, p = 1, we obtain that the Riemann problem (10) has one linear independent solution
of the form
Φ+
1 (z) =
1
(1− z)C+(z)
, z ∈ B+, Φ−1 (z) = −C
−(z)
z
, z ∈ B−,
which generate one linear independent solution of the equation (6), belonging to the space {c+, c−0 }.
Then the solution
Φ+(z) =
1
(1− z)A+(z)
, z ∈ B+, Φ−(z) = −A
−(z)
z
, z ∈ B−,
generates unique solution of the problem (6), (7). Let us prove this.
Consider the series
1
A+(z)
=
∞∑
n=0
anz
n, z ∈ B+ ∪ T, an =
1
2πi
∫
T
dt
A+(t)tn+1
, n ∈ Z+,
A−(z) =
0∑
n=−∞
anz
n, z ∈ B− ∪ T, an =
1
2πi
∫
T
A−(t) dt
tn+1
, n ∈ Z− ∪ {0}.
Note that from the equality A+(1) = 1 we have
1
A+(1)
=
∞∑
n=0
an = 1.
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PROJECTIVE METHOD FOR EQUATION OF RISK THEORY IN THE ARITHMETIC CASE 573
Expansion for Φ+(z) is of the following form:
Φ+(z) =
1
(1− z)A+(z)
=
∞∑
n=0
(
n∑
k=0
ak
)
zn, z ∈ B+.
Assuming5
ϕn =
n∑
k=0
ak, n ∈ Z+,
we obtain
lim
n→+∞
ϕn = lim
n→+∞
n∑
k=0
ak = 1.
Expansion for Φ−(z) is of the form
Φ−(z) = −A
−(z)
z
= −
0∑
n=−∞
anz
n−1 = −
−1∑
n=−∞
an−1z
n, z ∈ B−.
Assume
ϕn = −an−1, n ∈ Z−.
The series for A−(t), t ∈ T, converges at t0 = 1, since
A−(1) =
0∑
n=−∞
an = − c
α
+ µ.
This implies that
lim
n→−∞
ϕn = 0.
Theorem 2 is proved.
Especially simply the Wiener – Hopf method works when the random variables Tn and Zn have
the rational generating functions. This happens to be the case for such distributions as uniform
discrete, binomial, geometrical, negative binomial with entire exponent (all shifted in right in a
reasonable way). In these cases, the symbol A(t) is a rational function which can be factored in
explicit form.
It may be noted that the equation (6) and the Riemann problem is not equivalent each other. For
example, the solution of the Riemann problem (10)
5The formulas for the Taylor coefficients of analytic function in B+, having simple pole on T, can be considered as a
Cauchy type integral. If the function 1/A+(t) is, in addition, supposed Hölder-continuous at the point t0 = 1, then the
coefficients ϕn can be computed by the following formulas:
ϕn = − 1
2A+(1)
− 1
2πi
p.v.
∫
T
dt
A+(t)(1− t)tn+1
, n ∈ Z+, A+(1) = 1.
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574 V. A. CHERNECKY{
1
A+(t)
,−(1− t)A−(t)
t
}
does not generate a solution of the equation (6) since for the function
Φ−1 (z) = −(1− z)A−(z)
z
the condition Φ−1 (∞) = 0 is not valid. Thus, the homogeneous equation (6) has not any more solu-
tions in arbitrary other space except the solution (14). It opens the way to the search of approximate
solution of the problem (6), (7) reducing the equation (6) in the interval [0,∞) to an equivalent
nonhomogeneous equation in the space c+0 .
Example. Assume c = 2, and let Tn be shifted negative binomial random variable having
generating function
gT (t) =
49
64
t
(1− t/8)2
, α =
7
9
, gT (t−c) =
49t2
(8t2 − 1)2
,
and Zn be shifted negative binomial random variable with generating function
gZ(t) =
9
16
t
(1− t/4)2
, µ =
5
3
.
