fubini theorem for stable processes - uni-jena.de · supported by the academy hassan ii of sciences...

Fubini Theorem for Stable Processes

Youssef Ouknine

joint work with Mohamed Erraoui

March 2-13, 2009, Jena

Supported by the Academy Hassan II of Sciences andTechnology

Youssef Ouknine Faculte des Sciences Semlalia March 2-13, 2009, Jena

MotivationFubini-type techniques are used to establish identities in lawwith functionals of Levy processes.To calculate explicitly the Laplace transform of suchfunctionals.

HistoryStochastic Fubini theorem for quadratic functionals ofBrownian motion was first proved by Donati-Martin and Yor(1991).First extension of Stochastic Fubini theorem to symmetricstable process was established by Donati-Martin, Song and Yor(1994).Stochastic Fubini theorem for general Gaussian measures isproved by Deheuvels et al. (2004).

ObjectiveWe present new identities in law for quadratic functionals ofstable processes. In particular, our results provide atwo-parameter generalization of a celebrated identity in law,involving the path variance of a Brownian bridge, due toDonati Martin and Yor (1991).


A process X = {Xt, t ≥ 0} such that X0 = 0 is called a Levyprocess if it has stationary independent increments, and Xt iscontinuous in probability. The Levy exponent Ψ is given by

E (exp(iξXt)) = exp (−tΨ (ξ)) ,

where Ψ has the Levy-Khintchine representation

Ψ (ξ) =12σ2ξ2 − iγξ −

∫R

(eiξx − 1− iξx1{|x|≤1}

)ϑ (dx) ,

with γ ∈ R and ϑ is a positive Radon measure on R\ {0} (calledthe Levy measure) defined by

ϑ (A) = E [# {t ∈ [0, 1] : ∆Xt 6= 0, ∆Xt ∈ A}] , A ∈ B (R) ,

and verifying the integrability condition:∫R

(1 ∧ x2

)ϑ (dx) <∞.

Excellent reference on Levy processes is [4].Youssef Ouknine Faculte des Sciences Semlalia March 2-13, 2009, Jena

A Levy process (Xt)t≥0 is called α-stable if the Levy measuretakes the form

ϑ (dx) = (m1Ix<0 +m2Ix>0) |x|−(1+α) dx

for some positive constants m1 and m2. The following hold:

1 when 0 < α < 2, α 6= 1,

E [exp {iξ Xt}] = exp{− t |ξ|α

(1− iβ ξ

|ξ|tan

(πα

2

))},

where β ∈ [−1, 1]2 when α = 2, the above formula becomes

E [exp {iξ Xt}] = exp{− t ξ2

},

3 when α = 1,

E [exp {iξ Xt}] = exp {−t (|ξ| − iγξ)} ,

where γ ∈ R is the drift coefficient.


When β = 0, X is said to be symmetric α-stable. In this case theLevy exponent Ψ is given by

E (exp(iξXt)) = exp {− t |ξ|α}

Recall that

For α ∈ (0, 1) the paths of Xt are non-decreasing and theprocess is called stable subordinators.

For α ∈ (1, 2) X is a martingale.

For all α ∈ (0, 1) ∪ (1, 2) , Xt is a semimartingale.


Stable sheet

I denotes the class of all 2-dimensional sets in R2+ of the type

2×i=1

(si, ti].

f : R2+ −→ R, the increment f (I) of f over the set I ∈ I is

defined by

f

(2×i=1

(si, ti])

= f (t1, t2)− f (t1, s2)− f (s1, t2) + f (s1, s2)

λ denotes the 2-dimensional Lebesgue measure.


X ={Xt, t ∈ R2

+

}be stochastic process such that:

(i) The increments X (Ij) are independents ∀Ij ∈ I disjointsets, j ∈ {1, · · · , k} and ∀k ∈ N

(ii) For any I ∈ I and ξ ∈ R

E exp (iξ X (I)) = exp (−λ (I) |ξ|α) (1)


The starting point of this study is Fubini theorem for Stablemeasures.

Let (A,A, µ) and (B,B, ν) be two measurable spaces, with µ andν are σ-finite measures.

