LAWS OF EXCURSIONS ASSOCIATED TO ADDITIVE FUNCTIONALS

(1)

LAWS OF EXCURSIONS ASSOCIATED TO ADDITIVE FUNCTIONALS

Reçu le 07/01/2006– Accepté le 22/06/2006 Résumé

Soit (Pt)un semi-groupe droit de Borel, et soit (St) l’inverse continu à droite d’une fonctionnelle additive continue (Bt).

Soit

( )

t t R

Y

∈ un processus stationnaire à naissance et mort aléatoires, markovien de semi-groupe (Pt) sous la mesure de Kuznetsov Q associée à une mesure excessive. On définit, sous l’hypothèse que la mesure caractéristique

{ } ( )

1

0 ^t

B Q IY B dt

υ =

∫

_∈. ^{de (B}^t) est purement excessive pour le semi-groupe St

⎛

P

⎞

⎜ ⎟

⎝ ⎠, une fonctionnelle additive pour

( ) Y

t t R

∈ en fonction de (Bt) et on étudie les lois des excursions associées à l’ensemble regénératif constitué des temps de discontinuité de l’inverse continu à droite (Ut) de cette fonctionnelle additive. Plus précisément, si on note par

( ) Φ

t le processus

Ut

⎛

Y

⎞

⎜ ⎟

⎝ ⎠ et par H la

σ

-algèbre engendrée par Ht

(

^t^∈^R

)

^{où H}^t est la Q-complétion de H+t0 [ (Ht0) étant la filtration naturelle de

( ) Φ

t ], alors si T est un (Ht)-temps d’arrêt tel que

T T

U − ≠U ^est _T

T⁻

Φ ≠ Φ ,^{la loi}

conditionnelle de l’excursion chevauchant

T T

U −U

⎤ ⎡

⎥ ⎢

⎦ , ⎣ par rapport à H dépend uniquement de

Φ

_T^{et de}

Φ

_T−. Les lois conditionnelles des couples d’excursions ont été également étudiées. MSC: 60J25; 60J40; 60J55.

Mots clés: Standard process; Predictable process; Excursion; Additive functional; Conditional law; Exit measure; Kuznetsov process.

Abstract

Let (Pt) be a right borel semigroup and let (St) the right inverse of a continuous additive functional (Bt). Let

( )

t t R

Y

∈ be a right stationary process with random birth and death, Markov with semi group (Pt) under the Kuznetsov measure

Q

associated to an excessive measure. We define, under the assumption that the characteristic measure { }

( )

1

0 ^t

B Q IY B dt

υ =

∫

_∈. ^{of (B}^t) is purely excessive for the semigroup (Ps), an additive functional for

( )

t t R

Y

∈ in terms of (Bt) and we study the laws of excursions associated to the regenerative set which consists in times of discontinuity of the right inverse (Ut) of this additive functional. More precisely, if we note by

( ) Φ

t the process

Ut

⎛

Y

⎞

⎜ ⎟

⎝ ⎠ and by H the

σ

^-

algebra generated by Ht

( ^t ^∈ ^R )

^{where H}^t is the Q-completion of ⁰

H

t+ ( ^⎛⎜⎝

H

_t⁰^⎞⎟⎠ is the natural filtration of

( ) Φ

t ^), then if

T

^{is a}

( ) H

t -stopping time such that _T

U

T−

≠ U

^and

Φ ≠ Φ ,

_T− _T the conditional law of the excursion

straddling _T

U

T−

U

⎤ ⎡

⎥ ⎢

⎦

,

⎣ with respect to

H

depend only on

Φ

_T and T⁻

Φ

. Conditional laws of pairs of excursions are also considered.

Keywords: Standard process; Predictable process; Excursion; Additive functional; Conditional law; Exit measure; Kuznetsov process.

MSC: 60J25; 60J40; 60J55

n 1985 Kaspi

[ ]

⁶ constructs, for a standard process

X

, an additive functional

B

associated to a regenerative system

M

and gives, under the classical duality hypothesis, the probability measures

P

^{x y}^, allowing the law of excursions associated to

B

with respect to the

σ

-algebra

(

t

)

K = σ Z : ∈ t R

₊ , known to start at

x

end at

y (

t St

Z = X

where

S

t

=

^inf

{ u B :

u

> t }

⁾.

I

H. BOUTABIA Département de Mathématiques, Faculté des Sciences

Universté Badji-Mokhtar Annaba 23000, B.P.12.

