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Cramer-Lundberg model for some classes of extremal Markov sequences

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Abstract

We consider here the Cramer-Lundberg model based on generalized convolutions. In our model the insurance company invests at least part of its money, have employees, shareholders. The financial situation of the company after paying claims can be even better than before. We compute the ruin probability for $\alpha$-convolution case, maximal convolution and the Kendall convolution case, which is formulated in the Williamson transform terms. We also give some new results on the Kendall random walks.
arXiv:1901.05701v1 [math.PR] 17 Jan 2019
CRAMER-LUNDBERG MODEL FOR SOME CLASSES
OF EXTREMAL MARKOV SEQUENCES
B.H. JASIULIS-GOŁDYN 1, A. LECHAŃSKA2AND J.K. MISIEWICZ 2
Abstract. We consider here the Cramer-Lundberg model based
on generalized convolutions. In our model the insurance company
invests at least part of its money, have employees, shareholders.
The financial situation of the company after paying claims can
be even better than before. We compute the ruin probability for
α-convolution case, maximal convolution and the Kendall convolu-
tion case, which is formulated in the Williamson transform terms.
We also give some new results on the Kendall random walks.
Key words: Cramer-Lundberg model, ruin probability, generalized
convolution, Williamson transform, α-convolution, max-convolution,
Kendall convolution
Mathematics Subject Classification: 91B30, 60G70, 44A35, 60E10.
Contents
1. Introduction and the classical model description 1
2. Description of the proposed model 3
3. Basic information about generalized convolution 5
4. Random walk with respect to the generalized convolution 7
5. Model for αrandom walk 8
6. Model for -generalized convolution 10
7. Model for the Kendall random walk 12
7.1. Inversion formula and cumulative distribution functions 14
7.2. First safety condition for the insurance company 15
7.3. Ruin probability in the infinite time horizon 16
References 17
1. Introduction and the classical model description
The classical Cramér-Lundberg risk model was introduced by Lund-
berg in 1903 (see [19]) and developed by H. Crámer and his Stockholm
Date: January 18, 2019.
1Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384
Wrocław, Poland
2Faculty of Mathematics and Information Science, Warsaw University of Technol-
ogy, ul. Koszykowa 75, 00-662 Warszawa, Poland.
1
2 JASIULIS-GOŁDYN, LECHAŃSKA, MISIEWICZ
School at the beginning of the XX century (see e.g. [8]). A reach
information on actuarial risk theory, non-life insurance models and fi-
nancial models one can find in a huge number of papers and books
(e.g. [1, 3, 6, 10, 17, 20, 22, 23, 24]). The interested reader we want to
refer to the series of P. Embrechts papers and especially to the book
of P. Embrechts, C. Kluppelberg and T. Mikosch [9]. Between many
interesting results we can find there very important modifications of
Cramer-Lundberg models for heavy tailed distributions.
The basic model in this theory, called Cramer-Lundberg model or the
renewal model, has the following structure:
(a) Claim size process: the claim sizes (Xk)are iid positive random
variables with cumulative distribution function F, finite mean m=
EXkand variance σ2= VarXk.
(b) Claim times: the claims occur at the random instants of time
0< S1< S2< . . . a.s.
where the inter arrival times
T1=S1, Tk=SkSk1, k = 2,3,...
are iid with exponential distribution with mean ETk= 1;
(c) Claim arrival process: the number of claims in the time in-
terval [0, t]is denoted by
N(t) = sup {n>1: Sn< t};
(d) the sequences (Xk)and (Tk)are independent of each other.
The risk process {Kt:t>0}is defined by
Kt=u+βt
Nt
X
k=0
Xk,
where u>0denotes the initial capital, β > 0stands for the premium
income rate and Ktis the capital that company have at time t.
Notice that this model describes the following situation:
1. The insurance company is keeping all the money in a pocket.
2. All the incomes are coming from the customers payments.
3. There is no outcomes except for the individual claims of the
customers.
4. There is no cost or benefits coming from the company existence
and activities or from the money which the company have.
There are many modifications of this model. Some of them can be
found in [9]. In these modifications some of the defects of the Crámer-
Lundberg model are eliminated. However usually it means that much
more information about the insurance company policy is required for
description, while no one company is inclined to share all such infor-
mation with the outside world. In the next section we propose a family
CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 3
of models based on special class of Markov chains. The hidden infor-
mation can be coded in transition probabilities. This approach may
be more convenient than taking under considerations various different
company policy elements.
