Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability

318 Chapter 7

3.3.4 The M/SM model

Here, we keep a semi-Markov process

(, )

B

PF for the claim amount process and

we assume that the claim arrival process is a continuous time Markov process

(see Chapter 3, section 12.3), i.e.

A

F

is defined as:

0, 0,

()

1,0.

i

A

ij

x

Fx

ex

λ

−

<

⎧

=

⎨

−

≥

⎩

(3.50)

3.3.5 The M ' /SM Model

This is a variant of the preceding model, in which we set:

0, 0,

()

1,0.

j

A

ij

x

Fx

ex

λ

−

<

⎧

⎪

=

⎨

−

≥

⎪

⎩

(3.51)

In this case, the interarrival time distribution only depends on the future state,

and not on the past one, which is, as we already mentioned, more natural in risk

theory, but now the arrival process is no longer Markovian.

3.3.6 The SM(0)/SM(0) Model

Particular assumptions of this model are:

() (), () (),, ,, 0,

,, .

AAB B

ij j ij j

ij j

Fx Fx Fy Fyij Ixy

pijI

π

==∈>

=∈

(3.52)

So the claim types are independent and have the same distribution.

If we suppose that the initial type

0

J

has the same distribution, the processes

(, 1),(, 1)

nn

Xn Yn≥≥

become two renewal processes having as d.f. respectively

() (), () (),

AAB B

jj jj

jj

F

xFxFyFy

ππ

==

∑∑

(3.53)

but, due to the J-process, are not independent as:

(,,) ()().

AB

nn n jj j

PX xY yJ j F x F y

π

≤≤ == (3.54)

Let us recall that, in the terminology of Pyke (1962), the processes

(( , ), 0),(( , ), 0)

nn nn

JX n JY n≥≥ are called SMP of zero order of second type (see

Chapter 4, section 13.2).

3.3.7 The SM ' (0)/SM' (0) Model

Particular assumptions of this model are:

() (),() (),, ,,0,

,, .

AAB B

ij i ij i

ij j

Fx Fx Fy Fyij Ixy

pijI

π

==∈>

=∈

(3.55)

Insurance risk models 319

Here too, the claim types are independent and have the same distribution but

depending on the preceding claim.

If we suppose that the initial type

0

J

has the same distribution, the processes

(, 1),(, 1)

nn

Xn Yn≥≥ become two renewal processes having as d.f. respectively

() (), () (),

AAB B

jj jj

jj

F

xFxFyFy

ππ

==

∑∑

(3.56)

but, due to the J-process, not independent as:

(,,) ()().

AB

nn n jii i

i

PX xY yJ j F x F y

ππ

≤≤ ==

∑

(3.57)

Let us recall that, in the terminology of Pyke (1962), the processes

(( , ), 0),(( , ), 0)

nn nn

JX n JY n≥≥

are called SMP of zero order of first type.

3.3.8 The mixed zero order SM ' (0)/SM(0) and SM(0)/SM ' (0) models

These last two particular models are obtained by making the following two

choices:

a) for the SM '(0)/SM(0) model:

,, ,

() (), , 0,

ij j

AA

ij i

BB

ij j

pijI

Fx Fxi Ix

Fy Fyj Iy

π

⎧

=∈

⎪

=

∈≥

⎨

⎪

=

∈≥

⎩

(3.58)

b) for the SM(0)/SM ' (0) model:

,, ,

() (), , 0,

() (), , 0.

ij j

AA

ij j

BB

ij i

pijI

Fx Fxi Ix

Fy Fyj Iy

π

⎧

=∈

⎪

=

∈≥

⎨

⎪

=

∈≥

⎩

(3.59)

For the SM '(0)/SM(0) model, it is straightforward to see that:

(,,) () ()

BA

nn n jj ii

i

PX xY yJ j F y F x

ππ

≤≤ ==

∑

(3.60)

and for the SM(0)/SM ' (0) model:

(,,) () ().

AB

nn n jj ii

i

PX xY yJ j F x F y

ππ

≤≤ ==

∑

(3.61)

Here too, the processes ( , 1),( , 1)

nn

Xn Yn≥≥become two renewal processes

having the d.f given by relation (3.56) but furthermore they are independent.

Consequently, if one makes abstractions of the types of claims, these two last

models may be treated, at least for the ruin problem, as a G/G model

characterised by the d.f.

,

AB

F

F .

