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

26 Chapter 1

that there exists another function

1

ℑ

-measurable f so that relation (6.22) is still

true. Consequently, we have

1

() () , .

BB

fdPYdPB

ωω

=

∈ℑ

∫∫

(6.23)

From relations (6.22) and (6.23), we obtain

1

() () ,

BB

fdPE dPB

ωω

ℑ

=

∈ℑ

∫∫

(6.24)

so that

1

() (() ()) ,

BB

fdP f E dPB

ωωω

ℑ

=

−∈ℑ

∫∫

. (6.25)

As this relation holds for all B belonging to

1

ℑ

, it follows that

1

() ,a.s.EY f

ℑ

= ;

otherwise, there would exist a set B belonging to

1

ℑ

so that the difference

1

()

E

Yf

ℑ

− would be different from 0 and so also the integral

1

(() ())

B

f

EdP

ωω

ℑ

−

∫

(6.26)

in contradiction with property (6.25).

6.2 Conditioning: General Case

We can now extend the definition (6.17) to arbitrary sub-

σ

-algebras using

property (6.22) as a definition with the help of the Radon-Nikodym theorem,

Halmos (1974).

Definition 6.1 If

1

ℑ is a sub-

σ

-algebra of

ℑ

, the conditional expectation of the

integrable r.v. Y given

1

ℑ

, denoted by

1

()

E

Y

ℑ

or

(

)

1

EY

ℑ

, is any one r.v. of the

equivalence class such that:

(i)

1

()

E

Y

ℑ

is

1

ℑ -measurable,

(ii)

1

()() () , .

BB

EY dP Y dPB

ωω

ℑ

=∈ℑ

∫∫

(6.27)

In fact, the class of equivalence contains all the random variables a.s. equally

satisfying relation (6.27).

Remark 6.1 Taking Ω=

B

in relation (6.27), we get

)())(

1

YEYEE

=

ℑ

, (6.28)

a relation extending relation (6.17) to the general case.

Particular cases

(i)

1

ℑ is generated by one r.v. X .

Probability tools 27

This case means that

1

ℑ is the sub-

σ

-algebra of

ℑ

generated by all the inverse

images of X and we will use as notation

(

)

1

()

E

YEYX

ℑ

= , (6.29)

and this conditional expectation is called the conditional expectation of Y given

X.

(ii)

1

ℑ is generated by n r.v.

1

,...,

n

X

X.

This case means that

1

ℑ is the sub-

σ

-algebra of

ℑ

generated by all the inverse

images of

1

,...,

n

X

and we will use as notation

1

( ) ( ,..., )

n

E

YEYX X

ℑ

= , (6.30)

and this conditional expectation is called the conditional expectation of Y given

1

,...,

n

X

X . In this latter case, it can be shown (Loeve (1977)) that there exists a

version

1

( ,..., )

n

X

ϕ

of the conditional expectation so that

ϕ

is a Borel function

from

n

 to  and as such it follows that

1

( ,..., )

n

E

YX X is constant on each set

belonging to

1

ℑ

for which

11

( ) ,..., ( )

nn

X

xX x

ω

=

= , for instance. This justifies

the abuse of notation

(

)

11 1

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

nn n

E

YX x X x x x

ωωϕ

=== (6.31)

representing the value of this conditional expectation on all the

ω

’s belonging to

the set

{

}

11

: ( ) ,..., ( )

nn

X

xX x

ωω ω

==. Taking

B

=

Ω in relation (6.28), we get

()

11 11

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

n

nn nn

R

E

Y EYX xX xdPX xX x

ωω ωω

===≤≤

∫

, (6.32)

a result often used in the sequel to evaluate the mean of a random variable using

its conditional expectation with respect to some given event.

(iii) If

{

}

1

,ℑ=∅Ω, we get

1

()()

E

YEYℑ= and if

{

}

1

,, ,

c

BB

ℑ

=∅ Ω, then

1

()()

E

YEYBℑ= on B and

1

()()

c

E

YEYBℑ= on B

c

.

