Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability
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308 Chapter 7
2
3
() ,var() .
aa
ET T
σ
μ
μ
== (2.54)
Remark 2.3
(i) The more negative the correlation coefficient is, the riskier the situation is for
the company.
(ii) As a function of the correlation coefficient, extreme possibilities are:
1) "maximum"risk (1)
ρ
=−
2
2
()().
2
AB AB
σ
μμμσσ
+= ++ + (2.55)
2) "minimum" risk (
ρ
= 1)
2
2
()(),
2
AB AB
σ
μμμσσ
+= −+ − (2.56)
and if moreover the asset and liabilities volatilities are identical, we have
for 1:
ρ
=−
2
2
()2
2
AB A
σ
μ
μμ σ
+= −+ (2.57)
and for
ρ
=1
2
2
AB
σ
μ
μμ
+
=−. (2.58)
(iii) As the r.v. A(t)A
0
/B(t)B
0
has a log-normal distribution
2
(, )t
μσ
, we have:
2
22
2
0
0
2
2
0
0
()
,
()
()
var ( 1).
()
t
tt t
A
At
Ee
Bt B
A
At
ee e
Bt B
σ
μ
μσ σ
⎛⎞
+
⎜⎟
⎜⎟
⎝⎠
⎛⎞
=
⎜⎟
⎝⎠
⎛⎞
⎛⎞
=
−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
(2.59)
These results confirm a well-known fact for investors: the larger the risk is, the
larger the expectation of profit is too.
On the other hand, in all cases, the most dangerous situation happens when
0,
μ
≤ that is when
22
1
()().
2
AB AB
μ
μσσ
−≤ −
(2.60)
In this case, the manager must absolutely diminish this excess of risk with an
increase of
AB
μ
μ
− and a lowering of the volatility of the assets together with an
increase of the volatility of the liabilities.
Insurance risk models 309
3 SEMI- MARKOV RISK MODELS
In this paragraph, we will give a complete presentation of the so-called
homogeneous semi-Markov risk model (in short SMRM) first introduced by
Miller (1962) and fully developed by Janssen (1969b, 1970, 1977) and later
many other authors.
We will also develop special cases of interest which bring more tractable results.
3.1 The Semi-Markov Risk Model (or SMRM)
As we already know from section 1 of this chapter, any risk model is based on
three “basic” processes:
(i) the claim arrival process,
(ii) the claim amount process,
(iii) the premium income.
In general, the first two processes are stochastic processes and the last one
deterministic.
These processes are defined on a complete probability space
(
)
,,PΩℑ .
3.1.1 The general SMR Model
In the SMRM, the first idea was to introduce m possible types of claims
belonging to the set
{
}
1,...,Im= (3.1)
and later (see Janssen and Reinhard (1982)) this set was considered as an
environment parameter and in both cases as having influences on the three basic
processes given above.
Let
(
)
(
)
,1,,1
nn
Xn Yn≥≥ represent respectively the sequence of interarrival
times between two successive claims and the sequence of successive claim
amounts. The process
(
)
,1
n
Jn≥ will represent the successive type of claims or
environment states.
The basic assumption to get an SMRM is that:
1
( , , ( , , ,), 1,..., 1) ( , )
n
nnn kkk Jj
PJ jX xY y J X Y k n Q xy
−
=≤≤ = −= (3.2)
with
0000
,0,..
J
jX Y as
=
== (3.3)
This assumption means that the three-dimensional process
(( , , ), 0)
nnn
JXY n≥
is
what is called a two-dimensional (J-X)-process of kernel
Q, having the following
properties:
310 Chapter 7
(i) all the elements
ij
Q of Q are mass functions in two dimensions, null for x or y
negative,
(ii) the following limits exist:
,
1
lim ( , ) , , ,
1, .
ij ij
xy
m
ij
j
Qxy pij I
piI
→∞ →∞
=
=
∈
=∈
∑
(3.4)
Every such matrix
Q is called a two-dimensional semi-Markov kernel and the
corresponding (J-X-Y) process a two-dimensional J-X process or a two-
dimensional semi-Markov chain.
