Further examples of explicit Krein representations of certain subordinators

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Further examples of explicit Krein representations of

certain subordinators

Catherine Donati-Martin, Marc Yor

To cite this version:

Catherine Donati-Martin, Marc Yor. Further examples of explicit Krein representations of certain

subordinators. 2005. �hal-00008369�

ccsd-00008369, version 1 - 2 Sep 2005

representations of ertain subordinators

C. Donati-Martin

(1)

, M.Yor

(1),(2)

(1)

Laboratoire de Probabilités etModèles Aléatoires, Université ParisVI et VII, 4Pla e Jussieu- Case 188, F-75252ParisCedex 05

(2)

Institut Universitairede Fran e

Abstra t :Inapreviouspaper[1℄,wehaveshown thatthegamma subordi-nators may berepresented asinverse lo altimes of ertain diusions. In the present paper, we give su h representations for other subordinators whose Lévy densities are of the form

C

(sinh(y))

γ

,

0 < γ < 2

, and the more general family obtained from those by exponential tilting.

1.1

In this paper, we ontinue the program started in [1℄, that is : to represent asmany subordinators

(S

ℓ

, ℓ ≥ 0)

,i.e : in reasing Lévy pro esses, started at 0, aspossible asinverse lo altimes

(τ

ℓ

, ℓ ≥ 0)

of some parti ular

R

+

-valued diusion

(X

t

)

, su h that 0 is regular for itself (relatively to

X

). More pre isely, assume that:

E[exp(−λS

ℓ

)] = exp(−ℓΨ(λ)) ,

where:

Ψ(λ) =

Z

_∞

0

ν(dy)(1−e

−λy

₎

,and

ν(dy)

-theLévymeasureasso iated with

(S

ℓ

, ℓ ≥ 0)

-is of the form:

ν(dy) = h(y)dy ,

with

h(y) =

Z

_∞

0

dλ(x) e

−xy

,

for some positive

σ

-nite measure

λ(dx)

on

R

+

, then, it is known, as a onsequen e of Krein's theory ( f : Knight [5℄, Kotani-Watanabe [6℄), that there existsaunique diusion

(X

t

)

takingvaluesin

R

+

, su h thatitsinverse lo altime at0,

(τ

ℓ

, ℓ ≥ 0)

, isdistributed as

(S

ℓ

, ℓ ≥ 0)

.

Finding

X

when

ν

isgiven is alled (here)Krein representation problem. In our paper [1℄, we ould llinthe following

Table 1

h(y)

Generator of

(X

t

)

Distribution

C

y

α+1

L

−α

=

1

₂

d

2

dx

2

+

δ

₋₁

2x

d

dx

;

δ = 2(1 − α)

P

δ

(0 < α < 1)

C

y

α+1

e

−µy

L

µ

_↓

−α

= L

−α

+

√

2µ

K

b

′

α

(

√

2µx)

b

K

α

(

√

2µx)

d

dx

P

δ

_;µ↓

(0 < α < 1; µ > 0)

where

K

b

α

(y) = y

α

_K

α

(y), y > 0

C

y

e

−µy

L

µ

_↓

0

=

1

2

d

2

dx

2

+



1

2x

+

√

2µ

K

′

0

K

0

(

√

2µx)



_dx

d

P

2;µ↓

(µ > 0)

Infa t,the resultforthe rstrowgoesba k atleasttoMol hanov-Ostrovski [8℄, the result for the se ond row is dedu ed from that in the rst row with the help of the following dis ussion, whi h relates Ess her transforms (of subordinators) to Girsanov transforms(of diusions).

If

(X

t

)

t

≥0

is a

R

+

-valued diusion,whose inverse lo altime at0 :

τ

ℓ

= inf{t : L

t

> ℓ}, ℓ ≥ 0 ,

admits Lévy measure

ν(dy)

, and Lévy exponent

(Ψ(θ), θ ≥ 0)

, and if one denes :

ϕ

θ

↓

(x) = E

x

[exp(−θT

0

(X))] ,

then, there is another diusion, whi h we shall denote by

(X

θ

_↓

t

, t ≥ 0)

, with laws

(P

θ

↓

x

, x ≥ 0)

, su h that :

P

_x

θ

_|

↓

Ft

=

ϕ

θ

↓

(X

t

)

ϕ

θ

_↓

(x)

exp(Ψ(θ)L

t

− θt)

.