Algérie.

ﺺﺨﻠﻣ

.

( ) B

t ةﺮــﻤــﺘﺴﻤﻟا ﺔـــﻴﻌــﻤﺠﻟا لاوﺪــﻠﻟ ﻦــﻴﻤــﻴﻟا ﻦــﻣ سﻮــﻜﻌﻟا

( ) S

t ﻦﻜــﻴﻟو ،ﻦــﻴـــﻤﻴﻟا ﻦﻣ ةﺮـــﻣز ﻒﺼﻧ

( ) P

t ﻦﻜـــﺘﻟ ﻦﻜــﻴﻟو

( ) Y

t t R

∈ ﻦــﻴﻤــﻴﻟا ﻦــﻣ ﺮﻘــﺘﺴﻤــﻟا ﻂـــﻤﻨﻟا ةﺮــﻣﺰﻟا ﻒــﺼﻨﻟ ﺎﻌــﺒﺗ ﺔﻴﻓﻮآ رﺎﻤﻟا تﺎــﻴﻓﻮﻟاو ﺪــﻴﻟاﻮـــﻤﻟا ﺔــﻴﺋاﻮــــﺸﻌﻟ

( ) P

t سﺎـــﻴﻗ ﻖﻓو

Kuznetsov Q سﺎﻴﻗ قﻮﻔﻟ ﻖﻓاﺮﻤﻟا .

ﺰﻴﻤﻤﻟا سﺎﻴﻘﻟﺎﺑ ﺔﻘﻠﻌﺘﻤﻟا تﺎﻴﺿﺮﻔﻟا ﻖﻓو فﺮﻌﻧ

{ }

( )

1

0 t

B

Q I

Y

B dt

υ = ∫

_∈.

( ) B

t ـﻟ ﻲهو

ﺔﻴﻗﻮﻓ ﺔﻴﻓﺎﺻ ةﺮﻣﺰﻟا ﻒﺼﻨﻟ ﺔﻴﻌﻤﺠﻟا لاوﺪﻟا

( )

ل t t R

Y

∈ ﺖﻤﻟا ــﺑ ﺔﻘﻠﻋ

( ) B

t

( ) U

t و ﺔﻘﻓاﺮﻤﻟا ﺔﻴﺋاﻮﺸﻌﻟا ﺔﻠﺣﺮﻠﻟ ﻦﻴﻧاﻮﻘﻟا سرﺪﺗ

ﻦــﻳﺄﻨﻨﻨــﻴﻤــﻴﻟا ﻦــﻣ سﻮﻜــﻌﻠﻟ ﻊــﻄﻘﺘﻟا ﺔﻨﻣزأ ﻦــﻣ ﺔﻠﻜﺸﻣ ،ةﺪﻟﻮﺘﻣ ﺔﻋﻮﻤﺠﻤﻟ

( ^t ^∈ ^R )

H

t ﺪﻟﻮﻤﻟا ﺮﺒﺟ ـ

σ

ـﻨﻋ ل ـ

H

ـﺑ و Ut

⎛

Y

⎞

⎜ ⎟

⎝ ⎠

ﻂﻤــﻨﻠﻟ

( ) Φ

t ــﺑ ﺎـــﻧﺰــﻣر اذإ ﺪﻳﺪﺤﺘﻟﺎﺑ ،ﺔــﻴﻌــﻤﺠﻟا لاوﺪﻟا ﻩﺬﻬﻟ -

ﺚﻴﺤــﺑ ﻒﻗﻮــﺗ ﻦــﻣز

( ) H

t

T

ﻮه نﺎآ اذإ ﻪــﻧﺈــﻓ

( ) Φ

t ـﻟ ﻲﻌﻴﺒﻄﻟا نﺎﻳﺮﺴﻟا ﻲه 0

H

t

⎛ ⎞

⎜ ⎟

⎝ ⎠

0 )