320 Chapter 7

3.4 The Ruin Problem For The General SMRM

3.4.1 Ruin and Non-Ruin Probabilities

Using definition (1.22) for the lifetime T of the company,

{

}

inf : ( ) 0 ,Ttt

α

=< (3.62)

we know that the event “ruin” occurs before or at time t iff

Tt

≤

and of course

the complementary event called “non-ruin” iff T>t.

As we must now take into account the types of claims, we will use the following

notation for transient non-ruin and ruin probabilities, i.e. on the finite time

horizon

[]

0,t ,

(,) ( , () (0) ),

( , ) ( , ( ) (0) )( 1 ( , )).

ij

ij ij

ut PT tZt jZ i

ut PT tZt jZ i ut

φ

ϕ

=> = =

Ψ=≤ = ==−

(3.63)

The asymptotic non-ruin and ruin probabilities, i.e. on an infinite time horizon,

are defined as

() ( , () (0) ) lim (,),

() ( , () (0) ) lim (,)( 1 ()).

ij ij

t

ij ij ij

t

uPT Zt jZ i ut

uPT Zt jZ i ut u

φφ

ϕ

→∞

==∞ = ==

Ψ=<∞ = ==Ψ =−

(3.64)

The following results are trivial but useful:

(i) for every fixed t,, ,(,)

ij

ij I ut

φ

∀∈ is increasing in u and ( , )

ij

ut

Ψ

decreasing,

(ii) for every fixed u,, ,(,)

ij

ij I ut

φ

∀∈ is decreasing in t and ( , )

ij

ut

Ψ

increasing,

(iii) , , , : ( , ) ( ), ( , ) ( ).

ij ij ij ij

utijI ut u ut u

φ

∀∀∀ ∈ ≥ Ψ ≥Ψ (3.65)

As we already said in section 1.1.4, one of the most important problems in risk

theory is the optimal determination of the security loading

η

such that the

probability of ruin, transient or asymptotic, is larger than

1, 0

ε

−

>

being fixed,

a problem equivalent to the optimal determination of the solvency margin.

Often, the problem is solved for the stationary version, thus giving excessive and

therefore careful values for

η

.

As is often the case, if we are not interested in the value of Z(t), then we may

introduce the following ruin probabilities:

11

(,)(,),()(),

(,) (,), () ().

mm

iijiij

jj

mm

iijiij

jj

ut ut u u

φφφφ

==

Ψ=Ψ Ψ=Ψ

∑∑

(3.66)

Insurance risk models 321

Moreover, if we start with

π

as initial distribution for

0

J

, we define the last four

ruin and non-ruin probabilities:

11

(,)(,),()(),

(,) (,), () ().

mm

ii ii

ii

mm

ii ii

ii

ut ut u u

φπφφπφ

ππ

==

Ψ=ΨΨ=Ψ

∑∑

(3.67)

Finally, for the stationary model defined in section 3.2, we introduce these last

ruin and non-ruin probabilities:

'' '

00

'

1

() (1 () ( ) (),

() (), () 1 ().

uct

sAB

jllljll

ll

kk

k

m

sss s

j

uFtdtuctydFy

uuuu

φφ

πη

φφ φ

∞+

=

=−+−

=Ψ=−

∑∑

∫∫

∑

(3.68)

3.4.2 Change of Premium Rate

Let us start with a general SMRM of kernel (.,.)

ij

Q

⎡

⎤

=

⎣

⎦

Q and with c as premium

rate per unit of time.

First, we will show that, without loss of generality, we can always work with an

equivalent SMRM for which c=1.

Indeed, if such is not the case, let us introduce the following new r.v.:

''

00

,,1

nn

XXXcXn

=

=≥,a.s. (3.69)

The SM kernel

Q ' of the process

(

)

'

,,, 0

nnn

JXYn≥ is given by:

1

11

'''

'

( , ) ( , , ( , , ,), 1,..., 1),

(, ) ( , ).

n

nn

Jj n n n k k k

Jj Jj

QxyPJjXxYyJXYk n

x

QxyQ y

c

−

−−

==≤≤ = −

=

(3.70)

For the SMRM with kernel

Q ' , we have:

'' '

,,,,

ij ij ij ij ij ij

p

pa cab bij I== =∈ (3.71)

so that

''

,,,

AABB

iiii

cciI

ηηηη

=

=∈ (3.72)

and by relation (3.20)

'