P

x

|

Ft

(1.1)

whose inverse lo altime

(τ

ℓ

, ℓ ≥ 0)

under

P

θ

_↓

0

satises :

E

0

θ

↓

(exp −λτ

ℓ

) = exp(−ℓ(Ψ(λ + θ) − Ψ(θ)))

i.e : this inverse lo al time is the

θ

-Ess her transform of

(τ

ℓ

, ℓ ≥ 0)

under

P

0

:its Lévy measure (under

P

θ

↓

0

) is:

e

−θy

_ν(dy)

.

It isalsonoteworthy that,undersome adequaterestri tionoftheir domains, the innitesimalgenerators

L

θ

_↓

and

L

are related by :

L

θ

↓

= L +

_dx

d

(log(ϕ

θ

↓

(x)))

.

d

dx

Finally, the result for the third row was dedu ed by letting

α → 0

in the se ond row, while taking are of the hoi e

1

of the lo al times made for

L

−α , µ↓

. (A ompendiumof hoi es of lo altimes for Bessel-like diusionsis

made in [2℄).

1.3

In the present paper, we wish to omplete the pre eding Table 1, by onsidering the 3 parameter familyof Lévy measures on

R

+

:

ν

µ, α, k

(dy) = C



µ

sinh(µy)



α+1

exp(µky) dy

(1.2) 1

As iswell-known, thelo al timein astandardMarkovianset up,at agivenlevel,is uniqueuptoamultipli ative onstant,whi hforourstudies,needstobe hosen arefully.

(The "true" parameters are :

µ > 0

,

k

, and

α

; as before,

C

is simply there to ensure anadditionaldegreeof freedom, if ne essary).

In order that

ν

µ, α, k

(dy)

bea Lévy measure, i.e : it must satisfy

Z

_∞

0

(x ∧ 1) ν

µ, α, k

(dx) < ∞ ,

weneed :

0 ≤ α < 1 ; k < 1 + α .

Wenowre allthat, fromPitman-Yor[11℄formulae (16),p. 276), if

Q

δ, µ

z

,for

0 < δ ≡ 2(1 − α) < 2

, and

µ > 0

, denotes the distribution of the squared radial Ornstein-Uhlenbe k pro ess, with "dimension"

δ

, and parameter

µ

, started from

z

, i.e: the solution of :

dZ

t

= 2

p

Z

t

dB

t

+ (δ − 2µZ

t

)dt ;

Z

t

≥ 0 , Z

0

= z ,

then, under

Q

δ,µ

0

, the inverse lo al time

(τ

ℓ

, ℓ ≥ 0)

admits as its Lévy measure :

C



µ

sinh(µy)



α+1

exp(µ

δ

2

y) dy

(1.3)

whi his a parti ular ase of (1.2), with

k =

δ

2

= (1 − α)

.

In the next se tion, we shall show, essentially with the help of the re ipe (1.1), how to onstru t a diusion, indexed by the 3 parameters

(α, µ, k)

, whi hsolves the Krein representation problemfor

ν

µ, α, k

.

1.4

Some among the new diusions we are nding as solutions of Krein's problem are related to the diusions we found in [1℄ by time hanging. We rst dis overed this relationship by applying the analyti al identity:

W

0, β

(z) =

r

z

π

K

β

z

2



(1.4)

between

W

0, β

, a Whittaker fun tion with parameters

(0, β)

, and

K

β

(see Appendix). Thus, a part of our present dis ussion may be onsidered as givinga probabilisti interpretationto (1.4).

Wealso develop a similardis ussion forthe analyti alidentity

M

0, β

(z) = 4

β

Γ(β + 1)

√

z I

β

z

2



.

