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

6 Chapter 1

Remark 2.3 In fact, it is also possible to define the concept of d.f. in the discrete

case if we set, without loss of generality, on

0

(,2)

N

N , the measure P defined

from the sequence (2.22). Indeed, if for every positive integer k, we set

1

()

k

j

F

kp

=

∑

(2.34)

and generally, for any real x,

[

)

0, 0,

()

(), , 1,

x

Fx

Fk x kk

≤

⎧

=

⎨

∈

+

⎩

(2.35)

then, for any positive integer k, we can write

{

}

(

)

1,..., ( )PkFk= (2.36)

and so calculate the probability of any event.

3 RANDOM VARIABLES

Let us suppose the probability space ( , , )P

Ω

ℑ and the measurable space ( , )E

ψ

are given.

Definition 3.1 A random variable (in short r.v.) with values in E is an

application X from Ω to E such that

1

:()BXB

ψ

−

∀

∈∈ℑ, (3.1)

where X

-1

(B) is called the inverse image of the set B defined by

{

}

11

() : () , ()XB X BXB

ωω

−−

=

∈∈ℑ. (3.2)

Particular cases

a) If ( , )E

ψ

=( , )

β

 , X is called a real random variable.

b) If

(,) (,)E

ψ

β

=  , where  is the extended real line defined by

{}{}

+∞ −∞∪ ∪ and

β

the extended Borel

σ

-field of  , that is the minimal

σ

-field containing all the elements of

β

and the extended intervals

[

)

(

]

[

]

(

)

[

)(

][ ]

()

,, ,, ,, ,,

,,,,,,,, ,

aaaa

aaaaa

−∞ −∞ −∞ −∞

+∞ +∞ +∞ +∞ ∈ 

(3.3)

then X is called a real extended value random variable.

c) If ( 1)

n

En=> with the product

σ

-field

()n

β

of

β

, X is called an n-

dimensional real random variable.

d) If

()n

E

=  (n>1) with the product

σ

-field

()n

β

of

β

, X is called a real

extended n-dimensional real random variable.

Probability tools 7

A random variable X is called discrete or continuous according as X takes at most

a denumerable or a non-denumerable infinite set of values.

Remark 3.1 In measure theory, the only difference is that condition (2.11) is no

longer required and in this case the definition of a r.v. given above gives the

notion of measurable function. In particular a measurable function from ( , )

β



to ( , )

β

 is called a Borel function.

Let X be a real r.v. and let us consider, for any real x, the following subset of Ω :

{

}

:()

X

x

ωω

≤ .

As, from relation (3.2),

{

}

(

]

1

:() ( ,),

X

xX x

ωω

−

≤= −∞ (3.4)

it is clear from relation (3.1) that this set belongs to the

σ

-algebra ℑ.

Conversely, it can be proved that the condition

{

}

:()Xx

ωω

≤

∈ℑ, (3.5)

valid for every x belonging to a dense subset of  , is sufficient for X being a real

random variable defined on Ω . The probability measure P on ( , )

Ω

ℑ induces a

probability measure

μ

on ( , )

β

 defined as

{

}

(

)

:() : () .

B

BP X B

βμ ω ω

∀∈ = ∈ (3.6)

We say that

μ

is the induced probability measure on ( , )

β

 , called the

probability distribution of the r.v. X. Introducing the distribution function related

to

μ

, we get the next definition.

Definition 3.2 The distribution function of the r.v. X, represented by

X

F

, is the

function from

[]

0,1→ defined by

(

]

(

)

{

}

(

)

() , : ( ) .

X

F

xxPXx

μωω

=−∞= ≤ (3.7)

In short, we write

(

)

()

X

F

xPXx

=

≤ . (3.8)

This last definition can be extended to the multi-dimensional case with a r.v. X

being an n-dimensional real vector:

1

( ,..., )

n

X

XX

=

, a measurable application

from ( , , )PΩℑ to

(,)

nn

β

 .

Definition 3.3 The distribution function of the r.v.

