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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=Sk−Sk−1, k = 2,3,...

are iid with exponential distribution with mean ETk= 1/λ;

(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=Sk−Sk−1, k = 2,3,...

are iid random variables with exponential distribution, ETk= 1/λ;

(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:n∈N0}, which is a ⋄-Lévy process with the step distribution

Ui∼µ∈ 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(t−1).

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:n∈N0},

which is a ⋄-Lévy process with the step distribution Vi∼ν∈ 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(t−1).

(e) Independence assumption: the processes {N(t): n∈N0},

{Xn:n∈N0}and {Yn:n∈N0}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 u⊕Ynis the Markov process {Yn:n∈N0}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=u⊕Yn−Xn, where

Xn=U1⊕U2⊕ · ·· ⊕ Unand Yn=V1⊕V2⊕ · · · ⊕ Vnand ⊕denotes

adding in the generalized convolution sense, i.e Xn∼µ⋄nand Yn∼ν⋄n

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) = P∃s6t:Rs<0

= 1 −

∞

X

n=0

PRSk>0: k= 0,1,...,nPNt=n

= 1 −

∞

X

n=0

Pu⊕Yk> Xk:k= 0,1,...,n(λt)n

n!e−λt.

Calculating the same probability in the unbounded time horizon Q∞(u)

we see that

Q∞(u) = P∃t > 0 : Rt<0

= 1 −Pu⊕Yk> Xkfor all k∈N0.

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 λn→λ. 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) δ0⋄λ=λ;

(ii) (pλ1+ (1 −p)λ2)⋄λ=p(λ1⋄λ) + (1 −p)(λ2⋄λ)whenever p∈[0,1];

(iii) Ta(λ1⋄λ2) = (Taλ1)⋄(Taλ2);

(iv) if λn→λthen λn⋄η→λ⋄ηfor all η∈ P and λn∈ P+,

(v) there exists a sequence (cn)n∈Nof positive numbers such that the

sequence Tcnδ⋄n

1converges to a measure diﬀerent from δ0.

The set (P+,⋄)we call a generalized convolution algebra. A continuous

mapping h:P → Rsuch that

•h(pλ + (1 −p)ν) = ph(λ) + (1 −p)h(ν),

•h(λ⋄ν) = 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:

h≡1and h≡0. 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+∋λ−→ Φλ∈C([0,∞))

such that for all λ, ν, λn∈ P+the following conditions hold:

1. Φpλ+qν (t) = pΦλ(t) + qΦν(t), for p, q >0,p+q= 1;

2. Φλ⋄ν(t) = Φλ(t)Φν(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

λn→λ

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 Ωis 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:

δa∗δb=δa+b.

Here we have Ω(t) = e−tif 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

δa∗sδb=1

2δ|a−b|+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:

δa∗αδb=δc, c = (aα+bα)1/α, a, b >0.

The kernel of generalized ∗α-characteristic function is Ω(t) = e−tα.

3.3. ∞-convolution is deﬁned by

δa∨ δb=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

δx△αδ1=xαπ2α+ (1 −xα)δ1, x ∈[0,1],

where π2αis a Pareto measure with density 2αx−2α−11[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

δa⊗ωsδb=Lpa2+b2+ 2abθs,

where θsis absolutely continuous with the density function

fs(x) = Γ(s+ 1)

√πΓ(s+1

2)1−x2s−1

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:

δx⋄δ1=ϕ(x)δ1+xpλ1+ (c+ 1)(x−xp)λ2,

where λ1, λ2are probability measures absolutely continuous with re-

spect to the Lebesgue measure and independent of x. For example if

c= (p−1)−1then

λ1(du) = 2c

u3h(c+1)(p+1)u1−p+(c+1)(p−2)+cp(2p−1)u−2p−2i1[1,∞)(u)du,

and

λ2(du) = c2(p−2) + (p+ 1)u−p+1u−31[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:n∈N0}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) = δx⋄µ⋄(n−k)(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:Rn→Rsuch that