2 Solving Krein's problem for

ν

µ, α, k

2.1

We take up the notation in (1.3); in fa t, it is more onvenient to onsider the family of radial Ornstein-Uhlenbe k pro esses (and not their

squares), whi hweshall denote as

(R

δ, µ

(t), t ≥ 0)

and their laws

(P

δ, µ

r

,

r ≥ 0)

. It will be helpful, for the sequel, to have the following formula at hand, for the innitesimal generator

L

−α, µ

of

R

δ, µ

:

L

−α, µ

=

1

2

d

2

dx

2

+



δ − 1

2x

− µx



d

dx

(2.1)

It iswell-known (see, e.g., Pitman-Yor[9℄,p. 454, formula(6.b)) that there is the relationship :

R

δ, µ

(t) = e

−µt

R

δ



e

2µt

_{− 1}

2µ



,

_{t ≥ 0 ,}

(2.2) where, on the RHS,

(R

δ

(u), u ≥ 0)

denotes a

δ

-dimensional Bessel pro ess. Thus, we obtain:



e

2µT

0

_{− 1}

2µ

; P

δ, µ

x



(law)

=

(a)

(T

0

; P

δ

x

)

(law)

=

(b)

x

2

2γ

α

(2.3)

where, on the RHS,

γ

α

denotes a gamma variable with parameter

α

. [(a) follows from(2.2), while(b) is well-known,and goesba kto Getoor[3℄, see, e.g., Yor[13℄, forsome variants...℄.

Wethusdedu ethefollowingformulafrom(2.3),withthe helpofelementary omputations :

E

x

δ, µ

(exp(−θT

0

)) =

1

Γ(α)(µx

2

₎

2µ

θ

Z

_∞

0

t

α

−1+

2µ

θ

e

−t

(1 +

t

µx

2

)

θ

2µ

dt

=

Γ(α+

θ

2µ

)

Γ(α)

(µx

2

)

α−1

2

e

µ

x2

2

W

_{(1−α)− θ}

µ

2

,

α

2

(µx

2

₎

for

− 2αµ < θ

(2.4) where

W

a, b

denotes the Whittakerfun tion, with parameters

(a, b)

.

2.2 Wenow write :

ν

µ, α, k

(dy) = C



µ

sinh(µy)



α+1

exp(µky)dy

≡ C



µ

sinh(µy)



α+1

exp(µ

δ

2

y) exp(−θy)dy ,

where :

θ = µ



δ

2

− k



, k < 1 + α

.

A ordingtothepre eding omputation,wenowndthat thediusionwith innitesimal generator :

L

θ

−α, µ

↓

≡ L

−α, µ

+

d

dx

log



(µx

2

)

α−1

2

e

µ

x2

2

W

(1−α)− θ

_µ

2

,

α

2

(µx

2

)



.

d

dx

(2.5)

solves Krein's representation problem for

ν

µ, α, k

. (We note in fa t that :

(1 − α) −

θ

µ

2

=

k

2

,so that :

W

(1−α)− θ

_µ

2

,

α

2

(ξ) ≡ W

k

2

,

α

2

(ξ)

).

The ase where

k = 0

is parti ularlyinteresting, sin e, on one hand :

ν

µ, α,

0

(dy) = C



µ

sinh(µy)



α+1

dy ,

and on the other hand (see Appendix) :

W

0,

α

₂

(ξ) =

r

ξ

π

K

α

2



ξ

2



(2.6)

sothatthediusionwhi hsolvesKrein'srepresentationproblemfor

ν

µ, α,

0

(dy)

is the solution to:

dX

t

= dB

t

+

"

δ − 1

2X

t

+ µX

t

bK

′

_α

2

b

K

α

2

! 

µ

X

2

t

2

#

dt

(2.7)

Here, weneed to give some details about this omputation:

a) Wededu efromformula(2.5),intheparti ular ase

k = 0

,i.e:

θ =

δµ

2

with the help of formula(2.6), that :

L

θ

−α, µ

↓

= L

−α, µ

+





α

x

+ µx + µx

K

′

_α

2



µ

x

2

2



K

α

2

µ

x

2

2





.

d

dx

b) Now, trivially:

L

−α, µ

+ µx

d

dx

= L

−α

,

and, equally simply:

α

x

+ µx

K

′

_α

2



µ

x

2

2



K

α

2

µ

x

2

2

= µx

b

K

′

_α

2

b

K

α

2



µ

x

2

2



whi h translates into the sto hasti dierential equation form of for-mula (2.7).