1

( ,..., )

n

X

XX

=

, represented

by

X

F

, is the function from

n



to

[

]

0,1 defined by

8 Chapter 1

{

}

(

)

111

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

Xn nn

F

xxP X xX x

ωω ω

=≤≤. (3.9)

In short, we write

111

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

X

nnn

F

xxPXxXx

=

≤≤. (3.10)

Each component X

i

(i=1,…,n) is itself a one-dimensional real r.v. whose d.f.,

called the marginal d.f., is given by

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

i

Xi X i

Fx F x

=

+∞ +∞ +∞ +∞ . (3.11)

The concept of random variable is stable under a lot of mathematical operations;

so any Borel function of a r.v. X is also a r.v.

Moreover, if X and Y are two r.v., so are

{}{}

inf , ,sup , , , , ,

X

XY XY X Y X YX Y

Y

+−⋅, (3.12)

provided, in the last case, that Y does not vanish.

Concerning the convergence properties, we must mention the property that, if

(, 1)

n

Xn≥

is a convergent sequence of r.v.

−

that is, for all

ω

∈

Ω , the sequence

(())

n

X

ω

converges to ( )X

ω

−

, then the limit X is also a r.v. on Ω . This

convergence, which may be called the sure convergence, can be weakened to

give the concept of almost sure (in short a.s.) convergence of the given sequence.

Definition 3.4 The sequence (())

n

X

ω

converges a.s. to ()X

ω

if

{

}

(

)

:lim () () 1

n

PXX

ωωω

=

= . (3.13)

This last notion means that the possible set where the given sequence does not

converge is a null set, that is a set N belonging to

ℑ

such that

() 0PN

=

. (3.14)

In general, let us remark that, given a null set, it is not true that every subset of it

belongs to ℑ but of course if it belongs to

ℑ

, it is clearly a null set (see relation

(2.20)).

To avoid unnecessary complications, we will suppose from now on that any

considered probability space is complete, This means that all the subsets of a null

set also belong to ℑ and thus that their probability is zero.

4 INTEGRABILITY, EXPECTATION AND

INDEPENDENCE

Let us consider a complete measurable space ( , , )

μ

Ω

ℑ and a real measurable

variable X defined on Ω . To any set A belonging to

ℑ

, we associate the r.v.

A

I

,

called the indicator of A, defined as

Probability tools 9

1, ,

()

0, .

A

I

A

ω

∈

⎧

=

⎨

∉

⎩

(4.1)

If there exists partition ( , 1)

n

An≥ with all its sets measurable such that

() ( ), 1

nnn

AX aa n

ω

∈⇒ = ∈ ≥ , (4.2)

then X is called a discrete variable. If moreover, the partition is finite, it is said to

be finite. It follows that we can write X in the following form:

)()(

1

ωω

n

A

n

IaX

∑

∞

=

= . (4.3)

Definition 4.1 The integral of the discrete variable X is defined by

1

()

nn

n

X

daA

μμ

∞

=

Ω

=

∑

∫

, (4.4)

provided that this series is absolutely convergent.

Of course, if X is integrable, we have the integrability of

X

too and

1

()

nn

n

X

daA

μμ

∞

=

Ω

=

∑

∫

. (4.5)

To define in general the integral of a measurable function X, we first restrict

ourselves to the case of a non-negative measurable variable X for which we can

construct a monotone sequence ( , 1)

n

Xn≥ of discrete variables converging to X

as follows:

⎭

⎬

⎫

⎩

⎨

⎧

+

<≤

∞

=

∑

=

nn

k

X

k

n

I

k

X

2

1

2

:

1

2

)(

ω

. (4.6)

Since for each n,

1

() (),

1

0() () ,

2

nn

n

XX

ωω

+

≤

≤− ≤

(4.7)

the sequence ( , 1)

n

Xn≥ of discrete variables converges monotonically to X on

Ω .

Definition 4.2 The non-negative measurable variable X is integrable on Ω iff

the elements of the sequence

(, 1)

n

Xn≥

of discrete variables defined by relation

(4.6) are integrable and if the sequence

n

X

dP

Ω

⎛⎞

⎜⎟

⎝⎠

∫

converges.