Then
A(t) =
(t− 1)(64t5 − 448t4 + 560t3 + 247t2 − 8t− 16)
(8t2 − 1)2(t− 4)2
=
= A+(t)
1− t
t
A−(t), t ∈ T,
where
A+(t) =
(t− 2.168948920)(t− 5.160935390)
.5404356589(t− 4)2
,
A−(t) = −34.58788218
t(t2 + .5550511502t+ .09918770222)(t− .2251668399)
(8t2 − 1)2
,
and
Φ+(z) = .5404356589
(z − 4)2
(z − 2.168948920)(z − 5.160935390)(1− z)
, z∈ B+,
Φ−(z) = 34.58788218
(z2 + .5550511502z + .09918770222)(z − .2251668399)
(8z2 − 1)2
, z ∈ B−.
Expanding the functions Φ±(z) into series about z±1, respectively, we obtain the solution of the
problem (6), (7). Corresponding probabilities are adduced in the Table at the end of the paper.
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PROJECTIVE METHOD FOR EQUATION OF RISK THEORY IN THE ARITHMETIC CASE 575
3. Delayed renewal processes. In addition to the sequence {Tn}, n ∈ N, introduce a positive
integer-valued random variable S0 with some generating function gS0(z) =
∑∞
n=1
rnz
n and consider
the variables
Sn = S0 + T1 + T2 + . . .+ Tn
called the renewal epochs [6]. The renewal process {Sn} is called pure if S0 = 0 and delayed
otherwise. The expected number of renewal epochs on [0, n] equals
V (n) =
∞∑
k=0
P{Sk ≤ n}, n ∈ Z+,
and has the generating function
gV (z) =
gS0(z)
1− z
(
1 +
∞∑
k=1
[gT (z)]k
)
=
gS0(z)
(1− z)(1− gT (z))
, z ∈ B+. (16)
It is known, [6], that
lim
n→∞
[V (n+ 1)− V (n)] = α.
It follows from this that V (n) ∼ αn as n→∞. It is natural to ask whether gS0(z) can be chosen as
to get the identity V (n) = αn, n ∈ Z+, meaning a constant renewal rate.
Noticing that the generating function for the sequence {αn}, n ∈ Z+, is given by the formula
g{αn}(z) = α
∞∑
n=1
nzn =
αz
(1− z)2
, z ∈ B+,
and equating gV (z) = g{αn}(z), we obtain
gS0(z) =
αz(1− gT (z))
1− z
, z ∈ B+ ∪ T, (17)
which is the generating function of a proper probability distribution6 and so the answer is affirmative:
With the initial random variable S0 having generating function (17) the renewal rate is constant,
V (n) = αn, n ∈ Z+.
The following statement takes place:
The ordinary renewal process in arithmetic case is stationary if and only if the inter-arrival times
Tn have the shifted geometrical distribution with the generating function (8).
This shifted geometrical distributions is analog of exponential distribution for the nonarithmetic
case.
For the problem (6), (7), we consider the accompanying delayed stationary renewal process
{Sn}n∈Z+ with generating function (17) for S0. Then the generating function for the nonruin proba-
bilities in such process is built as follows:
Φs(t) = gS0(t−c) gZ(t) Φ+(t) =
αt−c(1− gT (t−c)) gZ(t) Φ+(t)
1− t−c
=
6Using the L’Hospital rule, we can prove that gS0(z) is extendable on T with gS0(1) = 1.
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576 V. A. CHERNECKY
=
α(1− gT (t−c)) gZ(t) Φ+(t)
tc − 1
, t ∈ T, (18)
similarly as it is done in the nonarithmetic case.
Using (10), we can exclude gT (t−c) from the last expression,
Φs(t) =
α(1− gZ(t))Φ+(t) + αΦ−(t)
tc − 1
, t ∈ T. (19)
Let
Φs(t) =
∞∑
u=−∞
ϕsut
u, t ∈ T.
Our immediate task is obtaining formulas for Φ±s (z), z ∈ B±. For this end we must solve the jump
problem for the function Φs(t) [7]
Φs(t) = Φ+
s (t) + Φ−s (t) ∈ {ĉ+, ĉ−0 }, t ∈ T,
where Φ+
s (t) has simple poles at the c th roots of unity, Φ−s (t) ∈W−, t ∈ T, and Φ−s (∞) = 0.
