Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability
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Discrete time SMP and numerical solution 157
11
11
(,) (1 (,)) (, )
(, ) ( ,) .
mt
iii iki
tsr sr
ks
mt
s
ik k
ks
Vst H st a b s a
bs V t
ϑ
ϑ
ϑ
ϑ
ψϑψ
ϑϑν
−−
==+
−
==+
=− +
+
∑∑
∑∑
(5.20)
To explain these results, as for the continuous case, we divide the evolution
equation in three parts. The meaning is the same given in the previous cases but
we use annuity formulas.
Let us just make the following comments:
The term
(1 ( , ))
ii
tsr
Hst a
ψ
−
−
represents the present value of the rewards
obtained without state changes. More precisely
(1 ( , ))
i
Hst−
is the probability
of remaining in the state i and
i
tsr
a
ψ
−
is the present value of a constant annuity
of ts
− payments
i
ψ
.
The term
11
(, )
mt
ik i
s
r
ks
bs a
ϑ
ϑ
ϑψ
−
==+
∑∑
gives the present value of the rewards t obtained
before the change of state.
The term
11
(, ) ( ,)
mt
s
ik k
ks
bs V t
ϑ
ϑ
ϑϑν
−
==+
∑∑
gives the present value of the rewards paid
or earned after the transitions and as the change of state happens at time
ϑ
, it is
necessary to discount the reward values at time s.
In the due environment we obtain:
1
11 11
() (1 ()) ( ) ( ) ( ) ,
mt mt
iiiikiikk
tr r
kk
Vt Ht a b a b V t
ϑ
ϑ
ϑϑ
ψϑψ ϑϑν
−
== ==
=− + + −
∑∑ ∑∑
(5.21)
11
1
11
(,) (1 (,)) (, )
(, ) ( ,) .
mt
iii iki
tsr sr
ks
mt
s
ik k
ks
Vst H st a b s a
bs V t
ϑ
ϑ
ϑ
ϑ
ψϑψ
ϑϑν
−−
==+
−−
==+
=− +
+
∑∑
∑∑
(5.22)
At last the evolution equations in the continuous case are the following:
()
0
1
0
1
11
() 1 () ( )
() ( ) ,
m
t
t
ii iik i
k
m
t
ik k
k
ee
Vt Ht Q d
QeVt d
δδϑ
δϑ
ψ
ϑψϑ
δδ
ϑϑϑ
−−
=
−
=
−−
=− +
+−
∑
∫
∑
∫
(5.23)
()
() ( )
1
()
1
11
(,) 1 (,) (, )
(, ) ( ,) .
m
ts s
t
ii iik i
s
k
m
t
s
ik k
s
k
ee
Vst H st Q s d
Qs e V td
δδϑ
δϑ
ψ
ϑψϑ
δδ
ϑϑϑ
−− − −
=
−−
=
−−
=− +
+
∑
∫
∑
∫
(5.24)
158 Chapter 4
5.3.2 Variable Interest Rate, Permanence And Transition Cases
Now we make the following assumptions:
a) rewards are variable in time,
b) rewards are given for permanence in the state and at a given transition,
c) the interest rate is variable.
Under these hypotheses, in the immediate case we get the following relations:
() ()
1111
11 11
() (1 ()) ( ) ( ) ( ) ( ) ( )
() () () ( ) ,
tmt
iii iki
hkh
mt mt
ik ik ik k
kk
Vt Ht h h b h h
bbVt
ϑ
ϑ
ϑϑ
ψν ϑψν
ϑγ ϑν ϑ ϑ ϑν ϑ
====
== ==
=− +
++−
∑∑∑∑
∑∑ ∑∑
(5.25)
() ()
1111
11 11
(,) (1 (,)) () (, ) (, ) () (, )
(, ) ( ) , (, ) ( ,) , .