The paper is organized as follows: In section 2 we present our model.
The considered random walks are very special - the transition probabil-
ities are defined by generalized convolution - however this restriction is
important only for this paper. Generalized convolutions are described
in Section 3, the construction of the random walk with respect to gen-
eralized convolution is given in Section 4. The detailed calculations
are given in the last 3 sections for the following examples: random
walk with respect to stable convolution, random walk with respect to
max-convolution and the Kendall random walk.
2. Description of the proposed model
In our model we assume that the insurance company invests at least
part of its money, have employees, shareholders which have to have
income and at each moment when the claim cames the company is
calculating the total claim amount, subtract from this all costs and
add benefits. Thus the corrected cost of the total outcome for claims is
not just simple sum of Xk. In fact, in this model the financial situation
of the company after paying Xkclaim can be even better than before.
The rich collection of generalized convolutions and freedom in choosing
claims distribution λshall give the possibility of adjusting model to the
real situation without precise information about company activities.
We propose here the following structure of the model:
(a) Claim times: the claims occur at the random instants of time
0< S1< S2< . . . a.s.
where the inter arrival times
T1=S1, Tk=SkSk1, k = 2,3,...
are iid random variables with exponential distribution, ETk= 1;
(b) Claim arrival process: the number of claims in the time in-
terval [0, t]is the Poisson process with the parameter λ > 0defined
by
N(t) = sup {n>1: Sn< t};
(c) Cumulated claims process: the total amount of money spent
on the first n-claims corrected by part of the incomes other than
premium and/or some of the costs is a discrete time Markov process
{Xn:nN0}, which is a -Lévy process with the step distribution
Uiµ∈ P+(see Section 3 and 4) and the transition probabilities
Pn(x, ·). The sequence (Ui)is i.i.d. The cumulative distribution
function for the measure µwe denote by F, its density by f, the
generalized characteristic function by Φµand we put H(t) = Φµ(t1).
4 JASIULIS-GOŁDYN, LECHAŃSKA, MISIEWICZ
(d) Cumulated income units: the total insurance premium col-
lected by the company up to the moment of n-th claim corrected by
part of the cost of the company activity and/or part of the income
from the investments is a discrete time Markov process {Yn:nN0},
which is a -Lévy process with the step distribution Viν∈ P+
(see Section 3 & 4) and the transition probabilities Πn(x, ·). The
sequence (Vi)is i.i.d. The cumulative distribution function for the
measure νwe denote by G, its density by g, the generalized charac-
teristic function by Φνand we put J(t) = Φν(t1).
(e) Independence assumption: the processes {N(t): nN0},
{Xn:nN0}and {Yn:nN0}are independent.
The risk process is defined by the following:
Rt=u
X
n=1
Yn1{N(t)=n}
X
n=1
Xn1{Nt=n},
where uYnis the Markov process {Yn:nN0}with the starting
point moved to u > 0in the generalized convolution sense (see e.g.
[5]). Notice that if N(t) = n, then we have Rt=uYnXn, where
Xn=U1U2⊕ · ·· ⊕ Unand Yn=V1V2⊕ · · · ⊕ Vnand denotes
adding in the generalized convolution sense, i.e Xnµnand Ynνn
respectively.
Let Qt(u)denotes the probability that the insurance company with the
initial capital u > 0will bankrupt until time t. Since the changes in
the process {Rt:t>0}can occur only at the moments Sn,n>0we
see that
Qt(u) = Ps6t:Rs<0
= 1
X
n=0
PRSk>0: k= 0,1,...,nPNt=n
= 1
X
n=0
PuYk> Xk:k= 0,1,...,n(λt)n
n!eλt.
Calculating the same probability in the unbounded time horizon Q(u)
we see that
Q(u) = Pt > 0 : Rt<0
= 1 PuYk> Xkfor all kN0.
Notice that Q(u)does not depend on the process {Nt}. This is nat-
ural since in our case this process is describing only the moments of
claims arrival and {Nt}is independent of the processes {Xn}and {Yn}.
Every continuous time Markov chain taking values in (whole!) N0
would give the same result. For abbreviation we introduce the fol-
lowing notation for probability that ruin does not occur:
δ(u) := Q(u).
CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 5
3. Basic information about generalized convolution
By P+we denote the set of probability measures on the positive half
line [0,). If λnconverges weakly to λwe write λnλ. For simplicity
we will use notation Tafor the rescaling operator (dilatation operator)
defined by (Taλ)(A) = λ(A/a)for every Borel set Awhen a6= 0, and
T0λ=δ0.
Following K. Urbanik (see [25]) we define
Definition 3.1. A commutative and associative P-valued binary op-
eration defined on P2
+is called a generalized convolution if for all
λ, λ1, λ2∈ P+and a>0we have:
(i) δ0λ=λ;
(ii) (1+ (1 p)λ2)λ=p(λ1λ) + (1 p)(λ2λ)whenever p[0,1];
(iii) Ta(λ1λ2) = (Taλ1)(Taλ2);
(iv) if λnλthen λnηληfor all η∈ P and λn∈ P+,
(v) there exists a sequence (cn)nNof positive numbers such that the
sequence Tcnδn
1converges to a measure different from δ0.
The set (P+,)we call a generalized convolution algebra. A continuous
mapping h:P Rsuch that
h(+ (1 p)ν) = ph(λ) + (1 p)h(ν),
h(λν) = h(λ)h(ν)
for all λ, ν ∈ P+and p(0,1), is called a homomorphism of (P+,).
Every convolution algebra (P+,)admits two trivial homomorphisms:
h1and h0. We say that a generalized convolution is regular
if it admits non-trivial homomorphism. If the generalized convolution
is regular then its homomorphism is uniquely determined in the sense
that if h1, h2are homomorphisms of (P+,)then there exists c > 0
such that h1(λ) = h2(Tcλ)(for details see [25]). It was also shown
in [25] that the generalized convolution is regular if and only if there
exists unique up to a scale function
P+λΦλC([0,))
such that for all λ, ν, λn∈ P+the following conditions hold:
1. Φ+(t) = pΦλ(t) + qΦν(t), for p, q >0,p+q= 1;
2. Φλν(t) = Φλ(tν(t);
3. ΦTaλ(t) = Φλ(at)for a>0;
4. the uniform convergence of Φλnon every compact set to a function
Φis equivalent with the existence of λ∈ P+such that Φ = Φλand
λnλ
The function Φλis called the -generalized characteristic function of the
measure λ. Let Ω(t) = h(δt). By properties 1 and 2 of the characteristic
6 JASIULIS-GOŁDYN, LECHAŃSKA, MISIEWICZ
function we see that
Φλ(t) = Z
0
Ω(xt)λ(dx),
thus the function is called the kernel of generalized characteristic
function (similarly as the function eit is the kernel of Fourier transform,
i.e. the classical characteristic function).
Examples. For details see [4, 7, 12, 16, 18, 21, 25, 26].
3.0. The classical convolution, denoted by is given by:
δaδb=δa+b.
Here we have Ω(t) = etif we consider this convolution on P+and
Ω(t) = eit if we consider it on the whole line.
3.1. Symmetric convolution on P+is defined by
δasδb=1
2δ|ab|+1
2δa+b.
The kernel of generalized characteristic function here is Ω(t) = cos(t).
3.2. By stable convolution αfor α > 0we understand the following:
δaαδb=δc, c = (aα+bα)1, a, b >0.
The kernel of generalized α-characteristic function is Ω(t) = etα.
3.3. -convolution is defined by
δaδb=δmax{a,b}.
This convolution admits existence of characteristic function, but its
kernel is not continuous: Ω(t) = 1[0,1](t).
3.4. The Kendall convolution αon P+,α > 0, is defined by
δxαδ1=xαπ2α+ (1 xα)δ1, x [0,1],
where π2αis a Pareto measure with density 2αx2α11[1,)(x). The
kernel of generalized characteristic function here is given by Ω(t) =
(1 (ts)α)+, where a+=afor a>0and a+= 0 otherwise.
3.5. The Kingman convolution ωson P+,s > 1
2, is defined by
δaωsδb=Lpa2+b2+ 2abθs,
where θsis absolutely continuous with the density function
fs(x) = Γ(s+ 1)
πΓ(s+1
2)1x2s1
2
+.
The kernel of generalized characteristic function here is given by the
Bessel function of the first kind with parameter connected with s.