Xn=fn(U1,...,Un):

4.2. for the ∗α-convolution Xn=Uα

1+···+Uα

n1/α,

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 δx⋄αδ1is

the distribution of the random variable

x⊕△α1(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:

a⊕ωsb(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

n−1+Uα

n1/α = (Uα

1+···+Uα

n)1/α , 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) = 1−e−γxα, 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

n−1+Vα

n1/α = (Vα

1+···+Vα

n)1/α , 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) = P∃s≤t:Rs60

in the special case t=∞, i.e.:

Q∞(u) = P∃t > 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>0∀n∈N

= 1 −Puα−

n

X

i=1 Uα

i−βαVα

i>0∀n∈N

= 1 −Psup

n>1

n

X

i=1 Uα

i−βαVα

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

i−βαVα

i)< uα+βαVα

1−Uα

1, Uα

1−βαVα

1< uαo

=

∞

Z0

γe−γ y

uα+βαy

Z0

Pnsup

n

n

X

i=2

(Uα

i−βαVα

i)< uα+βαy−xoFx(dx)dy

=β−αeγuα

βα

∞

Z

uα

γe

−γz

βα

z

Z0

δ(z−x) 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(uα−x)dF(x).

Integrating both sides of this equality over the set [0, t] with respect

to the measure with the density function αuα−1for u > 0we obtain:

δ(tα) = δ(0) + γα

βα

t

Z0

uα−1δ(uα)du−γα

βα

t

Z0

uα

Z0

uα−1δ(uα−x)dF(x)du.

The ﬁrst integral on the right hand side we denote by I1, second by I2.

Then

I1=α

t

Z0

uα−1δ(uα)du=

tα

Z0

δ(y)dy=

tα

Z0

δ(tα−x)dx.

In the second integral we change order of integration and then substi-

tute uα−x=r:

I2=

t

Z0

uα

Z0

αuα−1δ(uα−x)dF(x)du=

t

Z0

t

Z

x1

α

αuα−1δ(uα−x)dudF(x)

=

tα

Z0

tα−x

Z0

δ(r)d dF(x) = F(x)

tα−x

Z0

δ(r)drtα

x=0

+

tα

Z0

δ(tα−x)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(tα−x) (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(tα−x)(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

e−zsf(z)dz. Thus

∞

Z0

e−zs

z

Z0

δ(z−x)G(x)dxdz=

∞

Z0

∞

Z

x

e−zsδ(z−x) 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) = 1−γ

βαµα1

s+γ

βαb

δ(s)b

G(s),

thus

b

δ(s) = βα−γµα

(βα−γb

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 u⊕Ynhas Gn.

The ﬁrst safety condition for the insurance company is ERt>0, thus

we need to calculate EXtand E(u⊕Yt), 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!e−λt dx[F(x)n] = e−λt Z∞

0

x dx[exp{λtF (x)}]

=λt Z∞

0

x e−λtF(x)dF (x),

where F= 1 −Fis the survival function for Uk. In order to calculate

E(u⊕Yt)notice ﬁrst that the variable u⊕Ynis taking value uwith

probability G(u)n, thus

E(u⊕Yt) = λtue−λtG(u)+λt Z(u,∞)

x e−λtG(x)dG(x)

with the same notation G= 1 −G. Consequently the ﬁrst safety

condition in the case of ∞-generalized convolution is the following:

ue−λtG(u)+Z(u,∞)

x e−λtG(x)dG(x)>Z∞

0

x e−λtF(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(u⊕Yt) = λtu∨ae−λtG(u∨a)

Consequently the ﬁrst safety condition for the ∞-convolution is the

following:

u∨ae−λtG(u∨a)−Z∞

0

x e−λtF(x)dF (x)>0∀t > 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 k∈N, thus

δ(u) = Pu⊕Yk> Xkfor all k∈N=Pu∨a > Xkfor all k∈N

=P \

k∈N{X1< u ∨a, . . . , Xk< u ∨a}!