Table 2

h(y)

Generator of

(X

t

)

Distribution

C

y

α+1

e

−µy

L

µ

_↓

−α

= L

−α

+

√

2µ

K

b

′

α

b

K

α

(

√

2µx)

d

dx

P

δ

_;µ↓

(0 < α < 1; µ > 0)

_{(δ = 2(1 − α))}

C



sinh(µy)

µ



α+1

e

µ

δ

₂

y

L

−α, µ

≡ L

−α

− µx

dx

d

P

δ,µ

0 < δ = 2(1 − α) < 2; µ > 0

C



sinh(µy)

µ



α+1

L

θ

−α, µ

↓

≡ L

−α

+ µx

b

K

′

α

2

b

K

α

2



µ

x

2

2



d

dx

P

δ,µ;

δµ

₂

↓

θ =

δµ

₂

The rst rowis simplytaken fromTable 1(se ond rowthere).

As said above, the se ond row follows from Pitman-Yor [11℄. In the third row, we have written

L

θ

↓

−α, µ

for the innitesimal generator of the pro ess

whi his dened as: the radial Ornstein-Uhlenbe k pro ess, with dimension

δ = 2(1 − α)

, and drift parameter

(−µ)

, pushed downwards with parameter

θ =

δµ

2

. That this innitesimalgenerator may be expressed interms of

K

b

α

2

will bedis ussed after(2.7).

2.3 Weshall nowprovearemarkablerelationshipbetweenthe twofamilies of diusions whose innitesimal generators are found on the RHS of Table 2. This relationship explains pre isely why (Row 1) may be dedu ed from (Row 2),and vi e-versa.

Proposition.

The following relationship holds with :

θ =

δµ

2

:

X

2

_{−α, µ; θ↓}

(t) = X

−

α

2

;



µ2

8



↓



4

Z

t

0

X

_{−α, µ; θ↓}

2

(u) du



(2.8)

meaning that : startingfrom

X ≡ X

−α, µ; θ↓

on the LHS, there exists



X

−

α

2

;

µ2

8

(u), u ≥ 0



su h that the relationship (2.8) holds.

Comment about our notation

In formula (2.8), and possibly several times below, we have written

X

i

; θ↓

,

et ... instead of

X

θ

_↓

i

, for some index

i

. It seemed more appropriate here, be auseof the power 2on the left-side of (2.7).

Thereshouldbeno onfusionbetween thedierentdiusions

X

i, θ

and

X

i

; θ↓

.

Proof

Westart fromthe sto hasti dierentialequation satisedby

(X

−α, µ; θ↓

(t),

t ≥ 0)

as des ribed (impli itly)in Row 2of Table 2. Then, taking squares, we obtain :

X

t

2

= x

2

+ 2

Z

t

0

X

s

dB

s

+ δt + 2

Z

t

0

(µX

s

2

)

b

K

′

_α

2

b

K

α

2



µ

X

2

s

2



ds

Wenow dene

(Y

u

≡ Y (u), u ≥ 0)

via :

X

2

t

= Y



4

Z

t

0

X

2

s

ds



,

_{(t ≥ 0)}

and nd that

Y

satises:

Y

u

= x

2

+ β

u

+



δ

4

 Z

u

0

ds

Y

s

+

1

2

Z

u

0

µ

b

K

′

_α

2

b

K

α

2

µ

2

Y

s



ds

sin e

δ

4

=

b

δ − 1

2

, with

δ = 2 − α = 2(1 −

b

α

2

)

, we nd that

(Y

u

, u ≥ 0)

is pre isely the diusion with innitesimal generator

L

ν

_↓

−

α

2

, with

√

2ν =

µ

2

, i.e

ν =

µ

2

8

.



We now remark that the proof we have just given for the Proposition re-lies upon the identi ation of the innitesimal generator of the diusion

X

_{−α, µ;}

₍

δµ

2

)

↓

asgiven inTable 2;this identi ationwasobtained from an an-alyti alidentity between

W

0,

.

and

K

.

. (see formula(2.5)).

Wenowexplainandprovethe Propositionwithoutrelyingonsu hidentities, but ratheronabsolute ontinuityrelationshipsbetween thedierentlaws in-volved.

We now nd it a little more onvenient to refer to the laws

{Q

δ, µ

z

}

and the

main absolute ontinuity result we need is :

Q

δ, µ

_z

_|

Ft

= exp



−

µ

₂

(Z

t

− δt − z) −

µ

2

2

Z

t

0

Z

s

ds



.