Rewrite the formula (19) in the equivalent form
Φs(t) =
α(1− gZ(t))Φ+(t) + pc−1(t)
tc − 1
+
αΦ−(t)− pc−1(t)
tc − 1
, t ∈ T, (20)
where pc−1(t) is a specially chosen polynomial
pc−1(t) = pc−1t
c−1 + pc−2t
c−2 + . . .+ p0
with the coefficients pk, k = 0, 1, . . . , c − 1, satisfying the system of linear equations with the
Vandermonde matrix,
pc−1(εk) = αΦ−(εk), k = 0, 1, . . . , c− 1, (21)
where εk are the c th roots of unity.
Then from (20) we obtain
Φ+
s (t) =
α(1− gZ(t)) Φ+(t) + pc−1(t)
1− tc
∈ ĉ+, t ∈ T, (22)
Φ−s (t) =
αΦ−(t)− pc−1(t)
1− tc
∈ ĉ−0 , t ∈ T. (23)
Representing Φ−s (t) in the form
Φ−s (t) = −αΦ−(t)
tc
1
1− 1
tc
+
pc−1(t)
tc
1
1− 1
tc
, t ∈ T, (24)
we see that pc−1(t) is representable in the form
pc−1(t) = ϕs−1t
c−1 + ϕs−2t
c−2 + . . .+ ϕs−c
with the coefficients ϕs−k = pc−k, k = 1, 2, . . . , c, which are the first c coefficients of the expansion
of Φ−s (z) into series about 1/z, z ∈ B−.
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PROJECTIVE METHOD FOR EQUATION OF RISK THEORY IN THE ARITHMETIC CASE 577
In the simplest case c = 1, from the system (21), we obtain the relation
ϕs−1 = αΦ−(1) = α
( c
α
− µ
)
= 1− αµ
that is the analog of the known equality
ϕ(0) = 1− αµ
c
for the nonruin probability ϕ(u) in the nonarithmetic case at c = 1 [2, 13, 14].
Since the function Φ−s (t) is continuous on T, the Taylor-series expansion for it is not the problem.
The function Φ+
s (z) has the simple poles at the c th roots of unity. Therefore the expansion for
Φ+
s (z) is constructed in two steps. Since the numerator of (22),
A+
s (t) = α(1− gZ(t)) Φ+(t) + pc−1(t)
is continuous on T, we construct the expansion for it
A+
s (t) =
∞∑
n=0
asnz
n, z ∈ B+ ∪ T,
and then the expansion for Φ+
s (z) is constructed by the expansion of the ratio7
A+
s (z)
1− zc
=
∑∞
n=0 a
s
nz
n
1− zc
=
∞∑
n=0
ϕsnz
n.
Example (continued). The system (21) for ϕs−1 and ϕ−2 in our case has the form
ϕs−1 + ϕs−2 = αΦ−(1),
−ϕs−1 + ϕs−2 = αΦ−(−1),
the solution of which is ϕs−1 = .5348541581, ϕs−2 = .1688495471. Then
Φ+
s (z) =
α(1− gZ(z))Φ+(z) + .5348541581z + .1688495471
1− z2
,
Φ−s (z) =
.08000291975z + .01274595223
z2 + .7071067812z + .125
+
.4548512383z − .1089541931
z2 − .7071067812z + .125
.
Corresponding probabilities are adduced in Table at the end of the paper.
7If the function A+
s (t) is supposed Hölder-continuous at the c th roots of unity on T, then the coefficients ϕs
n can be computed
by the formulas
ϕs
n = −
∑c−1
k=0 A
+
s (εk)
2εn+1
k
−
1
2πi
p.v.
∫
T
A+
s (t) dt
(1− t)tn+1
, n ∈ Z+.