From a straightforward extension of the basic results of Chapter 3, section 2, we
get the following results:
(i) the process of successive claims
(
)
,0
n
Jn≥ is a homogeneous Markov chain
with I as state space and with
ij
p
⎡
⎤
=
⎣
⎦
P as transition matrix,
(ii) the processes
(( , ), 0),(( , ), 0)
nn nn
JX n JY n≥≥
are two SMP of kernels
,
AB
QQwhere for all i and j of I:
( ) ( , ), ( ) ( , ),
AB
ij ij ij ij
Qx Qx Qy Q y=+∞ =+∞ (3.5)
(iii) given the r.v.
,0
n
Jn≥
the two-dimensional r.v.
(,), 1
nn
X
Yn≥
are
conditionally independent and we have:
021
11
( , ) ( , ,..., , , )
(, )/ , 0,
() (), 0.
ij n n n n n
ij ij ij
ij
F
xy PX xY yJ J J iJ j
Qxy pp
UxUyp
−−
=≤≤ ==
>
⎧
⎪
=
⎨
=
⎪
⎩
(3.6)
(iv) From this last property, we see that given the r.v.
,0
n
Jn≥ , the r.v.
(, 1)
n
Xn≥ are conditionally dependent and similarly for the r.v. ( , 1)
n
Yn≥ and
moreover:
021
021
( ) ( , ,..., , , ) ( , ),
() ( , ,..., , , ) ( ,).
A
ij n n n n ij
B
ij n n n n ij
Fx PX xJ J J iJ j Fx
F
yPYyJ J J iJ jF y
−−
−−
=≤ ===+∞
=≤ ===+∞
(3.7)
Suppressing the conditioning relative to
n
J
, we get
021
021
021
( , ) ( , ,..., , ) ( , ),
() ( , ,..., , ) (, ),
( ) ( , ,..., , ) ( , ).
innnnijij
j
A
in nni
B
in nn i
Hxy PX xY xJ J J i pFxy
Hx PX xJ J J i Hx
Hy PY yJ J J i H y
−−
−−
−−
=≤≤ ==
=≤ ==+∞
=≤ ==+∞
∑
(3.8)
Now, we can introduce the means associated with the different conditional d.f.
defined above and we adopt the following notation:
Insurance risk models 311
00
00
11
(), (),
() , () .
AB
ij ij ij ij
mm
AA B B
i i ij ij j i ij ij
jj
axdFxbydFy
x
dHx pa ydH y pb
ηη
∞∞
∞∞
==
==
⎛⎞ ⎛⎞
== ==
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
∫∫
∑∑
∫∫
(3.9)
Now, we can first introduce the process
0
(, 1),
0
n
Tn
T
≥
=
(3.10)
defined as
1
,1
n
nk
k
TXn
=
=
≥
∑
, (3.11)
representing the successive claim time arrivals and secondly the process
0
(, 1),
0
n
Un
U
≥
=
(3.12)
defined as
1
,1
n
nk
k
UYn
=
=
≥
∑
, (3.13)
representing the successive total claim amount just after the arrivals of the
successive claims.
For the joint distribution of the process ( , , , 0)
nn n
JTUn≥ , we get:
()
0
(0)
00
(1)
() ( 1)
1
(,, ) (,),
(, ) () (),
(, ) (, ),
(, ) ( ', ') ( ', '), 1.
n
nnn ij
ij ij
ij ij
n
xy
nn
ij ij
k
PJ jT tU yJ i Q xy
Qxy UxUy
QxyQxy
Qxy Q xxyyQdxdyn
δ
−
−∞ −∞
=
=≤≤==
=
=
=−−>
∑
∫∫
(3.14)
Of course, for processes
(( , ), 0),(( , ), 0)
nn n n
JT n JU n≥≥
, both MRP, we have:
()
0
()
0
(, ) (),
(, ) ().
An
nn ij
Bn
nn ij
PJ jT tJ i Q t
PJ jU yJ i Q y
=≤==
=≤==
(3.15)
Remark 3.1 By analogy with the basic definitions for SMP given in Chapter 3,
section 2 (
Definition 2.1), the three-dimensional process (( , , ), 0)
nn n
JTU n≥ is
called a two- dimensional MRP of kernel
Q.