= lim

k→∞

PX1< u ∨a, . . . , Xk< u ∨a= lim

k→∞ Fku∨a.

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 u∨a

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{u⊕Yk> Xkfor all k∈N}

=P{u∨V1> X1, u ∨V1∨Y′

k> U1∨X′

kfor all k∈N}

=Z{u∨y>x}

P{u∨y∨Y′

k> x ∨X′

kfor all k∈N}dF (x)dG(y)

=Z{u∨y>x}

P{u∨y∨Y′

k> X′

kfor all k∈N}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{a∨Y′

k> b ∨X′

kfor all k∈N}. 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

1−u2

ab

,thus δ(u) = δ(0)

q1−u2

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) = p1−a/b, thus ﬁnally

δ(u) = s1−a

b

1−u2

ab

.

7. Model for the Kendall random walk

The Kendall random walk, i.e. random walk with respect to Kendall

convolution, {Xn:n∈N0}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. (Πk)i.i.d. random variables with distribution Pareto π2α;

where all these sequences are independent. Then the Kendall random

walk has the following representation: X0≡0,

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:n∈N0}and {Yn:n∈N0}

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) := δx△αδy(0, t) = 1−xy

t2α1{x<t,y<t}

=hΨx

t+ Ψ y

t−Ψx

tΨy

ti1{x<t,y<t},

δv△αµ(0, t) = P1(v, (0, t))

=hΨv

tF(t) + 1−Ψv

tH(t)i1{v<t},

δu△αν(0, t) = Π1(u, (0, t))

=hΨu

tG(t) + 1−Ψu

tJ(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 n∈N. Then

δu△αν△αn(0, t) = Gu,n(t) = hΨu

tGn(t) + 1−Ψu

tJn(t)i1{u<t}

=J(t)n−1hn(G(t)−J(t)) Ψ u

t+J(t)i1{u<t}.

and

δv△αµ△αn(0, t) = Fv,n(t) = hΨu

tFn(t) + 1−Ψu

tHn(t)i1{v<t}

=H(t)n−1hn(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

sα−1F(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

sα−1F(s)ds =α−1tαH(t).

Diﬀerentiating both sides with respect to twe obtain

F(t) = α−1t1−αd

dt (tαH(t)) = H(t) + α−1tH′(t),(∗∗)

thus also H′(t) = αt−1(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

sα−1Fn(s)ds

we see that

Fn(t) = H(t)n−1H(t)+nα−1tH′(t)=H(t)n−1H(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!e−λt =1 + λtF(x)−H(x)e−λt(1−H(x)).

Cumulative distribution function Fv,µ (t)of v⊕Xnis then given by

Fv,n(t) = α−1t1−αd

dt tα(1 −uαt−α)+Hn(t)

=1[u,∞)(t)Hn−1(t)H(t) + n(1 −uαt−α)+(F(t)−H(t)).

We need also to calculate the cumulative distribution function for the

variable u⊕Ynwith 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 u⊕Ynhas distribution

δu△αν△αnand the generalized characteristic function (1−(tu)α)+Φn

ν(t)

for n>1and δu△αν△α0=δuhas the distribution function Gu,0(t) =

CRAMER-LUNDBERG MODEL FOR EXTREMAL MARKOV SEQUENCES 15

1[u,∞)(t). Let J(t) = Φν(t−1), thus J′(t) = αt−1(G(t)−J(t)). Using

formula (∗∗) for µreplaced by δu⋄ν⋄nwe obtain for n>1

Gu,n(t) = α−1t1−αd

dt tα(1 −uαt−α)+Jn(t)

=1[u,∞)(t)Jn−1(t)J(t) + n(1 −uαt−α)+(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 u⊕Yt=P∞

n=0(u⊕Yn)1{Nt=n}we see that the c.d.f. Gu,t of u⊕Yt

is given by

Gu,t(x) =

∞

X

n=0

Gu,n(x)(λt)n

n!e−λt

=1[u,∞)(x)1 + λt1−uαx−αG(x)−J(x)e−λt(1−J(x)).