H

t+ ــﻟ ﺔﻠﻤﻜـﺗ

Q

- ﻲــه

H

ﻷ T ﻮ ﻔﻗ

U

T−

U

⎤ ⎡

⎥ ⎢

⎦

,

⎣

. ﻟا ﺔﻠــﺣﺮﻟ ﺔــﻴﻃﺮﺸﻟا ﻦﻴﻧاﻮﻘ

لﺎـــﺠﻤﻠﻟ ﺔﻳﺰﻔﻗ ﺔــﻴﺋاﻮــﺸﻋ

T⁻ T

Φ ≠ Φ ,

T T

U

−

≠ U

ﻚﻟﺬآ ﺎﻬﺘﺳارد ﻢـــﺗ ﺔـــﻴﺋاﻮـــﺸﻌﻟا تﻼـﺣﺮﻟا جاوزﻷ ﺔـــﻴﻃﺮﺸﻟا ﻦــﻴــﻧاﻮــﻘﻟاو

Φ Φ

و ﻰـــﻠﻋ ﻷا ﺪﻤﺘﻌﺗ

(2)

It was given in

[ ]

² , without duality, the measures

P

^{x y}^, in terms of the

Dt

⎛

F

⎞

⎜ ⎟

⎝ ⎠-predictable exit measures for a regenerative system consisting of the closure of the set of times that the regular points of an arbitrary continuous additive functional are visited. The purpose of this paper is to give the conditional laws of pairs of excursions for a Markov process with random birth and death

( ) ^Y

_{t t R}_∈

having the same semigroup as

X

. To this respect, we define an ^"additive functional ^" for

( ) ^Y

_{t t R}_∈ and we extend this result to

( ) ^Y

_{t t R}_∈ . Laws of pairs of excursions for

( ) ^Y

_{t t R}_∈ are discussed.

Preliminaries and notations

Let ^⎛⎜⎝

Ω, , , , , F F X

_t _t

θ

_t

P

^x^⎞⎟⎠ be the canonical realization for a borel standard semigroup

( ) P

t . We assume that the state space

E

is lusinian, and we note by

E

its

σ

-algebra of borel sets. The cemetery point

δ

is absorbent and outside of

E

. Let

( ) B

t be a continuous additive functional and let

R

be the perfect exact terminal time inf

{ u B :

u

>

⁰

}

. We note by

( )

{

^x

⁰ ¹ }

C = x P : R = =

the fine support of

( ) B

t . Let

F

^∗ be the universal completion of the

σ

- algebra

σ ( X

t

: ∈ t R

₊

)

. We consider the random homogeneous set

M = + { t R

o

θ

t

: ∈ t R

₊

}

, and its family of ^⎛^⎜⎜

F

Dt^⎞^⎟⎟

⎝ ⎠ -predictable exit measures

{ }

0 x

P

x E δ

⎛ ⎞

⎜ ⎟

⎝ ⎠ ∈ ∪ (we

assume that

R

is

F

^∗-measurable), where

D

_t is the random variable inf

{ ^s ^{> : ∈} ^{t s} ^M }

. Note that if

S

_t−

≠ S

_t, then

St t

D S

−

=

, hence the excursion associated to

t

is defined by:

( )( ) ( )( ) ( ) ( ) ( )

( ) ( )

if if

t St

S s t t

t R

t t

X s S S

e s k s

s S S

ω ω ω

ω θ ω

δ ω ω

− −

−

⎧

+

< −

= =⎨ ⎪

≥ −

o ⎪⎩

where

k

_t killing operator at

t

defined by:

( )( ) ( )

k

t

ω s = ω s

if

s < t

and

δ

if

s ≥ t

. We consider for

( ) ^{x y} ^, ^{∈ ×} ^{E E}

^{such that}

^x ^≠ ^y

^{, the}

measures

P

^{x y}^, on^⎛⎜⎝

Ω, F

^∗^⎞⎟⎠ ”defined by”:

( )

^where ⁰

( )

x y x x x

R R R

P

^,

= H k ∈. X = y H = P .; X ≠ x

Since ^⎛⎜⎝

Ω, F

⁰^⎞⎟⎠ is an U-space, and according to a classical lemma of Doob the measures

P

^{x y}^, can be chosen measurable for the pair

( ) ^{x y} ^,

^.

We associate to the right inverse

{ }

t u

S = u B : > t

of

( ) B

t , the following notations:

t St

Z = X

,

t St

M = F

and

St

t

θ

θ ⁼

. It is well known that the process

Z ^{= Ω, ,}

^⎛^⎜_⎝

F M Z

_t

^{, , ,}

_t

θ

_t

P

^x^⎞^⎟_⎠ is strong Markov with semigroup

t St

P

^⎛^⎜⎜

P

^⎞^⎟⎟

⎝ ⎠

⎛ ⎞

⎜ ⎟

⎝ ⎠

=

and takes values on

C C E

^∗

⎛ ⎞

⎜ ⎟

⎝

, ∩

⎠ (cf. Jacods

[ ]

⁵ ^).