tmt
iii iki
hs k s hs
mt mt
ik ik ik k
ks ks
Vst H st h sh b s h sh
bs s bs V t s
ϑ
ϑ
ϑϑ
ψν ϑ ψν
ϑγ ϑν ϑ ϑ ϑ ν ϑ
=+ = =+ =+
==+ ==+
=− +
++
∑∑∑∑
∑∑ ∑∑
(5.26)
In the due case we get:
11
0110
11 11
() (1 ()) ()() () ()()
( 1)()( ) ()()(),
tmt
iii iki
k
mt mt
ik k ik ik
kk
Vt H t b
bVt b
ϑ
τϑτ
ϑϑ
ψτντ ϑ ψτντ
ν
ϑϑϑ νϑϑγϑ
−−
====
== ==
=− +
+− −+
∑∑∑∑
∑∑ ∑∑
(5.27)
11
11
11 11
(,) (1 (,)) () (, ) (, ) () (, )
(, 1) (, ) ( ,) (, ) (, ) ( ).
tmt
iii iki
skss
mt mt
ik k ik ik
ks ks
Vst H st s b s s
sbsVt sbs
ϑ
τϑτ
ϑϑ
ψ
τν τ ϑ ψ τν τ
νϑ ϑ ϑ νϑ ϑγϑ
−−
===+=
==+ ==+
=− +
+− +
∑∑∑∑
∑∑ ∑∑
(5.28)
The evolution equations in the continuous case are the following:
()
()
0
00
()
0
1
() ()
000
1
() () ( ) ()
1() () ()() ,
m
t
d
iik kik
k
m
tt
dd
ii iki
k
Vt Q e V t d
H
tedQ edd
ϑ
θθ
δτ τ
ϑ
δτ τ δτ τ
ϑϑγϑϑ
ψ
θθ ϑψθθϑ
−
=
−−
=
∫
=−+
∫∫
+− +
∑
∫
∑
∫∫∫
(5.29)
()
()
()
1
() ()
0
1
(,) (, ) ( ,) ( )
1(,) () (,)() .
s
s s
m
t
d
iik kik
s
k
m
tt
dd
ii iki
ss
k
Vst Q s e V t d
Hst e d Q s e dd
ϑ
θ θ
δτ τ
ϑ
δτ τ δτ τ
ϑϑγϑϑ
ψ
θθ ϑψθθϑ
−
=
−−
=
∫
=+
∫∫
+− +
∑
∫
∑
∫∫∫
(5.30)
Discrete time SMP and numerical solution 159
5.3.3 Non-Homogeneous Interest Rate, Permanence And Transition Case
For our last case, we consider non-homogeneous rewards and interest rate. And
so basic assumptions are:
a) rewards are non-homogeneous,
b) rewards are given for premanence and transitions,
c) interest rate is non-homogeneous.
It can easily be verified that the evolution equations take the form:
()
()
111
11 11 1
(,) (1 (,)) (, ) (, ) (, ) ( ,) ,
(,) (,) , (,) (,)(,),
tmt
iii ikk
sks
mt mt
ik ik ik i
ks ks s
Vst H st s s b s V t s
bs s s bs s s
τϑ
ϑ
ϑϑτ
ψ
τν τ ϑ ϑ ν ϑ
ϑγ ϑν ϑ ϑ ψ τν τ
=+ = =+
= =+ = =+ =+
=− +
++
∑∑∑
∑∑ ∑∑ ∑
(5.31)
1
11
1
11 11 0
(,) (1 (,)) (, )(, ) (, ) (, ) (, )
(, 1) (, ) ( ,) (, ) (, ) (, ),
tmt
iii ikik
sks
mt mt
ik k ik i
ks ks
Vst H st s s s b s s
sbsVt bs ss
τϑ
ϑ
ϑϑτ
ψ
τν τ ν ϑ ϑγ ϑ
νϑ ϑ ϑ ϑ ψ τντ
−
===+
−
==+ ==+ =
=− +
+− +
∑∑∑
∑∑ ∑∑ ∑
(5.32)
()()
(,)
1
(,) (,)
1
(,) (,) (,) (,) 1 (,)
(, ) (, ) (, ) .