This distribution has an atom at uof the weight Gu,t(u+) = e−λt(1−J(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(u⊕Yt)α−EXα

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

αxα−1(1 −Ft(x)) dx

=Z∞

0

αxα−11−1 + λt

αxH′(x)e−λt(1−H(x))dx

=Z∞

0h(xα)′−xαe−λt(1−H(x))′idx

=xα1−e−λt(1−H(x))

∞

0= lim

x→∞ xα1−e−λt(1−H(x)).

In the similar way for absolutely continuous distribution of V1we obtain

Eu⊕Ytα=uαGu,t(u+) + Z∞

0

αxα−1(1 −Gu,t(x)) dx

=uαGu,t(u+) + Zu

0

αxα−1dx +Z∞

uxα−(xα−uα)e−λt(1−J(x))′dx

=uαGu,t(u+) + uα+xα−(xα−uα)e−λt(1−J(x))∞

u

=uαe−λt(1−J(u)) + lim

x→∞ xα−(xα−uα)e−λt(1−J(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,c−1](x)+1−1

2(cx)−α1(c−1,∞)(x),

and, assuming that 1[a,b]≡0for a > b we have

Gu,t(x) = 1 + λt

2(cx)α(1 −uαx−α)e−λt(1−1

2(cx)α)1[u,c−1](x)

+1 + λt

2(cx)−α(1 −uαx−α)e−λt

2(cx)−α1[u∨c−1,∞)(x).

Notice that in this case

Gu,t(u+) = e−λt(1−1

2(cu)α)if u6c−1,

e−λt

2(cu)−αif u > c−1.

Consequently, for u > c−1=α+1

αRxdG(x), which is a natural assump-

tion since the initial capital shall be signiﬁcant, we have

Eu⊕Ytα=uαe−λt

2(cu)−α+ lim

x→∞ xαh1−1−uαx−αe−λt

2(cx)−αi

=uαe−λt

2(cu)−α+uα+λt

2c−α.

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,c−1](x)+1−1

2(cx)−α1(c−1,∞)(x),

Since

EXα

t= lim

x→∞ xα1−e−λt(1−H(x))=λt

2c−α

we have

Eu⊕Ytα−EXα

t=uαe−λt

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 −Pu⊕Yk> Xkfor all k∈N=: 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=v⊕U1, Xv

2=v⊕U1⊕U2,···

and for the Markov sequence Ynstarting at the point u > 0we write

Yu

1=u⊕V1, Y u

2=u⊕V1⊕V2,···

Let

Λ(v, u) = Pu⊕Yk> v ⊕Xkfor all k∈N

=Pu⊕V1> v ⊕U1, u ⊕V1⊕V2> v ⊕U1⊕U2....

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, y1⊕V2> v ⊕U1⊕U2...δu△αν(dy1) =

Z∞

0Z{y1>x1}

Py1⊕V2> x1⊕U2, y1⊕V2⊕V3> x1⊕U2⊕U3...

δu△αν(dy1)δv△αµ(dx1)

=Z∞

0Z{y1>x1}

Λ(x1, y1)δu△αν(dy1)δv△αµ(dx1)

Now we have for v > u

Λ(v, u) = Z∞

⌊vZ∞

x

Λ(x, y)dΨu

yG(y) + 1−Ψu

yJ(y)δv△αµ(dx),

and if v6u

Λ(v, u) =

Z∞

⌊vΛ(x, u)J(u) + Z∞

u

Λ(x, y)dΨu

yG(y) + 1−Ψu

yJ(y)δv△αµ(dx)

=J(u)Z∞

⌊v

Λ(x, u)δv△αµ(dx)

+Z∞

⌊vZ∞

u

Λ(x, y)dΨu

yG(y) + 1−Ψu

yJ(y)δv△αµ(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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