Let

( ) ^K

_{t t R}_∈ ₊ be the filtration, where

K

_t is the intersection of the

P

^π-completions of the

σ −

algebra

0

K

t+ where

π

is in the set of all the bounded measures on

E

; ^⎛⎜⎝

K

_t⁰^⎞⎟⎠ is the natural filtration of the process

( ) Z

t

.

It was shown in

[ ]

² ^{that if}

^T

is a finite

( ) K

t

−

stopping time such that

S

_T−

≠ S

_T

a s . .

,then we have:

T

S T

F

⎛ ⎞−

K

−

⎜ −⎟

⎝ ⎠

=

(1)

and that if

S

_T−

≠ S

_T and

Z

_T−

≠ Z

_T

a s .

., then

( ( )

^T

)

^Z^T ^Z^T

^{( )}

P f e K = P

⁻^,

f

(2) for all positive and

F

^∗

−

measurable function

f

, where

( )

^x

( ) ( )

P A = ∫ P A µ dx

( µ

is an arbitrary law on

E ) .

Note that the formula

( )

² was proved by Kaspi

[ ]

⁸ ^under

the duality hypothesis.

Excursions of Kuznetsov processes

Let

W

be the set of applications

{ }

w R :

a

E ∪ δ

which satisfies the following properties: there exits an open subset of

R

on which

w

is

E

-valued right -continuous with left limits and out which equals

δ

. We note by

( )

t t R

Y

∈ the coordinate process on

W

.

(3)

Let ⁰

t R

G

t

∈

⎛ ⎞

⎜ ⎟

⎝ ⎠ be the natural filtration of

( )

t t R

Y

∈ and let

0 0

t R t

G =

_∈

VG .

Then the birth and the death times of

( ) Y

t t R

∈ are respectively:

{ } ( )

inf inf

sup sup

t

t R Y E t R Y E α

β

= ∈ : ∈ ∅ = +∞ ,

= ∈ : ∈ ∅ = −∞ .

We define the families of operators on

W

by:

( ) ( ) ( ) ( )

such that for

t t

W w s w s t s R t R

W W w s w s t s t R

τ τ

σ σ

: Ω = + ∈ , ∈

+

: = + , ∈

a a

Note that

X

_s

o τ

_t

= Y

_{t s}₊ ^on

{ Y

t

∈ E }

and

t u t u

σ σ o = σ

₊ ^for

t u , ∈ R

,

s ∈ . R

₊ Let

η

be an excessive measure with respect to

( ) P

t and let

Q

be the Kuznetsov measure on

W

that corresponds to

( ) P

t

⁽

η

⎛ ⎞

⎜ ⎟

⎝

,

⎠ cf.

[ ] [ ]

^{8 10 )}

^,

. We note by

G

_t and

G

the

Q

- completions of

G

_t⁰ and

G

⁰, and we assume that the semigroup

( ) P

t satisfies ^"les hypothèses droites de Meyer^". It follows by

[ ]

¹⁰ that the process

( )

t t t R t

Y W G G Y τ α β Q

= , , ,

∈

, , , ,

is stationary

(

^{i e}

. . σ

t

( ) Q = Q )

and strong Markov with semigroup

( ) P

t

.

For the generalization of formula

( )

² , we consider the additive functionals

B

and

S

given in the previous section. We also note by

B

the random measure on

W

, carried by

] ^{α β} ^, [

such that:

{ }

] ] on for all 0 and

s t t

B o τ = B t t s , + Y ∈ E s > t ∈ R

We assume that the characteristic measure { }

( )

1

0 ^t

B

Q I

Y

B dt

υ = ∫

_∈. ^of

^B

is purely excessive for the semigroup ^⎛⎜

P

_t^⎞⎟

⎝ ⎠ (i.e.

∫ P f x

t

( ) ( ) υ

B

dx →

⁰^as

^t ^{→ ∞}

if

υ

B

( ) f < ∞

⁾. It was shown in

[ ]

⁷ ^that

^Q

^a.e.

] ]

B α , < ∞ t

for all

t > . α

Let

( ) ^V

_{t t R}_∈ be the nondecreasing process defined on

W

by:

{ } { }

] ] on and on

t t

V = + α B α , t α < t V = α t ≤ α ,

and let

( ) ^U

_{t t R}_∈ be the right-continuous inverse of

( ) ^V

_{t t R}_∈ ^{that is:}

{ }

t inf u

U = u > : α V > t

We also note by

M

the closed random subset of

] ^{α β} ^, [

defined by:

M = < < ∪ + α t β { t R o τ

t

}

which verifies the following property of homogeneity ( cf.