(iv) Taking as r.v. Y the indicator of the event A, that is to say

()

1, ,

1

0, ,

A

ω

∈

⎧

=

⎨

∉

⎩

(6.33)

the conditional expectation becomes the conditional probability of A given

1

ℑ denoted as follows:

11

()(1())

A

PA E

ω

ℑ

=ℑ (6.34)

and then relation (6.27) becomes

()

11

() ( ),

B

PA dP PA B B

ω

ℑ

=∈ℑ

∫

∩ . (6.35)

Letting

B

=Ω in this final relation, we get

(

)

(

)

1

(),

E

PA PAℑ= (6.36)

28 Chapter 1

a property extending the theorem of total probability (6.14). If moreover, A is

independent of

1

ℑ

, that is to say, if for all B belonging to

1

ℑ

( ) ()()PA B PAPB

=

∩ , (6.37)

then we see from relation (6.34) that

(

)

1

() (),PA PA

ωω

ℑ

=∈Ω. (6.38)

Similarly, if the r.v. Y is independent of

1

ℑ

, that is to say if for each event B

belonging to

1

ℑ

and each set A belonging to the

σ

-algebra generated by the

inverse images of Y, denoted by

σ

(Y), the relation (6.37) is true, then from

relation (6.27), we have

(

)

1

()

E

YEYℑ= . (6.39)

Indeed, from relation (6.37), we can write that

()

1

()() () ,

1

( ) ( )

( ) ,

BB

B

EY dP Y dPB

EY

EY PB

EY dP

ωω

ℑ

=

∈ℑ

=

∫∫

∫

(6.40)

and so, relation (6.39) is proved. In particular, if

1

ℑ

is generated by the r.v.

X

1

,…,X

n

, then the independence between Y and

1

ℑ

implies, that

(

)

1

,..., ( )

n

E

YX X EY= . (6.41)

Relations (6.39) and (6.41) allow us to have a better understanding of the

intuitive meaning of conditioning. Under independence assumptions,

conditioning has absolutely no impact, for example, on the expectation or the

probability, and on the contrary, dependence implies that the results with or

without conditioning will be different, this fact meaning that we can interpret

conditioning as given additional information useful to get more precise results in

the case of dependence of course.

The properties of expectation, quoted in section 4, are also properties of

conditional expectation, true now a.s., but there are supplementary properties

which are very important in stochastic modelling. They are given in the next

proposition.

Proposition 6.1 (Supplementary properties of conditional expectation)

On the probability space (,,)PΩℑ , we have the following properties:

(i) If the r.v. X is

1

ℑ -measurable, then

1

(),..

E

XXasℑ= (6.42)

(ii) Let X be a r.v. and Y

1

ℑ -measurable, then

Probability tools 29

11

()(),..

E

XY YE X a sℑ= ℑ (6.43)

This property means in fact that

1

ℑ

-measurable random variables are like

constants for the classical expectation.

(iii) Since from relation (6.27), we have that () ,

E

YY

ℑ

=

taking

1

()YEY

ℑ

= , we

see that

11

(()) ()

E

EY EY

ℑℑ ℑ

=

, (6.44)

and of course since

11

(()) (),

E

EY E Y

ℑℑ ℑ

=

(6.45)

putting these last two relations together, we get

11 1

( ( )) (( )) ( )

E

EY EE Y EY

ℑℑ ℑℑ ℑ

=

= . (6.46)

This last result may be generalised as follows.

Proposition 6.2 (Smoothing property of conditional expectation) Let

12

, ℑℑ be

two sub-

σ

-algebras of ℑ such that

12

;

ℑ

⊂ℑ then it is true that

21 12 1

(()) (()) ()

E

EY EEY EY

ℑℑ ℑℑ ℑ

=

= , (6.47)

a property called the smoothing property in Loeve (1977).

A particular case of relation (6.47) is for example that

()

(

)

(

)

(

)

(

)

11 11 1

,..., ,...,

nn

E

EYX X X EEYX X X EYX==. (6.48)

This type of property is very useful for computing probabilities using

conditioning and will often be used in the following chapters.

Here is an example illustrating this interest for sums of a random number of

random variables with the so-called Wald identities.

Example 6.1 (Wald’s identities) Let ( , 1)

n

Xn≥ be a sequence of i.i.d. real

random variables and N a non-negative r.v. with integer values independent of

the given sequence. The random variable defined by

1

N

Nn

n

SX

=

∑

(6.49)

is called a sum of a random number of random variables and the problem to be

solved is the computation of the mean and the variance of this sum supposing

that the r.v. X

n

have a variance.