3.6. For every p>2and properly chosen c > 0the function h(δt) =
ϕ(t) = ϕc,p(t) = (1 (c+ 1)t+ctp)1[0,1](t)is the kernel of a Kendall
CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 7
type (see [21]) generalized convolution defined for x[0,1] by the
formula:
δxδ1=ϕ(x)δ1+xpλ1+ (c+ 1)(xxp)λ2,
where λ1, λ2are probability measures absolutely continuous with re-
spect to the Lebesgue measure and independent of x. For example if
c= (p1)1then
λ1(du) = 2c
u3h(c+1)(p+1)u1p+(c+1)(p2)+cp(2p1)u2p2i1[1,)(u)du,
and
λ2(du) = c2(p2) + (p+ 1)up+1u31[1,)(u)du.
4. Random walk with respect to the generalized
convolution
All the information contained in this section comes from [5], where the
Lèvy processes with respect to generalized convolution were defined
and studied. It was shown there that each such process is a Markov
process (in the classical sense) with the transition probabilities defined
by generalized convolution. We consider here only discrete time sto-
chastic processes of this kind.
Definition 4.1. A discrete time stochastic process {Xn:nN0}is
a random walk with respect to generalized convolution with the step
distribution µif it is the Markov process with the transition probabilities
Pk,n(x, dy) = δxµ(nk)(dy), n >k.
The consistency of this definition and the existence of the random walk
with respect to generalized convolution was shown in [5]. Notice
that in the case of classical convolution it is the simple random walk
with the step distribution µand it can be simply represented as Xn=
U1+···+Un, where (Uk)is a sequence of i.i.d. random variables with
distribution µ.
There are only two cases, when generalized convolution is represen-
tative, i.e. there exists a sequence of functions fn:RnRsuch that
Xn=fn(U1,...,Un):
4.2. for the α-convolution Xn=Uα
1+···+Uα
n1,
4.3. for the -convolution we have Xn= max{U1,...,Un}.
For other generalized convolutions rewriting convolution in the lan-
guage of the corresponding independent random variables is more com-
plicated (if possible) and requires assistance of some additional vari-
ables. For example we have
4.4. for the Kendall convolution for x[0,1] the measure δxαδ1is
the distribution of the random variable
xα1(ω) := 1{Q(ω)>xα}+1{Q(ω)6xα}Π2α(ω),
8 JASIULIS-GOŁDYN, LECHAŃSKA, MISIEWICZ
where Qhas uniform distribution on [0,1],Π2αhas the Pareto distri-
bution with the density π2αdescribed in example 3.4, Qand Π2αare
independent;
4.5. for the Kingman convolution and a, b > 0we can define:
aωsb(ω) := pa2+b2+ 2abθs,
where θsis absolutely continuous with the density function fsdescribed
in example 3.5.
5. Model for αrandom walk
For αgeneralized convolution on P+we have
Xn=Xα
n1+Uα
n1= (Uα
1+···+Uα
n)1, n >1,
where (Uk)are independent identically distributed random variables
with cumulative distribution function FUresponsible for the damage
claim values. By F=FUαwe denote the cumulative distribution func-
tion of Uα. We assume also that mα=EUα
1<.
We assume here that the variables Vk, responsible for the insurance pre-
mium during the time Tkare independent identically distributed with
the cumulative distribution function FV(x) = 1eγxα, for x > 0. This
assumption seems to be natural, since this is the distribution with the
lack of memory property (see [13]) for α-convolution. Consequently
Yn=Yα
n1+Vα
n1= (Vα
1+···+Vα
n)1, n >1.
Now we have
Rt=uα+βα
X
n=1
Yα
n1{Nt=n}1
α
X
n=1
Xα
n1{Nt=n}1
α
.
We want to calculate the ruin probability (see [3, 9]) for the insurance
company uby the time t:
Qt(u) = Pst:Rs60
in the special case t=, i.e.:
Q(u) = Pt > 0 : Rt60.
Since the ruin can occur only in the claims arrival moments i.e. in
the moments of jumps of the Poisson process Nt, thus it is enough to
consider RSn:
Q(u) = PRt= 0 for some t > 0= 1 PRSn>0nN
= 1 Puα
n
X
i=1 Uα
iβαVα
i>0nN
= 1 Psup
n>1
n
X
i=1 Uα
iβαVα
i< uα=: 1 δ(uα).
CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 9
Basically, the function δ(uα)we can calculate following the classical
calculations:
δ(uα) = Pnsup
n>2
n
X
i=2
(Uα
iβαVα
i)< uα+βαVα
1Uα
1, Uα
1βαVα
1< uαo
=
Z0
γeγ y
uα+βαy
Z0
Pnsup
n
n
X
i=2
(Uα
iβαVα
i)< uα+βαyxoFx(dx)dy
=βαeγuα
βα
Z
uα
γe
γz
βα
z
Z0
δ(zx) dF(x)dz.
In the last step in these calculations we substituted uα+βαy=z. Now
we calculate the derivative of both sides of this equality with respect
to duα:
dδ(uα)
duα=γ
βαδ(uα)γ
βα
uα
Z0
δ(uαx)dF(x).
Integrating both sides of this equality over the set [0, t] with respect
to the measure with the density function αuα1for u > 0we obtain:
δ(tα) = δ(0) + γα
βα
t
Z0
uα1δ(uα)duγα
βα
t
Z0
uα
Z0
uα1δ(uαx)dF(x)du.
The first integral on the right hand side we denote by I1, second by I2.
Then
I1=α
t
Z0
uα1δ(uα)du=
tα
Z0
δ(y)dy=
tα
Z0
δ(tαx)dx.
In the second integral we change order of integration and then substi-
tute uαx=r:
I2=
t
Z0
uα
Z0
αuα1δ(uαx)dF(x)du=
t
Z0
t
Z
x1
α
αuα1δ(uαx)dudF(x)
=
tα
Z0
tαx
Z0
δ(r)d dF(x) = F(x)
tαx
Z0
δ(r)drtα
x=0
+
tα
Z0
δ(tαx)F(x)dx.
The last equality we obtained integrating by parts. Since F(0)=0 we
obtain
δ(tα) = δ(0) + γ
βα
tα
Z0
δ(tαx) (1 F(x)) dx.
10 JASIULIS-GOŁDYN, LECHAŃSKA, MISIEWICZ
In order to calculate δ(0) notice first that ruin probability for the in-
surance company with infinite initial capital is zero, thus δ() = 1
and we have
1 = δ(0) + γ
βα
Z0
(1 F(x))dx=δ(0) + γ
βαµα.
Thus δ(0) = 1 γ
βαµαand we have
δ(tα) = 1 γ
βαµα+γ
βα
tα
Z0
δ(tαx)(1 F(x))dx.
For the convenience in further calculations we substitute tα=z. Let b
f
be the Laplace-Stietjes transform given by b
f(s) =
R0
ezsf(z)dz. Thus
Z0
ezs
z
Z0
δ(zx)G(x)dxdz=
Z0
Z
x
ezsδ(zx) dz G(x) dx
=
Z0
Z0
e(x+z)sδ(z) dz G(x) dx=b
δ(s)b
G(s).
Consequently we obtain:
b
δ(s) = 1γ
βαµα1
s+γ
βαb
δ(s)b
G(s),
thus
b
δ(s) = βαγµα
(βαγb
G(s))s.()
Since the Laplace-Stietjes transform uniquely determines function, we
finally have that in this case the ruin probability for the insurance
company with the initial capital uis equal Q(u) = 1 δ(uα)with the
function δobtained from the equation ().
6. Model for -generalized convolution
For the random walk with respect to the -convolution on P+we
have Xn= max{U1,...,Un}and Yn= max{V1,...,Vn}, where (Uk)
and (Vk)are independent sequences of i.i.d. positive random variables
with distributions µand νand the cumulative distribution functions F
and Grespectively. Consequently Xnhas the cumulative distribution
function Fn,Ynhas this function equal Gnand uYnhas Gn.
The first safety condition for the insurance company is ERt>0, thus
we need to calculate EXtand E(uYt), where
Xt=
X
n=1
Xn1{Nt=n}, Yt=
X
n=1
Yn1{N(t)=n}.
CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 11
We have
EXt=
X
n=0 Z
0
x(λt)n
n!eλt dx[F(x)n] = eλt Z
0
x dx[exp{λtF (x)}]
=λt Z
0
x eλtF(x)dF (x),
where F= 1 Fis the survival function for Uk. In order to calculate
E(uYt)notice first that the variable uYnis taking value uwith
probability G(u)n, thus
E(uYt) = λtueλtG(u)+λt Z(u,)
x eλtG(x)dG(x)
with the same notation G= 1 G. Consequently the first safety
condition in the case of -generalized convolution is the following:
ueλtG(u)+Z(u,)
x eλtG(x)dG(x)>Z
0
x eλtF(x)dF (x).