Q

δ

z

|

Ft

.

(2.9)

Here,

(Z

t

, t ≥ 0)

denotes the oordinate pro ess on the anoni al spa e

C(R

+

, R

+

)

.

Wenow ombine thisrelation(2.9) withthat ofthe "pushdownwards" with parameter

θ

, sothat,withnotations whi hweshall explainafterwritingthe formula:

Q

δ, µ

_z

_|

; θ↓

Ft

(2.10)

=

ϕ

θ

↓

(Z

t

)

ϕ

θ

_↓

(z)

exp(Ψ(θ)L

t

− θt) exp



−

µ

₂

(Z

t

− δt − z) −

µ

2

2

Z

t

0

dsZ

s



.

Q

δ

z

_|

_Ft

and wenote that for pre isely :

θ =

δµ

2

,this relationsimpliesas :

Q

δ, µ;

(

δµ

2

)

↓

z

|

Ft

=

exp −

µ

2

Z

t

ϕ

θ

↓

(Z

t

)

exp −

µ

2

z

ϕ

θ

_↓

(z)

exp



Ψ(θ)L

t

−

µ

2

2

Z

t

0

dsZ

s



.

Q

δ

z

|

Ft

(2.11) (The due explanation of the formula (2.10) is that we have ombined the "push-downwards" formula (1.1), relative to

{Q

δ, µ

z

}

, - i.e. the fun tion

ϕ

θ

↓

and

Ψ(θ)

are relative to that diusion - with the pre eding formula (2.9)). From now on, we keep :

θ =

δµ

2

.

2.4

Wenow onsiderwhatbe omesofformula(2.11),on ewetime hange bothsides withthe inverse of



4

Z

t

0

Z

u

du, t ≥ 0



,sothat, byaslightabuse of notation, the pro ess of referen e is now

( b

Z(h), h ≥ 0)

, with

Z

b

dened by :

Z

t

= b

Z



4

Z

t

0

Z

u

du



(2.12) Thus, weobtain :

b

Q

δ, µ;

(

δµ

2

)

↓

z

|

Fu

b

=

exp



₋

µ

₂

Z

b

u



ϕ

θ

↓

( b

Z

u

)

e

−

µ

2

z

ϕ

_θ

_↓

(z)

exp



Ψ(θ)b

L

u

−

µ

2

8

u



.

Q

b

δ

z

_|

_Fu

_b

(2.13)

From the well-known property of time hange for Besselpro esses (see [12℄, Chap.XI, Prop.1.11),

Q

b

δ

i.e. of dimension

δ = 2 − α

ˆ

, that is

Q

b

δ

z

= P

ˆ

δ

z

. Again, with obvious notation, the right-hand side of (2.13) may be written:

b

ϕ

µ2

8

↓

( b

Z

u

)

b

ϕ

µ2

8

↓

(z)

exp



b

Ψ



µ

2

8



b

L

u

−

µ

2

_u

8



.

Q

b

δ

z

|

Fu

b

,

and wedis overthat :











b

ϕ

µ2

8

↓

(z) = e

−

µ

2

z

ϕ

_θ

_↓

(z)

b

Ψ



µ

₈

2



= Ψ(θ)

(2.14) and

Q

b

δ, µ;

(

δµ

₂

)

↓

z

= P

ˆ

δ;

µ2

₈

↓

z

.

Again, let us explain, very mu h in the same spirit, e.g : the rst relation :

b

ϕ

µ2

8

↓

(z) = e

−

µ

2

z

ϕ

_θ

↓

(z)

in(2.14). This translates as:

b

E

z



exp −

µ

2

8

T

0

( b

Z)



= e

−

µ

2

z

E

_z



e

−

δµ

2

T

0

(Z)



(2.15)

where

Z

b

simply denotes a BES pro ess with dimension

b

δ

, and

Z

a pro ess with law

Q

δ, µ

z

. This may be well understood by onsidering the absolute ontinuity relationship (2.9), when we repla e

t

by

T

0

(Z)

. Then, it follows from that relationshipthat :

Q

δ,

−µ

z



exp



−

δµ

₂

T

0

(Z)



= e

µz

2

Q

δ

z



e

−

µ2

2

R

_T0

0

ds Z

s



= e

µz

2

Q

b

δ

z



e

−

µ2

8

T

0

( b

Z)



,

whi his pre isely (2.15).