Let{Xαµ (h) : h ∈ Lα(A,A, µ)

}and

{Xβν (k) : k ∈ Lβ (B,B, ν)

}be two stable symmetric processes, with α, β ∈ (0, 2] \ {1}


such that:

E[exp

{iλXα

µ (h)}]

= exp{−∫A |λh(a)|α µ (da)

}∀h ∈ Lα(A,A, µ),

and

E[exp

{iλXβ

ν (k)}]

= exp{−∫B|λk(b)|β ν (db)

}∀ k ∈ Lβ (B,B, ν) .

The following identity holds almost surely∫A

(∫Bφ (a, b)Xβ

ν (db))Xαµ (da) =

∫B

(∫Aφ (a, b)Xα

µ (da))Xβν (db) .

(2)


Theorem

Let φ : A×B → R be a bounded A⊗ B-measurable function. For

Yβ,α =∫A

∣∣∣∣∫Bφ (a, b)Xβ

ν (db)∣∣∣∣α µ (da)

and

Yα,β =∫B

∣∣∣∣∫Aφ (a, b)Xα

µ (da)∣∣∣∣β ν (db) ,

we have the following identity

(Yβ,α)1/γ Tγd= Yα,β, (3)

where γ = α/β and Tγ is a one-sided stable random variable withexponent γ, which is assumed to be independent of Yβ,α.


For α = β the identity in law (3) becomes∫A

∣∣∣∣∫Bφ (a, b)Xα

ν (db)∣∣∣∣α µ (da) d=

∫B

∣∣∣∣∫Aφ (a, b)Xα

µ (da)∣∣∣∣α ν (db) .

(4)

Proof.

Taking the characteristic functions of both sides of (2), for anyλ ∈ R, we obtain:

E[exp

(− |λ|α

∫A

∣∣∣∫B φ (a, b)Xβν (db)

∣∣∣α µ (da))]

=E[exp

(− |λ|β

∫B

∣∣∫A φ (a, b)Xα

µ (da)∣∣β ν (db)

)].

(5)

Taking u = |λ|α, as a new variable, the equality (5) becomes

E[exp

(−u∫A


∣∣∣α µ (da))]

=E[exp

(−u1/γ

∫B



)].


Proof.

On the other, it is easy to see that

E[exp

(−u∫A


∣∣∣α µ (da))]

=

E[exp

(−{u1/γ

(∫A


∣∣∣α µ (da))1/γ

}γ)]=

E[exp

(−u1/γ

(∫A


∣∣∣α µ (da))1/γ

Tγ

)],

Hence, we have obtained

E[exp

(−u1/γ

∫A


∣∣∣α µ (da))]

=

E[exp

(−u1/γ

(∫B



)1/γTγ

)],

which is equivalent to (3).Youssef Ouknine Faculte des Sciences Semlalia March 2-13, 2009, Jena

Special cases

Let % be a probability on [0, 1] and set At = % ([0, t]).

The function Ct = inf {s : As > t} ∧ 1 for t ∈ [0, 1], is increasingand right-continuous.

Moreover ACt ≥ t and CAt ≥ t for every t and

At = inf {s : Cs > t} .

We have ∫ t

0h(s)% (ds) =

∫ t

0h(s)dAs =

∫ At

0h(Cs)ds. (6)


Now we assume the following condition :

(H) : A is continuous and A0 = 0.

Note that, under condition H, we have

ACt ≥ t, % ([Ct− , Ct[) = 0 and supp (%) = {s ∈ [0, 1] : s = CAs} .


Proposition

We have∫ 1

0% (du)

∣∣∣∣{Xu −∫ 1

0Xs% (ds)

}∣∣∣∣α d=∫ 1

0du |XAu −AuX1|α

d=∫ 1

0dCu |Xu − uX1|α

(7)


Proof.

Let us consider the function φ : [0, 1]2 → R defined by

φ (u, s) =[I(s≤Cu) − (A1 −As)

].

On one hand, using integration by parts formula, we have∫ 1

0dXsφ (u, s) =

{XCu −

(X1A1 −

∫ 1

0AsdXs

)}

={XCu −

∫ 1

0Xs−dAs

}.