s
s s
m
t
sd
iik kik i
s
k
m
tt
sd sd
iiki
sss
k
Vst Q s e V t s d Hst
se d Qs se dd
ϑ
θ θ
δττ
ϑ
δττ δττ
ϑϑγϑϑ
ψθ θ ϑψθ θϑ
−
=
−−
=
∫
=++−
∫∫
+
∑
∫
∑
∫∫∫
(5.33)
6 GENERAL ALGORITHMS FOR DTSMRWP
In the previous section, we presented useful discrete time semi-Markov reward
processes as well as general global models for which the evolution equations can
be written in the matrix form
*
=
UV C. (6.1)
In the homogeneous case,
U is an infinite order lower-triangular matrix whose
elements are
mm× matrices and V and C are infinite order vectors whose
elements are m-dimensional vectors.
In the non-homogeneous case in (6.1)
U is an infinite order upper-triangular
matrix whose elements are
mm
×
matrices and V and C are infinite order
matrices whose elements are m-dimensional vectors.
Of course, matrices
U and C depend on the particular models presented in the
preceding section.
For real life applications, it is generally sufficient to study the problem on a finite
time horizon [0, ]T and then the infinite system (6.1) becomes a finite system
*
TT T
=UV C (6.2)
160 Chapter 4
where
T
U is a square lower triangular block matrix of order 1T + in the
homogeneous case and an upper triangular block matrix in the non-homogeneous
case. ,
TT
CV are respectively 1T
+
-dimensional vectors, in the homogeneous
case, and matrices, in the non-homogeneous case, whose elements are m-
dimensional vectors.
We will present briefly two general algorithms (homogenous and non-
homogeneous) solving all possible reward cases.
The main steps of these algorithms are the following:
(i) Homogeneous case
Input – selectors that choose among the SMRWP, the number of states and
the number of periods, the permanence and transition rewards, the fixed or
variable interest rate, the transition matrix
P and the matrix
T
F of waiting
time d.f.
Construct -
T
Q
Construct -
T
B
Construct -
T
H
Construct -
T
D
Construct – the permanence rewards
Construct – the transition rewards
Construct – the vector discount factors
Construct –
T
C , known terms
Solve - the system and find
T
V
(ii) Non-homogeneous case
Input – selectors, the number of states and the number of periods, the
permanence and transition rewards, the fixed, variable or non-homogeneous
interest rate, the transition matrix
P and the matrix
T
F waiting time d.f.
Construct -
T
Q
Construct -
T
B
Construct -
T
H
Construct -
T
D
Construct – the permanence rewards
Construct – the transition rewards
Construct – the matrix discount factors
Construct –
T
C , known terms
Solve - the system and find
T
V
Discrete time SMP and numerical solution 161
These algorithms are able to solve any DTSMRWP. They constitute a very
important tool for the application of semi-Markov reward processes in many
applied sciences and, in this book, mainly in Finance, Insurance and Reliability.
7 NUMERICAL TREATMENT OF SMRWP
7.1 Undiscounted Case
Let us consider relations (5.7) and (5.13):
()
00
11
() 1 () ( ) ( ) ( ) ,
mm
tt
i i i i ik ik k
kk
Vt Ht t Q d Q V t d
ψ
ψϑϑϑ ϑϑϑ
==
=− + + −
∑∑
∫∫
(7.1)
()
()
000
1
0
1
() 1 () ( ) ( ) ( )
() ( ) () .