If
Q is an extended SM matrix in two dimensions, this process is called a two-
dimensional MRW or extended SMC.
We finish this section with the following definition.
312 Chapter 7
Definition 3.1 The sequences (, 1),(, 1)
nn
Xn Yn≥≥ are conditionally
independent given the sequence (, 0)
n
Jn≥ iff
(, ) () (), , , , .
AB
ij ij ij
F
xy F x F y xy R ij I=∀∈∀∈ (3.16)
From the results given above in this subsection, we also have:
(3.16) ( , ) ( ) ( ) ( , ) ( ) ( )
(, ) () ().
AB A B
ij ij ij ij ij ij
AB
ij ij ij ij
Qxy FxQy Qxy QxFy
Qxy p FxFy
⇔= ⇔=
⇔=
(3.17)
This assumption seems quite reasonable in risk theory and moreover, it may be
useful to consider the following particular case:
() (),, , 0,
() (),, , 0.
AA
ij j
BB
ij j
Fx Fxij Ix
Fy Fyij Iy
=
∈≥
=
∈≥
(3.18)
The first type of condition (3.18) means that the d.f. of the interarrival time
between two consecutive claims uniquely depends upon the type of the future
claim and the second one that the d.f. of a claim amount uniquely depends upon
the type of this claim and not on the type of the preceding one.
3.1.2 The Counting Claim Process
Still using notation and concepts of Chapter 3, section 5, let us introduce the m+1
counting processes associated with the SMP of claim arrivals of kernel
A
Q :
()
(
)
( ), 0 , 1,..., , ( ), 0 ,
AA
j
Ntt j m Ntt≥= ≥ (3.19)
so that here, ( )
A
j
N
t represents the total number of claims of type i occurring on
(
]
0,t and ( )
A
N
t represents the total number of claims occurring on
(
]
0,t .
From now on, we will suppose that the MRP of kernel
A
Q
is ergodic with
1
( ,..., )
m
π
ππ
= as a unique stationary distribution related to P.
For
0
() ( () )
A
ij j
R
tENtJi==, we know from relations (6.15), (9.1) and (9.4) of
Chapter 3 that
()
0
() (),, ,
()
1
lim , , ,
1
,.
An
ij ij
n
ij
A
t
jj
AA
jj k k
k
j
R
tQtijI
Rt
ij I
t
jI
μ
μπη
π
∞
=
→∞
=
∈
=∈
=∈
∑
∑
(3.20)
From now on, we will drop the index A for the counting variables related to the
claim arrivals.
Insurance risk models 313
For the joint distribution of
(
)
() ()
(), ,
Nt Nt
Nt J T , using the semi-Markov property,
we successively get:
(
)
()
()
()
() () 0
0
10
10
()
0
() , ,
() , , ,0
,,
,,
(1 ( )) ( ).
Nt Nt
nn
nnnn
nnn
th
AAn
jij
PNt nJ jT t hJ i
PNt nJ jT t hJ i h t
PT t T J jT t hJ i
PT t hT tJ jJ i
Ht zdQ z
+
+
−
==≤−=
===≤−=≤≤
=≤< =≤−=
=≤− >==
=− −
∫
(3.21)
For h=0, we obtain:
(
)
()
() 0
0
() , (1 ( )) ( )
t
AAn
Nt j ij
PNt nJ jJ i H t z dQ z====−−
∫
(3.22)
and moreover, summing over j, we get:
()
()
0
0
1
() ()
0
111
() ( 1)
11
() (1 ( )) ( )
( ) ( )) ( )
( ) ( ).
m
t
AAn
jij
j
mmm
t
An A An
ij jk ij
jjk
mm
An An
ij ij
jk
PNt nJ i H t z dQ z
Qt QtzdQz
Qt Q t
=
===
+
==
=== − −
=− −
=−
∑
∫
∑∑∑
∫
∑∑
(3.23)
Using the following notation:
(
)
()
() 0
0
1
(, ) () , ,
(, ) () , (, ) ,
A
ij N t
m
AA
iij
j
Ptn PNt nJ jJ i
Ptn P Nt n J i P tn
=
====
⎛⎞
====
⎜⎟
⎝⎠
∑
(3.24)
we obtain from relations (3.23), the following recurrence formulas:
1
1
(, ) (, 1) (),
(, ) (, 1) (),
m
AAA
ij kj ik
k
m
AAA
ikik
k
Ptn P tn Q t
Ptn P tn Q t
=
=
=−•
=−•
∑
∑
(3.25)
with of course:
( ,0) (1 ( )),
(,0) 1 ().