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578 V. A. CHERNECKY
4. Nonruin probabilities for stationary process. Assume now that the inter-arrival times Tn
have the shifted geometrical distribution with the generating function (8), i.e., the renewal process is
stationary and hence Φs(t) = Φ(t). Then from (22) and (23) we obtain the equations for Φ±s (z):
Φ+
s (z) =
α(1− gZ(z)) Φ+
s (z) + pc−1(z)
1− zc
, z ∈ B+, (25)
Φ−s (z) =
αΦ−s (z)− pc−1(z)
1− zc
, z ∈ B−, (26)
whence
Φ+
s (z) =
ϕs−1z
c−1 + ϕs−2z
c−2 + . . .+ ϕs−c
1− zc − α(1− gZ(z))
, z ∈ B+, (27)
Φ−s (z) =
ϕs−1z
c−1 + ϕs−2z
c−2 + . . .+ ϕs−c
1− α− zc
, z ∈ B−. (28)
If we put z = 0 in (27), we obtain the formula8 for ϕs0,
ϕs0 = Φ+
s (0) =
ϕs−c
1− α
.
In the simplest case when c = 1, we obtain9
Φ+
s (z) =
1− αµ
1− z − α(1− gZ(z))
, z ∈ B+ ∪ T, (29)
and
ϕs0 =
1− αµ
1− α
.
This formula was derived by A. Mel’nikov in [15] by other method.
5. The projective method. Since to construct explicitly the factorization of the symbol frequently
is rather difficultly, the approximate methods have particular importance. For approximate solution
of fundamental equation of risk theory in arithmetic case we propose to use the projective method
(called in the literature also finite section method or natural reduction method) [16].
Consider discrete Wiener – Hopf equation in general case,
∞∑
j=0
ak−jξj = η+k , k ∈ Z+. (30)
Here it is assumed that
1)
∑∞
j=−∞
|aj | <∞,
2) η+ = {η+j }j∈Z+ is given and such that the equation (30) is solvable,
3) ξ+ = {ξj}j∈Z+ is the unknown vector in the space E+, where E+ is any of the spaces l+1
or c+0 .
8Remind that α = q < 1 and gZ(0) = 0.
9In [15] this formula is written in some cumbersome form.
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PROJECTIVE METHOD FOR EQUATION OF RISK THEORY IN THE ARITHMETIC CASE 579
For discrete Wiener – Hopf equation (30) the projective method consists in following [10 – 12,
16, 17]:
The equation (30) is replaced by the system of n+ 1 equations with n+ 1 unknown
n∑
j=0
ak−j ξ̃j = η+k , k = 0, 1, . . . , n, (31)
and solution of this curtained (reduced) system,
ξ̃(n) = {ξ̃0, ξ̃1, . . . , ξ̃n},
is considered as the approximation to the solution of initial equation (30).
If the condition of normality on the symbol,
A(t) =
∞∑
j=−∞
ajt
j 6= 0, t ∈ T,
is not fulfilled (as in our case), the projective process (31) in the space l+1 , generally, already does
not converge any more for all that η+ ∈ l+1 for which the equation (30) is solvable [16]. We are
interested by the conditions at which the projective method works for the problem (6), (7).
Let t0 = 1 be unique zero of the symbol A(t) on T and this zero is simple. Put
C(t) = A(t)
(
t−1 − 1
)−1
, t ∈ T.
The following theorem is the simple consequence of the Theorem 0.4 proved in [16].
Theorem 3 (convergence in l+1 ). Let C(t) ∈W1.
(a) If C(t) 6= 0 and indTC(t) = 0, then the projective method (31) converges in the space l+1 for
all η+ ∈ l+1,1.
(b) In addition, if C(t) ∈W1+δ and η+ ∈ l+1,1+δ, δ > 0, then it is valid the estimate
‖ξ − ξ̃(n)‖l+1 = O(n−δ), n→∞.
The following theorem is the simple consequence of Theorem 4.1 and Theorem 4.2, Ch. 11 [16].
Theorem 4 (convergence in c+0 ). Let C(t) ∈W, C(t) 6= 0.
(a) If indTC(t) = 0, then the projective method for the system (30) converges in the space c+0 .
(b) In addition, if C(t) ∈W1+δ, and η+ ∈ c+1+δ, δ > 0, then it is valid the estimate
‖ξ − ξ̃(n)‖c+0 = O(n−δ), n→ +∞.
6. Convergence of the projective method for the fundamental equation. Here it is convenient
to introduce the ruin probability
ψ+
u = 1− ϕ+
u , u ∈ Z+.