m
tt
iii iki
k
m
t
ik k ik
k
Vt Ht d Q dd
QVt d
ϑ
ψ
ττ ϑ ψττϑ
ϑϑγϑϑ
=
=
=− +
+−+
∑
∫∫∫
∑
∫
(7.2)
For non-homogeneous models, the simplest and the most difficult cases are given
by formulas (5.8) and (5.17), i.e:
()
1
1
(,) 1 (,) ( ) (, ) ( )
(, ) ( ,) ,
m
t
ii iiik
s
k
m
t
ik k
s
k
Vst H st t s Q s sd
Qs V td
ψ
ψϑϑϑ
ϑϑϑ
=
=
=− ⋅− + ⋅−
+
∑
∫
∑
∫
(7.3)
()
()
1
1
(,) 1 (,) (, ) (, ) (, )
(, ) ( ,) (, ) .
m
tt
iii iki
sss
k
m
t
ik k ik
s
k
Vst H st s d Q s s dd
Qs V t s d
ϑ
ψ
ττ ϑ ψ ττϑ
ϑϑγϑϑ
=
=
=− +
++
∑
∫∫∫
∑
∫
(7.4)
Relations (7.2) and (7.4) can be written also as follows:
()
000
1
00
11
() 1 () ( ) ( ) ( )
() ( ) () () ,
m
tt
iii iki
k
mm
tt
ik k ik ik
kk
Vt Ht d Q dd
QVt d Q d
ϑ
ψ
ττ ϑ ψττϑ
ϑ
ϑϑ ϑγϑϑ
=
==
=− +
+−+
∑
∫∫∫
∑∑
∫∫
(7.5)
()
1
11
(,) 1 (,) (, ) (, ) (, )
(, ) ( ,) (, ) (, ) .
m
tt
iii iki
sss
k
mm
tt
ik k ik ik
ss
kk
Vst Hst sd Qs sdd
Qs V td Qs s d
ϑ
ψ
ττ ϑ ψ ττϑ
ϑϑϑ ϑγϑϑ
=
==
=− +
++
∑
∫∫∫
∑∑
∫∫
(7.6)
and so both the homogeneous and non-homogeneous integral equations can be
written as follows:
162 Chapter 4
0
1
() () ( ) ( ) ,
m
t
ii ikk
k
Vt ct Q V t d
ϑ
ϑϑ
=
=+ −
∑
∫
(7.7)
1
(,) (,) (, ) ( ,) ,
m
t
ii ikk
s
k
Vst cst Q s V td
ϑ
ϑϑ
=
=+
∑
∫
(7.8)
where for relations (7.1) and (7.3), we have:
()
0
1
() 1 () ( ) ,
m
t
iiiiik
k
ct Ht t Q d
ψ
ψϑϑϑ
=
=− +
∑
∫
(7.9)
()
1
(,) 1 (,) ( ) (, ) ( ) ,
m
t
ii iiik
s
k
cst Hst t s Q s sd
ψ
ψϑϑϑ
=
=− ⋅− + ⋅−
∑
∫
(7.10)
and for relations (7.2) (7.4):
()
000
1
0
1
() 1 () ( ) ( ) ( )
() () ,
m
tt
iii iki
k
m
t
ik ik
k
ct Ht d Q dd
Qd
ϑ
ψ
ττ ϑ ψττϑ
ϑγ ϑ ϑ
=
=
=− +
+
∑
∫∫∫
∑
∫
(7.11)
()
1
(,) 1 (,) (, ) (, ) (, )
(, ) (, ) .
t
t
iiiikik
s
s
m
t
ik i
ss
k
cst Hst s d Q s s d
Qs s dd
ϑ
ψ
ττ ϑγ ϑϑ
ϑψττϑ
=
=− +
+
∫∫
∑
∫∫
(7.12)
As it can be seen, these last four equations have known terms that differ
substantially but the coefficients of the two homogeneous integral equations are
the same as in the non-homogeneous case.
Furthermore the integral equation (7.7) has the same coefficient as the equation
(2.1) and the equation (7.8) the same as the equation (2.2).
As for the discretization of the homogeneous and non-homogeneous semi-
Markov processes presented in section 2, we can consider the generic quadrature
formula (2.3).
Let us recall that h is the step measure, , , , , ,uk N ukN
≤
∈ ,w the weights
related to the quadrature formula.