A
ij ij i
A
ii
Pt Ht
Pt H t
δ
=−
=−
(3.26)
If we are only interested in one type of claim, say j, it suffices to consider the
delayed renewal process characterized by
(
)
,
AA
ij jj
GG. In this case we get
314 Chapter 7
0
(1) (1)
1(),0,
(() )
()(),1.
A
ij
j
AAnAn
ij jj jj
Gtn
PN t nJ i
GG G tn
−−
⎧
−=
⎪
===
⎨
•
−≥
⎪
⎩
(3.27)
For the means
0
() ( () )
A
ij j
H
tENtJi==, results (9.7) and (3.9) of Chapter 2
give:
()
1
() () (), ,
() ().
ij ij ij jj
n
jj jj
n
Ht Gt G H ti j
Ht H t
∞
=
=
+• ≠
=
∑
(3.28)
Finally,
Proposition 7.2 of Chapter 2 is interesting here to get asymptotic
normality:
2
3
() , ,
j
j
jj jj
t
t
Nt N
σ
μμ
⎛⎞
⎜⎟
⎜⎟
⎝⎠
≺
(3.29)
2
,
j
jj
μ
σ
being respectively the mean and the variance of the r.v. ()
A
n
Tjjdefined
in Chapter 3, section 6, the latter supposed to be finite.
3.1.3 The Accumulated Claim Amount Process
This is the process ( ( ), 0)Ut t≥ where:
()
()
1
() ( ).
Nt
nNt
n
Ut Y U
=
==
∑
(3.30)
We already know that the marginal distribution of U(t), for fixed t, is important
for insurance companies, as for example with a year as time unit, the value U(1)
represents the total expenses of the company for paying the claims in this year t.
Also, let us introduce the following marginal distributions:
() 0
(, ) ( () , ),
ij N t
M
ty PUt yJ jJ i
=
≤== (3.31)
so that
() 0
()
0
0
()
0
(, ) ( () , ),
(, ) (1 ( )) ( , ),
( , ) ( , ) (1 ( )),
ij N t
t
An
ij j ij
n
nA
ij ij j
n
Mty PUt yJ jJ i
M
ty H t z dQ zy
Mty Q zy Ht z
∞
=
∞
=
=≤==
=−−
=•−−
∑
∫
∑
(3.32)
where the convolution product only acts on the temporal variable.
In case of conditional independence, this last result becomes:
Insurance risk models 315
()
0
0
() ()
0
(, ) (1 ( )) ( ( ) ( )) ,
( , ) ( ( )) ( ( )) (1 ( )).
t
AABn
ij j ij ij ij
n
AnB n A
ij ij ij ij j
n
Mty Ht zdp FzFy
M
ty p F z F y H t z
∞
=
∞
=
=−−
=•−−
∑
∫
∑
(3.33)
Let us remark that for m=1, this last formula gives result (1.19) for Andersen’s
risk model.
3.1.4 The Premium Process
We will use the same approach as in section 1.1.2. From (3.29), we know that
1
()
lim , , ,
A
ij j
m
t
kk
k
Ht
ij I
t
π
πη
→∞
=
=
∈
∑
(3.34)
and it follows that for t large, we have:
0
1
(() ) ,, ,
i
j
m
kk
k
t
E
NtJ i ij I
π
πη
=
=
≈∈
∑
(3.35)
and so, approximately, the mean cost of the ( )
j
N
t claims of type j on
(
]
0,t is
B
j
kk
k
t
η
π
η
∑
, (3.36)
and finally, we get:
() 0
() .
B
j
j
j
Nt
A
kk
k
EU J i
π
η
π
η
=≈
∑
∑
(3.37)
This last relation shows that, whatever the initial state is, the total mean cost of
the claims is more or less than ct
with
.