By condition H, yields∫ 1

0dXsφ (u, s) =

{XCu −

∫ 1

0XsdAs

}.


Proof.

On other hand since A1 = 1 we obtain∫ 1

0dXsφ (s, u) =

∫ 1

0dXs

[I(u≤Cs) − (A1 −Au)

]= −XAu +AuX1.

Now using the equality (3), it yields∫ 1

0du∣∣∣{XCu −

∫ 10 XsdAs

}∣∣∣α d=∫ 1

0du |XAu −AuX1|α

d=∫ 1

0dCu |Xu − uX1|α ,

where we have used (6) in the last equality.


Remark

(i) Let (Bs, s ≤ 1) is a Brownian motion. It is well known that thefollowing identity holds∫ 1

0%(dt)

(Bu −

∫ 1

0%(ds)Bs

)2d=∫ 1

0B2%[0,t]dt, (8)

where(Bs, s ≤ 1

)is a standard Brownian bridge. It follows from

(6) that∫ 1

0%(dt)

(Bu −

∫ 1

0%(ds)Bs

)2d=∫ 1

0B2%[0,t]dt =

∫ 1

0B2sdCs.

This corresponds to the case α = 2 in formula (7) since standardBrownian bridge has (Bu − uB1; u ≤ 1) as a representation in law.


Remark

(ii) For p > 0, let %(du) = du p up−1, we have

At = tp and Ct = t1/p.

From formula (7), we obtain∫ 1

0du p up−1

∣∣∣∣{Bu − ∫ 1

0psp−1Bsds

}∣∣∣∣2 d=∫ 1

0du |Bup − upB1|2

d=1p

∫ 1

0duu

1p−1∣∣∣Bu∣∣∣2 .

Note that for p = 1 we obtain∫ 1

0du

∣∣∣∣{Bu − ∫ 1

0Bsds

}∣∣∣∣2 d=∫ 1

0du∣∣∣Bu∣∣∣2 ,

which is well known see ([2]).


Let µ (da) = ν (db) be the Lebesgue measure on [0, 1]2, so that

(A,A, µ) = (B,B, ν) =(

[0, 1]2 ,B(

[0, 1]2), dt ds

).

We consider now the random variables

Zα (s1, s2) =∫ ∫

[0,1]2φ (s1, s2, t1, t2)Xα (dt1, dt2) ,

and

Zβ (t1, t2) =∫ ∫

[0,1]2φ (s1, s2, t1, t2)Xβ (ds1, ds2) .

In this special case, the conclusion of Theorem 1 becomes(∫ ∫[0,1]2

|Zα (s1, s2)|β ds1ds2

)1/γ

Tγd=∫ ∫

[0,1]2

∣∣∣Zβ (t1, t2)∣∣∣α dt1dt2.


For α = β and

φ1 (s1, s2, t1, t2) = 1[0,s1](t1)1[0,s2](t2)− s1s2,

we obtain the following distributional identities:∫ 1

0

∫ 1

0|Xα (s1, s2)− s1s2Xα (1, 1)|α ds1ds2

=∫ 1

0

∫ 1

0

∣∣∣∣Xα

(2×i=1

(si, 1])−∫ 1

0

∫ 1

0t1t2X

α (dt1, dt2)∣∣∣∣α ds1ds2

=∫ 1

0

∫ 1

0

∣∣∣∣Xα

(2×i=1

(si, 1])−∫ 1

0

∫ 1

0Xα

(2×i=1

(ui, 1])du1du2

∣∣∣∣α ds1ds2


For α = β and

φ2 (s1, s2, t1, t2) = 1[0,s1](t1)1[0,s2](t2)− s11[0,s2](t2)

−s21[0,s1](t1) + s1s2.

we have ∫ 1

0

∫ 1

0|Xα (s1, s2)− s1Xα (1, s2)

−s2Xα (s1, 1) + s1s2Xα (1, 1)|α ds1ds2

=∫ 1

0

∫ 1

0

∣∣∣∣Xα

(2×i=1

(si, 1])−∫ 1

0Xα ((s1, 1]× (u2, 1]) du2

−∫ 1

0Xα ((u1, 1]× (s2, 1]) du1

+∫ 1

0

∫ 1

0Xα ((ui, 1]× (u2, 1]) du1du2

∣∣∣∣α ds1ds2.Youssef Ouknine Faculte des Sciences Semlalia March 2-13, 2009, Jena