We also know that N is such that
N
hY
=
and
[
]
0,Y is the integration interval.
Now, relations (7.7) and (7.8) can be discretized in the following way:
10
() () ( ) (),
mk
ii kl il
l
V kh c kh w V kh h Q h
τ
τ
ττ
==
⎛⎞
=+ −
⎜⎟
⎝⎠
∑∑
(7.13)
,,
1
(,) (,) (,)(,),
mk
iij ukili
lu
Vuhkh c uhkh w Q uh hV hkh
τ
τ
ττ
==
⎛⎞
=+
⎜⎟
⎝⎠
∑∑
(7.14)
but here, due to integral terms, the known term
c should also be discretized.
Relations (7.13) and (7.14) in matrix form become:
Discrete time SMP and numerical solution 163
0
() () () ( ),
k
k
kh kh w h kh h
τ
τ
τ
τ
=
=+ ∗−
∑
Vc QV
(7.15)
(,) (,) (,) (,),
k
uk
u
uh kh uh kh w uh h h kh
τ
τ
ττ
=
=+ ∗
∑
Vc QV
(7.16)
or equivalently:
0
1
() (0)()() ()( ),
k
kk
kh w kh kh w h kh h
τ
τ
τ
τ
=
−∗=+ ∗−
∑
VQVf QV
(7.17)
1
(,) (,) (,)
(,) (,) (,).
uu
k
uk
u
uh kh w uh uh uh kh
uh kh w uh h h kh
τ
τ
τ
ττ
=+
−∗
=+ ∗
∑
VQV
cQV
(7.18)
These last two relations take the form:
()
0
1
(0) ( ) ( ) ( ) ( ),
k
kk
wkhkhwhkhh
τ
τ
τ
τ
=
−∗=+ ∗−
∑
IQ V f Q V
(7.19)
()
1
(,) (,)
(,) (,) (,).
uu
k
uk
u
wuhuh uhkh
uh kh w uh h h kh
τ
τ
τ
ττ
=+
−∗
=+ ∗
∑
IQ V
cQV
(7.20)
Coefficient matrices of (7.19) and (7.20) are the same as those defined in systems
(2.6) and (2.7) and so Theorem 2.1 holds also in these cases.
The only difference is that the elements of the known terms are more difficult to
construct.
7.2 Discounted Case
In this financial environment, we will consider the most complicated
homogeneous and non-homogeneous discounted cases.
More precisely the CTHSMRWP related to relation (5.29) and the
CTNHSMRWP formula (5.33) will be tackled.
As before we report the two relations:
()
()
0
00
()
0
1
() ()
000
1
() () ( ) ()
1() () ()() ,
m
t
d
iik kik
k
m
tt
dd
ii iki
k
Vt Q e V t d
Ht e d Q e dd
ϑ
θθ
δτ τ
ϑ
δτ τ δτ τ
ϑϑγϑϑ
ψ
θθ ϑψθθϑ
−
=
−−
=
∫
=−+
∫∫
+− +
∑
∫
∑
∫∫∫
(7.21)
()()
(,)
1
(,) (,)
1
(,) (,) (,) (,) 1 (,)
(, ) (, ) (, ) .
s
s s
m
t
sd
iik kik i
s
k
m
tt
sd sd
iiki
sss
k
Vst Q s e V t s d Hst
se d Qs se dd
ϑ
θ θ
δττ
ϑ
δττ δττ
ϑϑγϑϑ
ψθ θ ϑψθ θϑ
−
=
−−
=
∫
=++−
∫∫
+
∑
∫
∑
∫∫∫
(7.22)
Relations (7.21) and (7.22) can be written as follows:
164 Chapter 4
()
00
00
() ()
00
11
() ()
000
1
() () ( ) () ()
1() () ()() ,
mm
tt
dd
i ik k ik ik
kk
m
tt
dd
ii iki
k
Vt Q e V t d Q e d
H
tedQ edd
ϑϑ
θθ
δτ τ δτ τ
ϑ
δτ τ δτ τ
ϑ
ϑϑ ϑ γϑϑ
ψ
θθ ϑψθθϑ
−−
==
−−
=
∫∫
=−
∫∫
+− +
∑∑
∫∫
∑
∫∫∫
(7.23)
()
(,) (,)
11
(,) (,)
1
(,) (,) (,) (,) (,)
1(,) (,) (,)(,) .