[ ] ^{4 ))} ^:

( M − ∩ ,∞ = t ) ] [ ⁰ M o τ

t

^on { Y

t

∈ E }

Remark:

1) If

α = −∞

,

{ u > : α V

u

> = t } %

and

U

_t

= +∞

, then

α > −∞

on

{ α < U

t

< β } .

2)

U

_t

= α

on

{ ^t ^≤ ^α } ^.

For

t ∈ R

, let

( )

0

t t t

t

Y

U

G

t

G

U

τ

t

τ

U

H

t

σ

u

u t

Φ = , = , = , = Φ : ≤

and

^H

⁰

⁼ ^σ ( ^H

^t⁰^:

^t ^∈ ^R )

. We note by

H

_t (resp.

H

) the

Q

-completion of

H

_t⁰+ (resp.

H

⁰

) .

Note that for all the following formulas, the

σ −

finiteness of

Q

is guaranteed by the argument used in

[ ] ¹ ^.

It is not hard to show that

( ) Φ

t has the same properties as

( ) Z

t and the following result hold.

Proposition:

1) The process

( ) U

t is right continuous, has left limits, and satisfies

U

_t

= U

_β for all

t ≥ β

Q a e . .

.

2)

( ) U

t is

( ) G

t -adapted.

3) For all

t ∈ R

and

s > 0

we have:

a)

U

_t

= + α S

_t₋_α

o τ

_α^on

{ −∞ < < α ^t }

b)

V

_{t s}₊

= + V

_t

B

_s

o τ

_t ^on

{ Y

t

∈ E }

and

t s t s t

U

+

= U + S o τ

^on

{ α < U

t

< β } .

4) On ^⎧^⎪^⎨⎪⎩

U

_t

U

_t−^⎫^⎪^⎬⎪⎭

U

_t−

U

_t

⎤ ⎡

⎥ ⎢

⎦ ⎣

≠ , ,

is a contiguous interval of

M

.

If _t

U ≠ U

t−

,

let

E

_t be the excursion associated to

B

defined by:

( )( ) ( ) ( ) ( )

( ) ( )

if 0 if

Ut

s t t

t

t t

Y w s U w U w

E w s

s U w U w δ

− −

−

+

≤ < −

= ⎨ ⎧⎪

≥ −

⎪⎩

According to the previous proposition, the process

( ) ^V

_{t t R}_∈ has got the same role as

B

for the process

(4)

( ) ^Y

_{t t R}_∈

^.

We say that

( ) ^V

_{t t R}_∈ ^{is an}^" additive functional ^"

for

( ) ^Y

_{t t R}_∈ . We have the extension of theorem 2

[ ] ²

^on

^W

^.

Theorem1:

1)The process

^{Φ =} ( W ^{,Φ , , , ,}

t

G G

t

τ

t

Q )

^{is strong}

Markov in the sense that for all

( ) G

t

−

stopping time

T

and

s > : 0

( )

(

T s T

)

s

( ) ^on {

T

}

Q f Φ

₊

G = P f ,Φ

_T

α < U < β

(3) for all function positive and

F

-measurable

f

.

2) Assume that

T

₁ is a finite

( ) H

t

−

stopping time such that

1 1

T T

U ≠ U

−and

1 1

T T⁻

Φ ≠ Φ

Q a e . .

.Then we have:

(

^T¹

)

^T¹ ^T¹

^{( )} ^on ^{

^T¹

^}

Q F E

^⎛^⎜⎜ ^⎞^⎟⎟

H P

⁻

F α U β

⎝ ⎠

=

Φ ,Φ

< <

(4) for all

F ≥ 0

,

F

^∗-measurable.

Proof : If

T ≡ t

is constant, the formula

( ) ³

follows from the Markov property of the process

( ) Y

t _{t R}

∈ at time

U

_t and the fact that

Φ =

_{t s}+

Z

_s

o τ

_t^and

τ

t s+

= θ τ

s

o

t^on

{ α < ^U

_t

< β } .

This formula is also true for

T

_n instead of

T

, where

( ) T

n is the decreasing dyadic approximation of

T

, which extends for a general

T

by the right continuity of the processes

( ) ( ) ^{Φ ,}

_t

^U

_t ^and

( ) τ

t

^.