Now, it is well known (see [3℄, [4℄, [10℄) that the Lapla e transform of

T

0

, under the distribution

P

ˆ

δ

z

of a Bessel pro ess, isgiven by:

b

ϕ

µ2

8

↓

(z) := E

δ

ˆ

z



exp(−

µ

2

8

T

0

)



= 2

1−

α

2

Γ(

α

2

)

−1

µz

2



α

2

K

α

2

µz

2



.

(2.16)

Using (2.14), we an re over the expression of

ϕ

θ

↓

for

θ =

δµ

2

obtained in (2.4) using the identity (2.6).

2.5

downwards arrows

↓

now hanged intoupwards arrows

↑

(for the denition ofthesepushed upwardsand downwards pro essesobtainedfromadiusion, see Pitman-Yor[10℄).

The analogue offormula(2.10) isnow :

Q

δ, µ

_z

_|

; θ↑

Ft

(2.17)

=

ϕ

θ

↑

(Z

t

)

ϕ

θ

_↑

(z)

exp(−θt) exp



−

µ

2

(Z

t

− δt − z) −

µ

2

2

Z

t

0

dsZ

s



.

Q

δ

z

|

Ft

and wenote again that, pre isely for :

θ =

δµ

2

, this relationsimplies as:

Q

δ, µ;

δµ

2

↑

z

_|

_Ft

=

exp −

µ

2

Z

t

ϕ

θ

_↑

(Z

t

)

exp −

µ

2

z

ϕ

θ

↑

(z)

exp



−

µ

2

2

Z

t

0

dsZ

s



.

Q

δ

z

_|

_Ft

(2.18)

(Wenote that this formulaiseven simpler than(2.11) sin e herethere is no lo altime ontribution).

Wenow ontinuetodevelop ananalogous dis ussiontothatmade in subse -tion (2.8).

Thus, we time- hange both sides of the absolute ontinuity relation (2.18) with the inverse of

4

Z

t

0

Z

u

du , t ≥ 0

, with

Z

b

, as dened from

Z

in (2.12). Weobtain :

b

Q

δ, µ; (

δµ

2

)↑

z

_|

_Fu

_b

=

exp



₋

µ

₂

Z

b

u



ϕ

θ

↑

( b

Z

u

)

exp −

µ

2

z

ϕ

θ

↑

(z)

exp



−

µ

2

_u

8



.

Q

b

δ

z

_|

_Fu

_b

With obviousnotation, this right-hand side may be written:

b

ϕ

µ2

8

↑

( b

Z

u

)

b

ϕ

µ2

8

↑

(z)

exp



−

µ

2

_u

8



.

Q

b

δ

z

_|

_Fu

_b

with :

b

ϕ

µ2

8

↑

(z) = e

−

µ

2

z

ϕ

_θ

↑

(z)

(2.19)

b

Q

δ, µ; (

δµ

2

)↑

z

isthe distribution of aBesselpro ess of dimension

δ = 2 − α

ˆ

with drift

µ

2

8

, i.e.

b

Q

δ, µ; (

δµ

2

)↑

z

= P

ˆ

δ;

µ2

₈

↑

z

.

Appendix):

M

_{0, −}

α

2

(ξ) = 4

−

α

2

Γ(1 −

α

2

)

p

ξ I

₋

α

2



ξ

2



;

while the ompanion formula of (2.16)is:

b

ϕ

µ2

8

↑

(z) :=

1

E

δ

ˆ

0



exp(−

µ

8

2

T

z

)

 = 2

−

α

2

Γ(1 −

α

2

)

µz

2



α

2

I

₋

α

2

µz

2



a well-known formula whi h goesba k to Kent [4℄, Pitman-Yor[10℄.