s s
s s
mm
tt
sd sd
iik k ik ik
ss
kk
m
tt
sd sd
ii iki
sss
k
Vst Qse V td Qse sd
H
st s e d Q s s e d d
ϑ ϑ
θ θ
δττ δττ
ϑ
δττ δττ
ϑ
ϑϑ ϑ γ ϑϑ
ψ
θθ ϑψθθϑ
−−
==
−−
=
∫∫
=+
∫∫
+− +
∑∑
∫∫
∑
∫∫∫
(7.24)
These equations take also the following form:
0
()
0
1
() ( ) ( ) (),
m
t
d
iik k i
k
Vt Q e V t d ct
ϑ
δτ τ
ϑϑϑ
−
=
∫
=−+
∑
∫
(7.25)
(,)
1
(,) (, ) ( ,) (,),
s
m
t
sd
iik ki
s
k
Vst Q s e V td cst
ϑ
δττ
ϑϑϑ
−
=
∫
=+
∑
∫
(7.26)
equations having the same structure.
This is also true for all the other cases with a fixed or variable intensity of interest
rate.
We can thus proceed to the discretization procedure as before, to get the
following relations:
()
1
10
1
() () ( ) () 1 () ,
mk
ii kl il h
l
V kh c kh w V kh h Q h r
τ
τ
τ
θ
ττ θ
−
==
=
⎛⎞
=+ − +
⎜⎟
⎝⎠
∑∑
∏
(7.27)
()
1
1
(,) (,)
(,) (,) 1 (,) .
ii
mk
uk l il h
lu u
Vuhkh cuhkh
wV hkhQuhh ru
τ
τ
τθ
ττ θ
−
== =
=
⎛⎞
++
⎜⎟
⎝⎠
∑∑ ∏
(7.28)
The functions
()
h
r
θ
and (, )
h
ru
θ
are the variable rates of interest obtained in this
way:
(1)
(1)
()
(,)
,1
(1 ( )) ,
1, 0,
,1
(1 ( , )) ,
1, .
h
h
h
h
d
h
ud
h
e
r
eu
ru
u
θ
θ
θ
θ
δτ τ
δττ
θ
θ
θ
θ
θ
θ
−
−
⎧
∫
⎪
≥
+=
⎨
⎪
=
⎩
⎧
∫
⎪
≥+
+=
⎨
⎪
=
⎩
(7.29)
In matrix form, the equations can be written in the following way:
()
1
0
0
() () 1 () () ( ),
k
kh
kh kh w r h kh h
τ
τ
τ
θ
θ
ττ
−
=
=
=+ + ∗−
∑
∏
Vc QV
(7.30)
Discrete time SMP and numerical solution 165
()
1
(,) (,) 1 (,) (,) (,)
k
uk h
u
u
uh kh uh kh w r u uh h h kh
τ
τ
τ
θ
θττ
−
=
=
=+ + ∗
∑
∏
Vc QV
(7.31)
or equivalently:
()
0
1
1
0
() (0) () ()
1() ()( ),
k
k
kh
kh w kh kh
wr hkhh
τ
τ
τ
θ
θ
ττ
−
=
=
−∗=
++ ∗−
∑
∏
VQVc
QV
(7.32)
()
1
1
(,) (,) (,)
(,) 1 (,) (, ) (,).