From this last definition, it follows that

()lim( )

n

E

XEX

=

, (4.8)

10 Chapter 1

where

1

:

1

22

()

2

nn

n

kk

n

X

k

Xd I

ω

ωμ μ

∞

+

⎧

⎫

≤<

⎨

⎬

=

Ω

⎩⎭

⎛⎞

⎜⎟

=

⎜⎟

⎝⎠

∑

∫

. (4.9)

To extend the last definition without the non-negativity condition on X, let us

introduce for an arbitrary variable X, the variables

X

+

and

X

−

defined by

{

}

{

}

() sup (),0, () inf (),0,XXX X

ωωω ω

+−

==− (4.10)

so that

X

XX

+

−

=

− . (4.11)

Definition 4.3 The measurable variable X is integrable on

Ω

iff the non-

negative variables

X

+

and

X

−

defined by relation (4.10) are integrable and in

this case

X

dXdXd

μ

μμ

+−

ΩΩ Ω

=−

∫∫ ∫

. (4.12)

Remark 4.1 a) If the integral of X does not exist, it may however happen that

, Xd Xd Xd Xd

μμμμ

+−−+

ΩΩΩΩ

⎛⎞ ⎛ ⎞

<∞ <∞ =∞ =∞

⎜⎟ ⎜ ⎟

⎝⎠ ⎝ ⎠

∫∫∫∫

. (4.13)

In these two cases, we say that the integral of X is infinite; more precisely, we

have

Xd Xd

μμ

ΩΩ

⎛⎞

=

−∞ = +∞

⎜⎟

⎝⎠

∫∫

. (4.14)

If A is an element of the

σ

-algebra

ℑ

, the integral on A is simply defined by

A

AA

X

dXId

μ

=

∫∫

. (4.15)

Of course, X being a non-negative measurable variable with an infinite integral, it

means that the approximation sequence (4.6) diverges to

+

∞

for almost all

ω

.

Now let us consider a probability space ( , , )P

Ω

ℑ and a real random variable X

defined on Ω . In this case, the concept of integrability is designed by expectation

represented by

(

)

()

E

XXdPXdP

Ω

==

∫

, (4.16)

provided that this integral exists. The computation of the integral

X

dP XdP

Ω

⎛⎞

=

⎜⎟

⎝⎠

∫∫

(4.17)

can be done using the induced measure

μ

on ( , )

β

 ,defined by relation (3.6)

and then using the distribution function F of X. Indeed, we can write

Probability tools 11

()

R

E

XXdPXd

μ

Ω

⎛⎞

==

⎜⎟

⎝⎠

∫

, (4.18)

and if F

X

is the d.f. of X, it can be shown that

() ()

X

R

E

XxdFx=

∫

, (4.19)

this last integral being a Lebesgue-Stieltjes integral. Moreover, if F

X

is absolutely

continuous with f

X

as density, we get

() ().

x

E

Xxfxdx

+∞

−∞

=

∫

(4.20)

If g is a Borel function, we also have (see for example Chung (2000), Royden

(1963), Loeve (1963))

(( )) ()

X

E

gX gxdF

+∞

−∞

=

∫

(4.21)

and with a density for X,

(( )) () ()

X

E

gX gxf xdx

+∞

−∞

=

∫

. (4.22)

The most important properties of the expectation are given in the next

proposition.

Proposition 4.1 (i) Linearity property of the expectation: If X and Y are two

integrable r.v. and a,b two real numbers, then the r.v. aX+bY is also integrable

and

()()().

E

aX bY aE X bE Y

+

=+ (4.23)

(ii) If

(, 1)

n

An≥ is a partition of

Ω

, then

1

()

n

A

E

XXdP

∞

=

∑

∫

. (4.24)

(iii) The expectation of a non-negative r.v. is non-negative.

(iv) If X and Y are integrable r.v., then

() ().

X

YEXEY

≤

⇒≤ (4.25)

(v) If X is integrable, so is

X

and

()

E

XEX≤ . (4.26)

(vi) Dominated convergence theorem (Lebesgue: Let (, 1)

n

Xn≥ be a sequence

of r.v. converging a.s. to the r.v. X integrable, then all the r.v. X

n

are integrable

and moreover

lim ( ) (lim ) ( ( ))

nn

E

XEX EX

=

= . (4.27)

12 Chapter 1

(vii) Monotone convergence theorem (Lebesgue): Let

(, 1)

n

Xn≥ be a non-

decreasing sequence of non-negative r.v; then relation (4.27) is still true

provided that

∞

+

is a possible value for each member.