Then the problem (6), (7) (only for u ∈ Z+) can be rewritten in the form of inhomogeneous equation
for ψ+
u :
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580 V. A. CHERNECKY
ψ+
u −
∞∑
v=1
qv
u+cv∑
k=1
pkψ
+
u+cv−k = 1−
∞∑
v=1
qv
u+cv∑
k=1
pk, u ∈ Z+, (32)
lim
u→+∞
ψ+
u = 0. (33)
It follows from results of S.Prössdorf [16] that the problem (32), (33) has an unique solution
in the spaces c+0 and l+1 . Using Theorems 3 and 4, we can receive the conditions on the random
variables Tn and Zn, and consequently, on the symbol A(t) and on the right part η+ = {η+u }u∈Z+ of
the equation (32) with
η+u = 1−
∞∑
v=1
qv
u+cv∑
k=1
pk, u ∈ Z+, (34)
for the convergence of projective process (31) for the problem (32), (33) in the spaces l+1 and c+0 .
For this end we must clarify the probabilistic nature of the right part η+ and its properties.
Theorem 5. The right part η+ of the equation (32) is the right tail of the random variable
U = Z1 − cT1.
Proof is obvious.
Corollary 1. If A(t) ∈Wk, then η+ ∈ l+1,k−1.
Proceeding from Theorem 5, Theorem 3 in the case of the problem (32), (33) can be formulated
in the following form:
Theorem 6 (convergence in l+1 ). Let the conditions of Assumptions be fulfilled.
(a) In addition, if the random variables Tn and Zn have finite variances then the projective
process (31) for the problem (32), (33) converges in the space l+1 .
(b) If the random variables Tn and Zn have finite (2 + δ)th moments, δ > 0, then it is valid the
estimate
‖ψ − ψ̃(n)‖l+1 = O(n−δ), n→ +∞.
Proof. The existence of finite variances for Tn and Zn implies A(t) ∈ W2, C(t) ∈ W1, and si-
multaneously η+ ∈ l+1,1, t ∈ T [16] (Ch. 5, Theorem 1.7). Besides we have C(t) 6= 0, indTC(t) = 0.
By Theorem 4 this implies the statement (a).
It follows from the existence of the finite (2 + δ)th moments for the random variables Tn and Zn
that A(t) ∈W2+δ, C(t) ∈W1+δ and η+ ∈ l+1,1+δ. By Theorem 4 this implies the statement (b).
Theorem 6 is proved.
Theorem 4 for the problem (32), (33) can be formulated as follows:
Theorem 7 (convergence in c+0 ). Let the conditions of Assumptions be fulfilled.
(a) The projective method (31) for the problem (32), (33) converges in the space c+0 .
(b) In addition, if the random variables Tn and Zn have finite (2 + δ)th moments, δ > 0, then it
is valid the estimate
‖ψ − ψ̃(n)‖c+0 = O(n−δ), n→ +∞.
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PROJECTIVE METHOD FOR EQUATION OF RISK THEORY IN THE ARITHMETIC CASE 581
Proof. (a) From existence of finite means for the random variables Tn and Zn it is follows that
A(t) ∈W1, and, consequently, C(t) =∈W [16]. By Theorem 4 this implies the statement (a).
The statement (b) is proved in a way analogous to that used for the proof of the statement (b) of
Theorem 6.
Theorem 7 is proved.
Note that the projective method converges in the space c+0 without superposition of somehow
additional conditions on the random variables Tn and Zn except the conditions of Assumptions. Such
quality of projective method removes the problem of “large” claims [5].