uu
k
uk h
u
u
uh kh w uh uh uh kh
uh kh w r u uh h h kh
τ
τ
τ
τ
θ
θττ
−
=+
=
−∗
=+ + ∗
∑
∏
VQV
cQV
(7.33)
We can also write:
(
)
()
0
1
1
0
(0) ( )
() 1 () () ( ),
k
k
kh
wkh
kh w r h kh h
τ
τ
τ
θ
θ
ττ
−
=
=
−∗
=+ + ∗−
∑
∏
IQ V
cQV
(7.34)
()
()
1
1
(,) (,)
(,) 1 (,) (, ) (,).
uu
k
uk h
u
u
wuhuh uhkh
uh kh w r u uh h h kh
τ
τ
τ
τ
θ
θττ
−
=+
=
−∗
=+ + ∗
∑
∏
IQ V
cQV
(7.35)
So, we have shown that in the most difficult discounted cases, we get equations
having the same coefficient matrices as the equations (7.19) and (7.20);
consequently, here too, Theorem 2.1 holds.
8. RELATION BETWEEN DTSMRWP AND SMRWP
NUMERICAL SOLUTIONS
This section is related to the relation between the numerical solution of
continuous and discrete time semi-Markov reward processes to show that, as in
the SMP case, this approximation formula leads to the related discrete time
process.
For simplicity, we just prove this result in the simplest numerical approach, i.e.
the rectangle formula.
With this method, it is possible to evaluate the integral using the values of the
function at the minimum of each of the discretization intervals or at the
maximum
Clearly, in the continuous case to distinguish between the due and the immediate
cases is meaningless and here we assume that reward payment times correspond
to the “times” of the evaluations so that we could obtain respectively, by means
of discretization, the due and the immediate cases. Here we only consider the
immediate case.
166 Chapter 4
8.1 Undiscounted Case
Taking into account relations (7.7) and (7.8), the general numerical solutions of a
non-discounted CTHSMRWP and CTNHSMRWP can be written respectively in
the following way:
01
() () ()( )
km
ii ill
l
V kh c kh h Q h V kh h
τ
τ
τ
==
=+ −
∑∑
, (8.1)
1
(,) (,) (,)(,).
km
ii ill
ul
V uhkh c uhkh h Q uh hV hkh
τ
ττ
==
=+
∑∑
(8.2)
In the special case of relations (5.7) and (5.8), the equations (8.1) and (8.2)
become:
()
11
01
() 1 () ()
()( ),
km
iiiili
l
km
il l
l
Vkh Hkh k h Q h
hQhVkhh
τ
τ
ψ
ττψ
ττ
==
==
=− +
+−
∑∑
∑∑
(8.3)
()
11
1
(,) 1 (,)( ) (,)( )
(,)(,),
km
ii iili
ul
km
il l
ul
V uh kh H uh kh k u h Q uh h u
hQuhhVhkh
τ
τ
ψ
ττ ψ
ττ
=+ =
==
=− − + −
+
∑∑
∑∑
(8.4)
where
i
ψ
represents the constant permanent reward paid at the end of each
period.
Substituting the differential by means of the increment, we get:
() ( )
()
11
01
() 1 () () (( 1))
( ) (( 1) ) ( ),
km
iiiilili
l
km
il il l
l
Vkh H kh k Q h Q h
QhQ hVkh h
τ
τ
ψ
τττψ
ττ τ
==
==
≅− + − −
+−−−
∑∑
∑∑
(8.5)
()
()
()
11
1
(,) 1 (,)( )
(,) (,( 1))( )
(,) (,( 1)) (,).
ii i
km
il il i
ul
km
il il l
ul
Vuhkh H uhkh k u
Quhh Quh h u
Quhh Quh hV hkh
τ
τ
ψ
τ
ττψ
τττ
=+ =
==
≅− −
+−−−
+−−
∑∑
∑∑
(8.6)
Using relations (1.11) and (1.12) and setting
1h
=
, these results can be written in
the form:
()
11 01
() 1 () () () ( ),
km km
iiiiliill
ll
Vk H k k b b Vk
ττ
ψ
ττψ τ τ
== ==
≅− + + −
∑∑ ∑∑
(8.7)