The formula

( ) ⁴

is argued in the same manner as

( ) ²

by using the formula

( ) ³⁰

^of

[ ] ¹

^.

Conditional laws of pairs of excursions

We consider now the time-reversed process

( )

_{( )}

t R

t t R

Y

t

Y

⁻

∈

⎛ ⎞

⎜ ⎟

∈

=

⎝ − ⎠ . It is an

E

-valued right continuous process with left limits on

^⎤ _⎦ ^{α β} ^, ^⎡ _⎣ ^{= − , −} ] ^{β α} [

, and which is equal to

δ

outside of

^⎤ ⎦ α β , ^⎡ ⎣

^{. As in}

[ ] ¹

and

[ ] ¹⁰

, we assume that

( ) Y

t is also Markov with respect to another standard semigroup

( ) P

t satisfying ^"les hypothèses droites de Meyer^", which implies the strong Markov property and the existence of exit systems. The measure

$ ( ) ^B ^Q ( ^Y

^t

^B ^t )

η = ∈ ; < < α β

is

( ) P

t -excessive and the stationarity of

( ) Y

t ^is

guarantee. Let

τ ^$

_t

^{, ,} G

_t

^B, S V ^$ ,

t

, U

_t and

E

_t be the analogous of

τ

_t

, , G B

_t ,

S , V U

_t

, ,

_t and

E

_t

corresponding to

( ) Y

t

.

As previously we assume that

Q a e . . B ^⎤ ⎦ α , < ∞ t ^⎤ ⎦

^{for all}

t > . α

For the process

( ) Ψ =

t

( ) Y

Ut and the random subset

{

t

}

M = < < ∪ + α t β t R

o

τ

$

of

^⎤ ⎦ α β , ^⎡ ⎣

, we have the analogous of theorem 1. In particular if we design by

H

and

H

_t^the

^Q

completions of

σ ( Ψ : ∈

_u

^u ^R )

and

σ ( Ψ : ≤

_u

^u ^t )

₊ respectively, and by

P

^{x y}^, the measure defined as

P

^{x y}^, in terms of the exit measures ⁰

P

^x of

M

for the canonical realization of

( ) P

t , we have the following formula:

( ( )

^T²

)

^T² ^T²

^{( )} ^on {

^T²

}

Q F E H = P

^{Ψ ,Ψ}⁻

F α < U < β

(5) for all finite

( ) H

t

−

stopping time

T

₂ such that

T2

U ≠

T2

U

⁻^and

Ψ ≠ Ψ

T₂⁻ T₂⁻

Q a e . .,

and for all positive function

F

^∗

−

measurable

F .

For the following theorem which gives the conditional law of pairs of excursions, we consider the family of probability measures

Q

^{x y z u}^{, , ,}

= P

^{x y}^,

⊗ P

^{z u}^,

.

Theorem 2:

Let

T

₁ (resp.

T

₂

)

as in

( ) ⁴ ( ^{resp 5} ^. ( ) )

. We assume that the following hypotheses are satisfied:

1)

σ

^⎛^⎜^⎜⎜⎝

U

T1−^⎞^⎟^⎟⎟⎠

∩ Λ ⊂ H

T2− 2)

( ) U

T2

H

T1

σ

₋

∩ Λ ⊂

₋

,

where

Λ = { α < − U

T²⁻

≤ U

T1−

< β } .

Then we have the following formula:

( )

(

_T¹ _T2

)

^T¹ ^T¹ ^T² ^T²

^{( )} on

Q H E E H H Q H

− −

Φ ,Φ ,Ψ ,Ψ

, ∩ = Λ

(6)

for all positive and

F

^∗

⊗ F

^∗-measurable function

H

. Proof:

We have to prove that:

( )

¹ ¹

^{( )}

² ²

( )

1 2

T T T T

T T

Q F E F E ZI Q P F P F ZI

⎛ ⎞ − −

⎜ ⎟

⎝ ⎠

Φ ,Φ

⎛ ⎞

⎛ ⎞ ⎜ Ψ ,Ψ ⎟

⎜ ⎟

⎜ Λ⎟ ⎜ Λ⎟

⎝ ⎠

=

⎝ ⎠ (7)

(5)

for all

F F ,

positive functions and

F

^∗-measurable, and for all positive random variable

Z

,

H ∩ H

-measurable.