Appendix : On the Whittaker and Bessel -M Donald fun tions

The following formulae involving these lassi al spe ial fun tions are found in Lebedev [7℄, towhi hwe referwith numberings su has :

(N)

∗

...

a) The Whittaker fun tions

M

k, µ

(z)

and

W

k, µ

(z)

are a pair of solutions of Whittaker's equation :

u

′′

+



−

1

4

+

k

z

+

(

1

₄

_{− µ}

2

₎

z

2



u = 0

(p. 279

∗

).

b)

W

k, µ

admitsthe integralrepresentation :

W

k, µ

(z) =

z

k

_e

−

z

2

Γ(µ − k +

1

2

)

Z

_∞

0

e

−t

t

µ

−k−

1

2



1 +

t

z



µ

−k+

1

2

dt

(see Problem 17

∗

,p. 279

∗

). )

W

0, µ

(z) =

r

z

π

K

µ

z

2



(see Problem 19

∗

,p. 279

∗

).

d) In terms of the onuent hypergeometri fun tion

Ψ

, there are the relations:

W

k, µ

(z) = z

µ+

1

2

e

−

z

2

Ψ



1

2

− k + µ, 2µ + 1; z



(see (9.13.16)

∗

, p. 274

∗

).

K

µ

(z) =

√

π(2z)

µ

e

−z

Ψ



µ +

1

2

, 2µ + 1; 2z



(see (9.13.15)

∗

, p. 274

∗

).

Taking

k = 0

inthe aboveformula for

W

k, µ

, one re overs ).

e) In terms of the onuent hypergeometri fun tion

Φ

, there are the relations:

M

k, µ

(z) = z

µ+

1

2

e

−

z

2

Φ



1

2

− k + µ, 2µ + 1; z



(see (9.13.16)

∗

, p. 274

∗

).

I

µ

(z) =

(z/2)

µ

Γ(µ + 1)

e

−z

_Φ



µ +

1

2

, 2µ + 1; 2z



(see (9.13.14)

∗

, p. 274

∗

).

Taking

k = 0

inthe aboveformula for

M

k,µ

, we obtain:

M

0, µ

(z) = 4

µ

Γ(µ + 1)

√

z I

µ

z

2



Referen es

[1℄ C. Donati-Martin,M. Yor : Some expli it Krein representations of ertain subordinators, in ludingthe Gammapro ess.

Submitted to Publ. RIMS,Kyoto, 2005.

[2℄ C. Donati-Martin,B. Roynette, P. Vallois, M. Yor: On onstants re-lated to the hoi e of the lo al time at 0, and the orresponding It measure for Besselpro esses with dimension

d = 2(1 − α), 0 < α < 1

. Preprint(2005).

[3℄ R.K. Getoor :The Brownian es ape pro ess. Ann. Prob. 7,p. 864-867(1979).

[4℄ J. Kent :Some probabilisti properties of Bessel fun tions. Ann. Prob. 6,p. 760-770(1979).

times of gap diusion.

Seminar onSto h. Pro esses, 1981.

Progr. Prob. Stat., 1, p.53-78, Birkhaüser (1981).

[6℄ S. Kotani, S.Watanabe : Krein's spe tral theory of strings and gener-alized diusion pro esses.

In : Fun tional Analysis inMarkov pro esses.

(Katata /Kyoto 1981), p.235-259.Springer, LNM 923 (1982)

[7℄ N. Lebedev :Spe ial Fun tions and their Appli ations. Dover(1972).

[8℄ S.A. Mol hanov-E. Ostrovski : Symmetri stable pro esses astra es of degenerate diusion pro esses.

Theo. Prob. Appl. 14, 1969, p.128-131.

[9℄ J. Pitman, M. Yor :A de omposition of Bessel bridges. Zeit. für Wahr., p.425-457,1982.

[10℄ J. Pitman, M. Yor :Bessel pro esses and innitely divisible laws. Pro eedings of the Durham Conferen e of July 1980.

In : "Sto hasti Integrals",Le t.Notes inMaths.,851, Springer(1981).

[11℄ J. Pitman, M. Yor : On the lengths of ex ursions of some Markov pro esses.

Séminaire Prob. XXXI, p. 272-286. Le t. Notes in Maths. 1655, Springer, Berlin (1997).

[12℄ D. Revuz, M. Yor :Continuous martingales and Brownian motion. Springer, Third Edition (1999).

[13℄ M. Yor: On ertain exponential fun tionalsof Brownian Motion. Paper

♯

1 in : Exponential Fun tionals of Brownian motion and related pro esses. Springer-Finan e(2001).