B
j
j
j
A
kk
k
c
π
η
π
η
≈
∑
∑
(3.38)
It follows that if we take this value
c
as constant premium rate per unit of time,
we have an asymptotically fair game between the insurance company and the
insured.
Let us point out that the value of this premium rate only depends on the means of
inter-arrivals and claim amounts and also the stationary distribution of the
embedded MC of successive claim or environment types. All of them may be
easily computed with the statistical data of observed claims.
316 Chapter 7
3.1.5 The Risk and Risk Reserve Processes
Definitions (1.20) and (1.21) are still valid for the SMRM so that the risk process
is defined by (U(t)
− ct ,t ≥ 0) and the risk reserve process by ( ( ), 0)tt
α
α
=
≥
with ( )t
α
=u+U(t) − ct where u is as usual the initial reserve or equity of the
company and c the loaded premium rate:
(1 ) .cc
η
=
+
(3.39)
3.2 The Stationary Semi-Markov Risk Model
Using a result from Chapter 3 section 11, the stationary version of the SMRM is
obtained if we take for
01
(, )
J
X
the following initial distributions:
0
110
0
() ,
(, ) (1(),
A
ii
A
kk
k
x
ij
A
ij
A
i
PJ i
p
PX xJ jJ i F zdz
πη
πη
η
==
≤=== −
∑
∫
(3.40)
so that
0
11
(1 ( )
(,) .
x
A
iij ij
i
A
kk
k
p
Fzdz
PX xJ j
π
πη
−
≤==
∑
∫
∑
(3.41)
Then we know that the process
() () 1 () 1
(( , , ), 0)
Nt Nt Nt
JJ T tt
++
−
≥ is stationary with:
() () 1 () 1
0
(, , ) (1().
x
jjk
A
Nt Nt Nt jk
A
ii
i
p
PJ jJ kT t x F zdz
π
πη
++
==−≤= −
∫
∑
(3.42)
The interest of stationary models in classical risk theory (take here m=1!) has
been investigated by Thorin (1975).
3.3 Particular SMRM With Conditional Independence
We know SMRM’s with conditional independence are entirely characterized by
the triplet of matrices ( , , )
AB
PFF where
,,,
AABB
ij ij ij
p
FF
⎡
⎤⎡⎤
⎡⎤
== =
⎣⎦
⎣
⎦⎣⎦
PF F (3.43)
as we know that in this case both claim arrival and claim amount processes are
SMP characterized respectively by ( , )
A
PFand ( , )
B
PF.
Insurance risk models 317
Paraphrasing, as before, Kendall‘s notation for queuing theory, we will denote
this model by SM/SM with respect to the order claim arrival process/claim
amount process.
3.3.1 The SM/G model
In this model, the claim arrival process is an SMP defined by the couple ( , )
A
PF
and for
B
F , we choose
() (), , , 0.
B
ij
Fy By ij Iy
=
∀∈ ≥ (3.44)
This means that the sequence ( , 1)
n
Yn≥ is a sequence of i.i.d. random variables
with B as common d.f.
3.3.2 The G/SM model
This model is symmetric to the preceding one: the claim amount process is an
SMP defined by the couple ( , )
B
PF and for
A
F , we choose
() (), , , 0.
A
ij
Fx Ax ij Ix
=
∀∈ ≥ (3.45)
This means that the sequence
(, 1)
n
Xn≥ is a sequence of i.i.d. random variables
with A as common d.f.
3.3.3 The P/SM model
This model is a particular case of the G/SM model where in addition the claim
arrival process is a Poisson process of parameter
λ
. It follows that
0, 0,
()
1,0,
x
x
Ax
ex
λ
−
<
⎧
=
⎨
−
≥
⎩
(3.46)
and of course that:
()
(() ) .
!
n
t
t
PNt n e
n
λ
λ
−
== (3.47)
Remark 3.2
Let us remark that the intersection of the first two models, called the
G/G model, gives Andersen’s model with
1, () (), () (),,
AB
ij ij
mFxAxFyByijI== =∈ (3.48)
and that the particular P/SM model with
1, ( ) ( ), ,
B
ij
mFyByijI
=
=∈ (3.49)
is identical to the Cramer-Lundberg model (see section 1)