From relation (6.28), we have that

(

)

(

)

()

NN

E

SEESN= (6.50)

and as, from the independence assumptions,

(

)

()

N

E

SN NEX= , (6.51)

30 Chapter 1

we also have

() ()(),

N

E

SENEX

=

(6.52)

the so-called the first Wald’s identity:

For the variance of S

N

, as

22

var( ) ( ) ( ( ))

NN N

SES ES=− , (6.53)

it suffices to evaluate

2

()

N

E

S

as we did above. From relation (6.28), we can write

that:

(

)

(

)

22

()

NN

E

SEESN=

. (6.54)

As

()

2

1

N

Ni

i

E

SN E X N

=

⎛⎞

⎜⎟

=

⎜⎟

⎝⎠

∑

, (6.55)

we obtain on the set

{

}

:()

N

n

ωω

=

:

()

2

1

2

11

2

var

var( ) ( ( )) .

n

Ni

i

nn

ii

ES N n E X

XEX

nXnEX

=

==

⎛⎞

==

⎜⎟

⎝⎠

⎛⎞

⎛⎞⎛⎞

=+

⎜⎟

⎜⎟⎜⎟

⎝⎠⎝⎠

⎝⎠

=+

∑

∑∑

(6.56)

Therefore, from relation (6.54),

222

22

() (var() (())

( ) var( ) ( )( ( ) ),

N

ES EN X N EX

EN X EN EX

=+

(6.57)

and thus, by relations (6.54) and the first Wald’s identity (6.53), we get:

22 2 2

var( ) ( ) var( ) ( )( ( )) ( ( )) ( ( ))

N

S EN X EN EX EX EN=+ − (6.58)

and finally we obtain the second Wald’s identity in the form

2

var( ) ( ) var( ) var( )( ( ))

N

SEN X NEX=+ . (6.59)

6.3 Regular Conditional Probability

The general definition (6.27) gives the conditional expectation of a r.v. Y given

1

ℑ by an implicit relation. Now the question is: can we define the conditional

expectation with an explicit relation? To give the answer, let us begin with the

conditional probability defined by relation (6.34); We know that the conditional

probability

(

)

1

PAℑ is a r.v., not unique but a.s. unique and this implies that

(

)

1

PAℑ as a set function on

ℑ

has a.s. the properties of probability measures:

(i)

()

1

() 1P

ω

Ωℑ = , (6.60)

Probability tools 31

(ii)

(

)

1

() 0 ,PA A

ω

ℑ≥∀∈ℑ (6.61)

(iii)

()

11

1

() (), , , ,

,, .

nnnij

n

PA PA A nAA

iji j

ωω

∞

=

⎛⎞

ℑ= ℑ ∈ℑ∀∩=∅

⎜⎟

⎝⎠

∀∀ ≠

∑∪

(6.62)

It is important to note here that the null events

123

,,

N

NN

, on which respectively

these last three properties are not true, are generally not identical, so that for each

ω

, the random set function

(

)

1

.()P

ω

ℑ from

ℑ

to

[

]

0,1 is not necessarily a

probability measure since, to be so, these three sets must be identical. That is

why we must introduce the concept of regular conditional probability (see Loeve

(1977) or Gikhman and Skorokhod (1980)).

Definition 6.2 The conditional probability

(

)

1

.()P

ω

ℑ is a regular conditional

probability if there exists a function p(.,.) from

ℑ

×Ω to

[

]

0,1 so that:

(i) for almost all

ω

of Ω , (., )p

ω

, as a set function on

ℑ

, is a probability

measure,

(ii) for every fixed event A belonging to

ℑ

, p(A,.) is

ℑ

-measurable and is a

version of the given conditional probability, that is a.s., we have

(

)

(,) ()pA P A

ω

=ℑ. (6.63)

The interest of such regular conditional probabilities is that we can express the

related conditional expectation of an integrable r.v. X a.s. as an integral with

respect to the measure (., )p

ω

:

()

1

() (')( ',)EX X pd

ω

ωωω

Ω

ℑ=

∫

. (6.64)

In many applications, it is sufficient to restrict the attention to all events of the

sub-

σ

-algebra generated by a r.v. X, with values in the measurable space ( , )E

ψ

,

and denoted by ( )

X

σ

. This means that we are only interested in the following

conditional probabilities:

(

)

1

, ( )PA A X

σ

ℑ∈

. (6.65)

If the conditional probability given

1

ℑ

is regular, we can then define the function

C from ( )X

σ

×

Ω to

[]

0,1 as

(

)

1

(,) (), ( ),CA PA A X

ωωσω

=

ℑ∈ ∈Ω

(6.66)

satisfying

(i) for almost all

ω

of Ω , (., )C

ω

, as a set function on ( )

X

σ

, is a probability

measure,

(ii) for every fixed event A belonging to ( )