Using the following distributional identity between processes{Xα

(2×i=1

(si, 1]), (s1, s2) ∈ [0, 1]2

}d={

Xα (1− s1, 1− s2) , (s1, s2) ∈ [0, 1]2},

with the change variable

(r1, r2) = (1− s1, 1− s2)

the above identities become


∫ 1

0

∫ 1

0|Xα (s1, s2)− s1s2Xα (1, 1)|α ds1ds2

=∫ 1

0

∫ 1

0

∣∣∣∣Xα (r1, r2)−∫ 1

0

∫ 1

0Xα (u1, u2) du1du2

∣∣∣∣α dr1dr2,and ∫ 1

0

∫ 1

0|Xα (s1, s2)− s1Xα (1, s2)

−s2Xα (s1, 1) + s1s2Xα (1, 1) |αds1ds2

=∫ 1

0

∫ 1

0

∣∣∣∣Xα (r1, r2)−∫ 1

0Xα (r1, u2) du2

−∫ 1

0Xα (u1, r2) du1 +

∫ 1

0

∫ 1

0Xα (u1, u2) du1du2

∣∣∣∣α dr1dr2.Youssef Ouknine Faculte des Sciences Semlalia March 2-13, 2009, Jena

Remark

It should be noted that the above identities are the extension of(7) to the two parameter case.


Integration by parts formula

It is well known that Theorem 1 has several applications. Namely,we give here two examples yielding integration by parts formula.

One parameter case

Let 0 ≤ a < b <∞, and f, g : [a, b]→ R+ be two continuousfunctions with f decreasing and g increasing. Let us now choose

A = B = [a, b]

µ (dx) = −df(x)+δb (dx) f(b) and ν (dy) = dg(y)+δa (dy) g(a).


Two parameters case

Let 0 ≤ a, c < b, d <∞, and f1, g1 : [a, b]→ R+ (resp.f2, g2 : [c, d]→ R+) be two continuous functions, with f1 (resp.f2) decreasing, and g1 (resp. g2) increasing. Let us now choose

A = B = [a, b]× [c, d]

µ (dx1, dx2) = {−df1(x1) + δb (dx1) f1(b)} {−df2(x2) + δd (dx2) f2(d)}

ν (dy1, dy2) = {dg1(y1) + δa (dy1) g1(a)} {dg2(y2) + δc (dy2) g2(c)} .


Particular cases

1 g1(a) = g2(c) = f1(b) = f2(d) = 0, we obtain∫ b

a

∫ d

cdf1(x1)df2(x2) |Xα (g1 (x1) , g2 (x2))|α

d=∫ b

a

∫ d

cdg1(y1)dg2(y2) |Xα (f1 (y1) , f2 (y2))|α

2 a = c = 0 and b = d = 1, g1(y) = g2(y) = y2 andf1(x) = f2(x) = log (1/x)∫ 1

0

∫ 1

0

∣∣∣∣∣ 1

(x1x2)1/αXα

(x2

1, x22

)∣∣∣∣∣α

dx1dx2

d=∫ 1

0

∫ 1

0

∣∣∣(y1y2)1/αXα (log (1/y1) , log (1/y2))∣∣∣α dy1dy2


.Deheuvels, G.Peccati and M.Yor: On quadratic functionals ofthe Brownian sheet and related processes, Stoch. Proc. Appl(2006), 116 (3), 493-538.

C.Donati-Martin and M.Yor: Fubini’s theorem for doubleWiener integrals and variance of the Brownian path, Annalesde l’Institut H. Poincare (1991), 181-200.

C. Donati-Martin, S. Song and M. Yor: Symmetric stableprocesses, Fubini’s theorem and some extension of theCiesielski-Taylor identities in law, Stochastics and StochasticsReports (1994), 50, 1-33.

K. Sato: Levy processes and infinitely divisible distributions,Cambridge Press, Cambridge (1999).


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