,,

jj jj

cjI

μμ

=

∈ (3.73)

and by relations (1.10) and (3.38)

1

'

B

ii

i

A

ii

i

c

cc

πη

=

∑



. (3.74)

322 Chapter 7

Taking into account the loading factor

η

, by relation (3.39), we know that the

value of c ' is given by:

'(1 )'cc

η

=

+



(3.75)

or by (3.74) and(3.39) again:

'(1 ) 1.

c

η

=

+=



(3.76)

As in this last relation the loading factor

η

is strictly positive, the last equality

shows that the ratio /cc



is strictly inferior to 1 and so by relations (3.74) and

(3.72) that the condition of having a strictly positive loading factor

η

is

equivalent to having, for the process

(

)

'

,,, 0

nnn

JXYn≥ , the condition

'

1.

B

ii

i

A

ii

i

πη

<

∑

(3.77)

3.4.3 General Solution Of The Asymptotic Ruin Probability Problem for a

general SMRM

From the results of the last subsection, let us consider a general ergodic SMRM

of kernel (.,.)

ij

Q

⎡

⎤

=

⎣

⎦

Q with c=1 and let us focus our attention on the process

(

)

(, ), 0

nn n

JY X n

−

≥ ; (3.78)

Clearly, this process can be seen as defining an SMRW of SM kernel

r

Q

given

by

{}

(,):

() ( , ).

r

ij ij

z

Qz Qd d

ξζ ξ ζ

ξζ

−≤

∫∫

(3.79)

In the special case of conditional independence, we get:

() ( ) ().

rBA

ij ij ij ij

Qz p Fz dF

ξ

+∞

−∞

=+

∫

(3.80)

We know that the position at epoch n of the SMRW is given by the partial sums

(, 0)

n

Sn≥ related to the random sequence (( , 0)

nn

YXn

−

≥ and introducing the

r.v. M defined by relation (20.1) of Chapter 3, that is

{

}

01

sup , ,..., ,...

n

MSSS= ,

we know that:

0

() ( ,lim ), 0,, ,

ij n

n

uPMu J jJiu ijI

φ

→∞

=≤ ==≥∈ (3.81)

and since the event non-ruin on

[

)

0,

∞

implies non-ruin at the first claim arrival

time, we have:

Insurance risk models 323

0, 0,

() ,,

() (),0.

u

r

ij

kj ik

k

u

uijI

uzdQzu

φ

−∞

<

⎧

⎪

=∈

⎨

−>

⎪

⎩

∑

∫

(3.82)

Summing over j, we get:

0, 0,

() , .

() (),0,

u

r

i

kik

k

u

uiI

uzdQzu

φ

−∞

<

⎧

⎪

=∈

⎨

−>

⎪

⎩

∑

∫

(3.83)

which is a Wiener-Hopf system of integral equations identical to the system (8.6)

of Chapter 5 as

0

() ( , ), 0,

i

uPMuJiu iI

φ

=

≤=≥∈ (3.84)

as in relation (20.6) of Chapter 3.

Considering now system (3.83) only for non-negative values of u, we know from

a result of Janssen (1970) mentioned in section 20 of Chapter 3 that this system

has a unique

P-solution iff

0.

r

kk

k

πη

<

∑

(3.85)

So, if this condition is not fulfilled, the SMRW

(

)

(,), 0

nn

JS n≥ drifts towards

+∞ and so ruin on

[

)

0,∞ is a certain event regardless of

0

J

, that is, in this case:

() 0, 0, .

i

uuiI

φ

=

≥∈ (3.86)

As

,,

rBA

kkk

kI

ηηη

=

−∈ (3.87)

condition (3.83) is equivalent to

()0

BA

kk k

k

πη η

−

<

∑

, (3.88)

which is equivalent to condition (3.77) always supposed to be fulfilled and

equivalent to the adjunction of a positive security loading to spoil the

asymptotically fair game “insurance company-policyholders” in favour of the

insurance company as, without this adjunction the ruin on an infinite horizon

time is certain.

Using the unicity theorem of Janssen (1970), it is clear that the following

relations are true:

() (),, , 0.

ij j i

uuijIu

φ

πφ

=

∈≥ (3.89)

We have thus proved the following theorem.

Theorem 3.1 For an ergodic SMRM, for every initial state i, the ruin on an

infinite time horizon is certain if condition (3.77) is not fulfilled.