Since

2 1 T2 1

T UT U UT

U

θ

τ $

−

=

−+ −

o τ $

− −^on

^Λ

, and since

1

UT

τ $

− ₋^is

T1

U

G

⎛ ⎞−

⎜ ⎟

⎜ −⎟

⎝ ⎠

−

measurable and

1 1

T T

D U

G

⎛ ⎞−

H

−

⎜ ⎟

⎜ −⎟

⎝ ⎠

= ,

then

2 T1

T

U

H

σ τ

− −

⎛ ⎞ ∩Λ ⊂

⎜ ⎟

⎝ $ ⎠

⁽

G

t^D

= G

D_t

⊃ G

t where

{ }

t

inf

D = s > : ∈ t s M

on

W

, for

t ∈ R

). By using the same argument, we prove that

σ

^⎛^⎜^⎜⎜⎝

Ψ , Ψ ∩ Λ ⊂

T2− T2^⎞^⎟^⎟⎟⎠

H

T1− ,

1 2

T T

U H

σ

^⎛^⎜^⎜⎜⎜⎝ −^⎞^⎟^⎟⎟⎟⎠ −

∩ Λ ⊂

and

σ

^⎛^⎜^⎜⎜⎝

Φ ,Φ ∩ Λ ⊂

T1− T1^⎞^⎟^⎟⎟⎠

H

T2−

.

The Markov property at time

T

₁ and the formula

( ) ⁵

implied that for all positive random variable

Z

₁,

T1

H

₋

−

measurable and for all positive and

F

0

−

measurable function

ϕ

:

( ) ( )

( ) ( ) ( )

1 2 1

1 1

2 1

1

T T

T T T

T T

Q F E F E Z I

Q P F F E Z I

ϕ τ ϕ τ

⎛ ⎞

⎜ ⎟

⎝ ⎠

−

⎛ ⎞

⎜ ⎟

⎜ Λ⎟

⎝ ⎠

Φ ,Φ

⎛ ⎞

⎜ ⎟

⎜ Λ⎟

⎜ ⎟

⎝ ⎠

=

and according to the fact that

H

is generated by

1 T

H

₋ and

T1

τ

, we have:

( )

¹ ¹

^{( )} ( )

1 2 2

T T

T T T

Q F E F E ZI Q P F F E ZI

⎛ ⎞ −

⎜ ⎟

⎝ ⎠

Φ ,Φ

⎛ ⎞

⎛ ⎞ ⎜ ⎟

⎜ ⎟

⎜ Λ⎟ ⎜ Λ⎟

⎝ ⎠

=

⎝ ⎠(8)

The formula

( ) ⁷

follows by using formulas

( ) ⁸

^and

( ) ⁵ ^.

REFERENCES

[1]- BOUTABIA H..-Lois conditionnelles des excursions d’un processus de Markov à naissance et mort aléatoires, Portugaliae Mathematica., Vol 56 fasc 2, 239-255, 1999.

[2]- BOUTABIA H.-Additive functionals and excursions of Kuznetsov processes, International Journal of Mathematics and Mathematical Sciences., 2031-2040 (2005).

[3]- GETOOR R.K.and SHARPE M.J.-Two results on dual excursions, Seminar on Stochastic Processes1981, Birkauser (1981), 31-52.

[4]- GETOOR R.K.-Killing a Markov process under a stationary measure involves creation, Ann.Prob.,16(1988).

[5]- JACODS.PA-Excursions of a Markov process induced by continuous additive functionals, Z.Wahrs.Verw.Geb.

44(1978),325-336.

[6]- KASPI H.-Excursion laws of Markov processes in classical duality, Ann.Prob., 33, (1985), 492-518.

[7]- KASPI H.-Random time changes for processes with random birth and death, Ann.Prob., 16, (1988), 586-599 [8]- KUZNETSOV S.E.-Construction of Markov processes with random times of birth and death, Prob.Th. Fiel 18 (1973), 571-575.

[9]-MAISONNEUVE B.-Systéme de sortie

Dt

⎛

F

⎞

⎜ ⎟

⎝ ⎠-

prévisibles, Prob.Th.Fields 80 (1989), 395-405.

[10]- MITRO J.B.-Exit Systems for Dual Markov

Processes, Z.Wahrs.Verw

Geb. 66(1984), 259-267.

LAWS OF EXCURSIONS ASSOCIATED TO ADDITIVE FUNCTIONALS