X

σ

, C(A,.) is

ℑ

-measurable,

32 Chapter 1

(iii) for every event A belonging to ( )

X

σ

and for every event B belonging to

ℑ

,

we have

(,)( ) ( )

B

CA Pd PA B

ωω

=

∫

∩ . (6.67)

C is called the conditional distribution of X given

1

ℑ

and the mixed conditional

distribution of X given

1

ℑ is defined as the function Q(.,.) from

ψ

×Ω to

[]

0,1

defined by:

{

}

(

)

1

(, ) ': ( ') (),QS P X S S

ω

ωω ωψ

=∈ℑ∈. (6.68)

The problem of the existence of regular conditional probability was solved by

Loeve (1977) or Gikhman and Skorokhod (1980). For our goal, let us just say

that this is the case if ℑ is generated by a finite or countable family of random

variables or if the space E is a complete separable metric space. In the particular

case of an n-dimensional real r.v. X=(X

1

,…,X

n

),we can now introduce the very

useful definition of the conditional distribution function of X given

1

ℑ defined as

follows:

(

)

{}

()

1

1111

11

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

': ( ') ,..., ( ') , .

nnn

nn

Fx x PX x X x

QX xX x

ω

ωωω

ℑ

=≤ ≤ℑ

=≤≤

(6.69)

Another useful definition concerns an extension of the concept of the

independence of random variables to the definition of conditional independence

of the n variables

1

,, .

n

X

X… For all (x

1

,…,x

n

) belonging to

n

 , we have the

following identity:

1

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

n

nk

k

Fx x Fx

ω

ℑℑ

=

∏

(6.70)

where of course we have

(

)

1

(,)

kkk

Fx PX x

ω

ℑ

=

≤ℑ (6.71)

according to the definition (6.69) with n=1.

Example 6.2 On the probability space ( , , )P

Ω

ℑ , let (X,Y) be a two-dimensional

real r.v. whose d.f. is given by

(, ) ( , ).

F

xy PX xY y

=

≤≤ (6.72)

As

2

 is a complete separable metric space, it is said above that there exist

regular conditional probabilities given the sub-

σ

-algebras ( )

X

σ

or ( )Y

σ

and so

the related conditional d.f. denoted by

{}

()

{}

(

)

:, :

XY YX

F

xYyFy Xx

ωω

== (6.73)

also exists. If moreover, the d.f. F has a density f, we can also introduce the

concept of conditional density for both functions

, ,

X

XY YX

F

FF, giving at the

Probability tools 33

same time an intuitive interpretation of conditioning in this special case. We

know that for every fixed (x,y):

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

f

xy x y xy x y Px X x xy Y y y

ο

ΔΔ + Δ Δ = < ≤ +Δ < ≤ +Δ (6.74)

where ( , , , ) 0xy x y

ο

Δ

Δ→ for ( , ) (0,0)xy

Δ

Δ→ and similarly for the marginal

density function of X:

() (, ) ( ),

X

f

xx xx PxX x x

ο

Δ+ Δ = < ≤ +Δ (6.75)

where

(, ) 0xx

ο

Δ→ for 0xΔ→ with of course:

() (, )

X

R

f

xfxydy=

∫

. (6.76)

Using formula (6.4), we thus obtain

()

(, ) (, , , )

() (, )

X

f

xy x y xy x y

Py Y y yx X x x

fxx xx

ο

Δ

Δ+ Δ Δ

< ≤ +Δ < ≤ +Δ =

Δ+ Δ

. (6.77)

Letting

x

Δ tend to 0, we get

()

0

(, )

lim

()

x

X

fxy

Py Y y yx X x x y

fx

Δ→

< ≤ +Δ < ≤ +Δ = Δ . (6.78)

This relation shows that the function

YX

f

defined by

()

(, )

()

YX

X

f

xy

fyx

f

x

= (6.79)

is the conditional density of Y, given X. Similarly the conditional density of X,

given Y, is given by

()

(, )

()

XY

Y

f

xy

fxy

f

y

= . (6.80)

Consequently, for any Borel subsets A and B of

 , we have

()()

()

1

() (,) ,

()

(( , ) ) ( , ) ( ) .