On the contrary, if this condition is satisfied, then for every initial state i:

() (),, , ,

ij j i

uuijIu

φπφ

++

=∈∈ (3.90)

with

324 Chapter 7

0

() () (), () (),, ,

ij ij i i

uUu u u uijIu

φφφφ

++

==∈∈

(3.91)

and by (3.81)

() ( ) (), ,, .

u

r

ikik

k

uuzdQzuiI

φφ

++

−∞

=− ∈∈

∑

∫

 (3.92)

Let us mention that the passage to the functions

ij

φ

+

is done because we are only

interested in the non-ruin probabilities for positive values of the reserve u, though

it may be that these probabilities are not necessarily zero for u negative.

Though there exist a lot of theoretical results on the Wiener-Hopf integral system

(3.92) like factorisation results and so on, there exist no explicit forms for non-

ruin probabilities for a general SMRM and so our approach is to see what

particular SMRM can be treated in such a way that right information can be

obtained by transferring the problem towards a model for which an explicit

solution exists or at least having satisfactory information.

Of course, for the general SMRM, the numerical approach remains very

important.

Remark 3.3 We know that for m=1, the GSMRM brings us back to the G/G

model.

So, for this one, the system (3.92) becomes:

() ( ) (), ,

() ( ) ().

u

r

uuzdQzu

Qz B udA

φφ

ξξ

++

−∞

+∞

−∞

=

−∈

=+

∫



(3.93)

3.5 The Ruin Problem For Particular SMRM

This section undertakes an analytical study of the particular SMRM introduced

above to get supplementary results for solving the ruin problem in these

particular cases.

3.5.1 The Zero Order Model SM(0)/SM(0)

In this case, we have from relation (3.79) that:

() ( ) (),

rBA

ij j j j

Qz F zdF

π

ξξ

+∞

−∞

=+

∫

(3.94)

such that we may write:

() ()

rr

ij j

Qz Qz= (3.95)

and the Wiener-Hopf system (3.92) becomes:

() ( ) (), ,, .

u

r

ikk

k

uuzdQzuiI

φφ

++

−∞

=− ∈∈

∑

∫



(3.96)

Insurance risk models 325

It follows that all the functions

i

φ

+

i=1,…,m are equal to a common function

φ

+

satisfying (3.96), that is:

() ( ) (), ,

() () ( ) (), .

u

r

rr B A

kjk

kk

uuzdQzu

Qz Q z F zd F z

φφ

πξ ξ

++

−∞

+∞

−∞

=− ∈

== + ∈

∫

∑∑

∫



(3.97)

In conclusion, relation (3.97) shows that the non-ruin probabilities for zero order

models SM(0)/SM(0) can be computed by an associated G/G model of kernel

.

r

Q

3.5.2 The Zero Order Model SM ' (0)/SM ' (0)

In this case, we have from relation (3.79) that:

() ( ) ()

( )

rBA

ij j i i

ji

Qz F zdF

Fz

π

ξξ

π

+∞

−∞

=+

=

∫

(3.98)

with

( ) ( ) ( ), 1,..., ,

BA

iii

Fz F zdF i mz

ξξ

+∞

−∞

=+ =∈

∫

 (3.99)

such that the Wiener-Hopf system (3.92) becomes:

() ( ) (),

( ( )) ( ).

u

ikki

k

u

kk i

k

uuzdFzu

uzdFz

φπφ

πφ

++

−∞

+

−∞

=−∈

=−

∑

∫

∑

∫



(3.100)

From this last relation, it follows that, using definition (3.67), we get:

() ( ) (), ,

() ( ) (), .

u

r

rBA

kk k

k

uuzdQzu

Qz F zd F z

φφ

πξ ξ

++

−∞

+∞

−∞

=− ∈

=+∈

∫

∑

∫



(3.101)

If we suppose to know the value of

φ

+

, the non-ruin probabilities

i

φ

+

are also

known from relation (3.100):

() ( ) (), 1,..., .

u

ii

uuzdFzim

φφ

++

−∞

=− =

∫

(3.102)

In conclusion, relation (3.101) and (3.102) show that the non-ruin probabilities

for zero order models SM '(0)/SM ' (0) can be computed by an associated G/G

model of kernel .

r

Q

3.5.3 The Model M/SM

Supposing that claim distributions depend only on the type of the occurred claim,

let us recall that for this model, we have:

326 Chapter 7

0, 0,

() () (),, .