(viii) If the sequence of integrable r.v. (, 1)

n

Xn≥ is such that

()

1

n

EX

∞

=

<

∞

∑

, (4.28)

then the random series

1

n

X

∞

=

∑

converges absolutely a.s. and moreover

11

() ( ())

nn

E

XEX EX

∞∞

==

⎛⎞

==

⎜⎟

⎝⎠

∑∑

, (4.29)

where the r.v. is defined as the sum of the convergent series.

Given a r.v. X, moments are special cases of expectation.

Definition 4.4 Let a be a real number and r a positive real number, then the

expectation

(

)

r

E

Xa−

(4.30)

is called the absolute moment of X, of order r, centred on a.

The moments are said to be centred moments of order r if a=E(X). In particular,

for

r=2, we get the variance of X represented by

2

(var( ))

X

σ

,

(

)

2

E

Xm

σ

=−

. (4.31)

Remark 4.2 From the linearity of the expectation (see relation (4.23)), it is easy

to prove that

22 2

()(())

E

XEX

σ

=− , (4.32)

and so

22

()

E

X

σ

≤ , (4.33)

and more generally, it can be proven that the variance is the smallest moment of

order 2 whatever the number

a is.

The next property recalls inequalities for moments.

Proposition 4.2 (Inequalities of Hölder and Minkowski) (i) Let X and Y be two

r.v. such that

qp

YX , are integrable with

11

1, 1,

p

pq

<

<∞ + = (4.34)

Probability tools 13

then:

()

(

)

()

(

)

11

()

pq

EXY E X E Y≤⋅. (4.35)

(ii)

Let X and Y be two r.v. such that ,,1 ,

pp

XY p

≤

<∞ are integrable, then

()()

(

)

()

(

)

11

1

.

pp

pp p

p

EX Y EX EY+≤ + (4.36)

If p=2 in the first part of this last proposition, then relation (4.36) gives the

Cauchy-Schwarz inequality

()

(

)

()

(

)

11

22

()EXY E X E Y≤⋅. (4.37)

The last fundamental concept we will now introduce in this section is that of

stochastic independence, or more simply independence.

Definition 4.5 The events

1

,..., ,( 1)

n

AAn> are stochastically independent or

independent iff

12

1

2,..., , 1,..., : : ( )

kk

m

kknn

k

mnn nnn nPA PA

=

⎛⎞

∀= ∀ = ≠ ≠ ≠ =

⎜⎟

⎝⎠

∏



∩

. (4.38)

For

n=2, relation (4.38) reduces to

12 1 2

()()()PA A PA PA

=

∩ . (4.39)

Let us remark that piecewise independence of the events

1

,..., ,( 1)

n

AAn> does

not necessarily imply the independence of these sets and thus not the stochastic

independence of these

n events. As a counter example, let us suppose we drew a

ball from an urn containing four balls called

b

1

, b

2

, b

3

, b

4

and let us consider the

three following events:

{

}

{

}

{

}

112213314

,, ,, ,AbbA bbAbb===. (4.40)

Then assuming that the probability of having one ball is

1

4

, we get

12 13 23

1

()()()

4

PA A PA A PA A

=

==∩∩∩, (4.41)

but as

123

1

()

4

PA A A

=

∩∩ (4.42)

too, we do not have the relation

123 1 2 3

()()()()PA A A PA PA PA

=

∩∩ , (4.43)

and so we have proved that independence in pairs does not imply the

independence of these three events.

We will now extend the concept of independence to random variables.