XY

Y

AA

Y

XY

AB B A

PX AY y f xydx fxydx

fy

P X Y A B f x y dxdy f x y dx f y dy

ω

∩

∈== =

⎛⎞

∈= =

⎜⎟

⎝⎠

∫∫

∫∫∫

∩

(6.81)

The last equalities show that the density of (X,Y) can also be characterised by

one marginal d.f. and the associated conditional density, as from relations (6.78)

and (6.81):

XY

YX XY

f

ff ff

=

×=×

. (6.82)

It is possible that conditional means exist; if so they are given by the relations

()

(

)

(

)

(

)

,

E

XY y f xy dx EYX x f yx dy== ==

∫∫



. (6.83)

The conditional mean of X (resp.Y) given Y=y (resp. X=x)can be seen as a

function of the real variable y (resp. x) called the regression curve of X (resp.Y)

given Y (resp. X). The two regression curves will generally not coincide and not

34 Chapter 1

be straight lines except if the two r.v. X and Y are independent because, in this

case, we have from relations (6.78) and (6.80) that

,

X

Y

XY YX

f

ff f

=

= (6.84)

and so:

(

)

(

)

(), ()

E

XY E X E YX EY==, (6.85)

proving thus that the two regression curves are straight lines parallel to the axes

passing through the “centre of gravity” (E(X), E(Y)) of the probability mass

in

2

 .

In the special case of a non-degenerated normal distribution for (X,Y) with vector

mean (m

1

,m

2

) and variance covariance matrix

2

112

2

21 2

σσ

σ

⎡

⎤

=

⎢

⎥

⎣

⎦

Σ , (6.86)

it can be shown that the two conditional distributions are also normal with

parameters

22

2

22 12

1

22

1

21 2 2

2

(),(1),

1

(),(1).

YX N x

XY N y

σ

μρ μσ ρ

σ

μμσρ

ρσ

⎛⎞

+− −

⎜⎟

⎝⎠

⎛⎞

+− −

⎜⎟

⎝⎠

≺

(6.87)

Thus, the two regression curves are linear.

7 STOCHASTIC PROCESSES

In this section, we shall always consider a complete probability space

(

)

,,ΩℑΡ

with a filtration F. Let us recall that a probability space

(

)

,,

Ω

ℑΡ is complete if

every subset of an event of probability 0 is measurable, i.e., in the

σ

-algebra ℑ ,

and so also of probability 0.

Definition 7.1 F being a filtration on the considered basic probability space

means that F is a family

(

)

,

t

tTℑ∈ of sub-

σ

-algebras of

ℑ

,the index set T being

either the natural set

{

}

0,1,..., ,...n or the positive real half-line

[

)

0,

∞

, such that:

0

(i) ,

(ii) ,

(iii) contains all subsets with probability 0.

st

tu

ut

st

>

<⇒ℑ⊂ℑ

ℑ= ℑ

ℑ

∩

(7.1)

Assumption (ii) is called the right continuity property of the filtration F. Any

filtration satisfying these three assumptions is called a filtration satisfying the

Probability tools 35

usual assumptions. The concept of filtration can be interpreted as a family of

amounts of information so that

t

ℑ

gives all the observable events at time t.

Definition 7.2 The quadruplet

(

)

(

)

,,, ,

t

tTΩℑΡ ℑ ∈ is called a filtered

probability space.

Definition 7.3 A random variable : T

τ

Ω

 is a stopping time i:

{

}

: : ( ) .

t

tT t

ωτω

∀∈ ≤ ∈ℑ

(7.2)

The interpretation is the following: the available information at time t gives the

possibility to observe the event given in (7.2) and to decide for example if one

stops the future observations after time t, or not. We have the following

proposition:

Proposition 7.1 The random variable

τ

is a stopping time if and only if

{

}

:() , .

t

ttT

ωτω

<

∈ℑ ∀ ∈

(7.3)

Definition 7.4 A stochastic process (or simply process) with values in the

measurable space

(

)

,E ℵ is a family of random variables

{

}

,

t

X

tT∈ (7.4)

where for all t:

(

)

:, , .

t

X

E measurableΩℑℵ−

This means, in particular, that for every subset B of the

σ

-algebra

ℵ

, the set

(

)

{

}

1

:()

tt

X

BXB

ωω

−

=∈

(7.5)

belongs to the

σ

-algebra ℑ.

Remark 7.1 If

(

)

(

)

,,E

β

ℵ= , the process is called a real stochastic process

with values in

 ; if

(

)

(

)

,,

nn

E

β

ℵ= , it is called a real multidimensional

process with values in

n

 .

If T is the natural set

{

}

0,1,..., ,...n , the process X is called a discrete time

stochastic process or a random sequence; if T is the positive real half-line

0,∞[),

the process X is called a continuous time stochastic process.

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

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