1,0,

i

AB

ij ij j

x

F

xFyByijI

ex

λ

−

≤

⎧

==∈

⎨

−≥

⎩

(3.103)

Instead of starting with the Wiener-Hopf system, we may resort to elementary

probabilistic reasoning to obtain the following relations:

00

1

() ( ) (), .

i

m

u

iiji j j

j

uped uxdBxiI

τ

λτ

φλτφτ

∞+

−

=

=+−∈

∑

∫∫

(3.104)

These relations are obtained by conditioning with the occurrence of the first

claim that occurs in (t,t+dt) and of type j.

Using the change of variables u

τ

ξ

+

= , we get:

()

0

1

() ( ) (), .

i

m

u

iiji jj

u

j

uped xdBxiI

ξ

λξ

φλξφξ

∞

−−

=

=−∈

∑

∫∫

(3.105)

These last relations show that the non-ruin probabilities

i

φ

are differentiable and

that:

'

0

1

() () ( ) (), .

m

iiiijj j

j

uu p xdBxiI

ξ

φφλ φξ

=

=− − ∈

∑

∫

(3.106)

To simplify, let us suppose that the d.f.

j

B

are differentiable with

j

b as

derivative; then, taking Laplace transforms of both members of this relation, we

get (using notation introduced in Chapter 2, section 5.1):

1

() (0) () () (),

m

ii ii iijjj

j

s

sspsbsiI

φφ λφλ φ

=

−= − ∈

∑





(3.107)

or

1

() (())()(0),.

m

iiijjijji

j

s

spbssiI

φλ δφφ

=

+−=∈

∑





(3.108)

With the following matrix notation:

() (), , () (),

iijj iij ijj

s

pb s s b s

λλδδ

⎡⎤ ⎡⎤

⎡⎤

===

⎣⎦

⎣⎦ ⎣⎦

M Λ b



(3.109)

and if ( )

s

φ



and ( )t

φ

represent respectively the column vectors of the functions

( ), 1,...,

i

s

im

φ

=



and ( ), 1,...,

i

ti m

φ

= , the last relation may be written in the form:

()(0),s

φφφ

+− =M Λ





(3.110)

and as

,=M ΛPb



(3.111)

we get:

()

(0).s

s

φφ

⎛⎞

−

−=

⎜⎟

⎝⎠

IPb

I Λ



(3.112)

Insurance risk models 327

With the matrix norm defined as the sum of the absolute values of all its

elements, the norm of the matrix ( )

s

A



defined as

1

() ( ())

s

=−A Λ IPb



(3.113)

is strictly inferior to 1 for s sufficiently large as a result of the fact that

lim ( ) .

s

→∞

=

A0



(3.114)

Consequently, for such values of s, ( )

s

−IA



is invertible and moreover

()

1

0

(()) ().

n

s

∞

−

=

−=

∑

IA A



(3.115)

By the inverse Laplace transform, relation (3.113) gives the value of the matrix

A:

() ( ())tt

=

−A Λ IPB (3.116)

where

B(t) is the diagonal matrix

0

( ) ( ( )),

( ) ( ) , 1,..., .

ij j

t

jj

tBt

tbzdzj m

δ

=

==

∫

B

(3.117)

By the inverse Laplace transform again, from relations (3.112), (3.116) and

(3.115), we find a theoretical explicit expression for the vector ,

ϕ

()

0

1

()

0

() () (0),

(0) ( )

u

n

nj

u

n

nj

utdt

tdt

φφ

φ

∞

=

−

∞

=

⎛⎞

=

⎜⎟

⎝⎠

⎛⎞

=

⎜⎟

⎝⎠

∑

∫

∑

∫

A

A1

(3.118)

where

1 is the m-dimensional vector with all components equal to1 as

lim ( ) .

u

φ

→∞

= 1

Remark 3.4

(i) For m=1, we get the result (1.60) for the P/G model.

(ii) For the stationary model introduced in section 3.2, we know from relation

(3.68) that

'''

1

0

'

0

1

() ( ) ()

l

uz

z

s

lll l l

ll

kk

k

upedzuzxdBx

λ

φπφ

πλ

∞

+

−

=+−

∑∑

∫∫

∑

, (3.119)

so that from relation (3.105), we get:

1

() ().

s

l

kk l

k

uu

π

φφ

πλ λ

−

=

∑

(3.120)

Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability

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