14 Chapter 1

Definition 4.6 (i) The n real r.v. X

1

,X

2

,…,X

n

defined on the probability space

(

)

,,PΩℑ are said to be stochastically independent, or simply independent, iff for

any Borel sets B

1

,B

2

,…,B

n

, we have

{}{}

()

1

:() :()

n

kk kk

k

PXB PXB

ωω ωω

=

⎛⎞

∈= ∈

⎜⎟

⎝⎠

∏

∩

. (4.44)

(ii)

For an infinite family of r.v., independence means that the members of

every finite subfamily are independent. It is clear that if X

1

,X

2

,…,X

n

are

independent, so are the r.v.

1

,...,

k

ii

X

X with

1

, 1,..., , 2,...,

kk

iiinkn

≠

≠= = .

From relation (4.44), we find that

11 11 1

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

n

nn nn n

PX x X x PX x PX x x x≤≤=≤ ≤∀∈. (4.45)

If the functions

1

, ,...,

n

X

XX

F

FF are the distribution functions of the r.v.

11

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

nn

X

XXXX= , we can write the preceding relation under the form

1

11 1

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

n

XnX Xn n

Fx x F x F x x x=⋅⋅ ∀ ∈. (4.46)

It can be shown that this last condition is also sufficient for the independence

of

11

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

nn

X

XXXX= . If these d.f. have densities

1

, ,...,

n

X

XX

f

ff, relation

(4.46) is equivalent to

1

11 1

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

n

XnXXn n

fx x f x f x x x=∀∈… 

. (4.47)

In case of the integrability of the n real r.v X

1

,X

2

,…,X

n

, a direct consequence of

relation (4.46) is that we have a very important property for the expectation of

the product of n independent r.v.:

11

()

nn

kk

E

XEX

==

⎛⎞

=

⎜⎟

⎝⎠

∏∏

. (4.48)

The notion of independence gives the possibility to prove the result called the

strong law of large numbers which says that if ( , 1)

n

Xn≥ is a sequence of

integrable independent and identically distributed r.v., then

..

1

()

n

as

k

X

EX

n

=

⎯⎯→

∑

. (4.49)

The next section will present the most useful distribution functions for stochastic

modelling.

5 MAIN DISTRIBUTION PROBABILITIES

Here we shall restrict ourselves to presenting the principal distribution

probabilities related to real random variables.

Probability tools 15

5.1 The Binomial Distribution

Let us consider a random experiment E such that only two results are possible: a

“success”(

S) with probability p and a “failure (F) with probability 1.qp=− If n

independent trials are made in exactly the same experimental environment, the

total number of trials in which the event

S occurs may be represented by a

random variable

X whose distribution

( , 0,..., )

i

p

in

=

with

(),1,...,

i

p

PX i i n

=

== (5.1)

is called a

binomial distribution with parameters (n,p). From basic axioms of

probability theory seen before, it is easy to prove that

, 0,...,

ini

i

n

p

pq i n

i

−

⎛⎞

==

⎜⎟

⎝⎠

, (5.2)

a result from which we get

() ,var() .

E

Xnp Xnpq

=

= (5.3)

The

characteristic function and the generating function, when it exists, of X

respectively defined by

() ( ),

() ( )

itX

X

tX

X

tEe

gt Ee

ϕ

=

(5.4)

are given by

() ( ),

() ( ).

it n

X

tn

X

tpeq

g

tpeq

ϕ

=+

(5.5)

Example 5.1

(The Cox and Rubinstein financial model) Let us consider a

financial asset observed on

n successive discrete time periods so that at the

beginning of the first period, from time 0 to time 1, the asset starts from value

S

0

and has at the end of this period only two possible values, uS

0

and dS

0

(

0

<d<1,u>1) respectively with probabilities p and q=1-p. The asset has the same

type of evolution on each period and independently of the past. In period

i, from

time

i–1 to time i, let us associate the r.v.

, 1,...,

i

in

ξ

=

defined as follows:

1, with probability ,

0, with probability .

i

p

q

ξ

⎧

=

⎨

⎩

(5.6)

The value of the asset at the end of period

n is given by the r.v. Y

n

defined as

nn

X

nX

n

Yud

−

= (5.7)

with

1nn

X

ξ

=

++ . (5.8)

It is clear that the r.v.

X

n

has a binomial distribution of parameters (n,p) and

consequently, we get the distribution probability of

Y

n

:

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

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