Usually we take the random variables Vkwith the distribution having
the lack of memory property, which in the case of -convolution is
given by the cumulative distribution function G(x) = 1(a,)(x)for
some a > 0(see [13] for details). In this case we have
E(uYt) = λtuaeλtG(ua)
Consequently the first safety condition for the -convolution is the
following:
uaeλtG(ua)Z
0
x eλtF(x)dF (x)>0t > 0.
Calculating the probability that the company will not bankrupt in the
unbounded time horizon we shall consider two cases. If the insur-
ance premium has distribution with the lack of memory property, i.e.
G(x) = 1(a,)(x), then we have Yk=V1=afor all kN, thus
δ(u) = PuYk> Xkfor all kN=Pua > Xkfor all kN
=P \
kN{X1< u a, . . . , Xk< u a}!
= lim
k→∞
PX1< u a, . . . , Xk< u a= lim
k→∞ Fkua.
We see that bankruptcy in unbounded time horizon is granted if only
the random variables Ukcan take any positive value, i.e. if F(x)<1
for all x > 0. However if the biggest possible claim is less than ua
then bankruptcy is impossible and δ(u) = 1.
12 JASIULIS-GOŁDYN, LECHAŃSKA, MISIEWICZ
If we assume that the cumulative distribution functions F, G are not
trivial then we have
δ(u) = P{uYk> Xkfor all kN}
=P{uV1> X1, u V1Y
k> U1X
kfor all kN}
=Z{uy>x}
P{uyY
k> x X
kfor all kN}dF (x)dG(y)
=Z{uy>x}
P{uyY
k> X
kfor all kN}dF (x)dG(y)
=Zu
0Zu
0
δ(u)dF (x)dG(y) + Z
uZy
0
δ(y)dF (x)dG(y),
where Y
k= max{V2,...,Vk+1}and X
k= max{U2,...,Uk+1}. For
a > b > 0let δ(a, b) = P{aY
k> b X
kfor all kN}. Thus we
can write:
δ(u) = δ(u)G(u)F(u) + Z
u
δ(y)F(y)dG(y).
If the distribution functions F, G have densities f, g then differentiating
both sides of the previous equation we obtain
δ(u) = δ(u)F(u)G(u) + δ(u)G(u)f(u).
Example. Assume that for 0< a < b we have
F(x) = x
a1(0,a](x) + 1(a,)(x), G(x) = x
b1(0,b](x) + 1(b,)(x).
Then for u(0, a)
δ(u)
δ(u)=
u
ab
1u2
ab
,thus δ(u) = δ(0)
q1u2
ab
.
For u(a, b)we have
δ(u) = δ(u)x
b1(a,b](u)thus δ(u) = const.
If u > b then evidently δ(u) = 1. Since in our case the function δis
continuous then δ(0) = p1a/b, thus finally
δ(u) = s1a
b
1u2
ab
.
7. Model for the Kendall random walk
The Kendall random walk, i.e. random walk with respect to Kendall
convolution, {Xn:nN0}for fixed α > 0can be described by the
recursive construction given below. We see that we can get here ex-
plicit formulas for Xn, but except the sequence (Un)we need also two
sequences of random variables (catalyzers of α-adding)
1. (Uk)i.i.d. random variables with distribution µ;
CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 13
2. (ξk)i.i.d. random variables with uniform distribution on [0,1];
3. k)i.i.d. random variables with distribution Pareto π2α;
where all these sequences are independent. Then the Kendall random
walk has the following representation: X00,
X1=U1, Xn+1 =Mn+11(ξn+1 > ̺n+1) + Πn+11(ξn+1 < ̺n+1 ),
where
Mn+1 = max{Xn, Un+1}, mn+1 = min{Xn, Un+1}, ̺n+1 =mα
n+1
Mα
n+1
.
This representation is especially helpful if we want to make computer
simulation of the Kendall random walk. For calculations however it is
more convenient to use the Markov properties and transition probabil-
ities given in Definition 4.1.
We consider here two Markov chains {Xn:nN0}and {Yn:nN0}
with transition probabilities given respectively as follows:
Lemma 7.1. For all x, y, t 0and µ, ν ∈ P+we have
h(x, y, t) := δxαδy(0, t) = 1xy
t2α1{x<t,y<t}
=hΨx
t+ Ψ y
tΨx
tΨy
ti1{x<t,y<t},
δvαµ(0, t) = P1(v, (0, t))
=hΨv
tF(t) + 1Ψv
tH(t)i1{v<t},
δuαν(0, t) = Π1(u, (0, t))
=hΨu
tG(t) + 1Ψu
tJ(t)i1{u<t}.