Exact and approximate nonruin probabilities from Example
u ϕu ϕ̃u ϕsu ϕ̃su
–10 .1233318169e–3 .1232544182e-3 .1121618097e-3 .1121656751e-3
-9 .5508161983e-3 .5507966082e-3 .5029119846e-3 .5029293048e-3
-8 .8270422905e-3 .8265289772e-3 .7554169265e-3 .7554429513e-3
-7 .3568781788e-2 .3568646637e-2 .3278631163e-2 .3278744079e-2
-6 .5339440364e-2 .5336180872e-2 .4908314995e-2 .4908484016e-2
-5 .2184827192e-1 .2184736324e-1 .2027173160e-1 .2027242975e-1
-4 .3250033924e-1 .3248103939e-1 .3018635667e-1 .3018739538e-1
-3 .1211703164 .1211644271 .1145153112 .1145192552
-2 .1782812446 .1781810546 .1688495471 .1688553493
-1 .5404356589 .5403995867 .5348541581 .5348725793
0 .7724782018 .7725048388 .7696659284 .7696924
1 .8920702933 .8921010541 .8907273961 .89075812
2 .9496515748 .9496843208 .9490234591 .949056191
3 .9766729524 .9767066317 .9763816202 .976415293
4 .9892229558 .9892570645 .9890882995 .9891224045
5 .9950269424 .9950612554 .9949647935 .9949991059
6 .9977063305 .9977407385 .9976776636 .9977120670
7 .9989423371 .9989767060 .9989291175 .9989635043
8 .9995123308 .9995470499 .9995062351 .9995409313
9 .9997751528 .9998084191 .9997723421 .9998057237
10 .9998963325 .9999325190 .9998950366 .9999315536
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582 V. A. CHERNECKY
Example (continued). Since Φ(t) ∈ {ĉ+, ĉ−0 }, t ∈ T, the approximation Φ̃+
(n)(t) ∈ ĉ+ for
Φ+(t) is selected in the form
Φ̃+
(n)(t) =
n∑
k=0
(
1− ψ̃(n)
k
)
tk +
tn+1
1− t
, t ∈ T.
Then the generating function Φ̃−(n)(t) for the vector ϕ̃−(n) is constructed by the formula10
Φ̃−(n)(t) =
(
−A(t)Φ̃+
(n)(t)
)−
, t ∈ T,
and the approximate generating function for the approximate probabilities ϕ̃s in delayed renewal
process is constructed using the formulas of Section 3, in which the exact solution Φ+(t) should be
replaced by the approximate solution Φ̃+
(n)(t).
The comparison of exact and approximate values of nonruin probabilities in the Table shows
sufficiently accuracy of the method.
1. Asmussen S. Ruin probabilities // Adv. Ser. Statist. Sci. and Appl. Probab. – Singapore: World Sci., 2000. – 2.
2. Bühlmann H. Mathematical methods in risk theory. – Berlin etc.: Springer, 1970.
3. Chernecky V. A. Exact non-ruin probabilities in arithmetic case// Theory Stochast. Processes. – 2008. – 14(30),
№ 3 – 4. – P. 39 – 52.
4. Dorogovtsev A.Ya., Silvestrov D. S., Skorokhod A. V., Yadrenko M. I. Probability theory: Collection of problems. –
Providence, Rhode Island: Amer. Math. Soc., 1997.
5. Embrechts P., Klüppelberg C., Mikosch T. Modeling extremal events for insurance and finance. – Berlin etc.: Springer,
1997.
6. Feller W. An Introduction to probability theory and its applications. – New York etc.: John Wiley & Sons, 1971. –
Vol. 2.
7. Gakhov F. D. Boundary-value problems (in Russian). – Moscow: Nauka, 1977.
8. Gakhov F. D., Chersky Yu. I. Equations of convolutions type (in Russian). – Moscow: Nauka, 1978.
9. Gel’fand I. M., Shilov G. E. Generalized functions. – New York: Harcourt, 1977. – Vol. 1.
10. Gokhberg I. C., Fel’dman I. A. Equations in convolutions and projective methods of their solution (in Russian). –
Moscow: Nauka, 1971.
11. Gokhberg I. C., Levchenko V. I. On convergence of projective method of solution of degenerated discrete Wiener –
Hopf equation // Mat. Issled. – 1971. – 6, № 4(22). – P. 20 – 36.
12. Gokhberg I. C., Levchenko V. I. On projective method for degenerated discrete Wiener – Hopf equation // Mat. Issled.
– 1971. – 7, № 3(25). – P. 238 – 253.