For proof of the Lemma see e.g. [2]. Moreover for any number of steps
the above formulas are generalized in the following way:
Lemma 7.2. Let µ, ν ∈ P+and nN. Then
δuαναn(0, t) = Gu,n(t) = hΨu
tGn(t) + 1Ψu
tJn(t)i1{u<t}
=J(t)n1hn(G(t)J(t)) Ψ u
t+J(t)i1{u<t}.
and
δvαµαn(0, t) = Fv,n(t) = hΨu
tFn(t) + 1Ψu
tHn(t)i1{v<t}
=H(t)n1hn(F(t)H(t)) Ψ u
t+H(t)i1{v<t}.
Proof.
14 JASIULIS-GOŁDYN, LECHAŃSKA, MISIEWICZ
7.1. Inversion formula and cumulative distribution functions.
The generalized characteristic function for the Kendall convolution is
the Williamson integral transform:
Φµ(t) :(1)
=Z
0
(1 (ts)α)+dF (s)
(2)
=F(1/t)tαZ1/t
0
sαdF (s)(3)
=αtαZ1/t
0
sα1F(s)ds.
Notice that the Williamson transform (see [11, 14, 15, 28]) is easy to
invert:
If µhas cumulative distribution function F, then for H(t) := Φµ(1/t)
using the formulation (3) we have
Zt
0
sα1F(s)ds =α1tαH(t).
Differentiating both sides with respect to twe obtain
F(t) = α1t1αd
dt (tαH(t)) = H(t) + α1tH(t),(∗∗)
thus also H(t) = αt1(F(t)H(t)). Applying this technique for the
c.d.f. Fnof Kendall random walk Xnwith the step variables (Uk)
H(t)n=αtαZ1/t
0
sα1Fn(s)ds
we see that
Fn(t) = H(t)n1H(t)+1tH(t)=H(t)n1H(t)+n(F(t)H(t)).
Since Xt=P
n=0 Xn1{Nt=n}we see that the c.d.f. Ftof Xtis given by
Ft(x) =
X
n=0
Fn(x)(λt)n
n!eλt =1 + λtF(x)H(x)eλt(1H(x)).
Cumulative distribution function Fv,µ (t)of vXnis then given by
Fv,n(t) = α1t1αd
dt tα(1 uαtα)+Hn(t)
=1[u,)(t)Hn1(t)H(t) + n(1 uαtα)+(F(t)H(t)).
We need also to calculate the cumulative distribution function for the
variable uYnwith the distribution Gu,n, where Ynis the Kendall
random walk with the steps (Vk)i.i.d. random variables with distribu-
tion νand distribution function G. We see that uYnhas distribution
δuαναnand the generalized characteristic function (1(tu)α)+Φn
ν(t)
for n>1and δuανα0=δuhas the distribution function Gu,0(t) =
CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 15
1[u,)(t). Let J(t) = Φν(t1), thus J(t) = αt1(G(t)J(t)). Using
formula (∗∗) for µreplaced by δuνnwe obtain for n>1
Gu,n(t) = α1t1αd
dt tα(1 uαtα)+Jn(t)
=1[u,)(t)Jn1(t)J(t) + n(1 uαtα)+(G(t)J(t)).
This distribution has an atom at the point uof the weight Gu,n(u+) =
J(u)nand the absolutely continuous part with the density
gu,n(t) = 1(u,)(t) [.].
Since uYt=P
n=0(uYn)1{Nt=n}we see that the c.d.f. Gu,t of uYt
is given by
Gu,t(x) =
X
n=0
Gu,n(x)(λt)n
n!eλt
=1[u,)(x)1 + λt1uαxαG(x)J(x)eλt(1J(x)).
This distribution has an atom at uof the weight Gu,t(u+) = eλt(1J(u)).
7.2. First safety condition for the insurance company. In the
classical theory the first safety condition for the insurance company
states that ERt>0for all t > 0. In our case we have
E(uYt)αEXα
t>0for all t > 0.
First we calculate EXα
tassuming that the distribution of U1is abso-
lutely continuous with respect to the Lebesgue measure (if this is not
the case we shall add the atomic part):
EXα
t=Z
0
xαdFt(x) = Z
0
αxα1(1 Ft(x)) dx
=Z
0
αxα111 + λt
αxH(x)eλt(1H(x))dx
=Z
0h(xα)xαeλt(1H(x))idx
=xα1eλt(1H(x))
0= lim
x→∞ xα1eλt(1H(x)).
In the similar way for absolutely continuous distribution of V1we obtain
EuYtα=uαGu,t(u+) + Z
0
αxα1(1 Gu,t(x)) dx
=uαGu,t(u+) + Zu
0
αxα1dx +Z
uxα(xαuα)eλt(1J(x))dx
=uαGu,t(u+) + uα+xα(xαuα)eλt(1J(x))
u
=uαeλt(1J(u)) + lim
x→∞ xα(xαuα)eλt(1J(x)).
16 JASIULIS-GOŁDYN, LECHAŃSKA, MISIEWICZ
If we consider as νthe distribution with the lack of memory property
for the Kendall convolution (see [13]) then for some c > 0we have
G(x) = min{(cx)α,1}, J(x) = 1
2(cx)α1[0,c1](x)+11
2(cx)α1(c1,)(x),
and, assuming that 1[a,b]0for a > b we have
Gu,t(x) = 1 + λt
2(cx)α(1 uαxα)eλt(11
2(cx)α)1[u,c1](x)
+1 + λt
2(cx)α(1 uαxα)eλt
2(cx)α1[uc1,)(x).
Notice that in this case
Gu,t(u+) = eλt(11
2(cu)α)if u6c1,
eλt
2(cu)αif u > c1.
Consequently, for u > c1=α+1
αRxdG(x), which is a natural assump-
tion since the initial capital shall be significant, we have
EuYtα=uαeλt
2(cu)α+ lim
x→∞ xαh11uαxαeλt
2(cx)αi
=uαeλt
2(cu)α+uα+λt
2cα.
For µwith the lack of memory property with c > 0in the Kendall
convolution algebra we have
F(x) = min{(cx)α,1}, H(x) = 1
2(cx)α1[0,c1](x)+11
2(cx)α1(c1,)(x),
Since
EXα
t= lim
x→∞ xα1eλt(1H(x))=λt
2cα
we have
EuYtαEXα
t=uαeλt
2(cu)α+uα>0,
i.e. the first safety condition holds.
7.3. Ruin probability in the infinite time horizon. Let Q(u)
be the ruin probability for our model:
Q(u) = 1 PuYk> Xkfor all kN=: 1 δ(u).
For the convenience we shall use the following notation: for the Markov
sequence Xnstarting in the point vwe will write
Xv
1=vU1, Xv
2=vU1U2,···
and for the Markov sequence Ynstarting at the point u > 0we write
Yu
1=uV1, Y u
2=uV1V2,···
Let
Λ(v, u) = PuYk> v Xkfor all kN
=PuV1> v U1, u V1V2> v U1U2....
CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 17
We need to calculate δ(u) = Λ(0, u). Thus
Λ(v, u) = Z
0
Py1> v U1, y1V2> v U1U2...δuαν(dy1) =
Z
0Z{y1>x1}
Py1V2> x1U2, y1V2V3> x1U2U3...
δuαν(dy1)δvαµ(dx1)
=Z
0Z{y1>x1}
Λ(x1, y1)δuαν(dy1)δvαµ(dx1)
Now we have for v > u
Λ(v, u) = Z
vZ
x
Λ(x, y)dΨu
yG(y) + 1Ψu
yJ(y)δvαµ(dx),
and if v6u
Λ(v, u) =
Z
vΛ(x, u)J(u) + Z
u
Λ(x, y)dΨu
yG(y) + 1Ψu
yJ(y)δvαµ(dx)
=J(u)Z
v
Λ(x, u)δvαµ(dx)
+Z
vZ
u
Λ(x, y)dΨu
yG(y) + 1Ψu
yJ(y)δvαµ(dx).
Since the above integral equations are very complicated and require the
use of additional mathematical tools we hope to consider this problem
in the future proposing it as an open question from the work written
here.
Acknowledgements. This paper is a part of project "First order
Kendall maximal autoregressive processes and their applications", which
is carried out within the POWROTY/REINTEGRATION programme
of the Foundation for Polish Science co-financed by the European Union
under the European Regional Development Fund.
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