13. Grandell J. Aspects of risk theory. – Berlin etc.: Springer, 1991.
14. Leonenko M. M., Mishura Yu. S., Parkhomenko V. M., Yadrenko M. I. Probability-theoretic and statistical methods
in econometrics and financial mathematics (in Ukrainian). – Kyiv: Informtekhnika, 1995.
15. Mel’nikov A. V. Risk analysis in finance and insurance. – Boca Raton etc.: Chapman & Hall/CRC, 2003.
16. Prössdorf S. Einige Klassen singulärer Gleichungen. – Berlin: Akad.-Verl., 1974.
17. Prössdorf S., Silbermann B. Ein Projectionsverfahren zur Lösung abstracter singulärer Gleichungen vom nicht
normalen Typ und einige seiner Anwendungen // Math. Nachr. – 1974. – 61. – S. 133 – 155.
Received 26.09.11,
after revision — 28.10.12
10By (·)− we mean a projection on the subspace of the boundary values Φ−(t), t ∈ T, of the functions analytical in
B−, Φ−(∞) = 0.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
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| id | umjimathkievua-article-2440 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:23:29Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/19/fbb8aa1d4a8f7c521bb2867339237019.pdf |
| spelling | umjimathkievua-article-24402020-03-18T19:15:36Z Projective method for equation of risk theory in the arithmetic case Проективний метод для рiвняння теорiї ризику в арифметичному випадку Chernetskii, V. A. Чернецький, В. А. We consider a discrete model of operation of an insurance company whose initial capital can take any integer value. In this statement, the problem of nonruin probability is naturally solved by the Wiener-Hopf method. Passing to generating functions and reducing the fundamental equation of risk theory to a Riemann boundary-value problem on the unit circle, we establish that this equation is a special one-sided discrete Wiener-Hopf equation whose symbol has a unique zero, and, furthermore, this zero is simple. On the basis of the constructed solvability theory for this equation, we justify the applicability of the projective method to the approximation of ruin probabilities in the spaces $l^{+}_1$ and $\textbf{c}^{+}_0$. Conditions for the distributions of waiting times and claims under which the method converges are established. The delayed renewal process and stationary renewal process are considered, and approximations for the ruin probabilities in these processes are obtained. Розглядається дискретна модель функцiонування страхової компанiї, початковий капiтал якої може набувати довiльного цiлого значення. У такiй постановцi проблема обчислення ймовiрностi стiйкостi компанiї природно розв’язується методом Вiнера – Хопфа. При переходi до твiрних функцiй i зведеннi фундаментального рiвняння теорiї ризику до граничної задачi Рiмана на одиничному колi з’ясовано, що розглядуване рiвняння є особливим одностороннiм дискретним рiвнянням Вiнера – Хопфа, символ якого має єдиний нуль i цей нуль є простим. На базi побудованої теорiї розв’язностi цього рiвняння обґрунтовано застосування проективного методу до апроксимацiї ймовiрностей банкрутства у просторах $l^{+}_1$ і $\textbf{c}^{+}_0$. Отримано умови на розподiли часiв очiкування вимог i розмiрiв виплат для збiжностi методу. Розглянуто процес вiдновлення iз запiзненням i стацiонарний процес вiдновлення, а також наближення для ймовiрностей банкрутства у цих процесах. Institute of Mathematics, NAS of Ukraine 2013-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2440 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 4 (2013); 565-582 Український математичний журнал; Том 65 № 4 (2013); 565-582 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2440/1646 https://umj.imath.kiev.ua/index.php/umj/article/view/2440/1647 Copyright (c) 2013 Chernetskii V. A. |
| spellingShingle | Chernetskii, V. A. Чернецький, В. А. Projective method for equation of risk theory in the arithmetic case |
| title | Projective method for equation of risk theory in the arithmetic case |
| title_alt | Проективний метод для рiвняння теорiї ризику в арифметичному випадку |
| title_full | Projective method for equation of risk theory in the arithmetic case |
| title_fullStr | Projective method for equation of risk theory in the arithmetic case |
| title_full_unstemmed | Projective method for equation of risk theory in the arithmetic case |
| title_short | Projective method for equation of risk theory in the arithmetic case |
| title_sort | projective method for equation of risk theory in the arithmetic case |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2440 |
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